Bad Lottery Math

[Fulano_De_Tal]

I just read the article at http://www.straightdope.com/mailbag/mlottery.html which asks, “If I win the lottery, should I ask for a lump sum or an annuity?”

In the answer, the basic idea is correct, that you need to apply calculations which take into account the time value of money. The math in the article, however, stinks. I’ll get to that in a minute. But first, I have to say that ignoring taxes (almost an afterthought in the article) is another very serious flaw.

Now back to the math. The article compares the 20-year annuity option on a $1,000,000 jackpot, to the offered $500,000 lump sum option. Ignoring taxes for the moment, it says that you would need a return of 8.92% on your $500k to have $1 million in twenty years. My calculations are very different. Even without compounding the interest (speding the $44,600 of interest each year), I make nearly $900,000 in interest alone over 20 years at that interest rate! Compounding it is even more drastic. At that interest rate, I come up with over $2.5 million in just 20 years.

No fancy financial calculator is needed - you can write out each payment period if you like. The first year is 500,000, and the second year is 500,000 * 1.0892. Then take that answer and multiply it by 1.0892.

Taxes complicate things more, and are essential to understanding why the annuity is not as big a rip-off as some people think (including some financial planners). The maximum federal tax rate is now at 35%. Most states have income tax around 5% or so (althout some have 0% and some are higher than 5%). Essentially, most people face a 40% tax bill on large payouts. The example used a $50,000 per year payout for the 20-year $1 million annuity. Most big jackpots are much larger, pushing people into the max tax bracket.

Let’s assume, therefore, a jackpot of $10,000,000, with a 20-year annuity of $500k per year. Cash value $5 million if taken in a lump sum. Mind you, the lump sum after taxes is just $3 million (60% left over). The annual payments would be $300,000 (60% of 500k) after tax if the annuity were taken. So after tax, the annual payments would equal a total of $6 million paid over 20 years.

Therefore, the real question should be this: can you invest your $3 million after-tax lump sum so that is equal to or greater than the $6 million (after tax) you receive over 20 years? (Plus any return on investments the annual payments might earn?) Remember, you also have to pay taxes each year on any interest you earn from your investments.

As it turns out, it is still only slightly better to take the lump sum. The is because the low rate of return on the annuity is augmented by the fact that it is tax-deferred year after year. (But introduces tax law risk.) Some financial planners have estimated that you need to make 4 percentage points more than the return on the annuity to compensate for this effect. So if the annuity pays 4% interest, you will need to make 8% per year or more with your investments with the lump sum to make it the wiser choice. This can be achieved, but you can see how the decision starts to be a close call.

Also consider that some or all of each annual payment can be invested too, making the total potentially larger than $6 million. Also consider that you will spend a lot or all of the income produced by the $3 million lump sum, lessening the effect of compounding. Both payouts end up about equal, taking all into consideration, unless you have a lot of discipline not to spend your windfall, and really good investments.

It ends up being more of a personal lifestyle choice than a math equation. Which would you rather have? (Figure life expectancy, risk tolerance, spending habits, self control, etc.) Would you spend half the lump sum right away instead of living off the interest?

[Rocketeer]

I suspect that this situation is an example of Rocketeer’s Law of Middle-Class Economics:

All your investment choices (except the obviously stupid ones) work out about the same.

[OtakuLoki]

Just putting the information here, rather than opening a new thread: The NYS lottery does allow the player to choose to play for either the lump sum payment or the 26 annual payments. But the choice has to be made at the time of purchasing the ticket, it’s not a decision that the gambler can make once he or she knows they’ve won.

[C_K_Dexter_Haven]

I’m travelling and on vacation, so I do not have access to calculator nor to the time/effort to think about this again. I felt the need to respond, but you’ll need to wait about two-weeks for a well-reasoned and complete response.

My gut recollection is that you’re not making the right comparison. The one choice is $50,000 paid each year for 20 years; the other choice is $500,000 paid today. Note that if you choose $50,000 each year for 20 years, and you invest that money each year, you have MORE than just $1,000,000 at the end of 20 years. To figure the present value of a 20-year string of payments of $50,000 does take either a calculator with financial functions or a lot of patience.

At 8.9% interest, the future value (at the end of 20 years) of $50,000 annual payments is around $2,714,000. Similarly, $50,000 invested at 8.9% interest for 20 years (no additional payments) gives roughly the same future value. I stand by my original numbers.

Note that there are underlying assumptions: that you live the full 20 years, and that you don’t spend any of the money until the 20 years are up.

I didn’t go into the tax comparison in the original report, because there are too many complexities and too many assumptions to be made. Basically, it depends on your level of other income… and other taxes (like state taxes.) I did say, in the original report, quite clearly:

Normally, tax on an annuity will be less than that on a lump sum, because of the lower rates on lower income brackets.

… which is what you’re saying, Fulano.

[ultrafilter]

@C K Dexter Haven

At 8.9% interest, the future value (at the end of 20 years) of $50,000 annual payments is around $2,714,000. Similarly, $50,000 invested at 8.9% interest for 20 years (no additional payments) gives roughly the same future value. I stand by my original numbers.

This makes sense, but the original article might be a little unclear. To wit:

Let’s say you win a jackpot of $1,000,000 from the California lottery. The lottery rules say that total will be paid out to you in 20 equal annual payments of $50,000 each. Alternatively, you can take the lump sum cash value, which is about half the face amount, about $500,000. Which should you take? To find out, we need to determine what kind of interest rate you’ll be getting with the annuity. Punching a few numbers into my calculator, I find that if we start with a cash value of $500,000 and expect to wind up with $1,000,000 after 20 years (in other words, the equivalent of a $50,000 annuity), the underlying annual interest rate has to be 8.92%.

I read that last sentence to mean that you were looking for the interest that made the future value of the lump sum equal to the face value of the annuity, not the future value of it. Maybe a little editing is in order?

Other than that, this is a good report. Kudos.

[C_K_Dexter_Haven]

Yeah, I see what you mean. When I’m back to civilization, I’ll give some thought to editing to be more clear. I guess I figured that any reader who was thinking about the math would know not to compare present value to future value, and I sort of wanted to skip the math details so as not to bore 95% of the readers. Thanks.