Answer to Question #195258 in Microeconomics for Narendra

Question

Please show all working out and explain

There are two riskless bonds which mature at t = 2. The first is a zero coupon bond that pays a balloon of $ 1,200. The other is a coupon bond with an annual coupon of $100 and a balloon payment of $990. The current yield on a riskless bond that mature in one year is 10%, the annualized yield on a two-year bond is also 10%.

(i) What is the duration of the zero coupon bond? And of the coupon bond? Explain its economic meaning.

(ii) Suppose the interest rate at t = 1 is 8%. Find the percentage price change for the zero coupon bond. Suppose next the interest rate at t=1 is 12%: find again the percentage change for the coupon bond. Compare the findings obtained in the two cases and explain them.

(iii) Compute then the percentage change for the coupon bond when the interest rate at t = 1 is 8% and then when it is 12%. Compare with the result obtained in the previous point (ii) and discuss.

Answer 1

i) The duration of the zero-coupon bond.

The weighted average of the time it takes to collect cash flows from a bond is known as Macaulay length. It is expressed as a number of years.

The length of a zero-coupon bond is equal to the bond's maturity since the payout occurs at the bond's maturity.

The coupon bond's length can be measured as follows:

The bond has a duration of 1.90 years.

The bond's current value is $991.24.

(II)

The effect of a change in the interest rate on the bond's price

At t=1, the bond's value is $\frac{1200}{0.909}=1090.8$

(For 1 year, the discounting factor is set at 10%.)

If the interest rate were to rise to 8%,

The bond's value would be $1200\times0.926=1111.2$

The bond's value has changed from $1111.2\ to\ 1090.8\ or\ 20.4\%$ . $\frac{20.4}{1090.8}\times100=1.87\%$

If the interest rate was raised to 12%,

The bond's value would be $1200$ $\times$ $0.893$ $=1071.6$

The bond's value has changed from $1071.6 \ to\ 1090.8\ or\ 19.2\%$. $\frac{19.2}{1090.8}\times100=1.76\%$ The market value of an existing bond declines as market interest rates rise. When interest rates in the economy fall, the market value of an existing bond rises. An inverse relationship is defined as the relationship between market interest rates and the market value of a bond.

iii)

Impact on the price of the coupon bond in the change in the interest rate

The market value of an existing bond declines as market interest rates rise. When interest rates in the economy fall, the market value of an existing bond rises. An inverse relationship is defined as the relationship between market interest rates and the market value of a bond.