How To Value An Annuity

Future Value of an

Annuity

After paying all your bills, you have $200 left each payday (at then end of each month) that you will put into savings in order to save up a down payment for a house. If you invest this money at 5% interest per year, compounded monthly, how much money will you have saved toward the down payment at the end of 5 years?

First, a little background: Definitions:

1. An annuity is a succession of equal payments made at equal periods of time. An ordinary annuity has payments due at the end of each time period.

2. A sequence is a list of numbers. 3. A series is the sum of the numbers in a

sequence.

4. A geometric sequence is a list of numbers, starting with a number, a, where each number is some constant r_{times the preceding number.} 5. A geometric series is the sum of a

geometric sequence.

Example:

1, 2, 4, 8 is a geometric sequence where a_{= 1and r = 2.}

1 + 2 + 4 + 8 is a geometric series.

In general, a geometric series has the form: a_{+ ar + ar}2_{+ ar}3_{+ . . . + ar}n

+ . . . The sum of the first n terms of a geometric series can be calculated by the formula:

) 1 ( ) 1 ( Sum − − = r r a n

So, if we invest $200 a month, using t m m r P A ⋅       + = 1 59 12 05 . 1 200       + 58 12 05 . 1 200       + 57 12 05 . 1 200       + the future maturity value of the first deposit is: of the second: of the third:

The sum of the future value of all the deposits is: 200 12 05 . 1 200 12 05 . 1 200 58 59 + +       + +       + _L

which is a geometric series, where

a_{= 200, r = , and n = 59.}      + 12 05 . 1

So the value of our annuity in 5 years is:

22 . 601 , 13 $ 1 12 05 . 1 1 12 05 . 1 200 ) 1 ( ) 1 ( 59 = −       +         −       + = − − r r a n

(rounded to the nearest penny).

Future Value of an Ordinary Annuity: Let r be the annual interest rate, m be the number of times per year that interest is compounded, and let . Also, R is the payment, and n the number of payments, made at the end of each interest period, n_{= m}.

t

_.

_{Then the future value of an annuity,} S_{, is:} m r i=

(

)













₊

₋

=

i

i

R

S

n

1

1

In the formula

(

)













₊

₋

=

i

i

R

S

n

1

1

(

)













₊

₋

i

i

n

₁

1

is sometimes denoted , read “s angle n at i”.

i n

s

_¬

Example: After 40 years of work, it’s time to retire. Your company invested $500 at the end of each month in an ordinary annuity with annual interest rate of 8% compounded monthly. If you take a lump-sum retirement payment, how much should it be?

(

)

      _{+ −} = i i R S n 1 1

where R = 500, , the number of payments, n, is (12 per year)(40 years) = 480, so your lump sum =

92 . 503 , 745 , 1 $ 12 08 . 1 12 08 . 1 500 480 =             −       + 12 08 . = i

Let’s say you work for the same company, and you want to have $2,000,000 when you retire after 40 years. The company will allow you to make additional monthly payments. How much more do you have to pay each month to retire after 40 years with 2 million?

(

)

      _{+ −} = i i R S n 1 1

where , n = 480, and now S =

2,000,000, and we need to solve for R where 12

08 . = i             −       + = 12 08 . 1 12 08 . 1 2000000 480 R ) 007831 . 3491 ( 2000000₌R

Simplifying, you have:

Divide to find R = 572.90 (to the nearest penny).

Keep as many decimal places as possible for greatest accuracy.

Classwork:

Find the future value of an ordinary annuity with payments of $150 made semiannually for 20 years in an account that pays 6% interest compounded semiannually.

Definition: A sinking fund is an annuity set up to reach a specified value at a specified time. Typically, sinking funds are set up by an organization to retire a debt.

Example: A department of

transportation wishes to build a bridge at a cost of $1.2 million. They plan to finance the bridge with the sale of bonds that will come due in 5 years and have a face value at that time of $2 million.

At the end of each fiscal year, the

department can invest part of their annual budgeted funds in a mutual fund that returns 10% per year compounded quarterly. How much do they need to budget each quarter in order to be able to repay the $2 million debt in 5 years? We can use the formula for the future value of an annuity.

(

)

26 . 294 , 78 $ ) 54465761 . 25 ( 2000000 ) 54465761 . 25 ( 4 1 . 0 1 4 1 . 0 1 2000000 1 1 20 = = =             −       + =       _{+ −} = R R R R i i R S n

Since we’re rounding the final answer two decimal places (the nearest penny), why hold on to 8 decimal places in the intermediate answer (25.54465761)? Could we round it to two decimal places and get the same answer?

26 . 294 , 78 $ 54 . 78308 $ 54 . 25 2000000 ≠ =

Hold on to as many decimal places as possible in intermediate calculations. When units are dollars, it is reasonable to round you final answer to 2 decimal

Annuity due:

An annuity due differs from an ordinary in that payments to an annuity due are made at the beginning of each period, not at the end of the period.

The future value, S, of an annuity due is:

(

)

_R i i R S n −       _{+ −} = + 1 1 1

where R is the payment, i is interest per period, and n is number of payments (same as for an ordinary annuity.