# What is the equation for an inflation adjusted annuity held in perpetuity?

What is the equation relating an initial investment, a fixed rate of return, an inflation rate, and a perpetual payment adjusted for that inflation rate?

• Does such a thing exist or is this theoretical? – JTP - Apologise to Monica Aug 4 '12 at 17:16
• @DilipSarwate the idea is that a portion of the periodic payment is reinvested thus increasing the principle. – Mike Deck Aug 4 '12 at 19:35
• @JoeTaxpayer It's real in as much as the rate of inflation stays constant and you can find an investment that will give you a constant rate of return in perpetuity. Obviously those are both ridiculous statements, but no more so than most of the other long term projections made in the financial world. – Mike Deck Aug 4 '12 at 19:45
• The PV of an (infinite) series of values increasing faster than inflation will be infinite. The reason $1/yr for perpetuity has a present value I can calculate is due to the time value of money. Even at .1%/yr, the PV only hits$1000. Of course division by zero yields infinity, which is meaningless. – JTP - Apologise to Monica Aug 4 '12 at 20:41
• @JoeTaxpayer You're assuming the discount rate is the inflation rate. But in this case it's not, it's a value grater than the inflation rate. I.e. the payments are growing at a rate smaller than the discount rate of the TVM calculation so things still stay finite. – Mike Deck Aug 4 '12 at 21:33

Let P denote the amount of the investment, R the rate of return and I the rate of inflation. For simplicity, assume that the payment p is made annually right after the return has been earned. Thus, at the end if the year, the investment P has increased to P*(1+R) and p is returned as the annuity payment.

If I = 0, the entire return can be paid out as the payment, and thus p = P*R. That is, at the end of the year, when the dust settles after the return P*R has been collected and paid out as the annuity payment, P is again available at the beginning of the next year to earn return at rate R. We have

P*(1+R) - p = P

If I > 0, then at the end of the year, after the dust settles, we cannot afford to have only P available as the investment for next year. Next year's payment must be p*(1+I) and so we need a larger investment since the rate of return is fixed. How much larger? Well, if the investment at the beginning of next year is P*(1+I), it will earn exactly enough additional money to pay out the increased payment for next year, and have enough left over to help towards future increases in payments. (Note that we are assuming that R > I. If R < I, a perpetuity cannot be created.) Thus, suppose that we choose p such that

P*(1+R) - p = P*(1+I)

Multiplying this equation by (1+I), we have

[P(1+I)]*(1+R) - [p*(1+I)] = P*(1+I)^2

In words, at the start of next year, the investment is P*(1+I) and the return less the increased payout of p*(1+I) leaves an investment of P*(1+I)^2 for the following year. Each year, the payment and the amount to be invested for the following year increase by a factor of (1+I). Solving

P*(1+R) - p = P*(1+I)

for p, we get

p = P*(R-I)

as the initial perpetuity payment and the payment increases by a factor (1+I) each year. The initial investment is P and it also increases by a factor of (1+I) each year. In later years, the investment is P*(1+I)^n at the start of the year, the payment is p*(1+I)^n and the amount invested for the next year is P*(1+I)^{n+1}.

This is the same result as obtained by the OP but written in terms that I can understand, that is, without the financial jargon about discount rates, gradients, PV, FV and the like.

EDIT: After reading one of the comments on the original question, I realized that there is a much more intuitive way to think about this. If you look at it as a standard PV calculation and hold each of the cashflows constant. Really what's happening is that because of inflation the discount rate isn't the full value of the interest rate. Really the discount rate is only the portion of the interest rate above the inflation rate. Hence in the standard perpetuity PV equation PV = A / r r becomes the interest rate less the inflation rate which gives you PV = A / (i - g).

That seems like a much better way to get to the answer than all the machinations I was originally trying.

I think I finally figured this out. The general term for this type of system in which the payments increase over time is a gradient series annuity. In this specific example since the payment is increasing by a percentage each period (not a constant rate) this would be considered a geometric gradient series.

According to this link the formula for the present value of a geometric gradient series of payments is:

P = A_1 [1 - (1 + g)^n(1 + i)^-n]/(i - g)


Where

P is the present value of this series of cashflows. A_1 is the initial payment for period 1 (i.e. the amount you want to withdraw adjusted for inflation). g is the gradient or growth rate of the periodic payment (in this case this is the inflation rate) i is the interest rate n is the number of payments

This is almost exactly what I was looking for in my original question. The only problem is this is for a fixed amount of time (i.e. n periods). In order to figure out the formula for a perpetuity we need to find the limit of the right side of this equation as the number of periods (n) approaches infinity.

Luckily in this equation n is already well isolated to a single term: (1 + g)^n/(1 + i)^-n}. And since we know that the interest rate, i, has to be greater than the inflation rate, g, the limit of that factor is 0.

So after replacing that term with 0 our equation simplifies to the following:

P = A_1 / (i - g)


Note: I don't do this stuff for a living and honestly don't have a fantastic finance IQ. It's been a while since I've done any calculus or even this much algebra so I may have made an error in the math.

• EDIT: Unfortunately I can't seem to get any of the Tex markup to render properly. This is the first time I've tried marking up mathematical formulae on a stackexchange site, so if anyone can fix the original revision of this question to render properly, please add a comment and let me know what I was doing wrong. – Mike Deck Aug 4 '12 at 21:03
• I think this site does not use MathJax. I am told that using MathJax adds a lot of expense, and so sites that do not need it, often choose to avoid the expense. Besides \$ is commonly used for other purposes on this sight. – Dilip Sarwate Aug 5 '12 at 2:42
• @DilipSarwate I see, makes sense. Thanks for the info. – Mike Deck Aug 5 '12 at 4:09

The question lacks specificity, i.e. when does the initial investment occur, now or one period from now? If now then it is a perpetuity due.

I will consider under 2 scenarios, A and B, relating to the size of the initial investment.

A. Assuming that the initial investment (C_0) occurs now and each payment thereafter has the relationship (1+g) with this investment then the relevant base equation is that for the present value of a growing perpetuity due, expressed in terms of C_0, i.e.

PVGPD= [C_0*(1+g)*(1+i)]/(i-g).

Now, to suit the question asked, we can see that i=fixed rate of return (f) and g = expected inflation rate (e) such that we can rewrite the equation as

PVGPD = [C_0*(1+e)*(1+i)]/(i-e].

We know that f = is a fixed nominal rate and must be adjusted for e to calculate the real rate (r) according to the equation f=(1+r)*(1+e)-1. Therefore

PVGPD = [C_0*(1+e)(1+(1+r)(1+e)-1)]/((1+r)*(1+e)-1-e]

Tidying up

PVGPD = {C_0*(1+r)(1+e)^2}/[r(1+e)]

PVGPD = [C_0*(1+r)*(1+e)]/r

B. Assuming that the initial investment (X) is not equal to each subsequent perpetual payment (C_1) then the relevant base equation is that for the the initial investment plus the present value of a growing perpetuity, i.e.

PVGP= X + [C_1/(i-g)]

Rewriting

PVGP = X + [C_1/(f-e)]

Substituting

PVGPD = X + {C_1/[(1+r)*(1+e)-1-e]}

Tidying up

PVGPD = X + C_1/[r*(1+e)]