Sure! Let's break down the problem step-by-step. --- ## **Problem Restatement** > **A $1,500,000 7% Value Bond has Annual Coupon Rate of 5% and 20 years until maturity. If yield on similar bonds is 7%, what is the price of the bond?** --- ## **Step 1: Identify Key Information** - **Face Value (FV):** $1,500,000 - **Coupon Rate:** 5% - **Annual Coupon Payment (C):** \( .05 \times 1,500,000 = \$75,000 \) - **Years to Maturity (n):** 20 - **Yield to Maturity (YTM or r):** 7% --- ## **Step 2: Write the Bond Price Formula** The price of a bond is the **present value of all future cash flows** (coupon payments + face value at maturity): \[ \text{Bond Price} = \left(\sum_{t=1}^{n} \frac{C}{(1+r)^t}\right) + \frac{FV}{(1+r)^n} \] Where: - \( C \) = annual coupon payment - \( r \) = yield to maturity per period (7% or .07) - \( n \) = number of periods (20) - \( FV \) = face value --- ## **Step 3: Calculate Present Value of Coupon Payments** This is a **present value of an ordinary annuity**: \[ PV_{\text{Coupons}} = C \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \] \[ PV_{\text{Coupons}} = 75,000 \times \left[\frac{1 - (1 + .07)^{-20}}{.07}\right] \] Calculate \((1 + .07)^{-20}\): \[ (1.07)^{-20} \approx .2584 \] So, \[ PV_{\text{Coupons}} = 75,000 \times \left[\frac{1 - .2584}{.07}\right] \] \[ PV_{\text{Coupons}} = 75,000 \times \left[\frac{.7416}{.07}\right] \] \[ PV_{\text{Coupons}} = 75,000 \times 10.5943 \approx \$794,572.50 \] --- ## **Step 4: Calculate Present Value of Face Value** \[ PV_{FV} = \frac{FV}{(1 + r)^n} \] \[ PV_{FV} = \frac{1,500,000}{(1.07)^{20}} \] \[ PV_{FV} = \frac{1,500,000}{3.8697} \approx \$387,664.30 \] --- ## **Step 5: Add Both Present Values** \[ \text{Bond Price} = PV_{\text{Coupons}} + PV_{FV} \] \[ \text{Bond Price} = \$794,572.50 + \$387,664.30 = \$1,182,236.80 \] --- ## **Final Answer** ### **The price of the bond is approximately \$1,182,237 (rounded to the nearest dollar).** --- #### **Summary Table** | Component | Value | |------------------|--------------| | PV of Coupons | \$794,573 | | PV of Face Value | \$387,664 | | **Total Price** | **\$1,182,237** | --- If you need this in Excel, use these cell references: - **C = B1 \* B2** - **PV of Coupons = C \* (1 - (1 + B4)^-B3)/B4** - **PV of Face Value = B1 / (1 + B4)^B3** - **Bond Price = PV of Coupons + PV of Face Value** Let me know if you need help setting this up in Excel or have any follow-up questions!