# Perpetuity > The perpetuity formula gives the amount of money needed today to pay a fixed amount every period forever, using only the payment and the interest rate. Page: https://formulabro.yunusemrevurgun.com/finance/perpetuity/ Category: Finance Published: 2026-10-23 Updated: 2026-10-23 ## Formula $$ PV = \frac{C}{r} $$ Read it out loud: "PV equals C divided by r. PV is the present value, the amount needed today. C is the payment made each period, and the payments go on forever. r is the interest rate per period, written as a decimal. So the present value is the payment divided by the rate." ## Before you start - **Perpetuity: payments that never end**: A **perpetuity** is a stream of equal payments that is assumed to continue forever, with the first payment one period from now. It is the endless version of an annuity. A bond that never repays its face value is an example of the same idea, though most real investments do end. - **Percent as a decimal**: A **percent** is a number out of 100, so 5 percent means 5 out of 100, or 0.05 as a decimal. The formula needs the rate as this decimal. Dividing by 0.05 is the same as multiplying by 20, which is why a 5 percent rate turns a yearly payment into a value twenty times as large. - **Dividing by a small number**: Dividing by a small decimal gives a large result. For example $1 \div 0.05 = 20$ and $1 \div 0.01 = 100$. The smaller the rate, the larger the amount needed to produce a given payment, because each dollar held in the fund earns only a small amount each period. - **Period and discounting**: A **period** is one equal step of time, such as a year, and the payment and the rate must refer to the same period. **Discounting** means finding the value today of money promised later. Each later payment is worth less than the one before it, so far-off payments count for very little, and the total stays finite. More notation help: https://formulabro.yunusemrevurgun.com/math-basics/ ## What each symbol means - $PV$: **Present value of the perpetuity**, the amount needed today to pay C every period forever, measured in dollars. It is the number the formula gives you, and it is finite even though the payments never stop. - $C$: **Payment per period**, the fixed amount paid at the end of each period, measured in dollars. The first payment comes one period from now. In the example C is 5000 dollars a year. - $r$: **Interest rate per period**, written as a decimal and matched to the payment period. Five percent per year is r = 0.05 for yearly payments. The rate must be above zero, because a rate of zero would mean dividing by zero. ## The formula piece by piece - $C$: $C$ is the payment made at the end of each period, a fixed amount in dollars. The formula uses the same $C$ for every period, so the whole stream is one repeated amount. - $r$: $r$ is the interest rate per period as a decimal. It is the share of the fund that is earned each period, so a fund of size PV earns $r \times PV$ of interest in each period. - $\frac{C}{r}$: Divide the payment by the rate. The fund must hold enough that its yearly interest equals the payment, and that means holding C divided by r. With C = 5000 dollars and r = 0.05, that is 100000 dollars. - $PV = \frac{C}{r}$: The whole formula in one sentence: the present value of the perpetuity equals the payment per period divided by the rate per period, which is the fund that pays C every period forever. ## What the formula is doing, and why A **perpetuity** is a series of equal payments that never ends. It sounds as if it should cost an infinite amount of money, but it does not. The present value is finite, because money promised far in the future is worth very little today: each later payment is discounted more heavily than the one before it. **Why divide by the rate?** Think about what the fund must earn. If a fund holds PV and earns the rate r each period, it earns $r \times PV$ of interest per period. To pay C every period and keep the fund's balance unchanged, the interest must exactly equal the payment, so $r \times PV = C$. Solving for PV gives $PV = \frac{C}{r}$. That is why the rate sits on the bottom: a smaller rate means the fund must be larger to earn the same interest. **Why does the balance stay the same?** The fund pays out exactly the interest it earns each period, so its balance never shrinks, and the payments can go on forever. Only the interest is spent. The fund itself stays whole. **Why the formula is a limit of the annuity formula.** The present value of an annuity with $n$ payments gets closer and closer to $\frac{C}{r}$ as $n$ grows very large, because the discount factor $(1 + r)^{-n}$ gets smaller and smaller. Payments far in the future count for almost nothing, so a long enough series behaves like a perpetuity. ## An everyday picture Think of a jar of money that only ever pays out the interest it earns. If the jar earns 5 percent a year and you take out exactly that 5 percent each year, the jar never gets smaller, so the payout can go on forever. The bigger the jar, the bigger the payout, and the smaller the rate, the bigger the jar needs to be. ## Worked example: Endowing a 5000 dollar a year scholarship forever A donor wants a scholarship that pays 5000 dollars at the end of every year, forever. The gift is held in an endowment fund that earns 5 percent per year, and the fund pays out only that year's interest. How much must the donor give today? In the formula, $C = 5000\ \text{dollars}$ per year and $r = 0.05$. The numbers are an illustration. 1. **Write down the inputs.** The scholarship pays 5000 dollars at the end of each year, so $C = 5000$ dollars and the period is one year. The fund earns 5 percent a year, which is $r = 0.05$ as a decimal. Both the payment and the rate refer to one year, so the formula can be used directly. 2. **Divide the payment by the rate.** The formula gives $PV = \frac{C}{r} = \frac{5000\ \text{dollars}}{0.05}$. Dividing by 0.05 is the same as multiplying by 20, because $1 \div 0.05 = 20$, so $5000 \times 20 = 100000$ dollars. Each dollar a year of payment needs 20 dollars held in the fund, because each dollar held earns only 0.05 dollars a year. 3. **Check the fund's interest.** The fund holds 100000 dollars and earns 5 percent of that in the first year: $0.05 \times 100000 = 5000$ dollars. That is exactly the scholarship payment, so the fund pays out 5000 dollars and still holds 100000 dollars at the end of the year. The next year the same thing happens, so the balance never falls and the payments never run out. 4. **Compare with a smaller rate.** If the fund earned only 4 percent, the same 5000 dollar payment would need $5000 \div 0.04 = 125000$ dollars. A lower rate means the fund must be bigger to produce the same interest, which is why the rate sits on the bottom of the formula. **Answer:** The donor must give about **100000 dollars** today. At 5 percent a year the fund earns exactly 5000 dollars each year, which pays the scholarship forever without touching the original 100000 dollars. A smaller rate of return would need a larger gift. ## Common mistakes - Using a payment per period with a rate given per year, such as a monthly payment with a yearly rate. Match the period first, for example by using 0.05 / 12 for a monthly rate. - Writing the rate as 5 instead of 0.05. Dividing by 5 gives a present value of 1000 dollars for a 5000 dollar payment, which is far too small. - Assuming the formula still applies when the first payment is due right away. The formula assumes the first payment is one period away. When payments start now, every payment comes one period sooner, so the answer is larger. - Thinking a perpetuity must be worth an infinite amount because it never ends. The payments go on forever, but each one is discounted more heavily than the one before, so the total present value is finite. ## Where you will meet it - Valuing an endowment or a trust that pays out a fixed sum every year, such as a scholarship fund. - Estimating the value of a preferred stock that pays a fixed dividend, when the dividend is assumed to continue indefinitely. - Pricing a fixed rent with no end date, where the payments are treated as continuing forever. - Quick approximation of a very long annuity, since the perpetuity value is what the annuity formula approaches as the number of payments grows. ## Questions people ask ### Can something really pay forever? In the formula, yes: it describes a payment that is assumed to go on without end. Real endowments and bonds have limits set by their terms, and interest rates can change over time, so the formula is a model of the idea rather than a promise. ### Why is the present value finite when the payments are infinite? Each payment is worth less the further away it is. Payments far in the future are discounted by huge factors, so their present values add up to a finite amount. The formula gives the total of those shrinking values in one step. ### How is this related to the present value of an annuity? A perpetuity is an annuity with endlessly many payments. As the number of payments grows, the annuity's present value gets closer and closer to $\frac{C}{r}$. ### What if the first payment is made right away? Then every payment arrives one period earlier, and the present value becomes $\frac{C}{r} \times (1 + r)$. Equivalently, the first payment C is counted in full today, and the rest is worth $\frac{C}{r}$ as a perpetuity starting one period later. ## Related formulas - [Present Value of an Annuity](https://formulabro.yunusemrevurgun.com/finance/present-value-of-annuity/) - [Present Value](https://formulabro.yunusemrevurgun.com/finance/present-value/) - [Gordon Growth Model](https://formulabro.yunusemrevurgun.com/finance/gordon-growth-model/) ## Sources - [Perpetuity, Wikipedia](https://en.wikipedia.org/wiki/Perpetuity) - [Present value, Wikipedia](https://en.wikipedia.org/wiki/Present_value) From Formula Bro (https://formulabro.yunusemrevurgun.com/), a side project developed by yunusemrevurgun.com.