Chapter 6.6, Problem 57E

To determine

To calculate: The rates of the investment.

Answer to Problem 57E

The required rates of return is $0.071$ .

Explanation of Solution

Given Information:

The given expression is $t=\frac{\ln 2}{\ln \left(1+r\right)}$ .

Where $t$ is time and $r$ is rate of return.

Also time is given $10$ years.

Formula used:

Property of logarithm:

If ${\ln }_{e}x=a$ then $x=e^a$ .

Calculation:

Consider the expression is $t=\frac{\ln 2}{\ln \left(1+r\right)}$ .

Substitute $10$ for $t$ and $0.693$ for $\ln 2$ in the above expression.

  $10=\frac{0.693}{\ln \left(1+r\right)}$

Simplify the above expression.

  $\begin{array}{l}10=\frac{0.693}{\ln \left(1+r\right)}\\ \ln \left(1+r\right)=\frac{0.693}{10}\\ \ln \left(1+r\right)=0.069\end{array}$

Use property of logarithm in the above expression.

  $1+r=e^{0.069}$

Substitute $2.718$ for $e$ in the above expression.

  $1+r={\left(2.718\right)}^{0.069}$

Simplify the expression.

  $\begin{array}{c}1+r={\left(2.718\right)}^{0.069}\\ 1+r=1.071\\ r=1.071-1\\ r=0.071\end{array}$

Hence, the require rate of return is $0.071$ .