You’ve just joined the investment banking firm of Dewey, Cheatum, and Howe. They’ve offered you two different salary arrangements. You can have $7,000 per month for the next two years, or you can have $5,700 per month for the next two years, along with a $31,000 signing bonus today. Assume the interest rate is 6 percent compounded monthly.

Requirement 1:
If you take the first option, $7,000 per month for two years, what is the present value? (Enter rounded answer as directed, but do not use rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Requirement 2:
What is the present value of the second option? (Enter rounded answer as directed, but do not use rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Making use of the present value concept is a useful way to compare cases where money is to be received in the future. Generally, the higher the present value, the better.

In this case, we are given two different salary arrangements, each one with an annuity involved; that is, in each arrangement we get an equal amount of money every month.
Since there is an annuity involved, we can make use of the next formula to calculate the present value of each case:

[tex]P=A[\frac{(1+i)^{n}-1 }{i(1+i)^{n} } ][/tex]

Where P: Present value (what we are required to calculate),

A: Annuity (the amount of money to be received every month; that is, $7,000 for the first option, and $5,700 for the second option),

i: Interest rate to be used (6%, or 0.06), and

n: Number of periods of time that will pass (in both cases, we are talking about 2 years, but since the 6% interest rate is compounded monthly, we have to consider the months involved, as shown below).

[tex]n=2 years*12\frac{months}{year}\\n=24months[/tex]

Now we can proceed to determine the present value of each case by substituting all the known values that we have.

Requirement 1: If you take the first option, $7,000 per month for two years, what is the present value?

[tex]P=A[\frac{(1+i)^{n}-1 }{i(1+i)^{n} } ]\\\\P=7000*[\frac{(1+0.06)^{24}-1}{0.06*(1+0.06)^{24}} ]\\\\P=7000*[\frac{(1.06)^{24}-1}{0.06*(1.06)^{24}} ]\\\\P=7000*[\frac{4.04893-1}{0.06*4.04893} ]\\\\P=7000*[\frac{3.04893}{0.24294} ]\\\\P=7000*[12.55036]\\\\P=87852.50269[/tex]

So, the present value of the first option, where we get $7,000 every month for 2 years, is P≈$87,852.50.

Requirement 2: What is the present value of the second option?

In this case, we get $5,700 per month for two years, plus a $31,000 bonus today.

Since we get this bonus exactly today, it adds directly into the present value, so we should now calculate the present value of the annuity (just like we did in the first case), and add it to the $31,000 bonus.

[tex]P=31000+A[\frac{(1+i)^{n}-1 }{i(1+i)^{n} } ]\\\\P=31000+5700*[\frac{(1+0.06)^{24}-1}{0.06*(1+0.06)^{24}} ]\\\\P=31000+5700*[\frac{3.04893}{0.24294} ]\\\\P=31000+5700*[12.55036]\\\\P=31000+71537.03791\\\\P=102537.03791[/tex]

So, the present value of the second option, where we get $5,700 every month for 2 years, along with a $31,000 bonus, is P≈$102,537.04.

It goes without saying that, given our calculations, the second option is much better than the first one.