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VIPSolutions · Accepted Answer

# Problem 7 Solution

---

## **Objective**

Find the price of a bond given:
 Face Value (\FV\)) \$1,000
- Coupon Rate = 5.2% (paid semiannually)
- Years to Maturity = 21 years
- Market Interest Rate (annual, compounded semiannually) = 3.3%

---

## **Step 1: Gather Data**

- Number of periods ($n$):  
  $ n = 21 \text{ years} \times 2 = 42 \text{ periods} $
- Coupon payment per period ($C$):  
  $ C = .052 \times 100 / 2 = \$26 $
- Market rate per period ($r$):  
  $ r = .033 / 2 = .0165 $

---

## **Step 2: Bond Price Formula**

The price of a bond is the present value of all future cash flows (coupons + face value):

$$
P = \left[ \sum_{t=1}^{n} \frac{C}{(1 + r)^t} \right] + \frac{FV}{(1 + r)^n}
$$

Or, using the present value of an ordinary annuity for coupons:

$$
P = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) + \frac{FV}{(1 + r)^n}
$$

---

## **Step 3: Insert Values**

$$
P = 26 \times \left( \frac{1 - (1 + .0165)^{-42}}{.0165} \right) + \frac{100}{(1 + .0165)^{42}}
$$

---

## **Step 4: Calculate Each Component**

### **A. Calculate $(1 + r)^{-n}$:**

$$
(1 + .0165)^{42} = (1.0165)^{42}
$$

Calculate $(1.0165)^{42}$:

$$
\ln(1.0165) = .016363
$$
$$
42 \times .016363 = .686246
$$
$$
e^{.686246} = 1.9862
$$

So:

$$
(1 + .0165)^{42} = 1.9862
$$
$$
(1 + .0165)^{-42} = 1 / 1.9862 = .5035
$$

---

### **B. Calculate Present Value of Coupons**

$$
PV_{\text{coupons}} = 26 \times \left( \frac{1 - .5035}{.0165} \right)
$$
$$
1 - .5035 = .4965
$$
$$
.4965 / .0165 = 30.1515
$$
$$
26 \times 30.1515 = 783.94
$$

---

### **C. Calculate Present Value of Face Value**

$$
PV_{\text{FV}} = \frac{100}{1.9862} = 503.50
$$

---

## **Step 5: Calculate Total Price**

$$
P = PV_{\text{coupons}} + PV_{\text{FV}}
$$
$$
P = 783.94 + 503.50 = 1287.44
$$

---

## **Final Answer**

**The price of the bond is:**

$$
\boxed{1287}
$$

*(Rounded to the nearest dollar, as requested.)*

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