Chapter 5, Problem 2QP

Summary Introduction

To calculate: The investment of cash flow that gives the higher present value at 6% and 22% rate of discount.

Introduction:

The present value of future cash flows that is discounted at a particular rate of discount is called the present value.

Answer to Problem 2QP

The cash flow X at 6% and 22% is $25,166.26 and $14,009.25 respectively and the cash flow Y at 6% and 22% is $23,168 and $15,750.02 respectively. Note that the present value cash flow is greater than at 6% in both the investment. At 6% rate of interest, Investment X is more valuable since it has the highest present value and at 22% interest, Investment Y has the highest present value.

Explanation of Solution

Given information:

Investment X provides Person X $3,700 in a year for nine years, whereas Investment Y provides $5,500 in a year for five years. The rate of discount are 6% and 22%.

Formula to calculate the present value annuity:

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

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

Compute the present value annuity for Investment X at 6%:

$\begin{array}{c}\text{Present value annuity at }6\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$3,700\left\{\frac{\left[1-\frac{1}{{\left(1+0.06\right)}^9}\right]}{0.06}\right\}\\ =\$3,700\left\{\frac{1-\frac{1}{1.689478959}}{0.06}\right\}\\ =\$3,700\left\{\frac{1-0.591898463}{0.06}\right\}\end{array}$

$\begin{array}{c}=\$3,4700\left\{\frac{0.408101536}{0.06}\right\}\\ =\$3,700\times 6.801692274\\ =\$25,166.26\end{array}$

Hence, the present value annuity for Investment X at 6% is $25,166.26.

Compute the present value annuity for Investment Y at 6%:

$\begin{array}{c}\text{Present value annuity at }6\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$5,500\left\{\frac{\left[1-\frac{1}{{\left(1+0.06\right)}^5}\right]}{0.06}\right\}\\ =\$5,500\left\{\frac{1-\frac{1}{1.338225578}}{0.06}\right\}\\ =\$5,500\left\{\frac{1-0.747258172}{0.06}\right\}\end{array}$

$\begin{array}{c}=\$5,500\left\{\frac{0.252741827}{0.06}\right\}\\ =\$5,500\times 4.212363786\\ =\$23,168\end{array}$

Hence, the present value annuity for Investment Y at 6% is $23,168.

Compute the present value annuity for Investment X at 22%:

$\begin{array}{c}\text{Present value annuity at }22\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$3,700\left\{\frac{\left[1-\frac{1}{{\left(1+0.22\right)}^9}\right]}{0.22}\right\}\\ =\$3,700\left\{\frac{1-\frac{1}{5.9874028}}{0.22}\right\}\\ =\$3,700\left\{\frac{1-0.167017325}{0.22}\right\}\end{array}$

$\begin{array}{c}=\$3,700\left\{\frac{0.832982674}{0.22}\right\}\\ =\$3,700\times 3.786284886\\ =\$14,009.25\end{array}$

Hence, the present value annuity for Investment X at 22% is $14,009.25.

Compute the present value annuity for Investment Y at 22%:

$\begin{array}{c}\text{Present value annuity at }22\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$5,500\left\{\frac{\left[1-\frac{1}{{\left(1+0.22\right)}^5}\right]}{0.22}\right\}\\ =\$5,500\left\{\frac{1-\frac{1}{2.702708163}}{0.22}\right\}\\ =\$5,500\left\{\frac{1-0.369999252}{0.22}\right\}\end{array}$

$\begin{array}{c}=\$5,500\left\{\frac{0.630000747}{0.22}\right\}\\ =\$5,500\times 2.863639762\\ =\$15,750.02\end{array}$

Hence, the present value annuity for Investment Y at 22% is $15,750.02.