Chapter 5, Problem 2QP

Present Value and Multiple Cash Flows.  Investment X offers to pay you $3,400 per year for nine years, whereas Investment Y offers to pay you $5,200 per year for five years. Which of these cash flow streams has the higher present value if the discount rate is 6 percent? If the discount rate is 22 percent?

Summary Introduction

To calculate: The investment of cash flow that gives the higher present value @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 $23,125.75 and $12,873.37 respectively and the cash flow Y at 6% and 22% is $21,904.29 and $14,890.93 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 $3,400 in a year for nine years, whereas Investment Y provides $5,200 in a year for five years. The rate of discount is given.

Time line for Investment X:

Time line for Investment Y:

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%:

$\begin{array}{c}\text{Present value annuity at }6\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$3,400\left\{\frac{\left[1-\frac{1}{{\left(1+0.06\right)}^9}\right]}{0.06}\right\}\\ =\$3,400\left\{\frac{1-\frac{1}{1.689478959}}{0.06}\right\}\\ =\$3,400\left\{\frac{1-0.591898463}{0.06}\right\}\end{array}$

$\begin{array}{c}=\$3,400\left\{\frac{0.408101536}{0.06}\right\}\\ =\$3,400\times 6.801692274\\ =\$23,125.75\end{array}$

Hence, the present value annuity for Investment X at 6% is $23,125.75.

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

$\begin{array}{c}\text{Present value annuity at }6\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$5,200\left\{\frac{\left[1-\frac{1}{{\left(1+0.06\right)}^5}\right]}{0.06}\right\}\\ =\$5,200\left\{\frac{1-\frac{1}{1.338225578}}{0.06}\right\}\\ =\$5,200\left\{\frac{1-0.747258172}{0.06}\right\}\end{array}$

$\begin{array}{c}=\$5,200\left\{\frac{0.252741827}{0.06}\right\}\\ =\$5,200\times 4.212363786\\ =\$21,904.29\end{array}$

Hence, the present value annuity for Investment Y at 6% is $21,904.29.

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

$\begin{array}{c}\text{Present value annuity at }22\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$3,400\left\{\frac{\left[1-\frac{1}{{\left(1+0.22\right)}^9}\right]}{0.22}\right\}\\ =\$3,400\left\{\frac{1-\frac{1}{5.9874028}}{0.22}\right\}\\ =\$3,400\left\{\frac{1-0.167017325}{0.22}\right\}\end{array}$

$\begin{array}{c}=\$3,400\left\{\frac{0.832982674}{0.22}\right\}\\ =\$3,400\times 3.786284886\\ =\$12,873.37\end{array}$

Hence, the present value annuity for Investment X at 22% is $12,873.37.

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

$\begin{array}{c}\text{Present value annuity at }22\%=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$5,200\left\{\frac{\left[1-\frac{1}{{\left(1+0.22\right)}^5}\right]}{0.22}\right\}\\ =\$5,200\left\{\frac{1-\frac{1}{2.702708163}}{0.22}\right\}\\ =\$5,200\left\{\frac{1-0.369999252}{0.22}\right\}\end{array}$

$\begin{array}{c}=\$5,200\left\{\frac{0.630000747}{0.22}\right\}\\ =\$5,200\times 2.863639762\\ =\$14,890.93\end{array}$

Hence, the present value annuity for Investment Y at 22% is $14,890.93.