Question

# There are two bonds in a portfolio. One is a 5-year zero-coupon bond with a face...

There are two bonds in a portfolio. One is a 5-year zero-coupon bond with a face value of \$5,000, the other is a 10-year zero-coupon bond with a face value of \$10,000. The Macaulay Duration of the portfolio is 7.89, the Modified Duration of the portfolio is 7.3015. If the price of the 10-year bond is \$3,999, what is the answer that is closest to the yield to maturity of the 5-year bond

The Duration of a zero-coupon bond is equal to its maturity.

Duration of 10 year zero-coupon bond = 10 years

Duration of 5year zero-coupon bond = 5years

w(10), w(5) are the weights of 10 year and 5 year bonds in the portfolio

w(10) + w(5) =1

Portfolio duration is the weighted average of the individual duration of bonds

10*w(10) + 5*w(5) = 7.89

10*w(10) + 5*(1-w(10)) = 7.89

w(10) = 2.89/5 = 0.578

w(5) = 1- 0.578 = 0.422

Weights are based on the market price of the bonds

w(10) = 3999/(3999+x)

0.578 = 3999/(3999+x)

where x is the price of the 5-year zero coupon bond

We get, x = 2919.68

Hence, for a 5-year zero coupon bond

Current price = 2919.68

Yield =

Yield of 5-year bond = (5000/2919.68)^(1/5) -1

Yield to maturity of 5-year bond = 0.11359 = 11.359%

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