Chapter 5, Problem 22CT

(a)

To determine

To calculate: The future value, if $\$1000$ is invested at $5\%$ compounded monthly for $8$ .

Answer to Problem 22CT

Solution:

The future value is $\$1033.82$ , if $\$1000$ is invested at $5\%$ compounded monthly for $8$ .

Explanation of Solution

Given information:

Principle amount $A=\$1000$ .

Compounded interest rate per annum $r=0.05$ .

Number of times the interest compounded per annum $n=12$ .

Amount compounded number of months $t=8=\frac{8}{12}$ years.

Formula used:

The amount $A$ after $t$ years due to a principal $P$ invested at an annual interest rate $r$ , expressed as a decimal, compounded $n$ times per year, is $A=P\cdot {\left(1+\frac{r}{n}\right)}^{nt}$ .

Calculation:

Use the future value formula with $A=\$1000$ , $r=0.05$ , $n=12$ and $t=\frac{8}{12}$ to find $A$ .

  $A=P\cdot {\left(1+\frac{r}{n}\right)}^{nt}$

  $\Rightarrow A=1000\cdot {\left(1+\frac{0.05}{12}\right)}^{\left(12\times \frac{8}{12}\right)}$

  $\Rightarrow A=1000\cdot {\left(1+\frac{0.05}{12}\right)}^{\left(8\right)}$

  $\Rightarrow A=1000\cdot {\left(1.004166667\right)}^{\left(8\right)}$

  $\Rightarrow A=1000\cdot \left(1.033823517\right)$

  $\Rightarrow A=\$1033.823517$

Thus, the future value is $\$1033.82$ , if $\$1000$ is invested at $5\%$ compounded monthly for $8$ .

(b)

To determine

To calculate: The principal amount, if you want to have $\$1000$ in $9$ months at $5\%$ compounded quarterly.

Answer to Problem 22CT

Solution:

The principal amount needed to get $\$1000$ after $9$ months at $5\%$ interest rate, if interest is compounded quarterly, is $\$963.42$ .

Explanation of Solution

Given information:

Future Value $A=\$1000$ .

Compounded interest rate per annum $r=0.05$ .

Number of times the interest compounded per annum $n=4$ .

Amount compounded number of months $t=9=\frac{9}{12}$ years.

Formula used:

The principal amount P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is $P=A\cdot {(1+\frac{r}{n})}^{-nt}$ .

Calculation:

Use the present value formula with $A=\$1000$ , $r=0.05$ , $n=4$ and $t=2$ to find $P$ .

  $P=A\cdot {(1+\frac{r}{n})}^{-nt}$

  $\Rightarrow P=1000\cdot {\left(1+\frac{0.05}{4}\right)}^{\left(-4\times \frac{9}{12}\right)}$

  $\Rightarrow P=1000\cdot {\left(1+\frac{0.05}{4}\right)}^{\left(-3\right)}$

  $\Rightarrow P=1000\cdot {\left(1.0125\right)}^{\left(-3\right)}$

  $\Rightarrow P=1000\cdot \left(0.9634183287\right)$

  $\Rightarrow P=963.4183287$

Hence, the principal amount needed to get $\$1000$ after $9$ months at $5\%$ interest rate, if compounded quarterly, is $\$963.42$ .

(c)

To determine

To calculate: The years required to double your money if you can invest it at $6\%$ compounded annually.

Answer to Problem 22CT

Solution:

The years required to double your money, if you can invest it at $6\%$ compounded annually, is $11.9$ .

Explanation of Solution

Given information:

Amount deposited $P$ .

Future Value $A=2P$ .

Rate of interest $r=0.06$ .

Number of times the interest compounded per annum $n=1$ .

Formula used:

The amount $A$ after $t$ years due to a principal $P$ invested at an annual interest rate $r$ , expressed as a decimal, compounded $n$ times per year, is $A=P\cdot {\left(1+\frac{r}{n}\right)}^{nt}$ .

Calculation:

Use the future value formula with $A=2P$ , $r=0.06$ , $n=1$ to find $t$ .

  $A=P\cdot {\left(1+\frac{r}{n}\right)}^{nt}$

  $\Rightarrow 2P=P\cdot {\left(1+\frac{0.06}{1}\right)}^{1t}$

Dividing both sides by $P$ ,

  $\Rightarrow 2={\left(1.06\right)}^t$

As $y={\log }_{a}x$ and $x=a^y,$ where $a>0$ and $a\ne 1,$ are equivalent,

  $\Rightarrow t={\log }_{1.06}2$

By using change of base formula,

  $\Rightarrow t=\frac{\ln 2}{\ln 1.06}$

  $\Rightarrow t=\frac{\ln 2}{\ln 1.06}$

  $\Rightarrow t=11.89566105$

  $\Rightarrow t\approx 11.9$

Thus, the years required to double your money, if you can invest it at $6\%$ compounded annually, is $11.9$ .