Chapter 12.4, Problem 30E

a.

To determine

Todraw a time line to show that to set up an annuity in perpetuity of amount R per time period, the amount must be invested now is $A_p=\frac{R}{1+i}+\frac{R}{{\left(1+i\right)}^2}+\frac{R}{{\left(1+i\right)}^3}+\frac{R}{{\left(1+i\right)}^4}+\cdot \cdot \cdot +\frac{R}{{\left(1+i\right)}^n}+\cdot \cdot \cdot$ .

Explanation of Solution

Given:

An annuity in perpetuity of amount R per time period.

Calculation:

The time line for an annuity in perpetuity of amount R per time period can be drawn as shown below.

  

That is,

  $A_p=\frac{R}{1+i}+\frac{R}{{\left(1+i\right)}^2}+\frac{R}{{\left(1+i\right)}^3}+\frac{R}{{\left(1+i\right)}^4}+\cdot \cdot \cdot +\frac{R}{{\left(1+i\right)}^n}+\cdot \cdot \cdot$ .

b.

To determine

To find the sum of the infinite series in part (a) to show that $A_p=\frac{R}{i}$ .

Explanation of Solution

Given:

From part (a), $A_p=\frac{R}{1+i}+\frac{R}{{\left(1+i\right)}^2}+\frac{R}{{\left(1+i\right)}^3}+\frac{R}{{\left(1+i\right)}^4}+\cdot \cdot \cdot +\frac{R}{{\left(1+i\right)}^n}+\cdot \cdot \cdot$ .

Calculation:

  $\begin{array}{c}{A}_p=\frac{R}{1+i}+\frac{R}{{\left(1+i\right)}^2}+\frac{R}{{\left(1+i\right)}^3}+\frac{R}{{\left(1+i\right)}^4}+\cdot \cdot \cdot +\frac{R}{{\left(1+i\right)}^n}+\cdot \cdot \cdot \\ =R\left[\frac{1}{1+i}+\frac{1}{{\left(1+i\right)}^2}+\frac{1}{{\left(1+i\right)}^3}+\frac{1}{{\left(1+i\right)}^4}+\cdot \cdot \cdot +\frac{1}{{\left(1+i\right)}^n}+\cdot \cdot \cdot \right]\\ =R\frac{\left(\frac{1}{1+i}\right)}{1-\left(\frac{1}{1+i}\right)}\\ =R\frac{\left(\frac{1}{1+i}\right)}{\left(\frac{i}{1+i}\right)}\\ =\frac{R}{i}\end{array}$

That is, $A_p=\frac{R}{i}$

c.

To determine

To find the amount of money that need to be invested now to provide an annuity in perpetuity of $5000 per year.

Answer to Problem 30E

  $\$50000$

Explanation of Solution

Given:

The annuity amount of $5000 per year and the rate 10% compounded annually.

Calculation:

From part (b), $A_p=\frac{R}{i}$ .

Substitute $R=5000$ and $i=0.10$ to find the value of $A_p$ .

  $\begin{array}{c}{A}_p=\frac{R}{i}\\ =\frac{5000}{0.10}\\ =\$50000\end{array}$

So, the amount to be invested now is $\$50000$ .

d.

To determine

To find the amount of money that need to be invested now to provide an annuity in perpetuity of $3000 per year if the interest is 8% compounded quarterly.

Answer to Problem 30E

  $\$\text{36393}\text{.56}$

Explanation of Solution

Given:

The annuity amount of $3000 per year and the rate 8% compounded quarterly.

Calculation:

The amount that needs to be invested if the interest is compounded quarterly is

  $\begin{array}{c}{A}_p=\frac{R}{{\left(1+\frac{i}{4}\right)}^4}+\frac{R}{{\left(1+\frac{i}{4}\right)}^{2\times 4}}+\frac{R}{{\left(1+\frac{i}{4}\right)}^{3\times 4}}+\frac{R}{{\left(1+\frac{i}{4}\right)}^{4\times 4}}+\cdot \cdot \cdot +\frac{R}{{\left(1+\frac{i}{4}\right)}^{n\times 4}}+\cdot \cdot \cdot \\ =R\frac{\left(\frac{1}{{\left(1+\frac{i}{4}\right)}^4}\right)}{1-\left(\frac{1}{{\left(1+\frac{i}{4}\right)}^4}\right)}\\ =R\frac{1}{{\left(1+\frac{i}{4}\right)}^4-1}\end{array}$

Substitute $R=3000$ and $i=0.08$ to find the value of $A_p$ .

  $\begin{array}{c}{A}_p=R\frac{1}{{\left(1+\frac{i}{4}\right)}^4-1}\\ =3000\cdot \frac{1}{{\left(1+\frac{0.08}{4}\right)}^4-1}\\ =\$\text{36393}\text{.5629}\\ \approx \$\text{36393}\text{.56}\end{array}$

So, the amount to be invested now is $\$\text{36393}\text{.56}$ .