1. **Problem Statement:** Calculate the discounted mean term (also called the Macaulay duration) of a bond redeemable at par in 10 years with annual coupons of 8%, at interest rates of 5%, 10%, and 15%. 2. **Formula and Explanation:** The discounted mean term (Macaulay duration) $D$ is given by: $$D = \frac{\sum_{t=1}^n t \cdot \frac{C}{(1+i)^t} + n \cdot \frac{F}{(1+i)^n}}{P}$$ where: - $n=10$ is the number of years - $C=0.08$ (8% coupon rate times par value 1) - $F=1$ is the redemption value (par) - $i$ is the interest rate - $P$ is the price of the bond, calculated as: $$P = \sum_{t=1}^n \frac{C}{(1+i)^t} + \frac{F}{(1+i)^n}$$ 3. **Calculate price $P$ and duration $D$ for each interest rate:** **At $i=0.05$ (5%):** Calculate $P$: $$P = \sum_{t=1}^{10} \frac{0.08}{(1.05)^t} + \frac{1}{(1.05)^{10}}$$ Using the formula for the present value of an annuity: $$\sum_{t=1}^{10} \frac{1}{(1.05)^t} = \frac{1 - (1.05)^{-10}}{0.05} = 7.7217$$ So coupon PV: $$0.08 \times 7.7217 = 0.6177$$ Redemption PV: $$\frac{1}{(1.05)^{10}} = 0.6139$$ Price: $$P = 0.6177 + 0.6139 = 1.2316$$ Calculate numerator for $D$: $$\sum_{t=1}^{10} t \cdot \frac{0.08}{(1.05)^t} + 10 \cdot \frac{1}{(1.05)^{10}}$$ Calculate $\sum t/(1.05)^t$: Using the formula for weighted sum of a geometric series: $$\sum_{t=1}^n t x^t = x \frac{1 - (n+1)x^n + n x^{n+1}}{(1-x)^2}$$ Here $x=1/1.05=0.95238$, $n=10$: $$\sum_{t=1}^{10} t (0.95238)^t = 6.8085$$ Multiply by coupon: $$0.08 \times 6.8085 = 0.5447$$ Add redemption term: $$10 \times 0.6139 = 6.139$$ Numerator: $$0.5447 + 6.139 = 6.6837$$ Duration: $$D = \frac{6.6837}{1.2316} = 5.43 \text{ years}$$ **At $i=0.10$ (10%):** Calculate $P$: $$\sum_{t=1}^{10} \frac{1}{(1.10)^t} = \frac{1 - (1.10)^{-10}}{0.10} = 6.1446$$ Coupon PV: $$0.08 \times 6.1446 = 0.4916$$ Redemption PV: $$\frac{1}{(1.10)^{10}} = 0.3855$$ Price: $$P = 0.4916 + 0.3855 = 0.8771$$ Calculate numerator: $$\sum_{t=1}^{10} t (1/1.10)^t = 5.7590$$ Coupon weighted sum: $$0.08 \times 5.7590 = 0.4607$$ Redemption term: $$10 \times 0.3855 = 3.855$$ Numerator: $$0.4607 + 3.855 = 4.3157$$ Duration: $$D = \frac{4.3157}{0.8771} = 4.92 \text{ years}$$ **At $i=0.15$ (15%):** Calculate $P$: $$\sum_{t=1}^{10} \frac{1}{(1.15)^t} = \frac{1 - (1.15)^{-10}}{0.15} = 5.0188$$ Coupon PV: $$0.08 \times 5.0188 = 0.4015$$ Redemption PV: $$\frac{1}{(1.15)^{10}} = 0.2472$$ Price: $$P = 0.4015 + 0.2472 = 0.6487$$ Calculate numerator: $$\sum_{t=1}^{10} t (1/1.15)^t = 4.9277$$ Coupon weighted sum: $$0.08 \times 4.9277 = 0.3942$$ Redemption term: $$10 \times 0.2472 = 2.472$$ Numerator: $$0.3942 + 2.472 = 2.8662$$ Duration: $$D = \frac{2.8662}{0.6487} = 4.42 \text{ years}$$ 4. **Summary of results:** - At 5% interest rate, discounted mean term $D = 5.43$ years - At 10% interest rate, discounted mean term $D = 4.92$ years - At 15% interest rate, discounted mean term $D = 4.42$ years 5. **Interpretation:** The discounted mean term decreases as the interest rate increases, reflecting that higher discount rates reduce the present value of distant cash flows more heavily. 6. **Graph sketch:** The graph of discounted mean term $D$ as a function of interest rate $i$ over $[0.05, 0.15]$ is a decreasing curve starting near 5.43 at 5% and falling to about 4.42 at 15%.