Learning Module 10
Interest Rate Risk and Return
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Calculate and interpret the sources of return from investing in a fixed-rate bond — coupon payments, reinvestment of coupons, and capital gain/loss on sale.
Describe the relationships among a bond's holding period return, its Macaulay duration, and the investment horizon — reinvestment risk vs. price risk, and the duration gap.
Define, calculate, and interpret Macaulay duration — the weighted average time to receipt of a bond's cash flows, and its role in balancing reinvestment and price risk.
1 · Sources of Return from Investing in a Fixed-Rate Bond
Three sources of return:
1. Coupon and principal payments — received on scheduled dates.
2. Reinvestment of coupons — interest earned on reinvested coupon payments.
3. Capital gain/loss — gain or loss on sale if the bond is sold prior to maturity.
Horizon yield: The annualized holding‑period rate of return based on the total return (reinvested coupons + sale price or redemption amount) and the purchase price.
YTM = Realized return if and only if:
1. The bond is held to maturity.
2. All payments are made as scheduled (no default).
3. Coupons are reinvested at the YTM.
Interest income: Return associated with the passage of time — includes coupon interest, reinvestment of coupons, and amortization of discount/premium.
Capital gain/loss: Return associated with changes in the value of the security due to changes in yield.
r = (Total Ending Value / Purchase Price)1/T − 1
Total Ending Value = Future value of reinvested coupons + Sale price (or redemption amount)
2 · Reinvestment Risk and Price Risk
Reinvestment risk: The risk that coupon payments are reinvested at lower interest rates than the original YTM. Matters more for investors with long‑term investment horizons.
Price risk: The risk that the bond's price declines when interest rates rise (if sold before maturity). Matters more for investors with short‑term investment horizons.
Offsetting relationship: When interest rates rise, reinvestment income increases (good) but bond prices fall (bad). When interest rates fall, reinvestment income decreases (bad) but bond prices rise (good).
Reinvestment income ↑
Bond price ↓
Reinvestment income ↓
Bond price ↑
When horizon = Macaulay duration,
reinvestment risk and price risk
3 · Macaulay Duration
Definition: The weighted average of the time to receipt of a bond's cash flows, where the weights are each cash flow's share of the bond's full price (present value).
Interpretation: The holding period that balances reinvestment risk and price risk for a one‑time instantaneous parallel shift in the yield curve.
Duration gap:
Duration gap = Macaulay duration − Investment horizon• Positive gap: MacDur > Horizon → Price risk dominates → Risk of rising rates
• Negative gap: MacDur < Horizon → Reinvestment risk dominates → Risk of falling rates
• Zero gap: MacDur = Horizon → Risks offset → Immunized against interest rate risk
Calculation:
MacDur = Σ [t × (PV of CFt / Full Price)]Properties:
• Zero‑coupon bond: Macaulay duration = time‑to‑maturity
• Coupon bond: Macaulay duration < time‑to‑maturity
• Perpetuity: Macaulay duration = (1 + r) / r
• Macaulay duration decreases as time passes (saw‑tooth pattern between coupon dates)
MacDur = Σt=1N [t × (PV(CFt) / Full Price)]
where PV(CFt) = present value of cash flow at time t
Price risk dominates. Investor is at risk from rising interest rates. Duration gap is positive.
Reinvestment risk dominates. Investor is at risk from fallinginterest rates. Duration gap is negative.
4 · Quick Reference — Key Concepts
Concept | Description |
|---|---|
Horizon Yield | Annualized holding‑period return based on actual reinvestment and sale price |
Reinvestment Risk | Risk of lower returns on reinvested coupons; dominates when horizon > Macaulay duration |
Price Risk | Risk of capital loss when selling before maturity; dominates when horizon < Macaulay duration |
Macaulay Duration | Weighted average time to receipt of cash flows; balancing point for interest rate risk |
Duration Gap | MacDur − Investment horizon; positive = price risk, negative = reinvestment risk |
Immunization | Setting investment horizon equal to Macaulay duration to offset interest rate risk |