Chapter 5, Problem 35QP

Summary Introduction

To select: The preferable salary arrangement offered by an investment bank.

Introduction:

The cash flow is the overall money that is transacted inside and outside the business mainly while affecting the liquidity of the business.

Answer to Problem 35QP

The first salary arrangement gives a present value annuity of $136,244.106and the second one gives$107,208.477. Additionally, while comparing the future value for both the arrangements, the first arrangement provides the highest value. As the first arrangement has the higher present and future value, Person X will prefer the first option.

Explanation of Solution

Given information:

Person X has joined an investment banking company. The company provides two various salary arrangements to Person X. The first one is that Person X will be provided $6,100 per month for the upcoming two years. The second arrangement is that Person X will be provided $4,800 per month for the upcoming two years including a bonus of $25,000 at present. The rate of interest is given at 7% compounded on a monthly basis.

Note: It is necessary to compare the 2 cash flows by finding the cash flows value at a common period. Hence, we have to compute the present value of every cash flow stream. As the cash flows are monthly, it is necessary to use the monthly rate of interest.

Formula to compute the monthly rate:

$\text{Monthly rate}=\frac{\text{Rate of interest}}{\text{Number of months}}$

Compute the monthly rate:

$\begin{array}{c}\text{Monthly rate}=\frac{\text{Rate of interest}}{\text{Number of months}}\\ =\frac{0.07}{12}\\ =0.0058\text{ or }0.58\%\end{array}$

Hence, the monthly rate is 0.58%.

Formula of present value annuity:

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

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

Compute the present value annuity for first arrangement:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$6,100\left\{\frac{\left[1-\frac{1}{{\left(1+0.0058\right)}^{12\times 2}}\right]}{0.0058}\right\}\\ =\$6,100\left\{\frac{1-\frac{1}{{\left(1.0058\right)}^{24}}}{0.0058}\right\}\end{array}$

$\begin{array}{c}=\$6,100\left\{22.33509931\right\}\\ =\$136,244.106\end{array}$

Hence, the present value annuity for the first salary arrangement is $136,244.106.

Compute the present value annuity for second arrangement:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$4,800\left\{\frac{\left[1-\frac{1}{{\left(1+0.0058\right)}^{12\times 2}}\right]}{0.0058}\right\}\end{array}$

$\begin{array}{c}=\$4,800\left\{\frac{1-\frac{1}{{\left(1.0058\right)}^{24}}}{0.0058}\right\}\\ =\$4,800\left\{22.33509931\right\}\\ =\$107,208.477\end{array}$

Hence, the present value annuity for the second salary arrangement is $107,208.477.

Note: The first choice is best in the above case.

Formula to compute the value of second option:

$\text{Value of second option}=\text{Present value annuity}+\text{Bonus}$

Compute the value of second option:

$\begin{array}{c}\text{Value of second option}=\text{Present value annuity}+\text{Bonus}\\ =\$107,208.477+\$25,000\\ =\$132,208.48\end{array}$

Hence, the value of second option is $132,208.48.

Formula to compute the difference in the value at present:

$\text{Difference in the value at present}=\left(\begin{array}{l}\text{Present value annuity in the first option}-\\ \text{Value in the second option}\end{array}\right)$

Compute the difference in the value at present:

$\begin{array}{c}\text{Difference in the value at present}=\left(\begin{array}{l}\text{Present value annuity in the first option}-\\ \text{Value in the second option}\end{array}\right)\\ =\$136,244.106-\$132,208.48\\ =\$4,035.63\end{array}$

Hence, the difference in the value at present is $4,035.63.

Note: Compute the future value for the two cash flows.

Formula to compute the future value annuity:

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

Where,

“C” denotes the annual cash flow or annuity payment,

“r” denotes the rate of interest,

“t” denotes the period.

Compute the future value annuity for the first arrangement:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$6,100\left\{\frac{\left[{\left(1+0.0058\right)}^{24}-1\right]}{0.0058}\right\}\\ =\$6,100\left\{25.67100962\right\}\\ =\$156,593.16\end{array}$

Hence, the future value annuity for the first arrangement is $156,593.16.

Compute the future value for the second arrangement:

$\begin{array}{c}\text{Future value}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}+\text{PV}{\left(1+r\right)}^t\\ =\$4,800\left\{\frac{\left[{\left(1+0.0058\right)}^{24}-1\right]}{0.0058}\right\}+\$25,000{\left(1+0.0058\right)}^{24}\\ =\$4,800\left\{25.68103159\right\}+\$28,745.15044\\ =\$152,014.102\end{array}$

Hence, the future value for the second arrangement is $152,014.102.

Note: The first option is again the best choice.

Formula to compute the difference in the future value:

$\text{Difference in the future value}=\left(\begin{array}{l}\text{Future value annuity in the first arrangement}-\\ \text{Future value in the second arrangement}\end{array}\right)$

Compute the difference in the future value:

$\begin{array}{c}\text{Difference in the future value}=\left(\begin{array}{l}\text{Future value annuity in the first arrangement}-\\ \text{Future value in the second arrangement}\end{array}\right)\\ =\$156,593.16-\$152,014.102\\ =\$4,579.06\end{array}$

Hence, the difference in the future value is $4,579.06.