11 you, or someone you know, is a regular lottery player, you probably already understand that you are 20 times more likely to be killed by a lightning bolt than to win a big lottery jackpot. How bad are the odds? Nearby you will find a table comparing your chances ot winning the Mega Millions Lottery to other events.
Big Game: Is It Worth the Gamble?_
Odds of winning Mega Millions jackpot 1:135,145,920* Odds of being killed by a venomous spider 1:57,018,763
Odds of being killed by a dog bite 1:11,403,753
Odds of being killed by lightning 1:6,479,405
Odds of being killed by drowning 1:690,300
Odds of being killed falling from a bed 1 388,411
or other furniture
Odds of being killed In a car crash 1:6,029
'Source: Virginia Lottery Web site. All other odds from the National Safety Council.
Sweepstakes may have different odds than lotteries, but the odds may not be much better. Probably the largest advertised grand prize ever was Pepsi's "Play for Billion," which, you guessed it, had a $1 billion (billion!) prize. Not bad for a day's work, but you still have to read the fine print. It turns out that the winner would be paid $5 million per year for the next 20 years, $10 million per year for years 21 to 39, and a lump sum $710 million In 40 years. From what you have learned, you know the value of the sweepstakes wasn't even close to $1 billion. In fact, at an Interest rate of 10 percent, the present value is about $70.7 million.
Lottery jackpots are often paid out over 20 or more years, but the winner often can choose to take a lump sum cash payment instead. For example, in 2007, an 84-year-old retired electrician won the $254 million Powerball lottery. He had the option of a single cash payment of $121 million or payments over the next 30 years. Which do you think he chose?
Some lotteries make your decision a little tougher. The Ontario Lottery will pay you either $2,000 a week for the rest of your life or $1.3 million now. (That's in Canadian dollars, by the way) Of course, there is the chance you might die in the near fulure, SO the lottery guarantees that your heirs will collect the $2,000 weekly payments until the 20th anniversary of the first payment, or until you would have turned 91, whichever comes first. This payout scheme complicates your decision quite a bit. If you live for only the 20-year minimum, the break-even interest rate between the two options is about 5.13 percent per year, compounded weekly. If you expect to live longer than the 20-year minimum, you might be better off accepting $2,000 per week for life. Of course, if you manage to invest the $1.3 million lump sum at a rate of return of about 8 percent per year (compounded weekly), you can have your cake and eat it too since the investment will return $2,000 at the end of each week forever! Taxes complicate the decision in this case because the lottery payments are all on an aftertax basis. Thus, the rates of return in this example would have to be aftertax as well,
As the accompanying Reality Bytes box shows, calculating present values is a vital step in comparing alternative cash flows. We will have much more to say on this subject in subsequent chapters.
How Much Is It Worth?
You are offered an investment that will pay you £200 in one year, $400 the next year, $600 the next year, and $800 at the end of the next year You can earn 12 percent on very similar investments What is the most you should pay for this one?
We need to calculate the present value of these cash flows at 12 percent. Taking them one at a lime gives:
$200 x 1/1.121 = $200/1 1200 = $ 178.57 $400 X 1/1,12? = $400/1 2544 = 318.88 S600 x 1/1.123 = $600/1 4049 = 427.07 + 35800 X 1/1.12' = $800/1.5735 = 508 41 Total present value = $1,432.93
If you can earn 12 percent on your money, then you can duplicate this investment's cash flows for $1,432.93, so this is the most you should be willing to pay
You are offered an Investment that will make three $5,000 payments The first payment will occur four years from today. The second will occur in five years, and the third will follow in six years. If you can earn 11 percent, what is the most this investment ts worth today7 What is the future value of the cash flows?
We will answer the questions In reverse order to illustrate a point. The future value of the cash flows in six years is:
$5,000 x 1.11= + 5,000 x 1 11 + 5,000 - $6,160 50 + 5,550 + 5,000
The present value must be
$16,710.50/1.11'1 = $8,934 12 Let's check this. Taking them one at a time, the PVs of the cash flows are
$5,000 x 1/1.1 f = $5,000/1.8704 = $2,673.20 $5,000 X 1/1.11F' = $5,000/1.6851 = 2,967.26 -$5,000 X 1/1,1 V = $5,000/1 5181 = 3,293.65 Total present value = $8,934.12
This is as we previously calculated The point we want to make is that we can calculate present and future values in any order and convert between them using whatever way seems most convenient The answers will always be the same as long as we stick with the same discount rate and are careful to keep track of the right number of periods.
How to Calculate Present Values with Multiple Future Cash Flows Using a Financial Calculator
To calculate the present value of multiple cash flows with a financial calculator, we will simply discount the individual cash flows one at a time using the same technique we used In our previous chapter, so this is not really new. There is a shortcut, however, that we can show yoi. We will use the numbers in Example 5 3 to illustrate
To begin, of course, we first remember to clear out the calculator! Next, from Example 5,3, the first cash flow is $200 to be received In one year and the discount rate is 12 percent, so we do the following:
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