Question $\backslash$#$3$: Present Value $[16$ Points$]$

Let's consider a scenario where soccer player, Kylian Mbappe is deciding between two contract offers from two teams: FC Barcelona and Paris Saint$-$Germain. You, as his sports agent, are tasked with advising Mbappe accordingly.

$($a$)$ Suppose that FC Barcelona makes the following contract offer:

Mbappe would receive a salary of $\backslash (43\backslash )$ million this year $(2025),$ a salary of $\backslash (48\backslash )$ million next year $(2026),$ a salary of $\backslash (51\backslash )$ million in two years $(2027),$ and a salary of $\backslash (68\backslash )$ million in three years $(2028).$ Additionally, Mbappe would receive a signing bonus of $\backslash (60\backslash )$ million payable this year $(2025).$

If the current interest rate is $\backslash (4\backslash \%\backslash ),$ what is the present value of the contract offer by FC Barcelona? $[4$ Points$]$

$($b$)$ Suppose that Paris Saint$-$Germain makes the following contract offer:

Mbappe would receive a salary of $\backslash (5\backslash )$ million this year $(2025),$ a salary of $\backslash (38\backslash )$ million next year $(2026),$ a salary of $\backslash (77\backslash )$ million in two years $(2027),$ and a salary of $\backslash (113\backslash )$ million in three years $(2028).$ Paris Saint$-$Germain would not pay a signing bonus to Mbappe if he signed with the team.

If the current interest rate is $\backslash (4\backslash \%\backslash ),$ what is the present value of the contract offer by Paris SaintGermain. Based on your answer from Parts $($a$)-($b$),$ would you recommend that Mbappe sign with FC Barcelona or with Paris Saint$-$Germain? $[5$ Points$]$

$($c$)$ Suppose the current interest rate is $\backslash (1\backslash \%\backslash )($instead of $4\backslash \%).$ Recalculate the present values of the FC Barcelona contract and the Paris Saint$-$Germain contract. Does your recommendation change? $[5$ Points$]$

$($d$)$ Based on your calculations from Parts $($a$)-($c$)$ what is the relationship between interest rates and present value? $[2$ Points$]$