Let's break down and solve the problem step-by-step:
### a. Present Value of the First Option
The first option offers you [tex]$7,200 per month for the next two years.
1. Identify the parameters:
- Monthly salary (\( Pmt \)): $[/tex]7,200
- Number of months ([tex]\( n \)[/tex]): 2 years [tex]\(\times\)[/tex] 12 months/year = 24 months
- Annual interest rate ([tex]\( r_{annual} \)[/tex]): 7%
2. Convert annual interest rate to a monthly rate:
[tex]\[
r_{monthly} = \frac{r_{annual}}{12} = \frac{0.07}{12} = 0.00583333
\][/tex]
3. Use the present value of an annuity formula:
[tex]\[
PV = Pmt \times \left( \frac{1 - (1 + r_{monthly})^{-n}}{r_{monthly}} \right)
\][/tex]
Plugging in the values, we get:
[tex]\[
PV = 7,200 \times \left( \frac{1 - (1 + 0.00583333)^{-24}}{0.00583333} \right) \approx 160,812.71
\][/tex]
Therefore, the present value of the first option is [tex]\( \$ 160,812.71 \)[/tex].
### b. Present Value of the Second Option
The second option offers you [tex]$5,900 per month for the next two years and a $[/tex]32,000 signing bonus today.
1. Identify the parameters for monthly payments:
- Monthly salary ([tex]\( Pmt \)[/tex]): [tex]$5,900
- Number of months (\( n \)): 24 months
- Monthly interest rate (\( r_{monthly} \)): 0.00583333 (same as above)
2. Calculate the present value of the monthly payments using the annuity formula as above:
\[
PV_{\text{payments}} = Pmt \times \left( \frac{1 - (1 + r_{monthly})^{-n}}{r_{monthly}} \right)
\]
Plugging in the values, we get:
\[
PV_{\text{payments}} = 5,900 \times \left( \frac{1 - (1 + 0.00583333)^{-24}}{0.00583333} \right) \approx 131,777.09
\]
3. Add the signing bonus:
- Signing bonus today is not discounted because it's paid right now, so it retains its value: $[/tex]32,000
4. Total present value of the second option:
[tex]\[
PV_{\text{option 2}} = PV_{\text{payments}} + \text{Signing bonus} = 131,777.09 + 32,000 = 163,777.09
\][/tex]
Therefore, the present value of the second option is [tex]\( \$ 163,777.09 \)[/tex].
### Summary
```
\begin{tabular}{|l|l|}
\hline
a. Value of first option & \[tex]$ 160,812.71 \\
\hline
b. Value of second option & \$[/tex] 163,777.09 \\
\hline
\end{tabular}
```
