Question: Steve wants to invest $250,000.00, and wants his two friends Mark and John to help him figure out how much money his new retirement fund might be worth in 30 years. They knew from their finance class that if you invest p dollars for n year at an annual interest rate if i percent in n years you would have p( 1+ i )ⁿ dollars. To help account for the potential variability in the investment, Steve 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 tiMeand 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%. Mark felt certain that this meant Steve 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, John said that he thought Mark was wrong. They way John looked at it, Steve 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 Mark, that would mean that Steve 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 Mark and Steve should have.

After listening to John’s argument, Mark said he thought John 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 John 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, John said that because Steve 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.

Who is right, Mark or John? Why?