**Future Value of Annuity**

**Future Value of Annuity** is a series of constant cash flows (CCF) over limited period time i.e. monthly rent, installment payments, lease rental. When a sequence of payments of some fixed amount are made in an account at equal intervals of time. There are two types of ordinary annuity:

## Ordinary Annuity or Deferred Annuity

If constant cash flow occur at the end of each period/year. Payment of car loan, mortgage loan and student loan are examples of ordinary annuity

## Future value of ordinary annuity (annual compounding)

The value of annuity at some future time evaluated at a given interest rate assuming that compounding take place one time in a year (Annually). Two methods for calculation:

### Formula

FVA_{n }= Future value of ordinary annuity for n years FVIFA = Future Value Interest Factor for Annuity

CCF = Constant Cash Flows i = Interest rate n = Number of years

** Illustration 1:** If an employee deposits Rs. 1,000 at the end of each year into his company’s plan which pays 7% interest compounded yearly, how much will he have in the account at the end of 5 years?

*Illustration 2*** :** What is the future value of a 4-year annuity, if the annual interest is 5%, and the annual payment is Rs. 1,000; calculate by scientific calculator and by spread sheet?

*Illustration 3*** :** What is the future value of income stream which provides Rs. 1,000 at the end of year one, Rs. 2,500 at the end of year two, and Rs. 5,000 during each of the year 3 through 5, if the discount rate is 4 percent?

## Future value of annuity (intra-year compounding)

The value of annuity at some future time evaluated at a given interest rate assuming that compounding take place more than one time in a year (Intra-year). Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. i/m and n*m. Two methods for calculation:

### Formula

*Illustration 4*** : **Mohammad Ali Corporation started making sinking fund deposits of Rs. 20,000 at end of year. Its bank pays 6% compounded semi-annually. What will the fund be worth at the end of that time by factor formula and general formula?

## Future value of annuity by continuous compounding

When interest is compounded “many times”, we say that the interest is compounded continuously. It can be done annually or intra yearly (No matter it is annually or intra yearly, answer will be same for all).

### Formula

*Illustration 5*** : **Ten annual payments of Rs. 400 are made into a national bank deposit account that pays 4 percent interest per year. What is the future value of this annuity at the end of 10

^{th}years if continuous compounding takes place

**(a)** Annually

**(b)** Semi annually?

## Annuity Due

If payments or receipts are made at the beginning of each year/period, the annuity is an annuity due. Rental payment for apartment and life insurance payments are typical example of this annuity

## Future value of annuity due (annual compounding)

The value of annuity due at some future time evaluated at a given interest rate assuming that compounding take place one time in a year (annually). Two methods for calculation

*Illustration 6*** :** An annuity makes 5 annual payments of Rs. 1,000 with the first payment coming today. What is the future value of this from now if the interest rate is 7%?

## Future value of annuity due (intra-year compounding)

The value of annuity due at some future time evaluated at a given interest rate assuming that compounding take place more than one time in a year (Intra Year). Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. i/m and n*m. Two methods for calculation

### Formula

*Illustration 7*** :** If 10 annual payments of Rs. 900 are made into saving account that pays 6 percent interest per year. What is the future value of this annuity, if compounding take place semi-annually. Solve this problem by factor formula and table?

** Illustration 8: **You deposit Rs. 17,000 each year for 10 years at 7%. Then you earn 9% after that. If you leave the money invested for another 5 years how much will you have in the 15th year?

>> Practice Future Value of Annuity MCQs.