When Alex Rodriguez moved to the Texas Rangers in $2001,$ he received a lot of attention for his $"$$$252$ million" contract $($the total of the payments promised was $$252$ million$).$ He later moved to

the Yankees, but assume the following in order to determine the value of his contract when he signed it:

Rodriguez earns $$16$ million in the first year, $$17$ million in years $2$ through $4,$$$19$ million in years $5$ and $6,$$$23$ million in year $7,$ and $$27$ million in years $8$ through $10.$ He also receives his $$10$ million

signing bonus spread equally over the first $5$ years $($ $$2$ million per year$).$ His deferred payments begin in $2011.$ The deferred payment amounts total $$33$ million and are $$5$ million, then $$4$ million,

then eight amounts of $$3$ million $($ending in $2020).$ However, the actual payouts will be different. All of the deferred payments will earn $4\%$ per year until they are paid. For example, the $$5$ million is

deferred from $2001$ to $2011,$ or $10$ years, meaning that it will actually be $$7\mathrm{.}4012$ million when paid. Assume that the $$4$ million payment deferred to $2012$ is deferred from $2002($each payment is

deferred $10$ years$).$

The contract is a $10-$year contract, but each year has a deferred component so that cash flows are paid out over a total of $20$ years. The contractual payments, signing bonus, and deferred

components are given in the following table:

$.$ Note that, by contract, the deferred components are not paid in the year they are earned, but instead are paid $($plus interest$)10$ years later.

Assume that an appropriate discount rate for Rodriguez to apply to the contract payments is $7\%$ per year.

a$.$ Calculate the true promised payments under this contract, including the deferred payments with interest.

b$.$ Draw a timeline of all of the payments.

c$.$ Calculate the present value of the contract.

d$.$ Compare the present value of the contract to the quoted value of $$252$ million. What explains the difference?