Patrick’s luck had changed overnight – but not hid skills at mathematical reason
ID: 3200236 • Letter: P
Question
Patrick’s luck had changed overnight – but not hid skills 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 management science 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 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 learnt 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 the 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 figured that if Patrick invested $250,000 for 30 years in an investment with a 10% annual return, then in 30 ya=ears he would have $4,362,31 (that is, $250,000 (1+.10) 30.
But after thinking about it a little, 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 stock, then some years the return might be higher than 10%, and in some years it would probably be lower. So to help account for the potential variability in the investment return, Patrick and his friends came up with a plan. They would assume he could find an investment that would produce a 17.5% annual return 70% of the time and a -7.5% return (or actually a loss) 30% of the time. Such an investment should produce an average an annual return of .7 (17.5%) + .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+.1) 30 =$4,362,351)
After sitting quietly and thinking about it for a while, Payton 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 .7(30) =21 years) and a -7.5% return in 30% of the 30 years (or .3(30) =9years). So, according to Peyton, that would mean Patrick should have $250,000(1+.175)21(1-.075) 9 = $3,664,467 after 30 years. But that is $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 calculations assume 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 (that is X+Y=Y+X and X × Y = Y × 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 in 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?
Explanation / Answer
I think Payton is right at this point. as 17.5% and -7.5% are the annual returns so Payton had divided the total time span in 70 and 30 percents and then calculated the total. But here the calculation is wrong. So, if Patrick makes an investment where he will get an interest of 10% each year then he will get after 30 years,
(1.1)^30*250,000=4,362,351
Again according to Payton if the money is invested in stock, after 30 years it will result in
(1.175)^21*(0.925)^9*250,000=3,664,467
So Patrick should invest in the first plan that is he should invest with an interest of 10% per year.