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$6.$ The Indianapolis Colts turn to the free$-$agent market to try to find a starting quarterback.

They find a promising free agent and sign him to a five$-$year contract. The contract has

the following provisions:

$$ The player receives a signing bonus of $$15$ million immediately.

$$ He receives a salary payment of $$20$ million at the end of each of the next five years.

$($That$$s the period during which he$$s obliged to play for the team.$)$

$$ He receives deferred payments of $$5$ million per year for another seven years,

beginning at the end of the sixth year after he signs the contract.

During a press conference at which the signing is announced, the player$$s agent describes

the agreement as a $$one hundred fifty million dollar contract.$$

A$.$ In what sense is the agent$$s description correct? Explain.

B$.1.$ In what sense is the agent$$s description incorrect? Explain.

Hint: It may be helpful to answer the next question $$ or$,$ at least,

to think about how you$$d approach answering it $$ before you try to answer

this question.

$2.$ Assume, here and below, that the market interest rate is $4$ percent. How much

would it cost the Pacers, in money paid up front, to arrange for a bank to make

the various payments, including the signing bonus, required by the contract?

Hints: What$$s the market price of a bond that returns these payments?

What$$s the present value of a payment received immediately?

What sort of credit instrument does the sequence of salary

payments look like? How is the sequence of deferred payments different?

The market value of an N$-$year annuity that$$s missing the first K annual

payments is equal to the market value of a normal N$-$year annuity that

provides the same annual payments, minus the market value of a normal

K$-$year annuity that also provides the same annual payments.

$3.$ How much would each of the five annual salary payments have to be

increased, leaving the other payments unchanged, so that the present value of

the contract would really be $$150$ million?

Hint: This is a two$-$step problem. First, figure out what the

present value of the salary payments would need to be$,$ in order for the

present value of the entire contract to be $$150$ million. Next, use the annuity

pricing formula, and some algebra, to identify the annual salary payment

that would make the present value of the salary payments equal to the

required value.

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