From Infinity to Reality: Mastering the Frequency of Compounding

When we study the Time Value of Money, we often start with single lump sums. But in the real world, finance is rarely that simple. Mortgages, car loans, retirement savings, and stock dividends all involve streams of cash flows that happen over and over again.

How do we value these complex streams?

At first glance, the starting point seems almost absurdly theoretical: the Perpetuity. This is a stream of cash flows that goes on forever. You might ask, “Why study this? Nothing lasts forever. I won’t live forever, and neither will my bank.”

Here is the secret: While a perpetuity seems like a theoretical abstraction, it is actually the mathematical “skeleton” for the most practical tools we use every day. If you can understand how to value a stream that lasts forever, you can easily use that logic to “build” an Annuity—like a 30-year mortgage or a 5-year car loan—without having to memorize complex formulas. We simply take the infinite stream and “cut it off” when we need it to stop.

In this post, we will walk through this logic step-by-step. We will start with the infinite and work our way down to the practical:

1. Perpetuity: The foundation (Infinite constant cash flows).
2. Deferred Perpetuity: Planning for the distant future.
3. Growing Perpetuity: Accounting for growth and inflation.
4. Annuity: The “difference between two perpetuities” (Real-world loans and savings).

Let’s start with the foundation.

1. Perpetuity

To handle multiple cash flows efficiently, we start with a special case that might seem theoretical but is surprisingly useful: an infinite set of cash flows.

The Concept

A Perpetuity is defined as a sequence of identical cash flows (\(C\)) that continues forever. Key characteristics include:

• The cash flow amount (\(C\)) is constant.
• The payments occur at regular intervals (typically at the end of every year).
• The time horizon is infinite (\(t \to \infty\)).

The Formula & Derivation

Calculating the Present Value (\(PV\)) of an infinite stream might seem impossible because you are adding up an infinite number of terms. However, as shown in the derivation, the math simplifies elegantly using a geometric series.

If we write out the present value of the cash flows, we get an infinite geometric series:

$$
PV = \frac{C}{(1+r)^1} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \dots \quad (1)
$$

To solve this, we can use a mathematical trick. First, divide the entire equation by \((1+r)\):

$$
\frac{PV}{(1+r)} = \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \frac{C}{(1+r)^4} + \dots \quad (2)
$$

Next, if we subtract equation (2) from equation (1), all terms on the right-hand side cancel out except for the very first one:

$$
PV – \frac{PV}{(1+r)} = \frac{C}{(1+r)}
$$

Solving for \(PV\) simplifies to the classic perpetuity formula:

$$
PV = \frac{C}{r}
$$

(Note: This assumes the first payment occurs one period from now, at \(t=1\).)

Example

Imagine a “Consol bond” (a type of perpetual bond issued by the British government in the past) or a preferred stock that promises to pay you $100 every year forever. If the current annual interest rate (or required rate of return) is 5%, how much is this stream of income worth today?

Using the formula:

$$
PV = \frac{$100}{0.05} = $2,000
$$

This tells us that receiving $100 forever (assuming a 5% discount rate) is financially equivalent to having $2,000 in cash today.

2. Deferred Perpetuity

Real-world financial planning isn’t always about immediate returns. Sometimes, we are setting up income streams for the future. This brings us to the deferred perpetuity.

The Concept

A deferred perpetuity is a series of constant, infinite cash flows—just like the standard perpetuity discussed above—but the payments do not begin immediately. They start at some specified date in the future.

The Calculation Trick

Calculating this requires a two-step process, and there is a common trap to avoid.

The standard perpetuity formula (\(PV = C/r\)) calculates the value of the cash stream exactly one period before the first payment is made.

If a perpetuity starts paying in three years, applying the standard formula tells you what that stream is worth at the end of year two. To find out what it is worth today (time zero), you must then discount that amount back for those remaining two years.

Example: The “Early Retirement” Fund

Let’s look at a personal finance scenario.

Imagine you are planning a retirement income stream. You want to set up a fund today that will pay you $12,000 a year forever, but you don’t need the income right away. You want the first payment to arrive exactly 5 years from today. The current interest rate is 4%.

How much money do you need to deposit today to make this happen?

Step 1: Find the value just before the payments start.

Since the first payment is in Year 5, the perpetuity formula gives us the total value of that stream at the end of Year 4.

$$
Value \ at \ Year \ 4 = \frac{\$12,000}{0.04} = \$300,000
$$

This means that four years from now, you need to have exactly $300,000 in the account to sustain those future $12,000 payments forever.

Step 2: Discount back to today.

Now, we need to figure out how much we need to invest today to have $300,000 sitting there in four years. We discount that lump sum back 4 periods at 4%.

$$
PV \ today = \frac{\$300,000}{(1 + 0.04)^4}
$$

$$
PV \ today = \frac{\$300,000}{1.16986}
$$

$$
PV \ today \approx \$256,442
$$

To secure that future income stream, you would need to invest approximately $256,442 today.

3. Growing Perpetuity

In the real world, cash flows rarely stay exactly the same forever. Prices rise due to inflation, and successful businesses grow their earnings. This brings us to the Growing Perpetuity.

The Concept

A growing perpetuity is an infinite stream of cash flows that are not constant but instead grow at a constant rate, \(g\), forever.

• The first payment is \(C\).
• The second payment is \(C(1+g)\).
• The third is \(C(1+g)^2\), and so on.

The Formula & Derivation

The math here follows a similar logic to the standard perpetuity. We start with the present value of the growing stream:

$$
PV = \frac{C}{(1+r)^1} + \frac{C(1+g)^1}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \dots \quad (1)
$$

To solve this, we use a slightly more advanced version of our previous trick. Instead of just dividing by \((1+r)\), we multiply the entire equation by the ratio of growth to return

$$
\left(\frac{1+g}{1+r}\right)
$$

When we subtract the new equation from the original, the middle terms cancel out, leaving us with a clean, powerful formula:

$$
PV = \frac{C}{r – g}
$$

(Note: This formula only works if the interest rate \(r\) is strictly greater than the growth rate \(g\). If \(g \ge r\), the value would be infinite!)

Example: Valuing a Dividend Stock

The most common application of this formula is valuing stocks for mature companies (often called the Gordon Growth Model).

Imagine you are considering buying shares in a stable utility company.

• The company is expected to pay a dividend of $4.00 next year.
• Because the company is solid, you expect this dividend to grow by 2% (\(g=0.02\)) every year forever to keep up with inflation.
• Given the risk of the stock, you require a 6% (\(r=0.06\)) annual return on your money.

What is the maximum price you should pay for this stock today?

$$
PV = \frac{\$4.00}{0.06 – 0.02}
$$

$$
PV = \frac{\$4.00}{0.04} = \$100
$$

According to the growing perpetuity model, this stock is worth exactly $100 to you today.

4. Annuity

Finally, we arrive at the most common financial structure you will encounter in daily life: the Annuity.

The Concept

While perpetuities last forever, most real-world cash flows do not. An annuity is simply a set of constant cash flows (\(C\)) that occur for a fixed, finite number of years (\(T\)). Common examples include car loans, mortgages, and retirement pension payouts.

The “Subtraction” Trick

You could sum up every individual cash flow, but that takes forever. Instead, we can use a clever logic derived from our previous sections.

Think of an annuity as the difference between two perpetuities:

1. Perpetuity A: Starts today (Time 1).
2. Perpetuity B: Starts in the future (Time \(T+1\)).

If you take Perpetuity A (which pays forever) and subtract Perpetuity B (which pays forever starting from year \(T+1\)), you are left with exactly the cash flows between year 1 and year \(T\)—which is our annuity!.

The Formula & Derivation

Using the logic above, the math becomes simple subtraction:

$$
PV = (PV \ of \ Perpetuity \ A) – (PV \ of \ Perpetuity \ B)
$$

We know the value of Perpetuity A is \(\frac{C}{r}\).

We know the value of Perpetuity B is also \(\frac{C}{r}\), but it is deferred by \(T\) years, so we must discount it by \((1+r)^T\).

$$
PV = \frac{C}{r} – \frac{C/r}{(1+r)^T}
$$

If we factor out the \(\frac{C}{r}\), we get the famous annuity formula used in finance textbooks everywhere:

$$
PV = \frac{C}{r} \left( 1 – \frac{1}{(1+r)^T} \right)
$$

Sometimes, the term in the brackets is shortened to \(A_r^T\), known as the Annuity Factor.

Example: Buying a Car

Let’s apply this to a loan. Suppose you can afford to pay $6,000 per year for the next 5 years to pay off a car. The bank charges an interest rate of 5%. How expensive of a car can you buy today?

• \(C = 6,000\)
• \(r = 0.05\)
• \(T = 5\)

$$
PV = \frac{6,000}{0.05} \left( 1 – \frac{1}{(1.05)^5} \right)
$$

First, we calculate the perpetuity value (the maximum possible value):

$$
\frac{6,000}{0.05} = 120,000
$$

Next, we calculate the adjustment factor (because you stop paying after 5 years):

$$
1 – \frac{1}{1.2763} \approx 1 – 0.7835 = 0.2165
$$

$$
PV = 120,000 \times 0.2165 \approx \$25,980
$$

So, with those payments, you can take out a loan for approximately $25,980 today.