Wednesday, March 28, 2012

The Lottery: Payments or Lump Sum?

I see that the Mega Millions Lottery  is offering (a) $359 million now or (b) $19.2 a year for 26 years.  The gross value of the payment stream is of course 19.2x26=$499.2 million.  Hey, do the math and you'll see that $499.2 tops $359 so take the deferred payments, right?

Well,   I gather that's that what the grownups want us to think.  Don't take it now or you'll blow it. Take the payments, you'll do better in the long run.

But of course it's not right.  Time is money and the present value of the payment stream is the nominal value discounted at the appropriate interest rate.

Anyway, try this.  Assume you get your first payment today and subsequent payments at the beginning of each of 25 more years.  So if you take the payments you are "paying"  $339.8 ($359-$19.2) million.  What is the implied discount rate (internal rate of return)?  Excel says 2.85* percent per year.

So, what is your discount rate?  If your rate is lower than 2.85percent, then the implied present value of the payment stream is higher than 339.8.  If lower, higher.  Such is the conventional wisdom.

And is this right?  Well, there's taxes.  Not really my department, though I think you have to pay tax on the lump sum payment at the front end, but on the payments only as they come in, which seems to tilt the advantage towards payments.  And I suppose with this kind of money, you may be able to get some Mitt Romney action.

Oh, wait--the "I" word, "inflation."  I gather that discount rate is based on the long-term treasury rate.  The rate is supposed to "impound" inflation.  But it looks to me like the inflation rate is already running  close to 3 percent--low by historical standards but still high enough to eat up all of the implied discount rate.  And what are the chances that inflation in the future will stay that low, huh?   Huh?

Short answer: take the money and run.  What say you?

[h/t: Buce's friend Bruce.]  

*See comments.


MBA said...

Hi, your Excel calculation fails to take into account compounding. The effective interest rate is less than 1.5%, not 2.6%.

Use a "future value of $1" accounting table, and you'll see that if $389,800,000 on-hand now is to grow to $540,000,000 after 25 years of compound interest, then the implied annual interest rate necessary is between 1% and 1.5%.

But your conclusion is correct: cash value option is better than the annuity payments because interest rates (and thus inflation) is unlikely to remain at the current historic lows.

Buce said...

Actually, there was a computational error in the original: the correct number is 2.85, not 2.6 (I must have used the wrong number of periods). But the formula does account for compounding. It's the Excel internal rate of return. To double check, discount each of the payments as PT(1.0285)^t, and sum. You'll get $339.80. Not sure what $389.8 has to do with it, nor $540.

If you ask: what is the implied rate of return when $339.8 grows to $499.2 over 25 years, the answer is 1.55 percent ((FV/PV)^(1/25)-1], but that is an entirely different question from the one I was undertaking to answer. I.e., I wasn't asking for the 25-year value, only for the DPV of payments from the time at which they were paid.