Annuities Explained: Constant vs. Growing Annuity

Annuities are an important concept in finance, particularly when valuing investments, retirement income, rental properties, and other assets that generate regular cash flows over a specified period.

At their core, annuities represent a series of cash flows received or paid at regular intervals. The key question is: What are those future cash flows worth today?

The answer depends on factors such as the amount received, the duration of the cash flows, the discount rate, and whether the payments remain constant or grow over time.

An annuity is a series of cash flows received or paid at regular intervals for a fixed number of periods. For example, a person might receive ₹1,00,000 every year for the next 10 years.

Annuities are commonly used in finance to calculate the present value of future income, retirement payments, loan repayments, and investment cash flows.

There are two important cases:

Scenario A: Constant annuity — the cash flow remains the same every period.
Scenario B: Growing annuity — the cash flow increases at a constant rate g every period.

Let’s understand both, with formulas and examples.

1. Scenario A: Constant cash flow (ordinary annuity)

Suppose you receive ₹ $C$ at the end of every year for $n$ years, and the discount rate is $r$ per year.

For example, you receive ₹10,000 at the end of each year for 5 years.

Constant Annuity: Cash Flow Timeline (5 Years)

Annual cash flow = ₹10,000

Year 0       Year 1       Year 2       Year 3       Year 4       Year 5
  |             |            |            |            |            |
  ₹0          ₹10,000      ₹10,000      ₹10,000      ₹10,000      ₹10,000

              ←────── Same cash flow every year ──────→

Note: Cash flows are received at the end of each year.

Present value of a constant annuity

The present value (PV) is the value today of all the future cash flows, after discounting each one at rate $r$

The cash flow received in year 1 is worth $C / (1 + r)$ today. The year 2 cash flow is worth $C / (1 + r)^{2}$ and so on.

Therefore,

P V = \frac{C}{1 + r} + \frac{C}{(1 + r)^{2}} + \dots + \frac{C}{(1 + r)^{n}}

Using the formula for a geometric series, we get:

PRESENT VALUE — CONSTANT ANNUITY

P V = C \frac{1 - (1 + r)^{- n}}{r}

Where:

$C$ = constant cash flow per period
$r$ = discount rate per period
$n$ = number of periods

Example

Suppose you receive ₹10,000 at the end of each year for 5 years, and the discount rate is 10%. $P V = 10,000 [\frac{1 - (1.10)^{- 5}}{0.10}]$

P V \approx ₹ 37,908

Although the total cash received over 5 years is ₹50,000, its value today is only about ₹37,908 because money received in the future is worth less than money received today.

Scenario B: Cash flow increases at a rate g

A growing annuity is a series of cash flows that increases at a constant growth rate $g$

every period for a fixed number of periods.

For example, suppose you receive ₹10,000 in Year 1, and the amount increases by 5% every year.

Growing Annuity: Cash Flow Timeline (5 Years)

Initial cash flow = ₹10,000 | Annual growth rate = 5%

Year 0       Year 1       Year 2       Year 3       Year 4       Year 5
  |             |            |            |            |            |
  ₹0         ₹10,000      ₹10,500      ₹11,025      ₹11,576      ₹12,155
                └──────── Cash flow increases by 5% every year ────────┘

Note: Cash flows are received at the end of each year.

How do we calculate the growing cash flows?

Let:

$C_{1}$ = cash flow received at the end of Year 1 $g$ = growth rate per period $r$ = discount rate per period $n$ = number of periods

Then the cash flows are:

\begin{aligned} C_{1} & = C_{1} \\ C_{2} & = C_{1} (1 + g) \\ C_{3} & = C_{1} (1 + g)^{2} \\ ⋮ \\ C_{n} & = C_{1} (1 + g)^{n - 1} \end{aligned}

The exponent is $n - 1$ because the first payment is already the starting cash flow; it has not yet grown.

Present value of a growing annuity

To calculate the present value, discount each cash flow to today:

P V = \frac{C_{1}}{1 + r} + \frac{C_{1} (1 + g)}{(1 + r)^{2}} + \dots + \frac{C_{1} (1 + g)^{n - 1}}{(1 + r)^{n}}

Simplifying this geometric series gives the growing annuity formula:

PRESENT VALUE — GROWING ANNUITY

P V = \frac{C_{1}}{r - g} [1 - {(\frac{1 + g}{1 + r})}^{n}]

This formula applies when $r \neq g$ , assuming payments occur at the end of each period.

$C_{1}$ = first payment, at the end of Year 1
$r$ = discount rate per period
$g$ = growth rate per period
$n$ = number of payments

Example

Suppose:

First annual payment: ₹10,000
Annual growth rate: $g = 5 %$
Discount rate: $r = 10 %$
Duration: $n = 5$

Substituting into the formula:

P V = \frac{10,000}{0.10 - 0.05} [1 - {(\frac{1.05}{1.10})}^{5}]

P V \approx ₹ 41,242

The present value is approximately ₹41,242, compared with ₹37,908 for the constant annuity paying ₹10,000 per year.

The growing annuity has a higher present value because the cash flows increase every year.

Important Assumptions When Using Annuity Formulas

Before applying these formulas, it is important to understand their assumptions.

Payment timing: The formulas used in this article assume that payments occur at the end of each period. If the first payment occurs immediately, at time zero, the present value calculation changes.

Consistent periods: The discount rate, growth rate, and payment frequency must be aligned. For annual cash flows, both rates should be annual rates.

Fixed duration: The formulas apply to annuities with a specified number of payments. They differ from perpetuity formulas, which assume that cash flows continue indefinitely.

Constant growth: A growing annuity assumes that cash flows increase at the same percentage rate in every period. If the growth rate varies from year to year, the cash flows generally need to be valued individually.

Appropriate discount rate: The discount rate should be appropriate for the risk and timing of the cash flows being valued.

Conclusion

Annuities provide a practical way to value a series of future cash flows. A constant annuity is appropriate when the cash flow remains unchanged throughout the period. A growing annuity is useful when the cash flow increases at a consistent rate over time.

In the next blog we will see how to compute retirement corpus using annuity and perpetuity.

Fiscroll

What is annuity?

1. Scenario A: Constant cash flow (ordinary annuity)

Present value of a constant annuity

Example

Scenario B: Cash flow increases at a rate g

How do we calculate the growing cash flows?

Present value of a growing annuity

Example

Important Assumptions When Using Annuity Formulas

Conclusion

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