Annuities and Perpetuities | Types AND Examples

Annuities and perpetuities are related to the discounting of cash flows that occur at some point in the future. Let’s discuss them below in detail with examples.

What is Annuity

An annuity is composed of constant payments divided over a fixed number of years, throughout the life of a person, or both. Famous types of annuities include mortgage payments, monthly rent, and insurance premiums.

Different Types of Annuities

There are two types of annuities:

1. Ordinary Annuity
2. Annuity Due

1. Ordinary Annuity

An ordinary annuity, also known as a deferred annuity, is composed of a series of constant payments made at the end of each period.

2. Annuity Due

An annuity due consists of a series of constant payments made at the beginning of each period. The formula for annual compounding at the end of each year is:

FV=CCF(1+i)n−1iFV = CCF \frac{(1+i)^n – 1}{i}FV=CCFi(1+i)n−1​

Where:

• n number of years
• i = interest rate

Example:
A rental payment of $15,000 to a landlord produces an annuity stream. The future value of an annuity is calculated as:

FV=constant cash flows×(1+i)n−1iFV = \text{constant cash flows} \times \frac{(1+i)^n – 1}{i}

Multiple Compounding

For multiple compounding periods, the formula is:

FV=CCF(1+i/m)m⋅n−1i/mFV = CCF \frac{(1+i/m)^{m \cdot n} – 1}{i/m}FV=CCFi/m(1+i/m)m⋅n−1​

When the future value of an annuity is calculated, the well-known interest rate formulas are used to calculate the present value of the annuity.

Present Value Formulas

Annual Compounding:

PV=FV(1+i)nPV = \frac{FV}{(1+i)^n}PV=(1+i)nFV​

Multiple Compounding:

PV=FV[1+(i/m)]m⋅nPV = \frac{FV}{[1 + (i/m)]^{m \cdot n}}PV=[1+(i/m)]m⋅nFV​

Where:

• $i$ = % interest per year
• m = number of compounding periods per year (monthly = 12, semi-annually = 2, quarterly = 4)
• n = number of years

Example of Annuity

Consider a financial decision: purchasing an asset outright or obtaining it on lease (installments).

• Car Market Value: $160,000
• Lease Rental Payment: $125,000 per year for 2 years
• Interest Rate: 20% p.a

Step 1: Calculate Future Value

FV=CCF(1+i)n−1iFV = CCF \frac{(1+i)^n – 1}{i}FV=CCFi(1+i)n−1​ FV=125,000(1+0.2)2−10.2=275,000(Yearly compounding)FV = 125,000 \frac{(1+0.2)^2 – 1}{0.2} = 275,000 \quad \text{(Yearly compounding)}FV=125,0000.2(1+0.2)2−1​=275,000(Yearly compounding)

Step 2: Calculate Present Value

PV=FV(1+i)n=275,000(1+0.2)2=190,972PV = \frac{FV}{(1+i)^n} = \frac{275,000}{(1+0.2)^2} = 190,972PV=(1+i)nFV​=(1+0.2)2275,000​=190,972

The final amount is approximately $65,000 more than the purchasing price of the car.

Monthly Payments:

If rental payments are $10,417 monthly, the future value is calculated as:

FV=CCF(1+i/m)m⋅n−1i/m,m=12FV = CCF \frac{(1+i/m)^{m \cdot n} – 1}{i/m}, \quad m = 12FV=CCFi/m(1+i/m)m⋅n−1​,m=12

The present value of this annuity, also called the intrinsic value of the annuity, helps compare leasing versus purchasing. The 20% annual rate shows that leasing incurs interest, making the total cost higher than purchasing.

What is Perpetuity

A perpetuity is an annuity with an infinite life, making constant payments. A retirement plan is a real-life example of perpetuity.

Future Value of Perpetuity:

FV=constant cash flowinterest rateFV = \frac{\text{constant cash flow}}{\text{interest rate}}FV=interest rateconstant cash flow​

Since perpetuities are ongoing, time is irrelevant in the formula.

Example of Perpetuity

Suppose Mr. Ali plans to retire at 60 and wants to receive $220,000 per year for life from a bank offering 10% p.a. interest.

PV=CCFi=220,0000.10=2,200,000PV = \frac{CCF}{i} = \frac{220,000}{0.10} = 2,200,000PV=iCCF​=0.10220,000​=2,200,000

This ensures Mr. Ali receives $220,000 per year indefinitely.

• The income comes from accrued interest on the initial investment.
• The principal remains untouched, and the payments are purely the yield of the investment.

Note: Inflation can reduce the real return over time, making the apparent income less valuable in real terms.