Annuities and perpetuities · RiskAtoms Academy

By the end you will be able to

1Value a level stream of cash flows (an annuity) with the annuity factor.
2Value a perpetuity and a growing perpetuity.
3Work out the level payment on a loan.
4Tell an ordinary annuity from an annuity due, and adjust the value.

Why it matters

Borrow A$600,000 over 25 years at 6% and the bank will ask for A$3,865.81 every month, for 300 months. Where does that exact number come from? From one factor, and you will be able to work it out by the end of this page.

Core explanation

Level streams of cash

Many financial products pay the same amount at regular intervals: mortgage and car loan repayments, rent, a fixed pension, the coupons on a bond. A stream like this is an annuity.

You already know how to value one: discount each payment and add them up (L1.3.03). Take $1,000 a year for 10 years at 5%. The first payment is worth $952.38 today, the second $907.03, and the last only $613.91. Add all ten and you get $7,721.73. That works, but for a 25-year monthly mortgage you would be adding 300 terms.

Because every payment is the same, there is a shortcut. The value today of 1 per period for T periods is the annuity factor. At 5% for 10 years it is 7.7217. Multiply by the payment and you are done: 1,000 × 7.7217 = 7,721.73.

What if the payments never stop? A stream that goes on for ever is a perpetuity. It sounds as if it should be worth an infinite amount, but far-off payments are worth almost nothing today. The value is simply the payment divided by the rate. Put $20,000 in an account at 5% and it pays $1,000 a year for ever without ever shrinking. So $1,000 a year for ever is worth $20,000 at 5%. The 10-year annuity above is already worth 39% of that.

Two variations come up all the time. If payments grow at a steady rate g, such as a dividend that rises each year, the stream is a growing perpetuity. And if payments come at the start of each period, like rent paid in advance, it is an annuity due. The usual case, with payments at the end, is an ordinary annuity.

Name that stream

0 / 5

A car loan repaid in 48 equal monthly instalments.
A preference share that pays a fixed dividend each year, with no end date.
A zero-coupon bond that repays 100 in five years and nothing before.
A company's dividend, expected to rise about 3% a year for the foreseeable future.
The coupons (not the face value) of a 10-year bond with a fixed annual coupon.

The formal version

We use the notation of L1.4.03: $C$ is the level payment, $y$ the rate per period and $T$ the number of payments. The first walkthrough turns the sum of discounted payments into the annuity factor.

Why does the sum collapse to that fraction? The Math Atom on geometric series gives the algebra. Here is a way to see it with no algebra at all, using perpetuities.

Growing perpetuity. If the first payment is $C_1$ , one period from now, and payments grow at rate $g$ each period, the growth eats into the discounting:

\mathrm{PV} = \frac{C_1}{y - g} \qquad (y > g)

With $g = 0$ it is the plain perpetuity again. If $g$ is at or above $y$ , the payments grow as fast as they are discounted and the formula breaks down.

Annuity due. Each payment arrives one period earlier, so each is discounted once less:

\mathrm{PV}_{\text{due}} = \mathrm{PV}_{\text{ordinary}} \times (1 + y)

Loan payments. A loan is an annuity seen from the lender's side: the amount lent is the present value of the payments. Rearrange to find the payment:

\text{Loan} = C \times \frac{1 - (1+y)^{-T}}{y} \quad\Rightarrow\quad C = \frac{\text{Loan}}{\text{annuity factor}}

The formulas, for reference

\mathrm{PV}_{\text{annuity}} = C \times \frac{1 - (1+y)^{-T}}{y} \qquad \mathrm{PV}_{\infty} = \frac{C}{y} \qquad \mathrm{PV}_{\text{growing}} = \frac{C_1}{y - g}

$y$ must match the payment period: for monthly payments use the monthly rate and count $T$ in months.

Try it

Start with $1,000 a year for 10 years at 5%. Stretch the number of payments to 100 and watch the running total flatten out just under the perpetuity line. Then switch to payments at the start of each period.

Try itAnnuity builder

Open the loan calculator →

Payment C$1,000

Rate y5.00%

Number of payments T10

Each period, paid

Value today

$7,722

Annuity factor7.7217

Total paid$10,000

Perpetuity value$20,000

Share of perpetuity38.6%

Each payment today, and the running total payment, worth today running total perpetuity C ÷ y

Bars: each $1,000 payment (outline) and its value today (fill), on their own scale. Line: the value today of all payments so far, against the left axis.

Worked example

The monthly mortgage payment

Back to the hook. A bank in Australia lends A$600,000 for 25 years at 6% a year, repaid monthly. Australian mortgage rates are quoted as a nominal annual rate, so the monthly rate is 6% ÷ 12. Try each step before revealing it.

Find the monthly payment

Step 1 of 5

1
Match the rate and the periods
Rate per month $y = 0.06 \div 12 = 0.005$ . Number of payments $T = 25 \times 12 = 300$ .

Two perpetuities, valued with the same idea:

Stream	Inputs	Formula	Value today
3½% undated gilt, per 100 face	C = 3.5, y = 4%	C ÷ y	87.50
Dividend, growing	C₁ = $2, g = 3%, y = 8%	C₁ ÷ (y − g)	$40.00
$1,000 a year, 10 years, paid in advance	y = 5%	Ordinary × (1 + y)	$8,107.82

Regulatory lens

Under IFRS 16, a company that leases an office or an aircraft records a lease liability equal to the present value of the lease payments it has not yet made. Level lease payments make that liability an annuity, valued exactly as above.

Looking ahead: insurers that sell retirement annuities hold long-dated bonds to match them, and Solvency II has special rules for that business (L4.6).

Sparks

SparkThe two-billion-dollar jackpot that was not

In November 2022 a US Powerball jackpot was advertised at about $2.04 billion. That headline is the total of 30 yearly payments, each 5% larger than the last: a growing annuity. Winners can instead take a cash lump sum, and this winner did, for about $998 million. The lump sum is roughly what the annuity is worth today.

SparkPerpetuities that ended

The UK government once had bonds with no repayment date, such as Consols and the 3½% War Loan from the First World War. Holders received a fixed coupon for as long as the government chose not to repay. In 2015 the Treasury repaid the last of them, with some of the debt tracing back to the 18th century.

Key takeaways

An annuity is equal payments for T periods: PV = C × annuity factor, where the annuity factor is (1 − (1 + y)^−T) ÷ y.
A perpetuity pays C for ever and is worth C ÷ y; a growing perpetuity is worth C₁ ÷ (y − g), as long as y > g.
An annuity is a perpetuity now minus a perpetuity starting at T.
A loan payment is the loan divided by the annuity factor, with the rate and periods matched to the payment frequency.
Paying at the start of each period (annuity due) multiplies the value by (1 + y).

Reaction

Check what stuck

Question 1 · objective 1

What is £1,000 a year for 5 years worth today at 6% a year? Payments at the end of each year. Answer in £.

Question 2 · objective 2

A share pays a fixed $50 a year for ever. At a required return of 4%, what is it worth today, in $?

Question 3 · objective 2

A company will pay a dividend of €3 next year, growing at 2% a year for ever. Investors require 7%. What is the stream worth today, in €?

Question 4 · objective 3

You borrow $20,000 for 4 years at 5% a year, repaid in 4 equal payments at the end of each year. How much is each payment, in $?

Question 5 · objective 4

Two annuities have the same payment, rate and number of payments. One pays at the start of each period. Its value is:

AThe same as the otherBHigher, by a factor of (1 + y)CLower, by a factor of (1 + y)DHigher by exactly one payment

Glossary

Annuity: A stream of equal payments made at regular intervals for a fixed number of periods.
Ordinary annuity: An annuity whose payments come at the end of each period. The default in formulas.
Annuity due: An annuity whose payments come at the start of each period, such as rent paid in advance.
Annuity factor: The value today of receiving 1 each period for T periods: (1 − (1 + y)^−T) ÷ y.
Perpetuity: A stream of equal payments that never ends. Its value today is C ÷ y.
Growing perpetuity: A never-ending stream whose payments grow at a constant rate g. Its value today is C₁ ÷ (y − g).
Amortising loan: A loan repaid in equal instalments that each cover the interest due and part of the original amount.

Deep Dives & Origins

Deep Dives

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill), chapter 2, on perpetuities and annuities. Beginner.
Koller, Goedhart and Wessels (McKinsey & Company), Valuation: Measuring and Managing the Value of Companies (Wiley), on the growing perpetuity as a continuing value. Intermediate.

Origins

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill).
IASB, IFRS 16 Leases (lease liability measurement).
UK Debt Management Office, redemption of undated gilts, 2015.
Multi-State Lottery Association, Powerball draw of 7 November 2022 (jackpot and cash value).

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