Level perpetuity

Exam FM – Financial Mathematics · SOA (Society of Actuaries) · 5 min read

Level Perpetuities: The Gift That Keeps on Giving

Welcome to one of the most elegant topics in Exam FM! So far, you have likely been dealing with annuities that have a set end date (like a 5-year car loan or a 30-year mortgage). But what happens if the payments never stop? That is the world of perpetuities.

While the idea of "forever" might sound intimidating mathematically, you will quickly find that perpetuities actually have much simpler formulas than regular annuities. In this chapter, we will explore why these infinite cash flows are so useful and how to master the calculations for your exam.

What exactly is a Perpetuity?

A perpetuity is an annuity where the payments continue forever. There is no "n" (number of periods) because time never ends in this scenario.

Real-World Analogy: Imagine a wealthy donor sets up a university scholarship. They want the scholarship to pay out $10,000 every year, forever, without ever touching the original "pot" of money. That scholarship fund is a level perpetuity.

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Did you know? The word "perpetuity" comes from the same root as "perpetual," meaning never-ending or unchanging. In finance, it represents a stream of cash that lasts as long as the institution (like a government or a university) exists.

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Key Prerequisite: The Concept of Convergence

Don't worry if you aren't a calculus expert! The reason a "forever" stream of money has a finite Present Value is because of the time value of money. A payment of $100 received 500 years from now is worth almost nothing today. Because those future payments become smaller and smaller in today's value, they eventually "sum up" to a specific, finite number.

1. Perpetuity-Immediate

A Perpetuity-Immediate is a series of level payments where the first payment occurs one period from today (at time $t=1$).

The Formula

The Present Value ($PV$) of a perpetuity-immediate with payments of $1$ at an effective interest rate $i$ is:

$ a_{\infty|i} = \frac{1}{i} $

If the payment amount is $P$ instead of $1$, the formula is simply:

$ PV = \frac{P}{i} $

Example: If you want to receive $500 at the end of every year forever, and the annual effective interest rate is 5%, how much must you invest today?

Step-by-step:
\n1. Identify the payment: $P = 500$
\n2. Identify the interest rate: $i = 0.05$
\n3. Apply the formula: $PV = \frac{500}{0.05} = 10,000$
\nAnswer: $10,000

Why does it work?

Think about it this way: If you have $10,000 in a bank account earning 5%, you will earn $500 in interest at the end of the year. If you withdraw only that $500 interest, you still have $10,000 left to earn another $500 next year. You can do this forever!

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Quick Review: For Perpetuity-Immediate, payments start at $t=1$. The formula is just Payment / Interest Rate.

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2. Perpetuity-Due

A Perpetuity-Due is a series of level payments where the first payment occurs immediately (at time $t=0$).

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The Formula

The Present Value ($PV$) of a perpetuity-due with payments of $1$ is:

$ \ddot{a}_{\infty|i} = \frac{1}{d} $

Remember that $d = \frac{i}{1+i}$. You can also think of this as:
\n$ \ddot{a}_{\infty|i} = \frac{1}{i} + 1 $ (The "Immediate" version plus one extra payment right now!)

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If the payment amount is $P$, the formula is:

$ PV = \frac{P}{d} $

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Example: You win a "Life-Long Prize" that pays $500 starting today and continues every year forever. If the annual effective interest rate is 5%, what is the value of this prize?

Step-by-step:
1. Identify the payment: $P = 500$
2. Identify the interest rate: $i = 0.05$
3. Calculate the discount rate: $d = \frac{0.05}{1.05} \approx 0.047619$
4. Apply the formula: $PV = \frac{500}{0.047619} = 10,500$
Alternatively: $PV = 500 + \frac{500}{0.05} = 500 + 10,000 = 10,500$

Memory Trick:
Immediate starts later, so it uses $i$.
Due starts today, so it uses $d$.

Common Pitfalls to Avoid

Even the best students can make these mistakes. Keep an eye out for them!

Rate Mismatch: Always ensure your interest rate $i$ matches the frequency of the payments. If payments are monthly, you must use a monthly effective interest rate.
The "One Period Before" Rule: The formula $PV = \frac{P}{i}$ gives the value of the perpetuity one period before the first payment. If the first payment is at $t=5$, the formula $\frac{P}{i}$ gives the value at $t=4$. You then have to discount that value back to $t=0$.
Confusing $i$ and $d$: Remember that for any annuity or perpetuity, the Due version (starting at $t=0$) is always worth more than the Immediate version (starting at $t=1$) because you get your money sooner!

Summary and Key Takeaways

The Formulas You Need:
1. Perpetuity-Immediate: $PV = \frac{P}{i}$
2. Perpetuity-Due: $PV = \frac{P}{d}$
3. Relationship: $\ddot{a}_{\infty|i} = a_{\infty|i} \times (1+i)$

Steps for Success:
1. Draw a Timeline: This is the most important step for any FM problem. Mark when the first payment occurs.
2. Identify the Rate: Make sure your $i$ matches the payment period.
3. Check the Timing: Is the first payment today ($t=0$) or at the end of the period ($t=1$)?
4. Apply and Adjust: Use the formula and then discount/accumulate the result to the specific time requested in the problem.

Don't worry if this seems tricky at first! Perpetuities are actually your friends because they get rid of the complex $(1+i)^{-n}$ term found in standard annuities. Once you master the timing, these will be the fastest points you earn on the exam.

Quick Tip for the Exam: If a question mentions a "very long time" or "indefinitely," it is often a hint to treat the cash flow as a perpetuity!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of a perpetuity in financial mathematics?

A perpetuity is an annuity where the payments continue forever, meaning there is no set end date.

Why is the Present Value of an infinite stream of payments a finite number?

Due to the time value of money, the value of future payments decreases over time; as time approaches infinity, their present value becomes negligible, allowing the total sum to converge to a finite value.

What is the timing of the first payment for a Perpetuity-Immediate?

The first payment occurs one period from today (at time $t=1$).

What is the formula for the Present Value of a level Perpetuity-Immediate?

$PV = \frac{P}{i}$, where $P$ is the payment amount and $i$ is the effective interest rate per period.

When does the first payment occur in a Perpetuity-Due?

The first payment occurs immediately (at time $t=0$).

What is the formula for the Present Value of a level Perpetuity-Due?

$PV = \frac{P}{d}$, where $P$ is the payment amount and $d$ is the effective discount rate per period.

What is the "One Period Before" rule for perpetuity formulas?

The standard perpetuity-immediate formula $\frac{P}{i}$ calculates the value of the stream exactly one period before the first payment occurs.

How must the interest rate be adjusted if payments occur with a different frequency than the annual effective rate?

The interest rate ($i$) must be converted to an effective interest rate that matches the frequency of the payments.

What is the relationship between the Present Value of a Perpetuity-Due and a Perpetuity-Immediate?

$\ddot{a}_{\infty|i} = a_{\infty|i} \times (1+i)$

Why is drawing a timeline essential for perpetuity problems?

A timeline helps identify the exact timing of the first payment, which determines whether to use the Immediate or Due formula and whether additional discounting/accumulation is needed.

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