With the annual McDonald’s Monopoly game underway yet again, I find myself dreaming of winning a million dollars. Because really, who wouldn’t want a million dollars magically added to their bank account, just because they ate a bunch of fast food?
The Monopoly game only offers the million dollars in the form of an annuity that pays out $50,000 every year for 20 years. But in other lotteries and sweepstakes, they often offer a lump sum option to get all the money up front, instead of receiving it as an annuity. So the question becomes, when is it better to get it as a lump sum?
Let’s say the jackpot is d dollars, with the annuity spreading it evenly over n years. For simplicity, let’s say that you somehow have a way of earning a fixed interest rate r, compounded annually. You will have received d/n the first year, which will be d(1+r)/n with interest. Then the second year, you receive another d/n, which also earns interest, but since the amount from the first year has earned interest, you actually have d(1+r)2/n + d(1+r)/n. Continuing this pattern, after n years you will have d((1+r)n + (1+r)n-2 + … + (1+r)) / n = d((1+r)n+1 – 1) / nr dollars.
Let’s assume that the lump sum is some fraction p of d. Then if we had gotten pd dollars in the beginning, after n years we would have pd(1+r)n dollars. Then we can find the breakeven point for p by setting these two values equal to each other. So we end up with p = ((1+r)n+1 – 1) / nr(1+r)n.
In the McDonald’s case, n = 20. Let’s assume that we can somehow get r = 0.07. If we plug those into the formula above, we get p = 0.58. So that means if the lump sum is at least 58% of the total amount, then it’s better to take the lump sum up front. It might seem like you’re getting a small fraction of the actual jackpot, but you end up ahead in the long run. (This is ignoring all the tax implications and so forth, and also assumes that you have a reliable interest rate, which is quite ridiculous.)
It’s quite annoying to have to do all this math. But if I won a million dollars, that’s one annoyance I would certainly live with.