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As a New Year's gift to yourself, you buy your roommate's 1976 Ford Pinto. She has given you the option of two payment plans. Under Plan A, you pay \(\$ 500\) now, plus \(\$ 500\) at the beginning of each of the next two years. Under Plan \(\mathrm{B}\), you would pay nothing down, but \(\$ 800\) at the beginning of each of the next two years. a. Calculate the present value of each plan's payments if interest rates are \(10 \%\). Should you choose Plan A or Plan \(\mathrm{B} ?\) b. Recalculate the present value of each plan's payments using a \(20 \%\) interest rate. Should you choose Plan \(\mathrm{A}\) or Plan B? c. Explain why your answers to (a) and (b) differ.

Short Answer

Expert verified
At 10% interest, choose Plan A. At 20% interest, choose Plan B. Higher interest rates increase the discounting of future payments.

Step by step solution

01

Calculate Present Value for Plan A (10% Interest Rate)

For Plan A, the payments are \(500 now and \)500 at the beginning of each of the next two years. Given the 10% interest rate, the present value (PV) of each payment is:- Year 0: \(500 (no discounting needed)- Year 1: \)500 discounted at 10% or PV = \( \frac{500}{(1+0.10)^1} = 454.55 \)- Year 2: $500 discounted at 10% or PV = \( \frac{500}{(1+0.10)^2} = 413.22 \)The total present value for Plan A is:\[PV = 500 + 454.55 + 413.22 = 1367.77\]
02

Calculate Present Value for Plan B (10% Interest Rate)

For Plan B, you pay \(800 at the beginning of the next two years. Calculate the present value at 10%:- Year 0: No payment made.- Year 1: \)800 discounted at 10% or PV = \( \frac{800}{(1+0.10)^1} = 727.27 \)- Year 2: $800 discounted at 10% or PV = \( \frac{800}{(1+0.10)^2} = 661.16 \)The total present value for Plan B is:\[PV = 727.27 + 661.16 = 1388.43\]Thus, with a 10% interest rate, Plan A has a lower PV.
03

Calculate Present Value for Plan A (20% Interest Rate)

Recalculate the present value for Plan A using a 20% interest rate:- Year 0: \(500 (no discounting needed)- Year 1: \)500 discounted at 20% or PV = \( \frac{500}{(1+0.20)^1} = 416.67 \)- Year 2: $500 discounted at 20% or PV = \( \frac{500}{(1+0.20)^2} = 347.22 \)The total present value for Plan A is:\[PV = 500 + 416.67 + 347.22 = 1263.89\]
04

Calculate Present Value for Plan B (20% Interest Rate)

Recalculate the present value for Plan B using a 20% interest rate:- Year 0: No payment made.- Year 1: \(800 discounted at 20% or PV = \( \frac{800}{(1+0.20)^1} = 666.67 \)- Year 2: \)800 discounted at 20% or PV = \( \frac{800}{(1+0.20)^2} = 555.56 \)The total present value for Plan B is:\[PV = 666.67 + 555.56 = 1222.23\]With a 20% interest rate, Plan B has a lower PV.
05

Compare Plans and Explanation of Differences

At 10%, Plan A (PV = 1367.77) is better than Plan B (PV = 1388.43). At 20%, Plan A (PV = 1263.89) is less favorable than Plan B (PV = 1222.23). The differences occur because higher interest rates increase the discounting effect on future payments, making later payments relatively less expensive in present value terms. Thus, as interest rates rise, Plan B becomes more attractive due to heavier discounting of its larger future payments.

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Key Concepts

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

Interest Rates
Interest rates are a crucial concept in understanding present value and its application to financial decision making. Essentially, an interest rate is the cost of borrowing money and is usually expressed as a percentage. It helps in determining how much more money will be needed later to repay a loan, or how much extra one will earn from an investment. In our Ford Pinto example, the interest rate is pivotal in computing present value. A 10% or 20% interest rate means that money you receive (or pay) in the future needs to be adjusted accordingly in today's terms to reflect this cost of borrowing. By applying an interest rate, we can calculate how much future payments are worth today, which helps in comparing different payment plans.
Discounting Future Payments
Discounting is the process of determining the present value of a payment or series of payments that will be received or paid in the future. This process involves using an interest rate to 'discount' future amounts to reflect their value in today's terms. In practical terms, discounting acknowledges that a dollar received today is worth more than a dollar received tomorrow because of the potential earning capacity of the dollar over that time. For instance, in Plan A of the exercise, you pay $500 now and $500 in each of the next two years. The future payments are discounted using the interest rate to calculate what they are worth today. This is done both for 10% and 20% interest rates, each affecting the calculations differently and, consequently, impacting the choice between Plan A and B.
Payment Plans
Payment plans involve structuring when and how payments for a purchase are made. For this exercise, we have two different plans for the car purchase.
  • Plan A involves an initial payment and two future payments, offering a spread-out approach.
  • Plan B postpones all payment to two equal future payments without an initial cost.
Deciding on a payment plan involves evaluating the present value of each plan. Plan A might be preferable if you can afford a down payment and prefer smaller future payments. Conversely, Plan B might suit someone who wants to delay costs and can handle larger future payments. The choice depends significantly on current financial situations and expected future cash flows.
Microeconomic Decision Making
Microeconomic decision making addresses how individuals and businesses make choices to maximize utility or profit with limited resources. When faced with financial choices, like different payment plans, individuals evaluate options based on cost, convenience, and how these fit within their financial goals or constraints. In microeconomics, understanding present value helps in comparing the economic impact of hiring resources today versus in the future. In our car example, the decision between Plan A and Plan B can be seen through a microeconomic lens. You'll weigh the opportunity cost of paying cash now or taking advantage of having the money available for other uses, versus the ease or difficulty of larger future payments, all factored through the lens of interest and present value concepts. These choices reflect everyday economic decisions, highlighting the intersection of personal finance and economic theory.

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Most popular questions from this chapter

A classmate offers to play the following game: He will roll a 10 -sided die; if it comes up between 1 and \(9,\) he will pay you \(\$ 10 ;\) if it comes up a \(10,\) he will pay you \(\$ 110 .\) a. If you are risk-neutral, and base all decisions on expected monetary value, what is the most you will pay to play this game? b. Your classmate Risa has a utility function that depends on wealth. Specifically, \(U=W^{2}\) If Risa bases her decisions on expected utility, what is the most she would pay to play this game? What can you ascertain about Risa's attitude toward risk?

Imagine that you have \(\$ 100\) of ill-gotten gains stashed in an offshore bank account. Lest the IRS get too nosey, you plan to leave that account idle until your retirement in 45 years. a. If your bank pays you \(3 \%\) annual interest, what will your account balance be upon retirement? b. If your bank pays you \(6 \%\) interest, what will your account balance be upon retirement? c. Does doubling the interest rate double your accumulated balance at retirement? More than double it? Less? Explain your answer.

Many college graduates feel as if their student loan payments drag on forever. Suppose that the government offers the following arrangement: It will pay for your college in its entirety, and in return you will make annual payments until the end of time. a. Suppose the government asks for \(\$ 6,000\) each year for all of eternity. If interest rates currently sit at \(4 \%,\) what is the present value of the payments you will make? b. Your college charges \(\$ 140,000\) for four years of quality education. Should you take the government up on its offer to pay for your college? What if your college charges \(\$ 160,000 ?\)

Perry the picker has stumbled across a piece of pottery in an antique shop. Because of its style, he believes it might be by Frog Woman, a famous Hopi artist. If he buys the pot for \(\$ 3,000\), he will be able to resell it for \(\$ 4,500\), provided it is a genuine Frog Woman pot. If it is not genuine, he will be forced to unload it for only \(\$ 1,000\). Perry estimates that there is a $$2 / 3$$ chance the pot is genuine. a. If Pemy bases his decision on expected monetary value, should he buy the pot? b. Perry's utility depends on his wealth: Specifically, \(U=W^{1 / 4}\) Compute Perry's utility at the possible levels of final wealth he might experience. c. If Perry bases his decision to buy on expected utility, what should he do? d. Suppose Perry spends a little bit too much time in the hotel bar, and buys the pot without doing the math first. At breakfast the next moming, Penny, another picker, notices the pot and decides to make an offer for it. What is the minimum Penny would have to offer to convince Perry to sell. (Remember - Perry still doesn't know whether the pot is genuine; an offer from Penny removes all risk.) e. What is Perry's risk premium?

Ricardo is considering purchasing an ostrich, which he can graze for free in his backyard. Once the ostrich reaches maturity (in exactly three years), Ricardo will be able to sell it for \(\$ 2,000\). The ostrich costs \(\$ 1,500\). a. Suppose that interest rates are \(8 \% .\) Calculate the net present value of the ostrich investment. Does the NPV indicate that Ricardo should buy the ostrich? b. Suppose that Ricardo passes on the ostrich deal and invests \(\$ 1,500\) in his next-best opportunity: a safe government bond yielding \(8 \% .\) How much money will he have at the end of three years? Is this outcome better or worse than buying the ostrich? c. Calculate the net present value of the ostrich if interest rates are \(11 \% .\) Does the NPV method indicate that Ricardo should buy the ostrich? d. If Ricardo passes on the ostrich deal and invests in a government bond yielding \(11 \%,\) how much money will he have at the end of three years? Is this outcome better or worse than buying the ostrich? e. Based on your answers to (b) and (d), how well does the NPV method capture the concept of opportunity cost?

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