# Duration Calculation Homework Assignment #3 Solutions **Assumptions:** - All bonds pay coupons semi-annually. - Face value (FV) = $1,000 unless otherwise specified. --- ## 1. **Duration and Modified Duration of a 2-Year 6% Bond (Yield = 5%)** ### **Step 1: Calculate the Bond Price** - Coupon (C) = 6% of $1,000 = $60/year = $30 per 6 months - Number of periods (n) = 4 (2 years × 2) - Periodic yield (y) = 5% / 2 = 2.5% = 0.025 \[ P = \sum_{t=1}^4 \frac{30}{(1 + 0.025)^t} + \frac{1000}{(1 + 0.025)^4} \] Let’s compute: - \( (1+0.025)^1 = 1.025 \) - \( (1+0.025)^2 = 1.050625 \) - \( (1+0.025)^3 = 1.076890625 \) - \( (1+0.025)^4 = 1.1038128906 \) \[ P = \frac{30}{1.025} + \frac{30}{1.050625} + \frac{30}{1.076890625} + \frac{30}{1.1038128906} + \frac{1000}{1.1038128906} \] \[ P = 29.27 + 28.57 + 27.86 + 27.19 + 906.73 = \$1,019.62 \] --- ### **Step 2: Calculate Macaulay Duration** \[ \text{Macaulay Duration} = \frac{1}{P} \sum_{t=1}^n \frac{t \cdot CF_t}{(1+y)^t} \] Where \( t \) is the period in half-years. \[ \text{Macaulay Duration} = \frac{1}{1019.62}[1 \times 29.27 + 2 \times 28.57 + 3 \times 27.86 + 4 \times (27.19+906.73)] \] \[ = \frac{1}{1019.62}[29.27 + 57.14 + 83.58 + 3735.68] = \frac{1}{1019.62}[3905.67] = 3.83 \text{ half-years} \] Convert to years: \[ 3.83/2 = 1.92 \text{ years} \] --- ### **Step 3: Modified Duration** \[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \text{periodic yield}} \] \[ = \frac{1.92}{1 + 0.025} = \frac{1.92}{1.025} = 1.87 \text{ years} \] --- ## 2. **Duration and Modified Duration of a 3-Year 5% Bond (Price = $972.91)** ### **Step 1: Find Yield to Maturity (YTM)** - Coupon = $25 every 6 months, n = 6 - Price = $972.91 Solve for \( y \): \[ 972.91 = \sum_{t=1}^6 \frac{25}{(1+y)^t} + \frac{1000}{(1+y)^6} \] This requires trial and error or a financial calculator. Let’s estimate \( y \): Trying \( y = 0.0275 \) (annual 5.5%): \[ \text{PV of coupons: } 25 \times \left[ \frac{1 - (1+0.0275)^{-6}}{0.0275} \right] = 25 \times 5.659 = 141.48 \] \[ \text{PV of principal: } \frac{1000}{(1.0275)^6} = 849.67 \] \[ \text{Total PV} = 991.15 \text{ (Too high)} \] Try \( y = 0.03 \) (annual 6%): \[ \text{PV of coupons: } 25 \times 5.417 = 135.43 \] \[ \text{PV of principal: } \frac{1000}{(1.03)^6} = 837.48 \] \[ \text{Total PV} = 972.91 \] So, YTM (annualized) = 6%. --- ### **Step 2: Calculate Macaulay Duration** \[ \text{Macaulay Duration} = \frac{1}{972.91}[1 \times 24.27 + 2 \times 23.57 + 3 \times 22.89 + 4 \times 22.22 + 5 \times 21.57 + 6 \times (20.92+746.06)] \] \[ \text{Dollar weights: } \] - \( t = 1: \frac{25}{1.03} = 24.27 \) - \( t = 2: \frac{25}{(1.03)^2} = 23.57 \) - \( t = 3: \frac{25}{(1.03)^3} = 22.89 \) - \( t = 4: \frac{25}{(1.03)^4} = 22.22 \) - \( t = 5: \frac{25}{(1.03)^5} = 21.57 \) - \( t = 6: \frac{25}{(1.03)^6} + \frac{1000}{(1.03)^6} = 20.92 + 837.48 = 858.40 \) \[ \text{Numerator} = 1 \times 24.27 + 2 \times 23.57 + 3 \times 22.89 + 4 \times 22.22 + 5 \times 21.57 + 6 \times 858.40 \] \[ = 24.27 + 47.14 + 68.67 + 88.88 + 107.85 + 5150.40 = 5487.21 \] \[ \text{Macaulay Duration (in periods)} = \frac{5487.21}{972.91} = 5.64 \text{ periods} \] Convert to years: \( 5.64/2 = 2.82 \) years --- ### **Step 3: Modified Duration** \[ \text{Modified Duration} = \frac{2.82}{1 + 0.03} = \frac{2.82}{1.03} = 2.74 \text{ years} \] --- ## 3. **2-Year 8% Bond, YTM = 8%** ### **Step 1: Duration** - Coupon = $40 every 6 months, n = 4, y = 8%/2 = 4% = 0.04 \[ P = \sum_{t=1}^{4} \frac{40}{(1.04)^t} + \frac{1000}{(1.04)^4} \] Values: - (1.04)^1 = 1.04 - (1.04)^2 = 1.0816 - (1.04)^3 = 1.124864 - (1.04)^4 = 1.169858 \[ P = \frac{40}{1.04} + \frac{40}{1.0816} + \frac{40}{1.124864} + \frac{40}{1.169858} + \frac{1000}{1.169858} \] \[ = 38.46 + 36.99 + 35.56 + 34.20 + 855.13 = \$1,000.34 \] (Pretty much at par.) \[ \text{Macaulay Duration} = \frac{1}{1000.34}[1 \times 38.46 + 2 \times 36.99 + 3 \times 35.56 + 4 \times (34.20 + 855.13)] \] \[ = \frac{1}{1000.34}[38.46 + 73.98 + 106.68 + 3555.32] = \frac{1}{1000.34}[3774.44] = 3.77 \text{ half-years} \] Convert to years: \( 3.77/2 = 1.89 \) years --- ### **Step 2: Modified Duration** \[ \text{Modified Duration} = \frac{1.89}{1.04} = 1.82 \text{ years} \] --- ### **Step 3: Price Change for 100 Basis Point Change in YTM** - 100 basis points = 1% = 0.01 \[ \Delta P \approx -\text{Modified Duration} \times \Delta y \times P \] \[ \Delta P \approx -1.82 \times 0.01 \times 1000 = -\$18.20 \] #### **Actual Price Change (for +1% YTM, i.e., y = 9%)** - New periodic yield = 9% / 2 = 4.5% = 0.045 \[ P_{new} = \sum_{t=1}^{4} \frac{40}{(1.045)^t} + \frac{1000}{(1.045)^4} \] - (1.045)^1 = 1.045 - (1.045)^2 = 1.092025 - (1.045)^3 = 1.141166 - (1.045)^4 = 1.192519 \[ P_{new} = 38.28 + 36.64 + 35.18 + 33.77 + 838.98 = \$982.85 \] \[ \Delta P = 982.85 - 1000.34 = -\$17.49 \] #### **For -1% YTM (y = 7%)** - New periodic yield = 7% / 2 = 3.5% = 0.035 \[ P_{new} = \sum_{t=1}^{4} \frac{40}{(1.035)^t} + \frac{1000}{(1.035)^4} \] - (1.035)^1 = 1.035 - (1.035)^2 = 1.071225 - (1.035)^3 = 1.108718 - (1.035)^4 = 1.147523 \[ P_{new} = 38.65 + 37.28 + 35.89 + 34.59 + 871.42 = \$1,017.83 \] \[ \Delta P = 1017.83 - 1000.34 = +\$17.49 \] --- ### **Summary Table** | Scenario | Approx. Change (\$) | Actual Change (\$) | |----------------------------|--------------------|--------------------| | +100 bps (YTM to 9%) | -18.20 | -17.49 | | -100 bps (YTM to 7%) | +18.20 | +17.49 | --- ## **Conclusion** - **Duration** is a measure of interest rate sensitivity. - **Modified duration** gives the approximate percentage change in price for a 1% change in yield. - Actual price changes are slightly less (in absolute value) than estimated by modified duration due to bond convexity. --- If you need detailed formulas, steps, or clarification, just ask!