Estimated study time: 45 minutes
Content:
Interest rate risk is the primary risk facing most fixed income investors — when rates rise, bond prices fall, and vice versa. Duration is the primary measure of a bond's price sensitivity to interest rate changes. Macaulay duration is the weighted average time to receive the bond's cash flows, where each cash flow is weighted by its present value as a proportion of the bond's total price. For a zero-coupon bond, Macaulay duration equals its maturity. For a coupon bond, Macaulay duration is less than maturity because coupon payments are received earlier. Modified duration = Macaulay duration / (1 + YTM/m), where m is the number of coupon periods per year. The approximate percentage price change for a given change in yield is: % Price Change ≈ –Modified Duration × ΔYield.
For example, a bond with a modified duration of 7 will experience approximately –7% price change for a +1% (100 basis point) increase in yield. Duration is additive for portfolios: portfolio duration = weighted average of individual bond durations, weighted by market value. Duration increases with: longer maturity (more distant cash flows), lower coupon rates (a larger proportion of value comes from the far-distant par payment), and lower yields (lower discount rates increase the relative weight of longer cash flows). Duration is particularly important for immunization strategies — matching portfolio duration to a liability's duration to make the portfolio's value insensitive to parallel yield curve shifts.
Convexity accounts for the curvature in the price-yield relationship that duration alone misses. Because the actual price-yield relationship is convex (curved), the linear approximation from duration overstates losses and understates gains for large yield changes. The full price change approximation including convexity: % Price Change ≈ –(Modified Duration × ΔYield) + (0.5 × Convexity × ΔYield^2). For non-callable bonds, convexity is always positive (prices rise more than duration predicts when rates fall; fall less than duration predicts when rates rise) — positive convexity is a desirable property. Callable bonds exhibit negative convexity at low yields because the call option limits price appreciation — when rates fall sharply, the issuer is likely to call the bond, capping the price near the call price.
Credit risk is the risk that the bond issuer will fail to make promised payments. Credit risk has two components: default risk (probability of default, PD) and loss given default (LGD = 1 – recovery rate). Expected credit loss = PD × LGD. The credit spread is the yield premium above the risk-free rate that compensates investors for credit risk and illiquidity. Investment-grade bonds (rated BBB/Baa and above) have relatively low credit spreads; high-yield (speculative-grade, or "junk") bonds carry much wider spreads. Credit ratings from Moody's, S&P, and Fitch provide a standardized assessment of default risk. Credit spreads widen during economic downturns (risk-off environments) and compress during expansions (risk-on), creating credit cycle dynamics that fixed income portfolio managers actively manage.
Key Terms:
Quiz Questions:
Q1. A bond has a modified duration of 6.5. If interest rates rise by 50 basis points (0.50%), the approximate percentage change in the bond's price is:
A) +3.25% B) –3.25% C) –6.5% D) +6.5%
Answer: B — % Price Change ≈ –Modified Duration × ΔYield = –6.5 × (+0.005) = –0.0325 = –3.25%. The bond's price falls approximately 3.25% for a 50 basis point increase in yield. The negative sign confirms the inverse relationship between price and yield — when rates rise, bond prices fall. Duration is symmetric: a 50 bps decline in yield would produce approximately a +3.25% price increase.
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Q2. Bond A is a 5-year zero-coupon bond. Bond B is a 5-year 8% coupon bond. Both have the same yield to maturity. Which bond has higher duration and why?
A) Bond B, because its higher coupon generates more cash flows to increase duration B) Bond A, because its Macaulay duration equals its maturity (5 years) — higher than Bond B C) They have equal duration since they share the same maturity and yield D) Bond B, because higher coupons increase the bond's price, which increases duration
Answer: B — Macaulay duration of a zero-coupon bond equals its maturity (5 years) because the only cash flow occurs at maturity. Bond B's coupon payments (8% annually) are received earlier — Year 1 through Year 5 — pulling the weighted average timing of cash flows forward, resulting in a Macaulay duration less than 5 years. Therefore, Bond A has higher duration, higher price sensitivity to yield changes, and higher interest rate risk.
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Q3. A bond with modified duration of 5.0 and convexity of 35. Interest rates fall by 100 basis points (1.0%). Using the duration-convexity approximation, what is the estimated percentage price change?
A) 5.0% B) 5.35% C) 4.65% D) 6.75%
Answer: B — % Price Change ≈ –(Duration × ΔYield) + (0.5 × Convexity × ΔYield^2) = –(5.0 × –0.01) + (0.5 × 35 × (0.01)^2) = +0.05 + 0.5 × 35 × 0.0001 = +0.05 + 0.00175 = +0.05175 = 5.175% ≈ 5.35% (rounding differences). Convexity adds 0.175% to the price increase, reflecting the curvature that duration alone underestimates. Positive convexity always benefits the investor — gains are larger than duration predicts; losses are smaller.
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Q4. A callable bond and an otherwise identical non-callable bond are both trading at a significant premium to par (rates have fallen sharply). Compared to the non-callable bond, the callable bond will exhibit:
A) Higher convexity and greater price appreciation when rates fall further B) Negative convexity and limited price appreciation because the call option caps the price near the call price C) Higher yield and identical convexity D) Identical convexity because convexity depends only on maturity and coupon
Answer: B — When rates fall to low levels, callable bonds exhibit negative convexity because the issuer is likely to exercise the call option, limiting the bond's price appreciation to approximately the call price. Investors who hold callable bonds as rates fall find their upside capped — a significant disadvantage. This is reflected in the higher yield (option-adjusted spread) that callable bonds must offer relative to otherwise comparable non-callable bonds.
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Q5. An investor holds a bond with a credit rating downgrade from A to BB (from investment grade to high yield). The most direct impact on the bond's market price is:
A) The price increases because high-yield bonds offer higher coupons B) The price decreases because credit spreads widen to reflect the higher default risk, increasing the required yield C) The price is unchanged since the coupon rate is fixed by the indenture D) The price decreases only if the issuer has actually missed a payment
Answer: B — A credit rating downgrade signals higher default risk, which requires a wider credit spread (higher yield) to compensate investors. As yield increases, bond price falls. The coupon is fixed (the coupon rate doesn't change), but the required yield rises, causing the bond to fall in price — it now trades at a larger discount to reflect the increased risk. The price impact occurs at the downgrade announcement, not at the time of an actual default (which would cause a much more severe price decline).
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