## One lump (sum) or two?

Nov 6

Written by: Charles Flock
11/6/2008 11:02 PM

When you win the lottery, should you take the lump sum distribution or the annuity?

The other night I was watching an episode of Life on NBC. It featured a con artist who had infiltrated an organization called the LWA—Lottery Winner’s Anonymous. This wasn’t just for any lottery winners. This was only for winners of \$20 million or more. One of the members had been murdered and the other members described one another by first name and how much they had won, and occasionally made reference as to whether or not the member had taken a lump sum payment. That got me to thinking about the math involved in choosing between the annuity and the lump sum payment.

For those of you unfamiliar with the lottery, the winner may have the option of accepting payment of the winning amount as a constant stream of payments over a fixed number of years or accepting a lump sum payment now. Apparently in New York, at least according to the lottery web site, you make that election at the time that you purchase the ticket, which I am sure is exactly what you are thinking about when you purchase the ticket. It also begs the question of how to make that decision since you don’t actually know what the alternatives are worth at the time that you purchase the ticket. But, we can at least look at a case and see what the implications might be.

In 2007 a Mahopac man won what was billed as \$75 million in the lottery. He received a lump sum settlement of \$45 million, and from that amount about \$14.7 million was deducted in taxes (you can see what current withholding tax rates apply by looking here http://www.nylottery.org/ny/nyStore/cgi-bin/ProdSubEV_Cat_202538_NavRoot_307.htm).

The first question seems to be, why was this promoted as \$75 million prize, if the man was actually paid \$45 million? The \$75 million represents the nominal value of all 26 annual payments starting from today. Basically, the payment amount is \$75 million divided by 26. How, then do we arrive at a value of \$45 million?

There are a few functions that enable us to be able to analyze these kinds of questions. The RATE function will tell us the rate that was used (either explicitly or implicitly) to turn that \$75 million into \$45 million. If we enter the following SELECT statement:

SELECT wct.RATE(26
,75000000/26
,-45000000
,0
,1)

It returns the following result:

----------------------
0.0465129019945254

(1 row(s) affected)

This means that the rate that the lottery used to discount the annuity was approximately 4.65%. The RATE function works by iteratively solving for a PV of the annuity cash flows minus the lump sum amount equal to zero. We can double-check this answer with the following SELECT:

SELECT wct.PV(0.0465129019945254
,26
,75000000/26
,0
,1)

which returns the following result.

----------------------
-45000000

(1 row(s) affected)

What does this 4.65% tell us? It should be telling us that the lottery commission is indifferent as to the lump sum decision at that rate. It can either pay \$45 million to the winner or it can buy an annuity which will earn (at least) 4.65% which will be used to pay the winner. Either way, the lottery commission will write a check for that amount.

What does it tell the winner? Theoretically, he is in the same position as the state. If he were to receive \$45 million, he should be able to purchase exactly the same annuity as the state and he would then have the same stream of payments. Or, he may believe that he can invest the money and earn a higher return than 4.65%, in which case he would be disposed to take the lump sum settlement.

Of course, he is not actually going to the have \$45 million to invest. The tax authorities took \$14.7 million, so he only has \$30.3 million to invest. Of course, the annuity payments would be subject to taxes as well. If the winner believes that tax rate will go up on the future, then he will be disposed to taking the lump sum. How can we evaluate this?

The tax rate is 14.7/45 or 32.67%. Assuming that the tax rate stays constant, the PV of the after tax amounts can be calculated by multiplying the annuity payment by 1 minus the tax rate, since that is the amount that the winner gets to keep, and is exactly the same as the after-tax lump sum amount:

SELECT wct.PV(0.0465129019945254
,26
,75000000/26 * (1-.3267)
,0
,1)

which returns the following result.

----------------------
-30298500

(1 row(s) affected)

You could have also just calculated the PV of the tax payments and subtracted it from the lump sum amount

SELECT wct.PV(0.0465129019945254
,26
,75000000/26 *.3267
,0
,1)

----------------------
-14701500

(1 row(s) affected)

45,000,000 – 14,701,500 = 30,298,500

What if we think that the tax burden will grow, 1% per year? In other words, what if the tax liability grew 1% per year (as opposed to the tax rate increasing 1% per year). This requires us to calculate the PV of an annuity where the payment is increasing 1% per year.

Calculating the value of an annuity where the cash payment amount is increasing at a constant rate, requires that we adjust the rate used for discounting purposes by the growth rate. Assuming that r represents the original rate and g represent the growth rate in the cash payments , this equation illustrates that adjustment

@rate = (1 + r / 1 + g) – 1

Since it is the tax payments that are growing, we need to calculate the PV of the tax payments. We can then adjust our SELECT statement to reflect that fact:

SELECT wct.PV(((1 + 0.0465129019945254)/ 1.01) - 1
,26
,75000000/26 * .3267
,0
,1)

which returns the following result.

----------------------
-16282342.7698086

(1 row(s) affected)

When we subtract this amount from the lump sum distribution, we see that the after-tax amount is \$28.7 million, or about a million and a half dollars less than the after-tax distribution, making it more favorable to take the lump sum distribution. We can also make adjustments for inflation or whatever other factors you might want to include in decision.

RATE and PV are important tools to have available when trying to compare the value of money across time periods or in different time periods. Both of these functions take future cash flows and evaluate them in terms of today. In a future post will look at FV and FV schedule which project the current values into the future.

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