Chapter 3, Problem 9P

To determine

The explanation of the choice made by each woman.

Answer to Problem 9P

Magdalen made a correct choice if the annual interest is greater than $25.99\%$.

Miriam made a correct choice if the annual interest is less than $25.99\%$.

Mary June made a correct choice if the annual interest rate is equal to $25.99\%$.

Explanation of Solution

Given:

Two different cash flows.

• $\$500$
• will be received today.

$\$1000$

will be received after 3 years.

Concept used:

There are two available options.

Either take

$\$500$ today or $\$1000$ after 3 years.

Calculation:

Write the expression to calculate the present value of $\$1000$

$PV=\frac{\$1000}{{\left(1+i\right)}^n}$

Here, interest is $i$, and number of years is $n$.

Explanation for the choice made by Magdalen:

Magdalen choose to take $\$500$ in spite of taking $\$1000$ after three years. Magdalen’s choice will be rational if the present value of

$\$1000$ received after three years is less than $\$500$.

$500>\frac{1000}{{\left(1+i\right)}^n}$ ...... (I)

Substitute, $3$ for $n$ in Equation (I) and calculate the interest rate.

$\begin{array}{l}500>\frac{1000}{{\left(1+i\right)}^3}\\ 500{\left(1+i\right)}^3>1000\\ {\left(1+i\right)}^3>\frac{1000}{500}\\ {\left(1+i\right)}^3>2\end{array}$

$\begin{array}{l}1+i>\sqrt[3]{2}\\ 1+i>1.2599\\ i>1.2599-1\\ i>0.2599\end{array}$

$i>25.99\%$

Therefore Magdalen made a correct choice if the annual interest is greater than $25.99\%$.

Explanation for the choice made by Miriam:

Miriam chooses to take $\$1000$ after 3 years in spite of $\$500$ today. Miriam choice will be rational if the present value of

$\$1000$ after 3 years is greater than the

$\$500$.

$500<\frac{1000}{{\left(1+i\right)}^n}$ ...... (II)

Substitute, $3$ for $n$ in Equation (II) and calculate the interest rate.

$\begin{array}{l}500<\frac{1000}{{\left(1+i\right)}^3}\\ 500{\left(1+i\right)}^3<1000\\ {\left(1+i\right)}^3<\frac{1000}{500}\\ {\left(1+i\right)}^3<2\end{array}$

$\begin{array}{l}1+i<\sqrt[3]{2}\\ 1+i<1.2599\\ i<1.2599-1\\ i<0.2599\end{array}$

$i<25.99\%$

Therefore Miriam made a correct choice if the annual interest is less than $25.99\%$.

Explanation for the choice made by Mary June Mary June choice is indifferent between taking $\$500$ today or $\$1000$ after three years can be considered a rational choice if the present value of $\$1000$ after three years is equal to $\$500$ today.

$500=\frac{1000}{{\left(1+i\right)}^n}$ ...... (III)

Substitute, $3$ for $n$ in Equation (III) and calculate the interest rate.

$\begin{array}{l}500=\frac{1000}{{\left(1+i\right)}^3}\\ 500{\left(1+i\right)}^3=1000\\ {\left(1+i\right)}^3=\frac{1000}{500}\\ {\left(1+i\right)}^3=2\end{array}$

$\begin{array}{l}1+i=\sqrt[3]{2}\\ 1+i=1.2599\\ i=1.2599-1\\ i=0.2599\end{array}$

$i=25.99\%$

Therefore, Mary June made a correct choice if the annual interest rate is equal to $25.99\%$.

Conclusion:

Magdalen made a correct choice if the annual interest is greater than $25.99\%$.

Miriam made a correct choice if the annual interest is less than $25.99\%$.

Mary June made a correct choice if the annual interest rate is equal to $25.99\%$.