Question

Suppose a​ ten-year, $1,000 bond with an 8.3% coupon rate and semiannual coupons is trading for $1,034.16.

a. What is the​ bond's yield to maturity​ (expressed as an APR with semiannual​ compounding)?

b. If the​ bond's yield to maturity changes to 9.8% ​APR, what will be the​ bond's price?

Answer 1

Answer to Part a.

Face Value = $1,000
Current Price = $1,034.16

Annual Coupon Rate = 8.30%
Semiannual Coupon Rate = 4.15%
Semiannual Coupon = 4.15% * $1,000 = $41.50

Time to Maturity = 10 years
Semiannual Period to Maturity = 20

Let Semiannual YTM be i%

$1,034.16 = $41.50 * PVIFA(i%, 20) + $1,000 * PVIF(i%, 20)

Using financial calculator:
N = 20
PV = -1034.16
PMT = 41.50
FV = 1000

I = 3.901%

Semiannual YTM = 3.901%
Annual YTM = 2 * 3.901%
Annual YTM = 7.80%

Answer to Part b.

Face Value = $1,000
Semiannual Coupon = $41.50
Semiannual Period to Maturity = 20

Annual YTM = 9.80%
Semiannual YTM = 4.90%

Price of Bond = $41.50 * PVIFA(4.90%, 20) + $1,000 * PVIF(4.90%, 20)
Price of Bond = $41.50 * (1 - (1/1.0490)^20) / 0.0490 + $1,000 * (1/1.0490)^20
Price of Bond = $41.50 * 12.568559 + $1,000 * 0.384141
Price of Bond = $905.74