Chapter 20, Problem 1P

(a)

To determine

Find the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding annually for 10 years.

Answer to Problem 1P

The future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding annually for 10 years is $\$19216.7$.

Explanation of Solution

Given data:

The present cost $\left(P\right)$ is $10000$,

The normal interest rate $\left(i\right)$ is $6.75\%$,

The number of years $\left(n\right)$ is $10\text{years}$,

The number of compound period $\left(m\right)$ is $1$.

Formula used:

Formula to calculate the future value is,

$F=P{\left(1+i\right)}^n$ (1)

Here,

$P$ is the present cost,

$i$ is the normal interest rate,

$n$ is the number of years.

Calculation:

To calculate the future value:

Substitute $10000$ for $P$, $6.75\%$ for $i$, and $10\text{years}$ for $n$ in equation (1) to find $F$.

$\begin{array}{c}F=\left(10000\right){\left(1+\left(6.75\%\right)\right)}^{10}\\ =\left(10000\right){\left(1+0.0675\right)}^{10}\\ =\$19216.7\end{array}$

Therefore, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding annually for 10 years is $\$19216.7$.

Conclusion:

Thus, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding annually for 10 years is $\$19216.7$.

(b)

To determine

Find the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding quarterly for 10 years.

Answer to Problem 1P

The future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding quarterly for 10 years is $\$19530$.

Explanation of Solution

Given data:

The present cost $\left(P\right)$ is $10000$,

The normal interest rate $\left(i\right)$ is $6.75\%$,

The number of years $\left(n\right)$ is $10\text{years}$,

The number of compound period $\left(m\right)$ is $4$.

Formula used:

Formula to calculate the future value is,

$F=P{\left(1+\frac{i}{m}\right)}^{nm}$ (2)

Here,

$P$ is the present cost,

$i$ is the normal interest rate,

$n$ is the number of years,

$m$ is the number of compound periods per year.

Calculation:

To calculate the future value:

Substitute $10000$ for $P$, $6.75\%$ for $i$, $10\text{years}$ for $n$, and $4$ for $m$ in equation (2) to find $F$.

$\begin{array}{c}F=\left(10000\right){\left(1+\frac{0.0675}{4}\right)}^{\left(10\right)\left(4\right)}\\ =\left(10000\right){\left(1+\frac{0.0675}{4}\right)}^{\left(40\right)}\\ =\$19530\end{array}$

Therefore, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding quarterly for 10 years is $\$19530$.

Conclusion:

Thus, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding quarterly for 10 years is $\$19530$.

(c)

To determine

Find the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding monthly for 10 years.

Answer to Problem 1P

The future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding monthly for 10 years is $\$19603.21$.

Explanation of Solution

Given data:

The present cost $\left(P\right)$ is $10000$,

The normal interest rate $\left(i\right)$ is  $6.75\%$,

The number of years $\left(n\right)$ is $10\text{years}$,

The number of compound periods per year $\left(m\right)$ is $12$.

Formula used:

Formula to calculate the future value is,

$F=P{\left(1+\frac{i}{m}\right)}^{nm}$ (3)

Here,

$P$ is the present cost,

$i$ is the normal interest rate,

$n$ is the number of years,

$m$ is the number of compound periods per year.

Calculation:

To calculate the future value:

Substitute $10000$ for $P$, $6.75\%$ for $i$, $10\text{years}$ for $n$, and $12$ for $m$ in equation (3) to find $F$.

$\begin{array}{c}F=\left(10000\right){\left(1+\frac{0.0675}{12}\right)}^{\left(10\right)\left(12\right)}\\ =\left(10000\right){\left(1+\frac{0.0675}{12}\right)}^{\left(120\right)}\\ =\$19603.21\end{array}$

Therefore, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding monthly for 10 years is $\$19603.21$.

Conclusion:

Thus, the future value for $\$10000\text{at}\text{6}\text{.75\%}$ compounding monthly for 10 years is $\$19603.21$.