Chapter 5, Problem 90RE

To determine

To find: The bond fund worth at maturity, the effective rate of interest, and the time requires to double the bond value.

Answer to Problem 90RE

The bond fund worth at maturity, the effective rate of interest, and the time requires to double the bond value are $\$20398.87$ , $4.04\%$ , and $17.5\text{years}$ simultaneously.

Explanation of Solution

Given information:

The initial cost of the bond is $\$10000$ , the maturity time is $18\text{years}$ , and the interest rate is $4\%$ compounded semi-annually.

The bond fund worth at maturityis calculated as,

  $\begin{array}{c}A=P{\left(1+\frac{r}{n}\right)}^{nt}\\ =\left(10000\right){\left(1+\frac{0.04}{2}\right)}^{2\times 18}\\ =\$20398.87\end{array}$

The effective rate of interest is calculated as,

  $\begin{array}{c}{r}_e={\left(1+\frac{r}{n}\right)}^n-1\\ ={\left(1+\frac{0.04}{2}\right)}^2-1\\ =0.0404\times 100\%\\ =4.04\%\end{array}$

The time required to double the bond worth is calculated as,

  $\begin{array}{l}A=P{\left(1+\frac{r}{n}\right)}^{nt}\\ 20000=\left(10000\right){\left(1+\frac{0.04}{2}\right)}^{2\times t}\\ 2={\left(1.02\right)}^{2t}\\ \ln 2=2t\ln \left(1.02\right)\end{array}$

Simplify further,

  $t=17.5\text{years}$

Thus, the bond fund worth at maturity, the effective rate of interest, and the time requires to double the bond value are $\$20398.87$ , $4.04\%$ , and $17.5\text{years}$ simultaneously.