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Chapter 6: Problem 47

You've just won the U.S. Lottery. Lottery officials offer you the choice of two alternative payouts: either \(\$ 2\) million today or \(\$ 4\) million 10 years from now. Which payout will you choose if the relevant discount rate is 0 percent? If it is 10 percent? If it is 20 percent?

Short Answer

Expert verified
For a 0% discount rate, choose Option 2 (\(PV_2 = \$ 4,000,000\)). For a 10% discount rate, choose Option 1 (\(PV_1 = \$ 2,000,000\)). For a 20% discount rate, choose Option 1 (\(PV_1 = \$ 2,000,000\)).

Step by step solution

01

Option 1: Present value of $2 million today

The present value of $2 million today is simply the same as its face value since there's no discounting to be applied. Therefore, the present value of Option 1 is: \(PV_1 = \$ 2,000,000\)
02

Option 2: Calculate present value of $4 million with 0% discount rate

In this case, we'll use the present value formula to calculate the present value of $4 million in 10 years with a 0% discount rate: \(PV_2 = \frac{\$ 4,000,000}{(1 + 0)^{10}} = \$ 4,000,000\)
03

Option 2: Calculate present value of $4 million with 10% discount rate

Now, we'll calculate the present value of $4 million in 10 years with a 10% discount rate: \(PV_2 = \frac{\$ 4,000,000}{(1 + 0.1)^{10}} \approx \$ 1,535,690\)
04

Option 2: Calculate present value of $4 million with 20% discount rate

Finally, we'll calculate the present value of $4 million in 10 years with a 20% discount rate: \(PV_2 = \frac{\$ 4,000,000}{(1 + 0.2)^{10}} \approx \$ 615,927\)
05

Make a choice for each discount rate

Now, let's compare the present values of the two options for each discount rate: - For 0% discount rate: \(PV_1 = \$ 2,000,000\) and \(PV_2 = \$ 4,000,000\). We would choose Option 2 because it has a higher present value. - For 10% discount rate: \(PV_1 = \$ 2,000,000\) and \(PV_2 \approx \$ 1,535,690\). We would choose Option 1 because it has a higher present value. - For 20% discount rate: \(PV_1 = \$ 2,000,000\) and \(PV_2 \approx \$ 615,927\). We would choose Option 1 because it has a higher present value.

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Key Concepts

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

Discount Rate
The discount rate is a critical concept in finance. It represents the interest rate used to determine the present value of future cash flows. This rate helps in adjusting the value of money from a future date to its current worth. Essentially, a higher discount rate means that future cash values are worth less today.

Think of the discount rate as the opposite of compound interest. Where compound interest grows your money over time, the discount rate reduces future sums to their present-day equivalent. This is crucial in making informed financial decisions.
  • Low Discount Rate: Keeps future values relatively high in present terms. Choosing higher future payouts could be beneficial.
  • High Discount Rate: Reduces future values significantly. Taking immediate cash might often be favorable.
In investment scenarios like the lottery example, the discount rate directly affects the attractiveness of future payouts. Assessing this rate helps determine better financial choices by comparing present vs. future value effectively.
Investment Decision
Making an investment decision involves evaluating various financial choices to maximize returns. In contexts like choosing between immediate or delayed payouts, understanding potential gains or losses is essential.

An investment decision should always be guided by the present value of options. Here's how you can approach it:
  • Calculate Present Values: Determine the present value of each option to compare them on the same financial footing.
  • Analyze Discount Rates: Recognize how different rates influence the future value, impacting decision making.
  • Consider Risk Tolerance: Higher potential gains might come with larger risks and vice versa.
To make a sound investment decision, always weigh the discounted future sums against current options. The use-case from the exercise illustrates that at a 0% discount rate, future payments hold their face value, making a large sum in the future more appealing. Yet, higher discount rates tilt the favor towards getting money now, as delayed values diminish significantly.
Time Value of Money
The time value of money is a fundamental concept in finance that acknowledges money's potential to earn over time. Essentially, a dollar today is worth more than a dollar received in the future, considering that money can be invested to generate returns.

This principle underpins why present values differ from future values. It affects personal finances, loans, mortgages, investments, and most financial decisions.
  • Present Value (PV): The current worth of a future sum, calculated using the discount rate.
  • Future Value (FV): The value of an investment or cash flow at a specific future date as it grows over time.
Understanding the time value of money helps in discerning why higher future sums may not always be the best choice. By applying this concept, as shown in the exercise, we see that at zero discount rate, waiting for a future sum didn't devalue it. However, with higher rates, the same future amount's worth today diminishes, favoring immediate payouts.

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Most popular questions from this chapter

Friendly's Quick Loans, Inc., offers you "three for four or I knock on your door." This means you get \(\$ 3\) today and repay \(\$ 4\) when you get your paycheck in one week (or else). What's the effective annual return Friendly's earns on this lending business? If you were brave enough to ask, what APR would Friendly's say you were paying?

This problem illustrates a deceptive way of quoting interest rates called add- on interest. Imagine that you see an advertisement for Crazy Judy's Stereo City that reads something like this: "\$1,000 Instant Credit! \(14 \%\) Simple Interest! Three Years to Pay! Low, Low Monthly Payments!" You're not exactly sure what all this means and somebody has spilled ink over the APR on the loan contract, so you ask the manager for clarification. Judy explains that if you borrow \(\$ 1,000\) for three years at 14 percent interest, in three years you will owe: $$\$ 1,000 \times 1.14^{3}=\$ 1,000 \times 1.48154=\$ 1,481.54$$ Now, Judy recognizes that coming up with \(\$ 1,481.54\) all at once might be a strain, so she lets you make "low, low monthly payments" of \(\$ 1,481.54 / 36=\) \(\$ 41.15\) per month, even though this is extra bookkeeping work for her. Is this a 14 percent loan? Why or why not? What is the APR on this loan? What is the EAR? Why do you think this is called add-on interest?

What is the present value of \(\$ 1,000\) per year, at a discount rate of 12 percent, if the first payment is received 8 years from now and the last payment is received 20 years from now?

You have \(\$ 1,100\) today. You need \(\$ 2,000 .\) If you earn 1 percent per month, how many months will you wait?

As discussed in the text, an annuity due is identical to an ordinary annuity except that the periodic payments occur at the beginning of each period and not at the end of the period (see Question 56). Show that the relationship between the value of an ordinary annuity and the value of an otherwise equivalent annuity due is: Annuity due value \(=\) Ordinary annuity value \(\times(1+r)\) Show this for both present and future values.
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