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Chapter 7: Problem 20

A rookie quarterback is in the process of negotiating his first contract. The team's general manager has offered him three possible contracts. Each of the contracts lasts for 4 years. All of the money is guaranteed and is paid at the end of each year. The terms of each of the contracts are listed below: The quarterback discounts all cash flows at 10 percent. Which of the three contracts offers him the most value?

Short Answer

Expert verified
Compute present values for each contract's cash flows and compare their totals.

Step by step solution

01

- Understanding the Problem

We are given three different contracts, each lasting 4 years. The cash flows for these contracts occur at the end of each year. We need to determine which contract has the highest present value, discounted at a rate of 10%.
02

- Present Value Formula

The present value (PV) of a future cash flow is calculated using the formula: \( PV = \frac{C}{(1 + r)^n} \)where \( C \) is the cash flow in each year, \( r \) is the discount rate, and \( n \) is the year number. We will apply this formula to each year's cash flow for all contracts.
03

- Contract 1 Analysis

Calculate the present value for each year's cash flow under Contract 1 and sum them up:- Year 1: \( PV = \frac{C_1}{(1 + 0.1)^1} \)- Year 2: \( PV = \frac{C_2}{(1 + 0.1)^2} \)- Year 3: \( PV = \frac{C_3}{(1 + 0.1)^3} \)- Year 4: \( PV = \frac{C_4}{(1 + 0.1)^4} \)
04

- Contract 2 Analysis

Calculate the present value for each year's cash flow under Contract 2 and sum them up:- Year 1: \( PV = \frac{C_1}{(1 + 0.1)^1} \)- Year 2: \( PV = \frac{C_2}{(1 + 0.1)^2} \)- Year 3: \( PV = \frac{C_3}{(1 + 0.1)^3} \)- Year 4: \( PV = \frac{C_4}{(1 + 0.1)^4} \)
05

- Contract 3 Analysis

Calculate the present value for each year's cash flow under Contract 3 and sum them up:- Year 1: \( PV = \frac{C_1}{(1 + 0.1)^1} \)- Year 2: \( PV = \frac{C_2}{(1 + 0.1)^2} \)- Year 3: \( PV = \frac{C_3}{(1 + 0.1)^3} \)- Year 4: \( PV = \frac{C_4}{(1 + 0.1)^4} \)
06

- Compare Present Values

Sum the present values calculated for each year for all three contracts. The contract with the highest sum is the most valuable in present value terms. Compare the total present value of Contract 1, 2, and 3 to determine which contract offers the most value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discount Rate
When we talk about the discount rate in financial contexts, we are referring to the interest rate used to determine the present value of future cash flows. This is crucial in contract valuation, especially in our exercise about the quarterback's contract choices.

Understanding the discount rate helps us see how the future cash flows are adjusted to reflect today's value. This adjustment is important because money today is often worth more than the same amount in the future due to factors like inflation and opportunity cost. In our example, the quarterback uses a discount rate of 10%.

The discount rate can greatly affect the perception of a contract's value. A higher discount rate means future cash flows are worth less today.
  • It can serve as a measure of risk: the higher the uncertainty of future cash flows, the higher the discount rate tends to be.
  • Affects decision-making in investments and contracts.
Cash Flow Analysis
Cash flow analysis involves studying the money flowing in and out of a business or contract over time. It鈥檚 a vital tool for evaluating investment opportunities like the ones our rookie quarterback needs to consider.

In the context of the contract valuation problem, each year鈥檚 cash flow, which is the guaranteed amount he will receive at the end of the year, needs to be analyzed and calculated for its present value. This gives us an idea of each contract's worth in today's terms.

To effectively conduct a cash flow analysis, follow these steps:
  • Identify all cash inflows - in this case, the payments made in each year of the contract.
  • Use the present value formula to adjust each inflow to its present value.
  • Sum all present values to determine the total worth of the contract today.
This analysis forms the basis for making informed financial decisions.
Contract Valuation
Contract valuation is determining how much a contract is worth today based on its future cash flows. This concept is crucial for our quarterback as he needs to decide which contract option gives him the most value.

To value a contract, consider the following factors:
  • Timing of Cash Flows: Cash expected sooner is more valuable than later.
  • Size of Cash Flows: Larger amounts naturally increase the contract's present value.
  • Discount Rate: Influences the present value calculation, as discussed previously.
Understanding these factors helps in comparing different contracts by calculating and summing the present values of each year鈥檚 cash flow.

Once you have all the present values, simply add them up for each contract. The contract that offers the highest sum indicates the greatest value. Thus, contract valuation simplifies the complex decision of choosing the best financial offer. This is the method used to determine the most valuable contract for our quarterback.

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Most popular questions from this chapter

Find the future values of the following ordinary annuities: a. \(\mathrm{FV}\) of \(\$ 400\) each 6 months for 5 years at a nominal rate of 12 percent, compounded semiannually. b. \(\mathrm{FV}\) of \(\$ 200\) each 3 months for 5 years at a nominal rate of 12 percent, compounded quarterly. c. The annuities described in parts a and b have the same amount of money paid into them during the 5 -year period, and both earn interest at the same nominal rate, yet the annuity in part b earns \(\$ 101.75\) more than the one in part a over the 5 years. Why does this occur?

The prize in last week's Florida lottery was estimated to be worth \(\$ 35\) million. If you were lucky enough to win, the state will pay you \(\$ 1.75\) million per year over the next 20 years. Assume that the first installment is received immediately. a. If interest rates are 8 percent, what is the present value of the prize? b. If interest rates are 8 percent, what is the future value after 20 years? c. How would your answers change if the payments were received at the end of each year?

Find the present value of \(\$ 500\) due in the future under each of the following conditions: a. 12 percent nominal rate, semiannual compounding, discounted back 5 years. b. 12 percent nominal rate, quarterly compounding, discounted back 5 years. c. 12 percent nominal rate, monthly compounding, discounted back 1 year.

An investment pays \(\$ 20\) semiannully for the next 2 years. The investment has a 7 percent nominal interest rate, and interest is compounded quarterly. What is the future value of the investment?

Assume that AT\&T's pension fund managers are considering two alternative securities as investments: (1) Security Z (for zero intermediate year cash flows), which costs \(\$ 422.41\) today, pays nothing during its 10 -year life, and then pays \(\$ 1,000\) after 10 years or (2) Security \(\mathrm{B}\), which has a cost today of \(\$ 1,000\) and which pays \(\$ 80\) at the end of each of the next 9 years and then \(\$ 1,080\) at the end of Year 10 a. What is the rate of return on each security? b. Assume that the interest rate AT\&T's pension fund managers can earn on the fund's money falls to 6 percent, compounded annually, immediately after the securities are purchased and is expected to remain at that level for the next 10 years. What would the price of each security change to, what would the fund's profit be on each security, and what would be the percentage profit (profit divided by cost) for each security? c. Assuming that the cash flows for each security had to be reinvested at the new 6 percent market interest rate, (1) what would be the value attributable to each security at the end of 10 years and (2) what "actual, after-the-fact" rate of return would the fund have earned on each security? (Hint: The "actual" rate of return is found as the interest rate that causes the \(\mathrm{PV}\) of the compounded Year 10 amount to equal the original cost of the security.) d. Now assume all the facts as given in parts b and \(c,\) except assume that the interest rate rose to 12 percent rather than fell to 6 percent. What would happen to the profit figures as developed in part b and to the "actual" rates of return as determined in part c? Explain your results.
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