Chapter 5, Problem 38QP

Summary Introduction

To think critically about: The relationship between the annuity value and the interest rate.

Introduction:

The total sum of interest that is due for a particular time is the interest rate. The rate of interest can be due for a period as a proportion of the sum borrowed or deposited and as the proportion of the sum lent.

The future sum of money that is worth today is described by the present value. The present value of the cash flows in the future with a particular discount rate is the present value of annuity.

Explanation of Solution

The relationship between the interest rate and the annuity value is as follows:

• If the interest rate maximizes, the present value of the annuity would decrease and the present value of annuity would increase, if the interest rate decreases.
• If the rate of interest increases, the future value of the annuity would increase and the future value of annuity would decrease if the interest rate decreases.

Summary Introduction

To calculate: The present value of annuity if the interest rate is 10%, a fall in the interest rate of 5%, and a rise in the interest rate of 15%

Answer to Problem 38QP

• The present value of annuity with an interest rate of10% is$61,445.67.
• The present value of annuity with an interest rate of 5% is $77,217.35.
• The present value of annuity with an interest rate of 15% is$50,187.69.

Explanation of Solution

Given information:

Person X purchased a ten-year annuity of $10,000 for a year at the current rate of interest of 10%. It is assumed that there is a sudden fall in the interest rate of 5% and rise in the interest rate of 15%.

Formula to calculate the present value of annuity:

$\text{Present value of annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}$

Note: C denotes the annual cash flow, r denotes the rate of exchange, and t denotes the period.

Compute the present value of annuity at 10% interest:

$\begin{array}{c}\text{Present value of annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.10\right)}^{10}}\right)\right]}{0.10}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1.10\right)}^{10}}\right)\right]}{0.10}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{2.59374246}\right)\right]}{0.10}\right\}\end{array}$

$\begin{array}{c}=\$10,000\left\{\frac{\left[0.61445671\right]}{0.10}\right\}\\ =\$10,000\times 6.144567106\\ =\$61,445.67\end{array}$

Hence, the present value of annuity at 10% is $61,445.67.

Compute the present value of annuity at 5% interest:

$\begin{array}{c}\text{Present value of annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$10,00\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.05\right)}^{10}}\right)\right]}{0.05}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1.05\right)}^{10}}\right)\right]}{0.05}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{1.628894627}\right)\right]}{0.05}\right\}\end{array}$

$\begin{array}{c}=\$10,000\left\{\frac{0.386086746}{0.05}\right\}\\ =\$10,000\times 7.721734931\\ =\$77,217.35\end{array}$

Hence, the present value of annuity at 5% is $77,217.35.

Compute the present value of annuity at 15% interest:

$\begin{array}{c}\text{Present value of annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.15\right)}^{10}}\right)\right]}{0.15}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1.15\right)}^{10}}\right)\right]}{0.15}\right\}\\ =\$10,000\left\{\frac{\left[1-\left(\frac{1}{4.045557736}\right)\right]}{0.15}\right\}\end{array}$

$\begin{array}{c}=\$10,000\left\{\frac{0.752815293}{0.15}\right\}\\ =\$10,000\times 5.018768626\\ =\$50,187.69\end{array}$

Hence, the present value of annuity at 15% is $50,187.69.