This is “Perpetuities”, section 7.2 from the book Finance for Managers (v. 0.1).
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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 EquationPV × interest rate = payment
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 GrowthPV now × withdrawal rate = payment at end of year
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.
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 is the payment we just received today, and is what we expect to receive in exactly one year. If we want to be more generic and describe any given year, than is year “n” and 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 Growthpayment at end of year = payment now + (g × payment now)
Here 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.