Regarding http://www.straightdope.com/mailbag/mlottery.html, when I plug $50,000 for 20 years into my calculator and calculate the PV using 8.92%, I get a PV of just over $459,000, not $500,000.
I think the correct discount rate for a 20 year annuity whose present value is $500,000 with payments of $50,000 is 7.75%.
For the 26 year annuity, where the payment would drop to $38,461, the discount rate is 6%, as far as I can tell.
Where did I go wrong? Or, could it be that Dex got it wrong?
You forgot that the payments are due at the beginning of each period, not the end. In excel, you can do this one of two ways:
=PV(0.0892,20,-50000,0,1)
=50000+PV(0.0892,19,-50000)
The last argument of “1” signifies that payments are due at the beginning of the period. The default value for this is 0, which means payments are due at the end of the period.
The second method of calculating just gives you the $50,000, then calculates 19 more payments at the end of each period. Same thing.
With each of these, I get $499,987.76. Close enough for government work. If Dex gave a few more decimal points on the interest rate, the result would be within a penny or two of $500,000.
Got it. Although payments at the end is way more common.
Yep, that’s why it is setup that way. But could you imagine payments at the end of the period for lottery winnings? “OK, Mr laid off factory worker, you can have either $500,000 right now, or $50,000 a year for the next 20 years, starting a year from today of course!”.
5cents is quite correct, which is why I chose beginning of period for my calcs for the lottery. And Ritter is quite correct, end of period is way more common for mortgages. It’s exactly the reverse logic – “OK, we’re loaning you $100,000 and the first payment is due NOW!”
I thought the example of mortgage would make the concept of annuity clear, and I didn’t want to confuse the Staff Report. Hence, I used beginning of period for all calcs, recognizing that was not really accurate for mortgages. I tried to side-step this additional calculation complexity, but I shoulda knowed that the Teeming M’s would never let me.
Thx Dex! It was a great report. I should have caught the beginning/ending difference (I do have an MBA, you’d think I’d know better, or maybe not). In my defense, it’s not just mortgages, but car loans, bonds, and immediate fixed annuities also typically use end of period. It’s really unusual to come across beginning of period in debt markets – you give me money and I give you some back immediately.
Since rates have come down so much, it’s amazing that states still use such high implied rates to calculate the number of payments. I thought that the period was always 20 years, so the move to 26 years probably reflects the low rate environment. Remember when savings bonds used to pay out in, what, 10 years? Those days are gone. Now it takes 20 years to be guaranteed to double in value. Just last year, it was 17 years.
As a life-long New Yorker, I can vouch for the fact that the NYS lotteries (what they call Lotto and Mega Millions) offer the choice between cash value and the annuity.
As to what happens if you don’t make a choice, I believe the default is cash value. Granted, that’s what I always pick anyway.
New Jersey, too, offers the choice. I don’t think there is a default; as far as I know, the machine kicks out the form if one is not made.
I figured NY probably did offer a choice, but they sure don’t publicize it well. The website didn’t indicate any choice… and didn’t give any information that would allow us to figure out the underlying interest rate being assumed. Since the Staff Report was a general “what’s the diff” kind of question, I didn’t see any reason to spend lots of time digging into the specifics of each state beyond a quick web-search.
Another New Yorker chiming in.
Most of the lotto agents around here push the “cash value” option over the annuity option. In the few times I’ve forgotten to specify, I’ve been given the cash value option.
I’m assuming that the amounts you’ve won have been fairly small (relatively speaking, of course.) While the math is the same regardless of the amount of the winnings, there’s administrative cost in paying the annuity, especially if the amount is only a couple hundred or so. Hence, the government prefers to spare itself the administrative costs and headaches on small amounts, and encourages you to take the lump sum.
If I’m wrong in my assumption, and you’ve in fact won the multi-million dollar pot “a few times”, then let’s become dear friends.
I didnt see any mention of the effect of TAXES on making your decision.
I suspect if you are getting $50,000 a year extra, you’ll be in a lower tax bracket than if you have the lump sum all in one year. True, one can use income averaging on the tax return to effectively spread the income out over several years, but the income is still going to be higher that way than if it is spread out over 20 or 26 years.
federal income tax info for 2003:
Joint Single
$0–$14,000 $0–$7,000 10.0%
14,000–56,800 7,000–28,400 15.0 %
56,800–114,650 28,400–68,800 25.0 %
114,650 –174,700 68,800–143,500 28.0%
174,700–311,950 143,500–311,950 33.0%
311,950 and up 311,950 and up 35.0%
( The california tax only changes about 1% in these two cases)
New York Lotto uses a progressive annuity with smaller payments at the begining that increase every year. This allows the state to get away with a smaller cash value for a given prize amout.
New York also taxes winnings at a high rate in addition to the federal tax. If a person who commutes between NY and NJ wants to buy a Megamillions ticket (sold in both states) , they will get a better deal in NJ since that state does not tax lottery winnings.
From whom are these annuities purchased? I would assume insurance companies, but I haven’t seen this published anywhere. Although the state(s) may guarantee a payment if the cash option is selected, will they guarantee an annuity from a third party if that company defaults?
Granted, there are two chances in Massachusetts that an insurance company will close its doors; slim and fat. They just raise rates, or move out of state if they can’t. Nevertheless, these are unsure economic times (aren’t they all?), and I would think that it would be better to have this bird in hand than to trust in Bush.
In the original article, SDSTAFF Dex wrote …
The second sentence isn’t a correct description of the present value of an annuity, and instead describes the present value of a lump sum payment of the face amount made at the very end of the annuity lifetime. The annuity doesn’t pay out all at once at the end, however. A correct description of the present value of the annuity is… It is the amount that you would have to deposit in an interest bearing account today such that if you were to make subsequent withdrawals matching both the amount and timing of each annuity payment, the last withdrawal would exactly empty the account.
The need to make payments along the way substantially raises the present value of the annuity relative to the definition given by Dex.
I don’t know if it has been factored in, or if it changes any of the numbers significantly, but the total period is only 25 years, 26 payments.
In Illinois, you don’t have to make a choice on lump sum vs. annuity until claiming the prize. However, if you wait too long, the annuity is the default choice. The state doesn’t really have any administrative costs on this, it’s all dumped onto the annuity provider.
Another thing to consider when taking the lump sum option is that you would have to pay federal taxes as well–so, if you won a $1m and decided to take the lump sum payment of $500k (assuming 50% here), you would actually take home roughly 55% of this amount, I think–I have never been in that tax bracket! Your $1 million winnings is really worth $275,000.
I’m suprised that Dex’s column makes no mention of inflation. It is a variable that swings the decision solidly in favor of the lump sum.
The vast majority of cash lottery winnings are converted by winners very rapidly.
In New York, and most states, you no longer have to make that choice at the time you purchase the ticket, but can decide when you claim your prize.
As for the information about New York Lottery payments, I found this 3 clicks into their website:
Though I’d agree it isn’t very obvious where to click to get there. Poor web design.
Randy, I don’t see how inflation enters into it. If you take the lump sum, you have to find an investment that will do as well as the implied rate of the annuity or better (without taking additional risk) to make it worth going that way.
So (I think I have this right), if your risk free investment earns a lower rate than the one implied by the annuity payments, then you’re better off with the annuity. I assume here that the annuity is backed by the state taxing power, which makes it essentially risk free.
Right now, a 26 year treasury bond would yield between 4% and 5%. If the annuity is implying a rate closer to 7%, I think you would be better off with the annuity, from a purely economic perspective (the story gets even better with the tax argument mentioned above).
Now, for your own enjoyment, you might rather have the money up front and have a blast in Vegas or something.
The only reason I could see inflation entering into it is if you want to invest the money in TIPS, which are protected from inflation. The real rate on a 26 year TIPS is 1.7% or 1.8%. Assuming a long term inflation average rate of around 3%, then the real rate of the annuity is a little under 4% (of course you wouldn’t be hedged if inflation came back in a big way). Still, it seems like the annuity rate is a good one.