Banking And Finance Mathematics | Duration, Convexity, DV01 and Interest-Rate Risk Mathematics

Duration, convexity and DV01 are the mathematical language that turns “interest rates moved” into “this position changes by approximately this much”. They connect bond pricing to risk measurement, hedging, asset-liability management, portfolio construction and bank treasury. This guide builds Macaulay duration, modified duration, effective duration, dollar duration, DV01/PVBP, convexity, key-rate duration, curve sensitivity, immunisation and hedging from the price function rather than treating them as formulas to memorise.

For readers searching for duration formula, modified duration, Macaulay duration, effective duration, DV01, PVBP, dollar value of a basis point, bond convexity, interest-rate risk, price-yield sensitivity, key-rate duration, duration gap, immunisation, bond hedging or fixed-income risk mathematics, the central proposition is that all of these measures are approximations or decompositions of one underlying object: how present value changes when the discounting environment changes.

Current professional materials use the same hierarchy. CFA Institute’s 2026 fixed-income curriculum separates yield-based duration and convexity from curve-based measures, while fixed-income texts such as Tuckman and Serrat organise DV01, duration and convexity as core tools for measuring and hedging rate risk. This article remains educational mathematics, not investment or hedging advice.

50-Second Router

• Duration: first-order sensitivity of price to yield or rates; Macaulay duration is also a present-value-weighted time.
• Modified duration: approximate percentage price change for a small change in yield.
• DV01/PVBP: approximate money change for a one-basis-point move.
• Convexity: second-order curvature correction when the price-yield relationship is not a straight line.
• Effective duration: reprice after rate shocks, especially useful when cash flows can change with rates.
• Key-rate duration: sensitivity to selected maturity points rather than one parallel move.
• Portfolio risk: dollar sensitivities add; percentage duration must be value-weighted.
• Hedging: match the sensitivities of the exposure and hedge, then test non-parallel and nonlinear scenarios.
• Immunisation: align present value and rate sensitivity with liabilities under explicit assumptions.
• Verification: always compare the approximation with exact repricing at shocked rates.

The Central Proposition: Risk Measures Are Derivatives of Value

Start with a bond or fixed-income portfolio whose value P depends on a yield y or, more realistically, on a vector of curve rates. If y changes slightly, calculus tells us that the price change can be approximated by the first derivative dP/dy. Duration rescales that derivative into a more interpretable form. Convexity adds the second derivative d²P/dy². DV01 expresses the first-order effect in currency for a one-basis-point shock.

This perspective immediately clarifies the hierarchy. Exact repricing is the underlying truth of the chosen valuation model. Duration is a tangent approximation. Convexity bends the tangent toward the curve. Key-rate measures replace one rate with several curve nodes. None of these quantities exists independently of the price function and shock definition.

Adrian’s rule is therefore: before asking for duration, ask “duration with respect to what?” Yield to maturity, a parallel zero-curve shift, a key-rate node, a credit spread and a floating-rate index are different risk factors. A number without its shock definition is incomplete.

1. Macaulay duration

Macaulay duration is the present-value-weighted average time of a bond’s cash flows under a yield-based framework. It gives a timing interpretation and forms the bridge to modified duration. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. D_Mac=Σ[t×PV(CF_t)]/P. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Treating Macaulay duration as maturity ignores early coupon cash flows. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into immunisation and modified duration. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

2. Modified duration

Modified duration is Macaulay duration divided by one plus the periodic yield under the standard discrete-compounding setup. It approximates proportional price sensitivity to a small yield change. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. D_mod=D_Mac/(1+y), with matching periodic units. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Using an annual yield in the denominator when duration was built from semi-annual periods creates a unit mismatch. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bond price sensitivity. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

3. Dollar duration

Dollar duration is the price-scaled form of duration. It translates percentage sensitivity into currency sensitivity for a position. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. DollarDuration=P×D_mod. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Comparing dollar duration across differently sized portfolios without normalising can mislead. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into DV01 and hedging. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

4. DV01

DV01 is the approximate currency value change for a one-basis-point change in the chosen rate factor. It speaks the operational language of rate desks and treasury. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. DV01≈P×D_mod×0.0001 in a simple yield-based approximation. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Some institutions report signed DV01 and others positive magnitude; sign convention must be stated. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into hedge ratios. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

5. PVBP

PVBP is price value of a basis point, commonly used as a close synonym for DV01 in many contexts. It turns a 1bp rate move into a price or value effect. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. PVBP≈|P(y−1bp)−P(y+1bp)|/2 under symmetric repricing. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Assuming every desk uses identical naming or sign convention creates reconciliation problems. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into portfolio aggregation. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

6. Convexity

Convexity is a second-order measure of the curvature of the price-yield relationship. It improves duration approximations for larger moves and explains asymmetric price responses. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. ΔP/P≈−D_modΔy+0.5C(Δy)^2. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Applying fixed-cash-flow convexity to a callable bond can misrepresent cash-flow optionality. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into nonlinear rate risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

7. Positive convexity

Positive convexity is the usual curvature of an option-free fixed-rate bond where price rises more for a yield fall than it falls for an equal yield rise. It creates favourable asymmetry relative to a tangent-line duration estimate. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. The second-order term is positive when C>0. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Calling positive convexity ‘free profit’ ignores price paid, carry and other risks. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bond comparison. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

8. Negative convexity

Negative convexity is a region where price appreciation is constrained as rates fall, often because cash flows change through embedded options or prepayment. It is central to callable bonds and mortgage-related instruments. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Effective convexity can become negative when P_down and P_up respond asymmetrically. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Using static promised cash flows ignores exercise behaviour. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into option-adjusted risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

9. Effective duration

Effective duration is a repricing-based sensitivity calculated from prices after upward and downward rate shocks. It accommodates instruments whose cash flows can change as rates move. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. D_eff=(P_-−P_+)/(2P_0Δy). All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. The result depends on the valuation model and shock definition. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into callable bonds and MBS. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

10. Effective convexity

Effective convexity is a repricing-based curvature measure. It extends effective duration using the same shocked-price framework. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. C_eff=(P_-+P_+−2P_0)/(P_0(Δy)^2). All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A shock that is too large or too small can contaminate numerical estimates. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into option-sensitive instruments. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

11. Yield duration

Yield duration is sensitivity to a change in the instrument’s own yield-to-maturity. It is simple and historically common. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Differentiate or bump the price-yield function. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. One YTM factor cannot capture non-parallel curve moves. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into single-bond analysis. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

12. Curve duration

Curve duration is sensitivity to a shift in an underlying benchmark or zero curve. It is closer to the economic rate factors used in modern valuation. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Reprice using shocked discount and projection curves. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Curve methodology and interpolation affect the result. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into portfolio risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

13. Key-rate duration

Key-rate duration is sensitivity to a local change at a selected maturity node. It decomposes where on the curve the rate risk lives. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. KRD_k≈−(P_k^-−P_k^+)/(2P_0Δr_k), sign conventions varying. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. The interpolation rule for a node shock influences neighbouring maturities. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into curve-shape hedging. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

14. Partial duration

Partial duration is sensitivity to a defined segment or factor of the curve. It generalises the idea of key-rate buckets. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Shock one factor while holding others according to a specified rule. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Partial measures are model-specific and should not be compared without definitions. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into multi-factor interest-rate risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

15. Spread duration

Spread duration is sensitivity to a change in credit or option-adjusted spread, holding the benchmark curve framework fixed. It separates spread risk from benchmark-rate risk. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Approximate −(1/P)dP/ds. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Credit spreads may move with rates and with each other, so independent shocks are approximations. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into corporate bonds. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

16. CS01

CS01 is currency sensitivity to a one-basis-point credit-spread move. It is the spread analogue of DV01. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. CS01≈P×SpreadDuration×0.0001. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Calling CS01 DV01 hides which risk factor moved. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into credit hedging. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

17. Duration contribution

Duration contribution is the amount each holding contributes to portfolio duration. It explains how portfolio risk is assembled from positions. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Contribution_i=w_i D_i for simple value-weighted duration. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Negative positions and derivatives require signed market-value and sensitivity treatment. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into portfolio construction. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

18. Portfolio duration

Portfolio duration is the market-value-weighted duration under compatible definitions and common shock assumptions. It compresses many instruments into one first-order sensitivity. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. D_p=Σw_iD_i. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Weighting durations that were computed against different rate factors creates a meaningless average. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into fixed-income portfolios. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

19. Portfolio DV01

Portfolio DV01 is the algebraic sum of individual position DV01s under the same factor definition. It is naturally additive in currency units. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. DV01_p=ΣDV01_i. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Offsetting total DV01 can hide large key-rate exposures. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into hedging. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

20. Hedge ratio

Hedge ratio is the amount of a hedging instrument needed to offset a chosen sensitivity. It converts risk measurement into an actionable mathematical relation. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Notional_hedge≈DV01_target/DV01_per_unit_hedge with sign chosen to offset. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A DV01 hedge neutralises only the chosen first-order factor, not basis or convexity risk. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into rate hedging. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

21. Duration gap

Duration gap is a comparison between asset and liability duration after value scaling. It summarises balance-sheet sensitivity in a simplified ALM model. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. A common form scales liability duration by L/A before subtracting from asset duration. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Deposit behaviour and nonparallel curves can make a simple gap incomplete. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bank interest-rate risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

22. Economic value of equity sensitivity

Economic value of equity sensitivity is the change in present value of assets minus liabilities under rate shocks. It turns duration-gap intuition into full discounted-cash-flow analysis. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. ΔEVE=ΔPV_assets−ΔPV_liabilities. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Assuming contractual cash flows for behavioural deposits can materially misstate risk. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bank ALM. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

23. Immunisation

Immunisation is a strategy designed to protect a target liability value against small rate changes under stated assumptions. It uses duration and often convexity matching. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Match PV and duration; seek favourable convexity conditions where feasible. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Immunisation is not permanent and must be rebalanced as time passes and rates move. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into liability-driven investing. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

24. Redington immunisation

Redington immunisation is a classical set of local conditions matching present value and first derivative while favouring asset convexity over liability convexity. It formalises small-shock protection. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. PV_A=PV_L, duration matched, asset convexity greater under the local framework. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. The result relies on the specified yield-shift model. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into actuarial finance. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

25. Cash-flow matching

Cash-flow matching is holding assets whose cash flows directly meet liability cash flows. It reduces reliance on reinvestment and duration approximations. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Match amount and date as closely as possible. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Perfect matching can be expensive or impossible in incomplete markets. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into liability management. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

26. Dedication

Dedication is a practical fixed-income approach that secures scheduled liabilities with matching or near-matching assets. It makes the timing dimension explicit. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Construct a ladder of assets to cover liability dates. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Credit and liquidity quality still matter. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into pensions and institutions. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

27. Parallel shift

Parallel shift is a scenario in which all relevant curve points move by the same number of basis points. It makes one duration number interpretable. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. r_k’ = r_k + Δ for all k. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Real curves frequently steepen, flatten or twist. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into first-order risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

28. Steepener shock

Steepener shock is a scenario that changes long-short slope. It reveals shape risk hidden by parallel duration. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Short nodes and long nodes receive different shocks. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A portfolio can be parallel-DV01 neutral and lose heavily under a steepener. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into key-rate analysis. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

29. Flattener shock

Flattener shock is a scenario that reduces long-short slope. It is the mirror family of steepener scenarios. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Define exact node shocks rather than relying on the label. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Bull and bear flatteners have different level moves even if slope narrows. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into scenario testing. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

30. Butterfly shock

Butterfly shock is a curvature scenario moving middle maturities relative to short and long ends. It probes convexity of the yield curve rather than bond-price convexity. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Shock middle tenors opposite wings under a defined weighting. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. The word convexity can refer to bond price curvature or curve-shape curvature; keep them separate. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into curve risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

31. Basis risk

Basis risk is the risk that the hedge instrument and exposure do not move together as assumed. It survives even after first-order sensitivity matching. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Residual=P&L_target+P&L_hedge under realised factor moves. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A government-bond hedge may not track a corporate spread move. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into hedging quality. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

32. Reinvestment risk

Reinvestment risk is uncertainty about the rate at which interim cash flows can be reinvested. It is the counterweight to price risk for coupon instruments. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Future accumulated value depends on reinvestment rates for coupons. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Duration alone is a price-sensitivity measure, not a complete realised-return model. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into horizon analysis. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

33. Horizon return

Horizon return is return measured over a chosen investment horizon including price change and cash flows. It connects duration to investor objectives. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. HPR=(cash income+ending value−starting value)/starting value. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Horizon return depends on the future curve and reinvestment, not just starting YTM. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into scenario analysis. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

34. Price-value function

Price-value function is the exact present-value mapping from rates to price. It is the object duration and convexity approximate. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. P(r)=ΣCF_tD_t(r). All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Calculating sensitivities without preserving the exact repricing function weakens validation. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into all rate-risk measures. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

35. Taylor expansion

Taylor expansion is the calculus framework behind duration and convexity. It explains why first- and second-order approximations work locally. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. ΔP≈P’Δy+0.5P”(Δy)^2. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Higher-order terms matter for large shocks or strongly nonlinear instruments. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into risk approximation. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

36. Finite difference

Finite difference is a numerical derivative estimated by repricing at nearby shocked inputs. It is widely used when analytic derivatives are inconvenient. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. P’≈[P(y+h)−P(y−h)]/(2h). All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Too-large h creates approximation error; too-small h can amplify numerical noise. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into effective risk measures. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

37. Analytic derivative

Analytic derivative is a closed-form derivative of a valuation formula where available. It can be fast and exact within the model. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Differentiate the present-value equation term by term. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Analytic elegance does not repair an inappropriate cash-flow model. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bond mathematics. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

38. Automatic differentiation

Automatic differentiation is a computational method that propagates exact derivative operations through valuation code. It can produce sensitivities efficiently for complex models. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Apply chain rule mechanically through the computational graph. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Implementation and model-risk controls remain necessary. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into quantitative systems. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

39. Bump-and-revalue

Bump-and-revalue is a direct sensitivity method that changes one input, reruns valuation and measures the difference. It is intuitive and model-agnostic. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Sensitivity≈[V(x+h)−V(x−h)]/(2h). All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Runtime cost and shock-design choices can be material. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into production risk engines. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

40. Sign convention

Sign convention is the chosen direction for reporting sensitivity. It determines whether a long bond’s DV01 is shown positive magnitude or signed negative derivative. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Document whether DV01 means −dP/dy×1bp or actual P&L for +1bp. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Unstated sign conventions generate hedge-direction errors. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into risk reporting. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

41. Basis point

Basis point is one hundredth of one percentage point. It is the standard small unit for rate and spread changes. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. 1bp=0.01%=0.0001 in decimal rate units. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Using 0.01 instead of 0.0001 creates a factor-100 error. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into DV01. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

42. Duration in years

Duration in years is a conventional unit that can obscure the derivative interpretation. It arises from rate-time scaling but behaves as a sensitivity coefficient. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. A modified duration of 5 implies roughly 5% price change for a 100bp yield move locally. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Interpreting ‘5 years’ as time to maturity is wrong. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into communication. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

43. Convexity units

Convexity units is a second-order scaling tied to squared rate changes. It becomes meaningful only with the formula convention used. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. The term multiplies (Δy)^2. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Copying convexity numbers across systems without scaling conventions can create errors. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into risk aggregation. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

44. Full price versus clean price

Full price versus clean price is the distinction between value including accrued interest and quoted clean price. Sensitivities are normally computed from the valuation basis appropriate to the system. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Dirty=Clean+AccruedInterest. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Using clean price in one step and full price in another changes scaled sensitivities. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bond risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

45. Coupon effect on duration

Coupon effect on duration is the tendency for higher coupon to shorten the weighted timing of value, all else equal. More value is returned earlier. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Macaulay duration weights earlier coupon PVs. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. The rule can be altered by unusual structures or options. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bond comparison. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

46. Maturity effect on duration

Maturity effect on duration is the tendency for longer maturity to increase duration for ordinary bonds, all else equal. More value sits farther in the future. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Zero-coupon duration equals maturity under Macaulay definition. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. For deep-discount or perpetual structures, relationships need careful interpretation. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into term risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

47. Yield effect on duration

Yield effect on duration is the tendency for higher yield to reduce the relative PV weight of distant cash flows. Distant cash flows are discounted more heavily. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Duration weights depend on y through PV. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Rules of thumb should not replace exact computation when structures differ. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into sensitivity analysis. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

48. Floating-rate duration

Floating-rate duration is the generally low benchmark-rate duration of a plain floater near reset, subject to spread and reset mechanics. Coupons reset toward current reference rates. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Effective duration may be estimated by curve bumping. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Credit spread duration can remain material even when rate duration is small. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into FRNs. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

49. Callable-bond duration

Callable-bond duration is rate sensitivity altered by issuer call optionality. Falling rates can increase call probability and cap price appreciation. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Use option-aware effective duration. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Yield-based modified duration from fixed promised cash flows can be misleading. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into embedded options. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

50. Mortgage duration

Mortgage duration is rate sensitivity affected by borrower prepayment behaviour. Falling rates can accelerate prepayment and shorten expected cash flows. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Model cash-flow changes under shocked curves. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Prepayment models create model risk and possible negative convexity. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into MBS. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

51. Inflation-linked duration

Inflation-linked duration is sensitivity that can be decomposed across real yields and inflation expectations/indexation. Nominal and real rate factors should not be conflated. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Use real cash-flow/indexation mechanics and appropriate real curve. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Nominal duration alone does not describe all inflation-linked risk. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into linkers. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

52. Currency-hedged bond risk

Currency-hedged bond risk is a combination of local rate, spread and FX-hedging curve exposures. A foreign bond can be hedged for FX yet retain cross-currency basis and rate risk. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Decompose PV by local discounting and hedge cash flows. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Calling it ‘just duration’ hides multiple curves. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into global fixed income. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

53. Liquidity-adjusted risk

Liquidity-adjusted risk is rate sensitivity considered together with the time and cost required to exit or hedge. A mark-to-market sensitivity does not guarantee executable liquidity. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Stress bid-ask and market depth alongside price shocks. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. DV01 is not a liquidity measure. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into risk management. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

54. Stress testing

Stress testing is full repricing under severe but plausible or exploratory scenarios. It catches nonlinearities and factor interactions beyond local sensitivities. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Shock curves, spreads, volatilities and behaviours jointly. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A stress test without a narrative or coherent factor map can be arbitrary. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into bank risk. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

55. Backtesting sensitivity assumptions

Backtesting sensitivity assumptions is comparing predicted first-order P&L with realised or full-revaluation P&L. It reveals when linear approximations are breaking down. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Residual=ActualOrFullP&L−SensitivityPredictedP&L. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Residual can reflect omitted factors, convexity, basis and data issues. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into model validation. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

56. P&L explain

P&L explain is attributing daily value change to rate, spread, carry, curve-shape and other drivers. It operationalises sensitivity models. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Approximate P&L=ΣSensitivity_i×Move_i plus higher-order/residual terms. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. Large unexplained residuals are control signals. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into trading and treasury. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

57. Risk limit

Risk limit is a quantitative bound on exposure such as DV01, key-rate risk or stress loss. It converts measurement into governance. The safest interpretation begins by naming the rate factor and the valuation basis before reading the number.

Mathematics. Compare current exposure with approved threshold. All yield changes must be expressed in compatible decimal units. A 25bp shock is 0.0025, not 0.25 and not 0.025. Price scaling, notional and clean/full-price conventions should also be explicit.

Failure mode. A limit on one scalar cannot guarantee bounded total risk. Jo’s check is to ask whether the reported sensitivity can be recreated by a small bump-and-revalue calculation. If not, the definition, units or implementation need investigation.

Connection. This feeds directly into risk control. Ryan would compare the local approximation with exact repricing; Mira would then widen the shock to see when nonlinearity becomes material. That progression keeps the risk measure tied to the valuation function it summarises.

Worked Example 1: Macaulay and Modified Duration

A two-year annual-pay par bond pays 5 in year 1 and 105 in year 2 with yield 5%. Present values are 5/1.05=4.7619 and 105/1.05²=95.2381, summing to 100. The PV weights are 0.047619 and 0.952381.

Macaulay duration =1×0.047619+2×0.952381=1.952381 years. Modified duration =1.952381/1.05≈1.85941. A 10bp yield rise therefore gives a first-order percentage price estimate of −1.85941×0.001≈−0.18594%.

Exact repricing at 5.10% provides the audit. The difference between exact and linear estimates is small for this modest shock but not zero because the price-yield curve is curved.

Worked Example 2: DV01

A S$2,000,000 full-value bond position has modified duration 4.7. Approximate DV01 magnitude =2,000,000×4.7×0.0001=S$940 per basis point.

For a +12bp yield shock, the duration-only P&L estimate is approximately −940×12=−S$11,280. The same result comes from −D_mod×P×0.0012.

Clara verifies both routes because equivalent formulas should agree. If one gives S$11,280 and the other S$112,800, a basis-point or percentage conversion is wrong.

Worked Example 3: Duration Plus Convexity

Suppose modified duration is 7.2 and convexity is 68. For a +50bp yield move, Δy=0.005. Duration term =−7.2×0.005=−3.60%. Convexity adjustment =0.5×68×0.005²=+0.085%. Combined estimate ≈−3.515%.

For a −50bp move, the duration term is +3.60% and the convexity term remains +0.085%, giving +3.685%. Positive convexity produces the familiar asymmetry.

Exact repricing should still be the final benchmark if the full model is available.

Worked Example 4: DV01 Hedge Ratio

A portfolio has DV01 +S$18,500/bp. A hedge instrument has DV01 +S$74/bp per S$100,000 face under the same shock convention. To offset the portfolio with a short hedge, required face is approximately 18,500/74×100,000=S$25.0 million.

The sign matters: the hedge must create −S$18,500/bp. After sizing, reprice both portfolio and hedge under +1bp, −1bp, +25bp and a nonparallel curve shock. A perfect one-factor hedge will not remain perfect under all scenarios.

This is the difference between matching a metric and eliminating risk.

Worked Example 5: Portfolio Duration

Portfolio A contains S$3 million of a bond with duration 2 and S$7 million of a bond with duration 8. With total value S$10 million, the value weights are 0.3 and 0.7. Portfolio duration is 0.3×2+0.7×8=6.2 under compatible definitions.

Portfolio DV01 magnitude is approximately 10,000,000×6.2×0.0001=S$6,200/bp. Alternatively, calculate each position’s DV01 and add: 3,000,000×2×0.0001=600 and 7,000,000×8×0.0001=5,600, total S$6,200.

The equality is a useful reconciliation check.

Worked Example 6: Why Total DV01 Can Hide Curve Risk

Portfolio X has +S$5,000/bp at the 2-year key rate and −S$5,000/bp at the 10-year key rate. Its summed DV01 can appear near zero. Yet a steepening in which the 2-year rate falls 20bp and the 10-year rate rises 20bp generates losses on both exposures under their respective signs.

This portfolio is parallel-neutral but slope-sensitive. The example shows why a scalar is insufficient when the risk factor is a curve.

A key-rate vector, scenario matrix or principal-component representation is needed to preserve shape information.

Worked Example 7: Duration Gap for a Simplified Bank

A bank has assets A=S$1 billion with duration 4.5 and liabilities L=S$900 million with duration 2.0. A common simplified duration-gap measure is DGAP=D_A−(L/A)D_L=4.5−0.9×2.0=2.7.

A positive duration gap means asset economic value is more rate-sensitive than liability value under the simplified parallel-shift model. A rate rise tends to reduce asset value by more than liability value, reducing economic value of equity locally.

But real bank ALM requires behavioural assumptions for deposits, multiple curves, optionality, nonparallel shocks and earnings effects. The gap is a teaching model, not a complete supervisory calculation.

Worked Example 8: Effective Duration With Changing Cash Flows

An option-embedded bond is worth 100 today. Repricing the model after a 25bp downward curve shock gives 101.10; after a 25bp upward shock gives 98.95. Effective duration =(101.10−98.95)/(2×100×0.0025)=4.30.

If the same bond were valued using fixed contractual cash flows, the duration might be higher because the option is ignored. The difference is economic information: the embedded option changes expected cash-flow timing as rates move.

This is why effective measures are model-dependent. The quality of the option or prepayment model matters.

Interest-Rate Risk Is More Than Duration

Duration captures one local first-order relationship. A complete rate-risk picture can include curve shape, volatility, basis, optionality, spread, liquidity, model and behavioural risks. The stronger the instrument’s nonlinear or path-dependent features, the less adequate one duration number becomes.

This does not reduce duration’s importance. It locates it correctly: duration is a highly useful coordinate in a larger risk space.

Ethan’s control question is: what P&L can occur without materially changing the reported duration? If the answer is “a lot under curve twists, spread widening or option exercise,” then additional risk dimensions need explicit limits.

Duration and the Yield Curve

Yield duration assumes a change in one bond yield. Curve duration assumes a shift in underlying rates. Key-rate durations create several local factors. Principal-component approaches create statistical level, slope and curvature factors. These are different coordinate systems for describing the same broad economic phenomenon: changes in discounting and projected cash flows across maturity.

There is no universally correct coordinate system independent of purpose. A portfolio manager comparing bonds may prefer modified duration; a bank treasury desk may need key-rate DV01; an options desk may require curve Greeks and volatility sensitivities; a regulator may prescribe defined shocks.

The model should be chosen by the decision, not by formula familiarity.

Duration and Credit Spreads

A corporate bond price can change because benchmark rates move or because the credit spread moves. Rate duration and spread duration seek to separate these effects. If both move together in stress, independent first-order estimates can miss interaction.

A bond can therefore have modest government-curve duration but large spread sensitivity, or vice versa. Hedging the interest-rate DV01 with government bonds does not neutralise issuer credit risk.

This is a practical illustration of basis risk: the hedge and exposure share some risk factors but not all.

Duration and Floating-Rate Products

A plain floating-rate note that resets frequently can have low sensitivity to benchmark-rate changes near a reset because future coupons adjust toward the reference rate. But its credit spread can still move, and caps, floors or delayed resets can add optionality.

A bank loan linked to SORA can similarly have contractual repricing that changes benchmark-rate duration, while funding, borrower credit and spread economics remain. “Floating rate” does not mean “risk free”; it changes the composition of the risk.

Convexity and Optionality

Positive convexity in an option-free bond comes from the mathematical form of discounting fixed cash flows. Embedded callability can remove some of that favourable curvature. Mortgage prepayment can do the same because borrowers refinance when rates fall, returning principal just when investors would prefer to keep a high coupon.

Put options can create the opposite protection by supporting value when rates rise or credit deteriorates, subject to contract details. The general rule is that when cash flows change with rates, convexity becomes a property of the full option-aware valuation model, not merely the static coupon schedule.

Immunisation Is Engineering, Not Prediction

The conceptual power of immunisation is that it seeks robustness without requiring a perfect rate forecast. By aligning present values and first-order sensitivities, an institution can reduce the effect of small rate changes on the ability of assets to fund liabilities. Convexity can provide second-order protection.

But immunisation is local and conditional. Time passes, cash flows occur, curves twist, spreads move and portfolio weights change. A once-immunised portfolio drifts away from its target and requires monitoring and rebalancing.

This is an excellent example of closed-loop financial mathematics: measure, hedge, observe drift, rebalance, verify.

A Professional Risk Workflow

1. Define the valuation model and cash-flow assumptions.
2. Name each rate and spread factor to be shocked.
3. Calculate exact base value on a full-price basis where appropriate.
4. Compute analytic or finite-difference first-order sensitivities.
5. Compute second-order sensitivities where nonlinearity is material.
6. Aggregate compatible dollar sensitivities across positions.
7. Retain key-rate or factor vectors rather than only one total.
8. Size hedges to the chosen risk factor and document sign convention.
9. Full-reprice under small, medium and large shocks.
10. Run nonparallel, spread and optionality scenarios.
11. Compare predicted sensitivity P&L with full-revaluation P&L.
12. Investigate residuals and update models, limits or hedges as needed.

Common Failure Modes

1. 100bp versus 1bp

A duration of 6 implies roughly 6% for a 100bp move, not a 1bp move. DV01 scales by 0.0001. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

2. Signed versus unsigned DV01

Two systems can report +940 and −940 for the same long bond depending on convention. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

3. Clean versus dirty price

Scaling duration by the wrong price basis changes currency sensitivity. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

4. Macaulay versus modified duration

They are related but answer different questions. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

5. Yield duration versus curve duration

One bond yield is not the same risk factor as a benchmark zero curve. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

6. Static duration on callable cash flows

Embedded options can change expected cash flows as rates move. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

7. Ignoring convexity for large moves

Linear approximations deteriorate as shocks grow. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

8. Ignoring key-rate exposures

Total DV01 can net to zero while slope risk remains large. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

9. Hedging rate risk with spread risk

A corporate bond and government hedge may diverge through credit spread. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

10. Assuming historical correlation is permanent

Hedge relationships can break in stress. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

11. No exact repricing benchmark

Sensitivities should be checked against the valuation function. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

12. No rebalance rule

Duration hedges decay as time, price and curve conditions change. The repair is to restate the risk factor, units, shock size and valuation basis, then rerun a small symmetric bump and compare the sensitivity prediction with full repricing.

Formula Map

Measure	Simplified form	Interpretation
Macaulay duration	Σ t×PV(CF_t)/P	PV-weighted average cash-flow time.
Modified duration	D_Mac/(1+y)	First-order proportional sensitivity to yield under matching periodic units.
Duration price change	ΔP/P≈−D_modΔy	Local linear approximation.
DV01 magnitude	P×D_mod×0.0001	Approximate currency value for 1bp.
Effective duration	(P_-−P_+)/(2P_0Δy)	Repricing-based duration.
Effective convexity	(P_-+P_+−2P_0)/(P_0Δy²)	Repricing-based curvature.
Duration + convexity	ΔP/P≈−DΔy+0.5CΔy²	Second-order price approximation.
Portfolio duration	Σw_iD_i	Value-weighted duration when definitions are compatible.
Portfolio DV01	ΣDV01_i	Additive currency first-order sensitivity.

Authoritative Reference Map

• CFA Institute 2026 — Yield-Based Bond Duration Measures and Properties
• CFA Institute 2026 — Yield-Based Bond Convexity and Portfolio Properties
• CFA Institute 2026 — Curve-Based and Empirical Fixed-Income Risk Measures
• Tuckman & Serrat — DV01, Duration, and Convexity
• Bukit Timah Tutor — Bond Mathematics
• Bukit Timah Tutor — Yield Curve Mathematics

Connected Banking And Finance Mathematics Route

• Complete Banking And Finance Mathematics System
• Bonds, Bond Pricing and YTM
• Yield Curves, Spot and Forward Rates
• How Banks Measure Interest-Rate Risk
• Finance & Banking Algorithms

Applied Case Study 1: A S$50 million SGS portfolio

Situation. A treasury portfolio contains several Singapore Government Securities across the 2-, 5-, 10- and 30-year regions. Total DV01 is useful for headline reporting, but the real risk sits across tenor buckets. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Calculate full value, modified/effective duration and key-rate DV01 for each position. Sum currency sensitivities by tenor, then run parallel, steepener, flattener and butterfly shocks. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. A hedge that neutralises total DV01 with one 10-year instrument can leave short- and long-end residuals. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 2: A corporate bond hedged with government bonds

Situation. The corporate bond contains benchmark-rate and credit-spread sensitivity; the government hedge contains primarily benchmark-rate exposure. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Match rate DV01 first, then preserve a separate CS01/spread-duration report. Stress widening spreads while rates remain unchanged. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. A good government-rate hedge cannot remove issuer-specific credit risk. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 3: A callable bond

Situation. Falling yields make the issuer’s call option more valuable and can limit price appreciation. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Use an option-aware model to reprice under up/down curve shocks and calculate effective duration and convexity. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Static modified duration from promised maturity cash flows can overstate upside sensitivity. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 4: A mortgage portfolio

Situation. Borrowers can prepay, changing cash-flow timing when rates move. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Shock the curve, update the prepayment model, regenerate cash flows, then reprice. Compare effective duration with a static-cash-flow measure. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Negative convexity can emerge because falling rates shorten expected maturity. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 5: A bank with non-maturity deposits

Situation. Deposits have no contractual maturity matching their behavioural economic life. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Model deposit beta, decay and effective repricing assumptions; calculate EVE sensitivity across scenarios. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. The duration is model-driven and should be governed, challenged and stress-tested. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 6: A liability immunisation portfolio

Situation. An institution must pay S$20 million in seven years. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Match present value and duration with available fixed-income assets; seek adequate convexity and monitor the target as time passes. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. A parallel-shift immunisation can fail under curve twists, spread moves or asset credit events. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 7: A duration-neutral barbell

Situation. Short and long bonds are combined to match the duration of an intermediate bullet. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Match market value and duration, then compare convexity and key-rate exposures under nonparallel shifts. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Same duration does not mean same P&L distribution. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 8: A floating-rate note

Situation. The coupon resets quarterly to a reference rate plus spread. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Estimate benchmark-rate effective duration around reset and separate spread duration. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Low benchmark duration does not mean low credit or liquidity risk. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 9: A cross-currency bond hedge

Situation. An SGD investor owns a USD bond and uses FX forwards. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Decompose USD rate DV01, credit CS01, FX delta and cross-currency/funding basis exposures. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. One duration statistic cannot describe the multi-market position. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 10: A risk-limit breach

Situation. A desk exceeds its 10-year key-rate DV01 limit even though total DV01 is within limit. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Trace positions contributing to the 10-year bucket, evaluate hedge alternatives and full-reprice proposed corrections. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Headline neutrality can hide concentrated curve risk. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 11: A large rate shock

Situation. A 200bp scenario is applied to a long-duration bond portfolio. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Compare duration-only, duration-plus-convexity and exact full-repricing losses. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. The larger the move, the more important curvature and model nonlinearity become. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Applied Case Study 12: Daily P&L explain

Situation. Rates moved differently across the curve and credit spreads widened. The objective is to identify the smallest set of sensitivities that preserves the economics without pretending the position is one-dimensional.

Method. Use yesterday’s sensitivities times observed factor moves as first-order explain, add carry/convexity, then compare with actual full-revaluation P&L. Adrian maps the cash flows, Jo names the factors, Aisha verifies the units, and Ryan compares sensitivity-based P&L with exact repricing.

Boundary. Large residuals are information: they may signal omitted factors, nonlinearities, model changes or data errors. Mira then asks what unhedged factor could dominate in stress. This final question prevents a successful local hedge from being described as total risk removal.

Final Principle

Duration is the slope, convexity is the curvature, DV01 is the money scale, and exact repricing is the reference model that keeps all three honest.

The strongest fixed-income risk analysis therefore moves in layers. First value the cash flows. Then compute local sensitivity. Then add curvature. Then decompose the curve. Then stress the model beyond the local neighbourhood. Every layer preserves more of the underlying price function.

That sequence is useful from school financial mathematics through university fixed income, bank treasury and professional risk management because it is not tied to one formula. It is a method for turning changes in assumptions into changes in value.

Deep Practice Lab 1: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 2: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 3: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 4: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 5: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 6: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 7: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 8: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 9: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 10: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 11: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 12: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 13: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 14: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 15: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 16: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 17: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 18: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 19: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 20: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 21: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 22: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 23: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 24: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 25: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 26: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 27: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 28: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 29: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 30: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 31: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 32: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 33: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 34: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 35: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 36: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 37: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 38: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 39: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 40: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 41: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 42: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 43: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 44: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 45: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 46: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 47: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 48: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 49: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 50: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 51: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 52: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 53: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 54: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 55: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 56: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 57: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 58: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 59: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 60: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 61: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 62: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 63: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 64: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 65: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 66: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 67: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 68: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 69: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 70: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 71: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 72: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 73: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 74: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 75: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 76: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 77: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 78: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 79: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 80: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 81: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 82: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 83: Test hedge decay

Size a DV01 hedge today. Advance time by one month, update prices and durations without changing the curve, and calculate the new residual DV01. The hedge ratio will generally drift because cash-flow timing and value change.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 84: Separate rate and spread duration

Take a corporate bond modelled as benchmark curve plus spread. Bump the benchmark curve while holding spread fixed, then bump spread while holding benchmark fixed. Compare rate DV01 and CS01. Finally shock both together to reveal interaction.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 85: Interrogate sign conventions

Write the same long-bond sensitivity three ways: derivative dP/dy, signed P&L for +1bp, and positive DV01 magnitude. Show that apparently opposite signs can represent the same economics when conventions are stated.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 86: Exact repricing versus duration

Choose a fixed-rate bond and compute its exact base price. Shock yield by ±1bp, ±25bp, ±100bp and ±200bp. Compare the exact percentage change with modified-duration prediction. Plot or tabulate the residual. Then add convexity and show how the second-order approximation changes the residual.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.

Deep Practice Lab 87: Build a DV01 ladder

Take four bonds at different maturities. Calculate each position’s DV01 and assign it to 2y, 5y, 10y and 30y buckets. Construct two portfolios with the same total DV01 but different bucket vectors. Apply a steepener shock to prove that aggregate equality does not imply equal risk.

Complete the lab with a written prediction before calculating. Ben should state whether price should rise or fall; Clara should state the approximate scale; Ethan should define the verification tolerance. After full repricing, explain any residual rather than merely reporting it.

Then change the shock definition. Replace a parallel yield move with a key-rate or spread move and record which reported sensitivity ceases to be relevant. This is how a learner develops factor discipline instead of formula reflex.