Question

A bond has 8 years until maturity, has a coupon rate of 8%, and sells for $1100.  

1. What is the yield to maturity if interest is paid once per year?
2. What is the yield to maturity if interest is paid semi-annually?
3. What is the current yield on the bond?

Answer 1

a)

Coupon = 8% * 1000 = 80

Yield to maturity = 6.37%

Keys to use in a financial calculator:

FV 1000

PV -1100

N 8

PMT 80

CPT I/Y

2)

Coupon = (8% * 1000) / 2 = 40

Number of periods = 8 * 2 = 16

Yield to maturity = 6.38%

Keys to use in a financial calculator:

2nd P/Y 2

FV 1000

PV -1100

N 16

PMT 40

CPT I/Y

3)

Current yield = (Annual coupon / price) * 100

Current yield = (80 / 1100) * 100

Current yield = 7.27%

Answer 2

a. Yield to maturity (YTM) when interest is paid once a year

Assume that the yield to maturity is \(r\), the face value of the bond is \(F = 1000\) USD (usually the face value of the bond is \(1000\) USD, and this assumption is made if the question does not mention it), the annual interest payment is \(C=1000\times8\% = 80\) USD, the term is \(n = 8\) years, and the current price is \(P = 1100\) USD.

According to the bond yield to maturity calculation formula \(P = C\times\frac{1-(1 + r)^{-n}}{r}+\frac{F}{(1 + r)^{n}}\), that is, \(1100=80\times\frac{1-(1 + r)^{-8}}{r}+\frac{1000}{(1 + r)^{8}}\).

Using trial and error:

- When \(r = 6\%\):

\(80\times\frac{1-(1 + 0.06)^{-8}}{0.06}+\frac{1000}{(1 + 0.06)^{8}}\)

\(=80\times\frac{1 - 0.627412}{0.06}+\frac{1000}{1.593848}\)

\(=80\times\frac{0.372588}{0.06}+ 627.412\)

\(=80\times6.2098+627.412\)

\(=496.784+627.412 = 1124.196\)

- When \(r = 7\%\)：

\(80\times\frac{1-(1 + 0.07)^{-8}}{0.07}+\frac{1000}{(1 + 0.07)^{8}}\)

\(=80\times\frac{1 - 0.582009}{0.07}+\frac{1000}{1.718186}\)

\(=80\times\frac{0.417991}{0.07}+582.009\)

\(=80\times5.9713+582.009\)

\(=477.704+582.009=1059.713\)

Through interpolation:

\(r=6\%+\frac{1124.196 - 1100}{1124.196 - 1059.713}\times(7\% - 6\%)\)

\(=6\%+\frac{24.196}{64.483}\times1\%\)

\(\approx 6.375\%\)

### b. Yield to maturity (YTM) when interest is paid semiannually

Semi-annual interest payment amount\(C_{semi}=1000\times8\%\div2 = 40\) USD, term\(n_{semi}=8\times2 = 16\) periods, bond face value\(F = 1000\) USD, current price\(P = 1100\) USD.

Assume that the six-month maturity yield is \(r_{semi}\), according to the formula \(P = C_{semi}\times\frac{1-(1 + r_{semi})^{-n_{semi}}}{r_{semi}}+\frac{F}{(1 + r_{semi})^{n_{semi}}}\), that is, \(1100 = 40\times\frac{1-(1 + r_{semi})^{-16}}{r_{semi}}+\frac{1000}{(1 + r_{semi})^{16}}\).

Using trial and error:

- When \(r_{semi}=3\%\):

\(40\times\frac{1-(1 + 0.03)^{-16}}{0.03}+\frac{1000}{(1 + 0.03)^{16}}\)

\(=40\times\frac{1 - 0.623167}{0.03}+\frac{1000}{1.604706}\)

\(=40\times\frac{0.376833}{0.03}+623.167\)

\(=40\times12.5611+623.167\)

\(=502.444+623.167 = 1125.611\)

- When\(r_{semi}=4\%\):

\(40\times\frac{1-(1 + 0.04)^{-16}}{0.04}+\frac{1000}{(1 + 0.04)^{16}}\)

\(=40\times\frac{1 - 0.533908}{0.04}+\frac{1000}{1.872981}\)

\(=40\times\frac{0.466092}{0.04}+533.908\)

\(=40\times11.6523+533.908\)

\(=466.092+533.908 = 1000\)

By interpolation:

\(r_{semi}=3\%+\frac{1125.611 - 1100}{1125.611 - 1000}\times(4\% - 3\%)\)

\(=3\%+\frac{25.611}{125.611}\times1\%\)

\(\approx 3.204\%\)

Annual yield to maturity\(YTM = 2\times r_{semi}=2\times3.204\% = 6.408\%\)

### c. Current Yield

The current yield calculation formula is\(Current\ Yield=\frac{Annual\ Interest\ Payment}{Bond\ Price}\).

The annual interest payment is \(C = 80\) dollars, and the bond price is \(P = 1100\) dollars, so the current yield is \(=\frac{80}{1100}\approx7.27\%\) .

In summary, a. about \(6.375\%\); b. about \(6.408\%\); c. about \(7.27\%\) .