8.4 Duration, Convexity, and Interest Rate Risk

Interest rate risk in one sentence

Interest rate risk is the risk that changes in market yields change a bond's price. For a fixed-rate option-free bond, higher yields cut price and lower yields raise price. Duration and convexity turn that relationship into usable estimates, support comparison across bonds, and underpin immunization and asset-liability matching in later levels.

Macaulay and modified duration

Macaulay duration (MacDur) is the PV-weighted average time to receive a bond's cash flows, where each weight is that cash flow's present value as a share of price. A zero-coupon bond's MacDur equals its maturity because all cash flow arrives at the end.

Modified duration (ModDur) converts MacDur into price sensitivity: ModDur = MacDur / (1 + periodic yield). The first-order price estimate is:

%DeltaPrice ~= -ModDur x DeltaYield.

Example: ModDur = 5.2 and yield rises 0.50% (50 bps). %DeltaPrice ~= -5.2 x 0.0050 = -2.6%. The minus sign reflects the inverse price-yield relation. The money duration (dollar duration) = ModDur x full price; multiplied by the yield change it gives the dollar price change. A related figure, the price value of a basis point (PVBP), is the price change for a 1 bp yield move.

Effective duration

Effective duration estimates sensitivity from full revaluation when the benchmark yield curve shifts up and down:

Effective duration = (PV_minus - PV_plus) / (2 x PV_0 x DeltaCurve),

where PV_minus is the price if the curve falls, PV_plus if it rises, and DeltaCurve is the decimal shift. It is required when cash flows can change with rates: callable, putable, and mortgage-backed securities all need effective duration because option exercise reshapes the cash flows. For an option-free bond under a small parallel shift, modified and effective duration are close.

Worked example: a bond is priced at 99.50 today (PV_0). If the curve falls 25 bps it would be worth 100.80 (PV_minus); if the curve rises 25 bps it would be worth 98.30 (PV_plus). Effective duration = (100.80 - 98.30) / (2 x 99.50 x 0.0025) = 2.50 / 0.49750 = 5.03. The same arithmetic, applied to bond values rather than yields, also produces effective convexity = (PV_minus + PV_plus - 2 x PV_0) / (PV_0 x DeltaCurve^2), which for a callable bond can be negative.

A key contrast: modified duration takes the bond's own yield (YTM) as the input variable, whereas effective duration takes a shift in the entire benchmark curve, which is why effective duration is the proper measure once cash flows are no longer fixed.

Convexity

Duration is a tangent line to a curved price-yield function, so it is least accurate for large yield moves. Convexity measures that curvature. The two-term estimate is:

%DeltaPrice ~= (-ModDur x DeltaYield) + (0.5 x Convexity x DeltaYield^2).

Convexity is positive for option-free bonds, so price gains when yields fall exceed price losses when yields rise by the same amount, which is desirable. Callable bonds can show negative convexity as yields fall, because a likely call caps price appreciation near the call price. Putable bonds show favorable convexity as yields rise, because the put supports price.

Drivers of duration

A zero-coupon bond has the highest duration for a given maturity; an amortizing bond has lower duration than an otherwise similar bullet; an FRN has very low duration near reset dates.

Two curve-risk refinements appear at Level I. Key rate (partial) durations measure sensitivity to a shift at a single point on the yield curve while other points hold fixed, which is useful for non-parallel (steepening or flattening) moves that a single effective duration cannot describe. Empirical duration is estimated from observed price and yield data using regression rather than from a pricing model; it often differs from analytical duration for high-yield and option-embedded bonds because credit spreads and benchmark rates can move together or in opposite directions.

For a portfolio, duration is approximately the market-value-weighted average of the constituent bonds' durations, so adding a long-dated low-coupon bond raises portfolio duration more than adding a short floater.

Structured aid: duration and convexity formulas

Exam focus

Watch yield units: 100 bps = 0.0100, 50 bps = 0.0050, 25 bps = 0.0025. If duration is 7 and yields rise 25 bps, the duration-only estimate is -7 x 0.0025 = -1.75%. When an item mentions callable debt, mortgage prepayment, or changing cash flows, choose effective duration. When two bonds share a maturity but differ in risk, compare coupon, yield, amortization, and embedded options. For large yield moves the two-term duration-plus-convexity estimate beats duration alone; for small moves duration may suffice.

Finally, remember that adding convexity always raises the predicted price relative to the duration-only estimate, whether yields rise or fall, because the 0.5 x Convexity x DeltaYield^2 term is positive for positively convex bonds.