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 reduce price and lower yields increase price. Duration and convexity turn that relationship into useful estimates.

For CFA Level I, duration is both a concept and a calculation. It helps compare risk across bonds with different coupons, maturities, and yields. It also supports portfolio immunization and asset-liability matching in later study.

Macaulay and modified duration

Macaulay duration is the weighted average time to receive the bond's cash flows, where weights are each cash flow's present value as a proportion of bond price. A zero-coupon bond has Macaulay duration equal to its maturity because all cash flow arrives at the end.

Modified duration converts Macaulay duration into price sensitivity. With periodic yield y and Macaulay duration MacDur, Modified duration = MacDur / (1 + y). The approximate percentage price change is: %DeltaP = -Modified duration x Delta yield.

If modified duration is 5.2 and yield rises by 0.50 percent, the approximate price change is -5.2 x 0.005 = -2.6 percent. The minus sign reflects the inverse price-yield relation.

Effective duration

Effective duration estimates price sensitivity using bond values when the benchmark yield curve shifts up and down. It is especially useful when cash flows can change with rates. Callable, putable, mortgage-backed, and other option-embedded bonds often need effective duration.

The formula is: Effective duration = (PV down - PV up) / (2 x PV0 x Delta curve). PV down is the price if yields fall. PV up is the price if yields rise. Delta curve is the decimal yield change.

For an option-free bond, modified and effective duration are often close if the yield change is small and the curve shift is parallel. For bonds with embedded options, effective duration captures expected option exercise and changing cash flows.

Convexity

Duration is a line tangent to a curved price-yield relationship. Convexity measures the curvature. The duration-only estimate is usually less accurate for larger yield changes. Adding convexity improves the approximation.

The common approximation is: %DeltaP = -Duration x Delta yield + 0.5 x Convexity x (Delta yield)^2. Convexity is positive for most option-free bonds. Positive convexity means price gains when yields fall are larger than price losses when yields rise by the same amount.

Callable bonds can show negative convexity when rates fall. As yields decline, the issuer becomes more likely to call the bond, which limits price appreciation. Putable bonds can have favorable behavior when yields rise because the investor's put option supports value.

Drivers of duration

Longer maturity usually increases duration because more cash flow arrives later. Lower coupon usually increases duration because less cash flow arrives early. Lower yield usually increases duration because later cash flows receive relatively more present value weight.

Amortizing bonds often have lower duration than otherwise similar bullet bonds because principal returns sooner. Floating-rate notes usually have low duration near reset dates because coupon payments adjust to market rates.

Structured aid: duration and convexity formulas

Exam focus

Watch yield units. A 100 basis point change is 0.0100. A 50 basis point change is 0.0050. If duration is 7 and yields rise 25 basis points, the duration-only estimate is -7 x 0.0025 = -1.75 percent.

When an item mentions callable debt, mortgage prepayment, or changing cash flows, think effective duration. When it asks why two bonds with the same maturity have different risk, compare coupon, yield, amortization, and embedded options.

Convexity questions often ask which approximation is more accurate. For small yield changes, duration may be adequate. For larger changes, duration plus convexity is better. Positive convexity is generally desirable because it improves upside versus downside for parallel yield changes.