13.6. Annuities http://www.ck12.org
Recall that a geometric series with initial valueaand common ratiorwithnterms has sum:
a+ar+ar^2 +···+arn−^1 =a·^11 −−rrn
So, a geometric series with starting valueRand common ratio( 1 +i)has sum:
FV=R·^1 −(^1 +i)n
1 −( 1 +i)
=R·^1 −(^1 +i)n
−i
=R·(^1 +i)n− 1
iThis formula describes the relationship betweenFV(the account balance in the future),R(the annual payment),n
(the number of years) andi(the interest per year).
It is extraordinarily flexible and will work even when payments occur monthly instead of yearly by rethinking what,
R,iandnmean. The resulting Future Value will still be correct. IfRis monthly payments, theniis the interest rate
per month andnis the number of months.
Example A
What will the future value of his IRA (special type of savings account) be if Lenny saves $5,000 a year at the end of
each year for 35 years at an interest rate of 4%?
Solution:R= 5 , 000 ,i= 0. 04 ,n= 35 ,FV=?
FV=R·(^1 +i)n− 1
i
FV= 5 , 000 ·(^1 +^0.^04 )(^35) − 1
0. 04
FV=$368, 281. 12
Example B
How long does Mariah need to save if she wants to retire with a million dollars and saves $10,000 a year at 5%
interest?
Solution:FV= 1 , 000 , 000 ,R= 10 , 000 ,i= 0. 05 ,n=?
FV=R·(^1 +i)
n− 1
i
1 , 000 , 000 = 10 , 000 ·(^1 +^0.^05 )
n− 1
0. 05
100 =(^1 +^0.^05 )
n− 1
0. 05
5 = ( 1 + 0. 05 )n− 1
6 = ( 1 + 0. 05 )n
n=ln 1ln 6. 05 ≈ 36. 7 years