The problem asks to calculate the price of a corporate bond given its face value, coupon rate, yield to maturity (YTM), and time to maturity. The bond has a face value of $1000$, pays a coupon rate of $8\%$ annually, paid quarterly, has a YTM of $4.5\%$, and 20 years left to maturity.

1. Problem Description

The problem asks to calculate the price of a corporate bond given its face value, coupon rate, yield to maturity (YTM), and time to maturity. The bond has a face value of $1000$, pays a coupon rate of $8\%$ annually, paid quarterly, has a YTM of $4.5\%$, and 20 years left to maturity.

2. Solution Steps

First, we need to find the quarterly coupon payment. The annual coupon payment is $0.08 * 1000 = 80$. Since the coupon is paid quarterly, the quarterly coupon payment is $80 / 4 = 20$.

Next, we need to find the quarterly yield. The annual YTM is $4.5\%$, so the quarterly yield is $0.045 / 4 = 0.01125$.

The number of quarters until maturity is $20 * 4 = 80$.

Now we can use the bond pricing formula:

$Price = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^n}$

Where:

$Price$ = Bond Price

$C$ = Coupon payment per period

$r$ = Discount rate per period (YTM per period)

$n$ = Number of periods

$FV$ = Face Value of the bond

In our case:

$C = 20$

$r = 0.01125$

$n = 80$

$FV = 1000$

$Price = \sum_{t=1}^{80} \frac{20}{(1+0.01125)^t} + \frac{1000}{(1+0.01125)^{80}}$

We can simplify the summation using the present value of an annuity formula:

$PV = C * \frac{1 - (1+r)^{-n}}{r}$

$PV = 20 * \frac{1 - (1+0.01125)^{-80}}{0.01125} = 20 * \frac{1 - (1.01125)^{-80}}{0.01125} \approx 20 * \frac{1 - 0.42635}{0.01125} \approx 20 * \frac{0.57365}{0.01125} \approx 20 * 50.99111 \approx 1019.822$

The present value of the face value is:

$PV_{FV} = \frac{1000}{(1.01125)^{80}} = \frac{1000}{2.34537} \approx 426.35$

$Price = 1019.822 + 426.35 \approx 1446.172$

3. Final Answer

The price of the bond is approximately $1446.17$.