8.4 Annuities: Future Value

R e g u l a r D e p o s i t s A n d F i n d i n g T i m e

AnnuityA series of payments or investments made at regular intervals.

A simple annuity is an annuity in which the payments coincide with the compounding period. An ordinary annuity is an annuity in which the

payments are made at the end of each interval.

F r o m p r e v i o u s l e s s o n s , y o u h a v e l e a r n e d h o w t o fi n d . . .

A = P (1 + r) n

Future Value of Compounding InterestFuture value, A, of the amount invested (in dollars $) at the beginning, $P, at the end of n compounding periods.

P = Amount Invested r = Interest Rate Per Period n = Number Of Compounding Periods

S n = a (r n - 1)

r - 1

Future Value of an Annuity (Formula #1)Future value of ALL investments until the LAST compounding period.

a = Amount Invested Each Period r = Interest Rate Per Period n = Number Of Compounding Periods

The formula for the Sum of a Geometric Series can be used to determine the future

value of an annuity.

Future Value of an Annuity (Formula #2)Future value of annuity in which $R is invested at the end of each n compounding periods earning i% of

compound per interval is:

FV = R x[(1 + i) n - 1 i ]

Now that we know how to find FV, we can now find the values of:

R The regular payment of an annuity required to reach future value

n The number of compounding periods to reach future value

t The term (number of years to pay off) of an annuity.

1 1ni

FV Ri

Example 1

Sam wants to make monthly deposits into an account that guarantees 9.6 %/a compounded monthly. He would like to have $500 000 in the account at the end

of 30 years. How much should he deposit each month?

First, we must calculate i and n according to the compounding period :

i = 9.6% = 0.096 / 12

= 0.008

n = 30 yrs = 30 x 12

= 360

FV = $500 000

We are now solving for r :

r = $ ?

500 000 = R x [(1 + 0.008) 360 - 1 0.008 ]

500 000 = R x 2076. 413

(500 000) 2076.413 = R x

(2076.413) 2076.413

r = $ 240.80

Now we are able to solve for R, or the amount Sam should be depositing each month:

Sam would have to deposit $240. 80 into the account each month in order to have $500 000 at the end of 30 years.

Example 2

Nahid borrows $95 000 to buy a cottage. She agrees to repay the loan by making equal monthly payments of $750 until the balance is paid off. If Nahid is being charged 5.4%/a compounded monthly, how long will it take her to pay off the

loan?

First, we must calculate i according to the compounding period :

i = 5.4% = 0.054 / 12

= 0.0045

PV = $95 000

R = $750

We are now solving for n :

n = ? yrs

Take a look at your handout for solution.