Problem 1: Are you better off playing the lottery or saving the money? Assume you can buy one ticket for $5, draws are made monthly, and a winning ticket correctly matches 6 different numbers of a total of 49 possible numbers.
The probabilities: In order to win, you must pick all the numbers correctly. Your number has a 1 in 49 chance of being correct. Your second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options.
But the order in which you pick them does not matter, so you actually have a few more ways to win. You can pick 6 different numbers in exactly 6 x 5 x 4 x 3 x 2 x 1 = 720 orders of choice. Any one of those orders would still win the lottery.
Putting this all together, your ticket has 720/10,068,347,520 = 1/13,983,816 chance of winning. This equates to a .000000071 percentage chance.
If you played one ticket every month from age 18 to age 65, you would have 47 x 12 = 564 plays. Your odds of not ever winning would be calculated using a binomial distribution to be .9999599568, meaning your chances of winning would be 1 – .9999599568 = .0000400432.
So, if the lottery winnings averaged $10 million over this time period, your expected return would be less than .0000400432 x $10 million = $400.43.
(It's less than $400.43 because your 564 plays are spread out over the next 47 years, so the present value of these future plays would be significantly less than if you were able to play all 564 immediately. The $400.43 assumes you play all 564 plays today, which makes it the highest possible expected value.)
A) What would your $400.43 be worth if you invested it at 1% real interest for 47 years?
B) If, instead, you wrote down your 6 numbers on a piece of paper, and deposited your $5 in a bank at 1% real interest, how much would you have at the end of the first year?
C) If you did this every year for 47 years, how much would you have at age 65?
D) If you earned 5% real interest on your deposits, how much would you have at age 65?
E) Which option would make you better off at age 65? How many times better off?
Problem 2: Assume the yield curve on "plain vanilla" default-free bonds is flat at 5%, and you are thinking of buying a default-free bond. Specifically, you're thinking of buying a bond issued by Risklessco, a company considered to be default-free by all major bond rating firms.
You will select one of the following three bonds, all identical except for the special features listed:
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Face Value
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Maturity
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Coupon Rate (Paid Annually)
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Yield to Maturity
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Special Features
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Price
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A
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1000
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20 years
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5.5%
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5%
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None
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?
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B
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1000
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20 years
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5.5%
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5%
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Callable
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Par
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C
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1000
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20 years
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5.5%
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3.5%
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Callable and Convertible into Risklessco Stock
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?
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A. Why is the yield on bonds A and B 5%? Why is the yield on bond C different?
B. What would be the price of Bond A?
C. If bond C is considered identical to bond B except for the conversion privilege, what is the value of the conversion privilege? Does the conversion privilege benefit the issuer of the bond or the purchaser? Is this consistent with the price you calculated for bond C?
D. Who does the callability provision benefit, the issuer or the purchaser? Is this consistent with the price you calculated for bond A?