Patrick’s luck had changed overnight – but not his skill at mathematical reasoning. The day after
graduating from college he used the $20 that his grandmother had given him as a graduation gift
to buy a lottery ticket. He knew his chances of winning the lottery were extremely low and it
probably was not a good way to spend this money. But he also remembered from the class he took
in business analytics that bad decisions sometimes result in good outcomes. So he said to himself,
“What the heck? Maybe this bad decision will be the one with a good outcome.” And with that
thought, he bought his lottery ticket.
The next day Patrick pulled the crumpled lottery ticket out of the back pocket of his blue jeans and
tried to compare his numbers to the winning numbers printed in the paper. When his eyes finally
came into focus on the numbers they also just about popped out of his head. He had a winning
ticket! In the ensuing days he learned that his share of the jackpot would give him a lump sum
payout of about $500,000 after taxes. He knew what he was going to do with part of the money,
buy a new car, pay off his college loans, and send his grandmother on an all-expenses paid trip to
Hawaii. But he also knew that he couldn’t continue to hope for good outcomes to arise from more
bad decisions. So he decided to take half of his winnings and invest it for his retirement.
A few days later, Patrick was sitting around with two of his fraternity buddies, Josh and Peyton,
trying to figure out how much money his new retirement fund might be worth in 30 years. They
were all business majors in college and remembered from their finance class that if you invest p
dollars for n years at an annual interest rate of i percent then in n years you would have p(1 + i)n
dollars. So they figure that if Patrick invested $250,000 for 30 years in an investment with a 10%
annual return, then in 30 years he would have $4,362,351 (i.e., $250,000(1 + 0.10)30). But after
thinking about it a little more, they all agreed that it would be unlikely for Patrick to find an
investment that would produce a return of exactly 10% each and every year for the next 30 years.
If any of this money is invested in stocks, then some years the return might be higher than 10%
and some years it would probably be lower. So to help account for the potential variability in the
investment returns Patrick and his friends came up with a plan; they would assume he could find
an investment that would produce an annual return of 17.5% seventy percent of the time and a
return (or actually a loss) of -7.5% thirty percent of the time. Such an investment should produce
an average annual return of 0.7(17.5%) + 0.3(-7.5%) = 10%. Josh felt certain that this meant
Patrick could still expect his $250,000 investment to grow to $4,362,351 in 30 years (because
$250,000(1+ 0.10)30 = $4,362,351). After sitting quietly and thinking about it for a while, Peyton
said that he thought Josh was wrong. The way Peyton looked at it, Patrick should see a 17.5%
return in 70% of the 30 years (or 0.7(30) = 21 years) and a -7.5% return in 30% of the 30 years (or
0.3(30) = 9 years). So, according to Peyton, that would mean Patrick should have $250,000 (1 +
0.175)21(1 + 0.075)9 = $3,664,467 after 30 years. But that’s $697,884 less than what Josh says
Patrick should have. After listening to Peyton’s argument, Josh said he thought Peyton was wrong
because his calculation assumes that the “good” return of 17.5% would occur in each of the first
21 years and the “bad” return of -7.5% would occur in each of the last 9 years. But Peyton
countered this argument by saying that the order of good and bad returns does not matter. The
commutative law of arithmetic says that when you add or multiply numbers, the order doesn’t
matter (i.e., X + Y = Y + X and X x Y = Y x X). So Peyton says that because Patrick can expect
21 “good” returns and 9 “bad” returns and it doesn’t matter in what order they occur, then the
expected outcome of the investment should be $3,664,467 after 30 years.
Patrick is now really confused. Both of his friends’ arguments seem to make perfect sense
logically—but they lead to such different answers, and they can’t both be right. What really worries
Patrick is that he is starting his new job as a business analyst in a couple of weeks. And if he can’t
reason his way to the right answer in a relatively simple problem like this, what is he going to do
when he encounters the more difficult problems awaiting him the business world? Now he really
wishes he had paid more attention in his business analytics class.
So what do you think? Who is right, Joshua or Peyton? And more importantly, why?
