Annuities & perpetuities are related to the discounting of cash flows that occur in some future time. Let’s discuss them below one by one in detail with examples.
What is Annuity
Annuity is composed of constant payments which are divided over a fix number of years or throughout the life of the person or both.
The famous kinds of annuities are mortgage payments, monthly rent or insurance premiums etc.
Different Types of Annuities
There are two types of annuities which are as follow
- Ordinary Annuity
- Annuity Due
Ordinary Annuity
Ordinary annuity is also known as deferred annuity composed of series of constant payments at the end of each period.
Annuity Due
A series of constant payments due at the beginning of each period is called annuity due. Annual Compounding (at the end of each year):
Following is the formula of annual compounding at the end of each year.
FV = CCF (1+i)n -1
For example, rental payment of $15,000 to the landlord produces an annuity stream. The future value of annuity is described as follow:
Future value of annuity = constant cash flows x (1+i)n -1 / i
Where n = no of years
i = Interest rate
Multiple Compounding
Future value of annuity = CCF (constant cash flow) * (1+i/m)m/n -1 / i / n.
When the future value of annuity is calculated then famous interest rate formula is used to calculate the present value of annuity.
Annual Compounding (at the end of each year)
PV = FV (1+i)n Where n = life of annuity in number of years
Multiple Compounding
PV = FV / [ 1+(i/m)mxn ] Where i = % Interest per year
More than once per year i.e. monthly (m=12), semi-annually (m=2), quarterly (m=4)
n = no of years
Example of Annuity
There is a financial decision of either purchasing certain asset or obtain it on lease (Installments). A car has current market value of 160,000.
If that car is obtained on lease than certain fixed rental payments are required to be paid to leasing company at some fixed interest rate.
The car is used by the person but the ownership remains to the leasing company until all the rental payments are made to the leasing company. The main decision is to either purchase the car or get it on lease.
The leasing company charges lease rental payment of $125,000 every year for two years at the nominal rate of interest of 20%.
If time value of money is not taken into account, then total $250,000 is being paid to the leasing company after two years.
First of all the present value of the investment is calculated by using time value of money concept. For this purpose, the future value should be calculated by using annuity formula.
FV = CCF [(1+i)n -1] / i
FV = 125,000[(1+0.2)2-1] / 0.2
FV = 275,000 (Yearly compounding)
If the amount is deposited into bank annually at a rate of 20%, $275,000 is obtained at the end of the second year.
Now the present value of the future value needs to be calculated by previous interest rate formula.
PV = FV / (1+i)n
PV = 275,000 / (1+0.2)2
PV = 190,972
The final amount is 65,000 (App.) more than the purchasing price of the car.
The above example is not formed on realistic assumptions because of the fact that the car lease rental payments are monthly paid rather than on annual basis.
There is rental payment of $10,417 every month for 2 years. In this case the periodic interest rate (i/m) is used.
A technical question can arise from in the light of new calculation that by paying $10,417 monthly equal 125,000 for year than what is the difference?
The answer can be simply understood from the concept of time value of money. Following formula is used in the calculation of future value of annuity on monthly basis.
FV = CCF {[(1+i/m)mxn -1] / (i/m)}
In this case m = 12, so
FV = 125,000 {[( 1+0.2/12)12×2 – 1] / (02/12)
Now the present value of the future value of the annuity is calculated below.
PV = FV / (1+i)n
The present value of annuity has another name which is intrinsic value of annuity. The above-mentioned technique is helpful in comparing the money paid to the leasing company with the car’s market value.
The results assist in making the decision to either purchase the car or get it on lease. The 20% p.a rate exhibits that there is payment of 20% interest on leasing the car which will make its cost higher than its purchase value.
What is Perpetuity
The annuity that has infinite life making constant payments is referred to as perpetuity. The retirement plan is best real-life example of perpetuity.
For example, a person needs to save particular amount of money and invested that money in some kind of investment or security that will provide him steady & consistent rate of return on each month or quarter which is continuous payment throughout the life of that individual.
As the life of person is not fully sure therefore the formula of perpetuity is simpler than annuity
Future value of perpetuity = constant cash flow / interest rate.
It is assumed that the perpetuity is ongoing & never ending so time is dropped out of the equation because it is irrelevant.
Example of Perpetuity
The best example of perpetuity is related to the retirement planning case.
Suppose Mr. Ali has planned to retire at the age of 60 years and receives $220,000 per year from his bank as long as he lives.
How much amount of money should be deposited by Mr. Ali into his bank which is offering 10% p.a interest so that he will receives $220,000 per year throughout his remaining life.
The formula for this query is below
PV = CCF / i
PV = 220,000 / 0.10
PV = 2,200,000
It is clear that Mr. Ali will receive $220,000 per year for rest of his life and the receiving amount neither decreases nor finishes.
The main concept behind that is Mr. Ali will receive the income as accrued interest on the investment that he has made earlier.
The receiving amount is not the part of the investment made by Mr. Ali but instead it is the yield on his investment.
It is an attractive idea for earning money but there is one big issue which the inflation that erodes the money with the passage of time.
The real return is much lower due to inflation rate which is subtracted from the apparent income.
