Duration, Convexity & Interest-Rate Risk

Learning objectives

By the end you can:

1. Build a bond's full period-by-period cash-flow table from its coupon, yield, maturity, and coupon frequency, and compute its Macaulay duration (the present-value-weighted average time to cash flow) entirely from that table.
2. Convert Macaulay duration to modified duration, state the linear price-sensitivity approximation it powers (%ΔP ≈ −ModDur × Δy), and explain precisely why that formula requires the bond's cash flows to be fixed and yield-independent.
3. Compute money duration (dollar/rupee duration) and PVBP/DV01 for a position of a given face value, and explain (permanently) why DV01 is a currency amount per basis point and never a percentage.
4. Explain why a callable bond or an agency MBS pool has no fixed cash-flow stream to differentiate, and compute effective duration and effective convexity for such an instrument by bump-and-reprice, given a small set of scenario prices.
5. Derive convexity from a bond's cash flows in closed form, combine it with modified duration in the two-term Taylor shock estimate, and quantify the approximation error against an exact reprice for both a small and a large yield move, in both directions.
6. Distinguish positive convexity (option-free bonds) from negative convexity (callable bonds, MBS) in cash-flow terms, and explain why negative convexity makes both the gain on a rally and the loss on a sell-off worse than a linear duration estimate predicts.
7. Build a simplified key-rate duration (KRD) profile for a bond or a small portfolio, verify that key-rate durations sum to the total effective duration under a parallel shift, and use the profile to predict (then verify by exact reprice) a portfolio's response to a non-parallel curve move that a single duration number gets wrong.
8. Build a duration- and present-value-matched (Redington) immunization portfolio against a single liability, verify the asset-convexity-≥-liability-convexity safety condition, and explain why an immunized position drifts and must be rebalanced.
9. Size a DV01-neutral hedge of a bond position using a second instrument's own DV01, a second bond in a US Treasury setting, an interest-rate swap referencing INR OIS/MIBOR in an India setting.
10. (Productivity objective: R10 duality.) Recognize why duration/convexity/DV01 arithmetic is one of the highest AI-over-trust zones in the program, and apply this module's adapted verification discipline, independent recomputation by a second method, plus a check against the named market convention, before any AI-produced risk figure is used to size a real hedge.

The duality, stated once (R10). Objectives 1–9 are the understanding objective the mastery gate rewards: you earn the pass by building every one of these numbers from a blank sheet and defending them. Objective 10 is the productivity payoff you keep, knowing exactly how far to trust a tool's arithmetic in a domain where a wrong number costs real money, not an awkward citation. A tool can accelerate the calculation. It cannot buy you the gate, and in this branch more than almost any other, it cannot be allowed to be the calculation either.

Prerequisites & connections

Builds on. M1.08's effective-interest machinery, you already know how a bond's issue price, coupon, and market yield relate, and how a discount or premium amortizes to par over the bond's life; this module does not re-derive any of that. It starts from a bond's price and asks how that price moves when the yield that produced it moves. M7.03 introduced Macaulay and modified duration once, in a single worked comparison of a 2-year and a 10-year bond, to make the point that a higher discount rate crushes long-duration, terminal-value-heavy assets hardest, that intuition, and the curve-shape vocabulary (normal/inverted, the four bull/bear-steepener/flattener moves) built alongside it, are assumed here, not re-taught. This module supplies the full closed-form and numerical toolkit M7.03 only sketched: money duration, PVBP/DV01, cash-flow convexity, effective duration for option-embedded instruments, key-rate duration, immunization, and hedging. E11.01 built the credit side of a bond's risk, spread duration, the price sensitivity of a corporate bond to a change in its own credit spread. Hold the boundary sharp from the first page: this module's duration is interest-rate duration, sensitivity to the risk-free curve M7.03 taught you to read. E11.01's spread duration is a different, additive sensitivity, to the same bond's credit spread. A corporate bond carries both, computed with structurally similar arithmetic, but they move for different reasons, a central-bank decision moves the first; a rating action or an earnings miss moves the second, and a number for one tells you nothing about the other.