Perpetuities

This is “Perpetuities”, section 7.2 from the book Finance for Managers (v. 0.1). For details on it (including licensing), click here.

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7.2 Perpetuities

PLEASE NOTE: This book is currently in draft form; material is not final.

Learning Objectives

Define a perpetuity.
Calculate the PV of a perpetuity.
Explain how the PV of a perpetuity is derived.

As discussed in the previous section, finding the NPV for a fixed set of cash flows will always work. But what if the cash flows were to continue forever? If the payments are constant over time, we can intuit what the value must be for the entire set, instead of figuring out each one separately: if I put $200 into a bank account that promises me 5% per year, and I just withdraw the interest each year without ever touching the principal, how much will I get each year? The first year, the interst will be 5% of $200, or $10. Since I withdraw the interest, my second year balance is also $200, so my interest is again $10. In fact, I (or my heirs) can withdraw $10 a year forever as long as I keep the original $200 in the bank.

Figure 7.3 Perpetuity Timeline

Note that this is essentially the same as the bank offering the following: pay $200 today, and the bank will pay the customer $10/year. Therefore, at 5% interest, $10/year forever must be the equivalent of exactly $200 today! This arrangement, with constant periodic payments lasting forever, is called a perpetuityConstant periodic payments lasting forever.. The periodic payment must be the interest rate times the present value, since it is analogous to our bank deposit example above.

Equation 7.1 Perpetuity Equation

PV × interest rate = payment

PV × r = PMT

PV = \frac{P M T}{r}

Note that the interest rate and the payment must be quoted in the same period; if the payments are monthly, than the monthly rate should be used. Also, the first payment is assumed to be at the end of the year; if the first payment was immediate, then the value would be higher by exactly the amount of one payment.

The fact that a stream of payments lasting forever has a specific finite value today is a surprising result for some people. The key is to realize that, because of the time value of money, each successive payment contributes a smaller and smaller amount to the PV.

Figure 7.4 Contribution of Payments over Time to Value of Perpetuity

Returning to the bank deposit example: if I don’t withdraw the full 5% of the $200, instead only withdrawing 4% (that is, 1% less), then I won’t receive as much money this year ($8 vs. $10), but my principal balance will grow by 1% to $202. If I continue to withdraw only 4%, my second year’s withdraw will be $202 × 4% = $8.08, which happens to be exactly 1% higher than my first year’s withdrawal. Withdrawing 1% less causes both my principal to grow by 1% at the end of the year, and my following year’s payment to grow by the same 1%.

Figure 7.5 Perpetuity with Growth Timeline

If I consistently withdraw less than the interest rate (the difference, interest rate minus withdrawal rate, being the growth rate, g), I can model the relationship using the following modified perpetuity equation:

Equation 7.2 Perpetuity with Constant Growth

PV now × withdrawal rate = payment at end of year

PV now × (interest rate – growth rate) = payment at end of year

P V_{n} \times (r - g) = P M T_{n + 1}

P V_{n} = \frac{P M T_{n + 1}}{(r - g)} {for g < r}

Note that we are assuming that some money is withdrawn in each year, meaning that g must be less than r. And, when g equals zero, this equation reduces to our earlier equation.

What Do These Subscripts Mean?

Since our payments are changing over time, we use subscripts (the little “n” and “n+1” in the above equation) to differentiate one payment from another. Almost always, subscripts will denote time periods for us, such that today is 0, one time period in the future is 1, and so on. Unless we specifically say otherwise, the default time period will be years, so $P M T_{0}$ is the payment we just received today, and $P M T_{1}$ is what we expect to receive in exactly one year. If we want to be more generic and describe any given year, than $P M T_{n}$ is year “n” and $P M T_{n + 1}$ is the year immediately following year “n”.

Since we know that our growth rate, g, is constant, given any payment we can find the following year’s payment by increasing it by g.

Equation 7.3 Payment Growth

payment at end of year = payment now + (g × payment now)

P M T_{n + 1} = P M T_{n} + (g \times P M T_{n}) = P M T_{n} (1 + g)

P V_{n} = \frac{P M T_{n + 1}}{(r - g)} = \frac{P M T_{n} (1 + g)}{(r - g)} {for g < r}

Here $P M T_{n}$ is the payment just paid, so technically its value isn’t included in the PV; we are only including it since we can use it to find the payment due next year.

Key Takeaways

A perpetuity is a constant stream of payments lasting forever, but a finite PV can be calculated.
The perpetuity equation can be extended to account for cases where the payment grows over time, provided that the rate of growth is less than the interest rate.

Exercises

What is the present value of $50 per year forever, if the current interest rate is 10%? What if only the first payment in a year’s time is $50, but then the payments grow at 2% per year?
I buy a perpetuity from a bank today for $200. In five years, after collecting each year’s payment, I then offer to sell the perpetuity back to the bank. If interest rates haven’t changed, what should the perpetuity be worth at that time? If interest rates have risen, should the value of the perpetuity rise or fall?