Question

Below is a list of prices for $1,000-par zero-coupon Treasury securities of various maturities. An 11% coupon $100 par bond pays an semi-annual coupon and will mature in 1.5 years. What should be the YTM on the bond? Assume semi-annual interest compounding for this question. Maturity (periods) Price of $1,000 par bond

1 943.4

2 873.52

3 770

Answer 1

Price of:

1-Period Maturity Bond = $ 943.4, 2-Period Maturity Bond = $ 873.52 and 3-Period Maturity Bond = $ 770

All periods are assumed to be semi-annual periods and let the annualized spot rates be 2s1,2s2 and 2s3 for 1-period,2-period and 3-period respectively.

943.4 = 1000 / (1+s1)

s1 = [(1000/943.4) - 1] = 0.06 or 6%

1-Period Annual Rate = 6 x 2 = 12 %

873.52 = 1000 / (1+s2)^(2)

s2 = [(1000/873.52)^(1/2) - 1] = 0.06995 or 6.995 %

2-Period Annual Rate = 6.995 x 2 = 13.99%

770 = 1000 / (1+s3)^(3)

s3 = [(1000/770)^(1/3) - 1] = 0.09103 or 9.103%

3-Period Annual Rate = 9.103 x 2 = 18.21%

Coupon Paying Bond:

Annual Coupon Rate = 11%, Coupon Frequency: Semi-Annual, Maturity = 1.5 years, Par Value = $ 100

Semi-Annual Coupon = 0.11 x 100 x 0.5 = $ 5.5

Bond Price = 5.5 / (1.06) + 5.5 / (1.06995)^(2) + 5.5 / (1.09103)^(3) + 100 / (1.09103)^(3) = $ 91.227891 ~ $ 91.23

Let the YTM be 2y

Therefore, 91.23 = 5.5 x (1/y) x[1-{1/(1+y)^(3)}] + 100 / (1+y)^(3)

Using EXCEL's goal seek function/hit and trial method/a financial calculator to solve the above equation, we get:

y = 0.08962 or 8.962 %

YTM = 2 x y = 2 x 8.962 = 17.92 %