Certainly! Let’s answer your question in clear, step-by-step fashion, with explanations and a concise summary at the end. --- ## **Step 1: Introduction to Concepts** **Bond Duration** measures the sensitivity of a bond’s price to changes in interest rates. **Modified duration** tells us the approximate percentage change in price for a 1% change in yield. **Convexity** refines this estimate, accounting for the fact that the price-yield relationship is not perfectly linear. When interest rates rise, bond prices fall, and vice versa. By calculating duration and convexity, we can estimate how much a bond’s price will change for a given change in yield. --- ## **Step 2: Calculate the Bond’s Price (at 3.8% Yield)** **Given:** - Principal (Face Value) = $1,000 - Annual Coupon = $38 - Yield (YTM) = 3.8% - Maturity = 10 years **Bond Price (P):** \[ P = \sum_{t=1}^{10} \frac{38}{(1+0.038)^t} + \frac{1000}{(1+0.038)^{10}} \] Let's calculate: \[ P = 38 \times \left(\frac{1-(1+0.038)^{-10}}{0.038}\right) + 1000 \times (1+0.038)^{-10} \] \[ (1+0.038)^{-10} \approx 0.6922 \] \[ \text{Coupon PV: } 38 \times \frac{1-0.6922}{0.038} = 38 \times 8.0947 = 307.60 \] \[ \text{Principal PV: } 1000 \times 0.6922 = 692.20 \] \[ \text{Total Price: } 307.60 + 692.20 = \boxed{999.80} \] This is very close to par, as expected when coupon rate equals yield. --- ## **Step 3: Calculate Macaulay & Modified Duration** **Macaulay Duration (D):** \[ D = \frac{1}{P} \sum_{t=1}^{10} \frac{t \times CF_t}{(1+y)^t} \] Where \( CF_t \) is the cash flow at time \( t \). Let’s compute: - For years 1-9, \( CF_t = 38 \) - For year 10, \( CF_{10} = 1038 \) \[ D = \frac{1}{999.80} \left[ \sum_{t=1}^{9} \frac{t \times 38}{(1.038)^t} + \frac{10 \times 1038}{(1.038)^{10}} \right] \] Let's calculate each part: First, find the present value for each year: | Year | PV(Cash Flow) | t × PV(Cash Flow) | |------|---------------|------------------| | 1 | 38 / 1.038^1 = 36.62 | 36.62 × 1 = 36.62 | | 2 | 38 / 1.038^2 = 35.28 | 35.28 × 2 = 70.56 | | 3 | 38 / 1.038^3 = 34.00 | 34.00 × 3 = 102.00 | | 4 | 38 / 1.038^4 = 32.77 | 32.77 × 4 = 131.08 | | 5 | 38 / 1.038^5 = 31.59 | 31.59 × 5 = 157.95 | | 6 | 38 / 1.038^6 = 30.45 | 30.45 × 6 = 182.70 | | 7 | 38 / 1.038^7 = 29.36 | 29.36 × 7 = 205.52 | | 8 | 38 / 1.038^8 = 28.31 | 28.31 × 8 = 226.48 | | 9 | 38 / 1.038^9 = 27.30 | 27.30 × 9 = 245.70 | | 10 | 1038 / 1.038^10 = 718.85 | 718.85 × 10 = 7188.50 | Sum of PVs: \( \approx 999.80 \) (matches price as before) Sum of t × PVs: \[ S = 36.62 + 70.56 + 102.00 + 131.08 + 157.95 + 182.70 + 205.52 + 226.48 + 245.70 + 7188.50 = 8547.11 \] \[ \text{Macaulay Duration} = \frac{8547.11}{999.80} = 8.55 \text{ years} \] **Modified Duration:** \[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + y} \] \[ = \frac{8.55}{1.038} = 8.24 \] --- ## **Step 4: Calculate Bond Convexity** **Convexity Formula:** \[ \text{Convexity} = \frac{1}{P} \sum_{t=1}^{10} \frac{CF_t \cdot t(t+1)}{(1+y)^{t+2}} \] Let's compute: - For years 1–9: \( CF_t = 38 \) - For year 10: \( CF_{10} = 1038 \) Let's compute the sum: | Year | Numerator | Denominator | Value | |------|-----------|-------------|-------| | 1 | 38 × 1 × 2 = 76 | (1.038)^{3} = 1.119 | 76 / 1.119 = 67.92 | | 2 | 38 × 2 × 3 = 228 | (1.038)^{4} = 1.161 | 228 / 1.161 = 196.38 | | 3 | 38 × 3 × 4 = 456 | (1.038)^{5} = 1.205 | 456 / 1.205 = 378.00 | | 4 | 38 × 4 × 5 = 760 | (1.038)^{6} = 1.250 | 760 / 1.250 = 608.00 | | 5 | 38 × 5 × 6 = 1140 | (1.038)^{7} = 1.298 | 1140 / 1.298 = 878.27 | | 6 | 38 × 6 × 7 = 1596 | (1.038)^{8} = 1.347 | 1596 / 1.347 = 1185.31 | | 7 | 38 × 7 × 8 = 2128 | (1.038)^{9} = 1.399 | 2128 / 1.399 = 1522.51 | | 8 | 38 × 8 × 9 = 2736 | (1.038)^{10} = 1.453 | 2736 / 1.453 = 1883.65 | | 9 | 38 × 9 × 10 = 3420 | (1.038)^{11} = 1.509 | 3420 / 1.509 = 2266.08 | | 10 | 1038 × 10 × 11 = 113,580 | (1.038)^{12} = 1.567 | 113580 / 1.567 = 72,491.51 | Sum = 67.92 + 196.38 + 378.00 + 608.00 + 878.27 + 1185.31 + 1522.51 + 1883.65 + 2266.08 + 72491.51 = 81,477.63 \[ \text{Convexity} = \frac{81,477.63}{999.80} = 81.51 \] --- ## **Step 5: Estimate Change in Price (Yield Increases by 0.9%)** \[ \Delta y = +0.009 \] \[ \frac{\Delta P}{P} = -\text{Modified Duration} \times \Delta y + \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2 \] \[ = -8.24 \times 0.009 + 0.5 \times 81.51 \times (0.009)^2 \] \[ = -0.0742 + 0.5 \times 81.51 \times 0.000081 \] \[ = -0.0742 + 0.5 \times 0.00660 \] \[ = -0.0742 + 0.00330 = -0.0709 \] So, the **predicted percentage change in price is -7.09%**. \[ \text{Predicted new price} = 999.80 \times (1 - 0.0709) = 999.80 \times 0.9291 = \boxed{929.07} \] --- ## **Step 6: Actual New Price at 4.7% Yield** Now, recalculate the price at y = 4.7% (0.047): \[ P_{4.7\%} = 38 \times \left(\frac{1-(1+0.047)^{-10}}{0.047}\right) + 1000 \times (1+0.047)^{-10} \] \[ (1+0.047)^{-10} \approx 0.6556 \] \[ \text{Coupon PV: } 38 \times \frac{1-0.6556}{0.047} = 38 \times 7.334 = 278.69 \] \[ \text{Principal PV: } 1000 \times 0.6556 = 655.60 \] \[ \text{Total Price: } 278.69 + 655.60 = \boxed{934.29} \] **Actual price at 4.7% yield is $934.29** --- ## **Step 7: Interpretation & Comparison** - **Predicted price using duration & convexity:** $929.07 (7.09% drop) - **Actual price at 4.7% yield:** $934.29 (6.56% drop) - **Difference:** $934.29 - $929.07 = $5.22 The duration/convexity estimate slightly **overestimates** the price drop. This is normal: duration and convexity work best for small yield changes; as the yield change grows, the estimate becomes less accurate. --- ## **Step 8: Final Summary** **In summary:** We calculated the bond’s modified duration (8.24) and convexity (81.51), then used these to predict that a 0.9% yield increase would reduce the bond’s price by about 7.09% to $929.07. The actual price at the higher yield is $934.29 (a 6.56% drop), showing that duration and convexity provide a good but slightly conservative estimate for larger yield changes. --- **What we did in each step:** - **Step 1:** Introduced key concepts (duration, convexity, bond price sensitivity). - **Step 2:** Calculated the bond’s price at the starting yield. - **Step 3:** Computed Macaulay and modified duration. - **Step 4:** Calculated convexity. - **Step 5:** Used duration and convexity to estimate price change for a 0.9% yield rise. - **Step 6:** Calculated the bond’s actual price at the new yield. - **Step 7:** Compared the predicted and actual price changes, explained the result. - **Step 8:** Summarized findings concisely. --- **In a nutshell:** We found that duration and convexity provide a practical estimate of bond price sensitivity to yield changes, though actual price moves may be slightly smaller for larger rate shifts.