Questions: Duration and Interest Rate Sensitivity Applications

Question 1

A 10-year zero-coupon bond and a 10-year 8%-coupon bond have the same maturity. Which has greater price sensitivity to a 1% rise in yields, and why?

• A. The coupon bond — it pays more total cash, so yield changes affect it more
• B. The zero-coupon bond — all its cash flow is at maturity, giving it the longest possible duration for a 10-year instrument
• C. They are identical — same maturity means identical price sensitivity
• D. The coupon bond — higher coupon payments amplify the effect of yield changes

Answer: (B)

Duration measures interest rate sensitivity, and duration ≤ maturity for coupon-paying bonds. A zero-coupon bond has duration equal to its maturity (10 years) because there is only one cash flow at the end — nothing to pull the weighted average earlier. A 10-year coupon bond has duration of perhaps 7–8 years because early coupon payments arrive sooner and reduce the time-weighted average. Since ΔP/P ≈ −D_mod × Δy, the zero-coupon bond drops more in price for the same yield increase. Option C is the most common misconception: equating maturity with duration.

Question 2

A pension fund must pay $100 million in 15 years. To immunize this liability against interest rate risk, the fund manager should:

• A. Buy bonds with maturities of exactly 15 years, since maturity matches the liability date
• B. Hold only short-term bonds to minimize duration and thus minimize all interest rate risk
• C. Construct a bond portfolio with duration equal to 15 years, so assets and liabilities respond equally to yield changes
• D. Match the total face value of bonds to $100 million, since face value determines the final payment

Answer: (C)

Immunization requires matching the *duration* of assets to the *duration* of liabilities — not their maturities. When asset duration equals liability duration, a given change in yields affects both sides by approximately the same amount, preserving the funded ratio. A 15-year coupon bond has duration less than 15 years; the manager must blend instruments to reach exactly 15-year duration. Option B is wrong: short-duration assets expose the fund to reinvestment risk and will not grow to match the liability if rates fall.

Question 3

A bond with a higher coupon rate (most else equal) will have a longer duration and therefore greater price sensitivity to interest rate changes.

• T. True
• F. False

Answer: False

Higher coupon rates *reduce* duration, not increase it. Larger early coupon payments shift more of the bond's total cash flow toward the present, pulling the time-weighted average forward and shortening duration. A high-coupon bond is *less* sensitive to interest rate changes than a low-coupon bond of the same maturity. The zero-coupon bond has the longest duration of any bond with a given maturity — precisely because it has no early payments to reduce sensitivity.

Question 4

Duration provides only an approximation of bond price changes because the actual price-yield relationship is convex, not linear.

• T. True
• F. False

Answer: True

Duration captures the first-order (linear) sensitivity: ΔP/P ≈ −D_mod × Δy. But the true price-yield curve is convex — for a given yield change, a bond falls less than duration predicts when yields rise and gains more than duration predicts when yields fall. For small yield changes, the linear approximation is excellent. For larger changes or for precision hedging, the convexity correction (the second-order term) becomes necessary. Positive convexity is generally favorable — it represents an asymmetric advantage compared to a purely linear instrument.

Question 5

Why does a duration mismatch between long-duration assets and short-duration liabilities (such as long-term mortgages funded by deposits) create a risk when interest rates rise rapidly?

Model answer: When rates rise, the market value of long-duration assets falls sharply — approximately duration × rate change × asset value. Short-duration liabilities lose much less value because their cash flows are near-term and barely discounted by higher rates. The result is that the asset side shrinks relative to the liability side, eroding the institution's equity. If the duration gap is large enough and the rate rise severe enough, equity can fall to zero — insolvency.

This is the mechanism behind the 2023 US banking crisis. Banks like Silicon Valley Bank held portfolios of long-duration bonds (10–15 year duration) funded by demand deposits (near-zero duration). A 2–3 percentage point rate rise caused asset market values to fall 15–30%, while deposit values were unchanged. Duration gap risk transforms from academic theory to existential threat when combined with leverage and liquidity mismatch.