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Infinite Horizons

The overall critique of infinite horizon forecasting is sound, but when the AARP informs us that "the Trustees provide no information as to how far this 'infinite horizon' extends" one becomes a little puzzled. It extends forever that's what makes it the infinite horizon. Since even Mark Schmitt was parenthetically confused by the math of infinite horizon forecasting ("how they stop at $10 trillion is a mystery to me -- infinity is infinity") it's perhaps worth an explanation. The forecast is for the present value of the deficit when extended out to infinity. This means that deficits in future years are discounted by some percentage rate that compounds annually. To calculate the present value of some future number you use the formula P=F/(1+r)n Where F is the future value, r is the discount rate, and n is the number of years. As n approaches infinite, (1+r)n approaches infinite as well, so P approaches zero. Thus the summation of the values for every n has a finite limit. It does, that is, as long as you assume that the un-discounted value of F is grows slower than (1+r)n which the Trustees obviously do believe is the case for Social Security.

UPDATED: Edited to remove an error pointed out in comments.

UPDATED AGAIN: Note the more detailed and, er, accurate explanations of the math in the comments. Apologies, it's been a long time since I've taken a math class.

January 11, 2005 | Permalink

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Comments

As n approaches infinite, (1+r)^n approaches infinite as well, so F approaches zero. Thus the summation of the values for every n has a finite limit.

OK, that makes no sense at all to me. You're assuming n approaches infinity, OK, but is the value of F what's being calculated here? Or is it the value of P? What values are being summed?

Posted by: Haggai | Jan 11, 2005 3:51:33 PM

"so F approaches zero"
you mean P, right?

Posted by: o | Jan 11, 2005 3:51:48 PM

Please don't depend on a philosophy graduate for math advice - particularly financial math!

Yes, an infinite series can have a finite sum.

The derivations for these formulas are available in any Finance, Engineering Economics, or College Algebra textbook. Or you can just use the formulas themselves with no understanding from any MBA text ;-)

Cranky

Posted by: Cranky Observer | Jan 11, 2005 3:56:31 PM

What do you mean, "even Mark Schmitt"?? You have obviously mistaken me for a well-educated -- i.e., Harvard-educated -- person.

Posted by: Mark Schmitt | Jan 11, 2005 4:00:25 PM

I know, I've taught this stuff in college math classes before. What confuses me is what's being calculated here in the first place. Are they just taking the present value of the deficit and computing the limit of the future value, as time approaches infinity? And if there's an infinite sum being calculated, where's the series?

Posted by: Haggai | Jan 11, 2005 4:00:30 PM

Looks like this still needs some more editing to me:

Thus the summation of the values for every n has a finite limit.

The summation of what values? You skip from P = F/(1+r)^n to a summation. Are you summing those values for n = 1, 2, 3...infinity? That might start to make some sense.

Posted by: Haggai | Jan 11, 2005 4:08:38 PM

tanks for the correction!

Posted by: o | Jan 11, 2005 4:15:38 PM

Lets us clean this up a bit.

Let Fn be the deficit in year n. Assume a constant discount rate (say 5% or 0.05). Then the present value, Pn, of the deficit in year n, Fn, is [A^B is A raised to the exponent B, and exponents are done exponents before divides]:

Pn = Fn/(1 + 0.05)^n

Suppose the deficit this year is F0 = 100, P0 is

P0 = 100/(1.05)^0

Anything raised to the power 0 is 1 so the present value of this year's deficit is this year's deficit (not a surprise).

Suppose the estimated deficit next year is again 100. P1 is

P1 = 100/(1.05)^1 = 100/1.05 = 95.24

So the present value of next years deficit is 95.24 and the present value of the deficits for this year and next is 195.24.

The present value for the next 50 years (this year and the next 49) is

SUM(i=0,49)Pi = SUM(i=0,49)Fi/(1.05)^1

If all the Fi are the same, 100, this is 1916.87 (I used my computer).

For the next 1000 years, again assuming all the deficits are 100, the sum is 2100 (not much bigger).

What is going on is 1.05^i gets really large when i gets even a little big. When 1 is 50, it is a little over 11, when i is 100, 131.5, when i is 500, over 30 billion.

the sum to inifinity is just the sum with no stopping. If the deficits are all the same, there is a formula (which I cannot remember now) which gives the value exactly. (For Fi=100, it is not much bigger than 2100, the value at 1000 years.)

If we know the deficits are bounded (all less than a specific value), we know the sum is less than the sum with that value. As long as the deficits are growing no faster than x^n for some n (i.e. no faster than x^5 grows or even no faster than x^10,000 grows), then the inifinite sum is finite. That is because the denominator (1.05)^n is growing exponentially, and exponential growth will always exceed polynomial growth eventually. Only if the deficits themselves grow exponentially will the infinite sum be infinite.

Now, you have to look at the assumptions for future deficits. I assume those calsulating the values give them somewhere. They almost certainly are not growing exponentially. In fact, they may fall as we baby boomers kick off.

Posted by: David Margolies | Jan 11, 2005 4:16:49 PM

"To calculate the present value of some future number you use the formula P=F/(1+r)n Where F is the future value, r is the discount rate, and n is the number of years."

Isn't this more properly: to calculate the value of an income stream, you use the formula PV = F/(1+r)^n where F is the recurring amount/payment/whatev, r is the discount rate, and n is the number of periods.

Posted by: praktike | Jan 11, 2005 4:23:51 PM

"If the deficits are all the same, there is a formula (which I cannot remember now) which gives the value exactly."

The formula is known as the Gordon Constant Growth model, which is P = [F0 * (1 + g)] / (k - g), where F0 is today's deficit, g is an assumed constant annual growth rate, and k is the discount rate. So, if we're assuming no growth (real or nominal), a nominal discount rate of 5%, and a constant annual deficit of 100, the present value into infinity is simply 100 / .05, or 2000.

Posted by: Aziz | Jan 11, 2005 4:26:30 PM

This doesn't affect this particular infinite sum, but it is not true that just because the summands go to zero, the infinite sum itself must be bounded (finite). (This general statement is implied by the "thus" in Matt's admittedly garbled statement "As n approaches infinite, (1+r)^n approaches infinite as well, so F approaches zero. Thus the summation of the values for every n has a finite limit.")

For example, the sum of all numbers of the form (1/n) (this is called the "harmonic series" and for purposes of this counterexample n is any positive integer -- it can start whereever you like, so long as the series includes all the integers bigger than n -- it's the behavior of the infinitely lengthy tail that matters), is infinite, even though as n grows larger, (1/n) goes to zero.

Posted by: David | Jan 11, 2005 4:29:42 PM

OK, thanks David Margolies, that's starting to make some sense. A few basic problems I had with MY's post were that:

(a) I didn't know where the future values were coming from; I assume this involves population-based actuarial modeling, of course.

(b) I didn't realize he was talking about summing the present value for each future year into infinity.

But that does make sense. Now, here's something very wrong with the mathematical explanation in MY's post:

As n approaches infinite, (1+r)^n approaches infinite as well, so P approaches zero. Thus the summation of the values for every n has a finite limit.

No, no, no! That logic is false. This is an instance of the (tempting, but incorrect) claim that if an infinite sequence of numbers approaches zero (in this case, the present values are forming an infinite sequence, i.e. just a list that goes on forever, as n goes to infinity), then the infinite sum of that sequence must be a finite number. That is false. Among many others, the infinite sum:

1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ...

does NOT approach a finite limit, even though the infinite sequence of 1/2, 1/3, 1/4, ... 1/n, ... approaches zero as n goes to infinity.

In order to conclude that the infinite sum of the values Fn/(1+r)^n approaches a finite limit, you have to know something more precise about how Fn behaves as n approaches infinity. It is not enough to know/assume that Fn grows slower than (1+r)^n. To be sort of cryptic, Fn has to grow "slow enough" in comparison to (1+r)^n for that claim to be true.

Posted by: Haggai | Jan 11, 2005 4:30:59 PM

Ah, good, commenter David right before me catches the mathematical faux pas as well. Good stuff. :)

Posted by: Haggai | Jan 11, 2005 4:32:00 PM

Another david says:

"For example, the sum of all numbers of the form (1/n) (this is called the "harmonic series" and for purposes of this counterexample n is any positive integer -- it can start whereever you like, so long as the series includes all the integers bigger than n -- it's the behavior of the infinitely lengthy tail that matters), is infinite, even though as n grows larger, (1/n) goes to zero."

Quite right. Also, SUM(i=1, infinity)1/Pi (where Pi is the i'th prime number) also diverges (sums to infinity), so you can be way sparser than 1/n for all n after a certain point, or even 1/k*n for fixed k and all n over a certain point, and still diverge.

From my earlier post, I have

SUM(i=0,49)Pi = SUM(i=0,49)Fi/(1.05)^1

where of course I meant

SUM(i=0,49)Pi = SUM(i=0,49)Fi/(1.05)^i

Aziz gave the formula and calculated discount rate 5%, constant deficit 100, infinite sum is 2000, which is correct. I had shown a calculation for i=1000 of 2100, the extra 100 being accumulated round off error.

Posted by: David Margolies | Jan 11, 2005 4:38:09 PM

Haggai writes:

"In order to conclude that the infinite sum of the values Fn/(1+r)^n approaches a finite limit, you have to know something more precise about how Fn behaves as n approaches infinity."

Quite so, but we know the denominator is SocSec calculations grows exponentially. As I said in my first post, so long as the numerator has polynomial growth, the sum will converge (sum to a finite number). A polynomial growth of the deficit is a reasonable assumption IMO. I am sure the people doing the estimate say what the assume somewhere.

(polynomial/exponential is sufficient for convergence, but not necessary. There is no necessary formula, as far as I know.)

Posted by: David Margolies | Jan 11, 2005 4:45:06 PM

Ah, integral calculus. Used to be my favorite subject... Well, all right, who am I kidding - I hated it like poison.

Posted by: abb1 | Jan 11, 2005 4:46:04 PM

Quite so, but we know the denominator is SocSec calculations grows exponentially. As I said in my first post, so long as the numerator has polynomial growth, the sum will converge (sum to a finite number). A polynomial growth of the deficit is a reasonable assumption IMO. I am sure the people doing the estimate say what the assume somewhere.

Ah, yes, that would definitely do it.

(polynomial/exponential is sufficient for convergence, but not necessary. There is no necessary formula, as far as I know.)

Actually, the condition of the nth term approaching zero (I'm assuming that n is the variable we're summing over) is a necessary condition of the sum converging. The common mistake that MY was made was in assuming that it was sufficient!

Posted by: Haggai | Jan 11, 2005 4:52:34 PM

I wish I still remembered that little algebra trick where you subtract and get the formla for an infinite series.

Posted by: praktike | Jan 11, 2005 4:56:06 PM

Ask and ye shall receive, praktike!

Posted by: Haggai | Jan 11, 2005 4:58:12 PM

Praktike:

S = x + x^2 + x^3 + x^4 + ...
Sx = x^2 + x^3 + x^4 + ...

subtract:

S(1-x) = x
S = x/(1-x)

that's the basic idea

Posted by: baba | Jan 11, 2005 4:58:39 PM

The ridiculous nature of the estimate is in its arbitrariness.

What is the proper value of r? The average interest rate we get on our saving accounts or that we pay in mortgage loans? Should we look at Treasury bills or Treasury bonds? Should we look at the current interest rates or the average of the last 10 years?

Mind you, getting 1 dollar every year is worth 20 dollars with 5% discount rate and 50 dollars with 2% discount rate.

The second issue is if the PRESENT value is a good measure. Probably a better measure would be accumulated debt compared to GDP, the limit ratio. If the limit is not bounded, then the policy is unsustainable. Even then one has to ask several "what if"s, like what would happen if we alter the formula for the future benefits in, say, 2040. I read a claim that we are doomed unless we act before 2011 -- and even that does not sound all that urgent.

Posted by: piotr | Jan 11, 2005 5:01:08 PM

Yeah, that's basically it. But one must be proper and take limits, as in the explanation I linked to. :)

Posted by: Haggai | Jan 11, 2005 5:01:16 PM

"that" in my previous comment = baba's comment

Posted by: Haggai | Jan 11, 2005 5:02:26 PM

Pyotr wrote

"What is the proper value of r? The average interest rate we get on our saving accounts or that we pay in mortgage loans? Should we look at Treasury bills or Treasury bonds? Should we look at the current interest rates or the average of the last 10 years?"

I do not see this as such a problem We are interested in real interest rates (nominal interest rate less inflation), and that moves around less. Taking a high end value of the real rate can be used quite safely. (Because wages and tax revenues, and also benefits, to first order, track inflation, real interest rates are the right measure.)

"The second issue is if the PRESENT value is a good measure. Probably a better measure would be accumulated debt compared to GDP, the limit ratio."

This seems to me to be a much better point. The 10 trillion present value is about 100% of GDP, so debt is 160% of GDP now (leaving off future general fund deficits and medicare deficit). That is high but far from unreasonable nor unheard of (I think is was much worse just after WW II). So from this point of view, SocSec is not a problem. The general fund and medicare are, but we all knew that.

"I read a claim that we are doomed unless we act before 2011 -- and even that does not sound all that urgent."

Quite so, even if the claim were true!

Posted by: David Margolies | Jan 11, 2005 5:16:12 PM

Haggai: On the sufficient vs. necessary question, I believe David M. was saying that he doesn't know of any test that is simultaneously necessary and sufficient for convergence. If anyone is interested, Mathworld has a list of convergence tests, available at:
http://mathworld.wolfram.com/ConvergenceTests.html

Posted by: JBL | Jan 11, 2005 5:23:09 PM

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