Expert: Josh - 5/3/2009

Question
At age 35 louise starts an IRA to save for retirement. she deposits $50 at the end of each month. IF she can count on an APR of 8% how much will she have when she retires 30 years later at age 65? compare the value of the IRA to the total amount of her deposits over the same period.

Answer
Holly

The formula may be constructed based on an understanding of geometric series. It will be shown that the formula consists of two components. People working in finance like to call these (i) capital accumulation (this is just the regular compound interest formula) and (ii) future value of a series.

If we make the right observations, the formula can be easily derived. We don't need any fancy terms. Let "a" be the amount deposited at the end of each month. Let "r" be the interest rate for each payment period. For an annual interest rate of P% that is calculated monthly, r=(P/100)/12. In your question, r=0.08/12.

If we look at the amount accumulated at the end of each month:

Component (i)           (ii)

t=1,  a
t=2,  a(1+r)           +  a
t=3,  a((1+r)+1)*(1+r) +  a
     ...we split this into two terms using linearity
   = a(1+r)^2         +  a(1+r)+a
     ...we keep multiplying the expression by (1+r) each month and add "a" (this represents the monthly deposit)
t=4,  a(1+r)^3         + a(1+r)^2+a(1+r)+a

By induction, when t=n, we have
a(1+r)^n + {a[(1+r)^(n-1)+...+(1+r)^2+(1+r)+1]}

Note: the exponent notation x^n indicates that x is multiplied by itself n times. For example, 2^3 = 2*2*2 = 8.

The first component (i) corresponds to a(1+r)^n [the usual compound interest formula]. The second component inside the curly brackets is just a geometric sum, it is equivalent to a(G^(n-1)-1)/(G-1), where G=1+r. It may be further simplified to a((1+r)^(n-1)-1)/r.

================================
Summary: The formula is given by
a(1+r)^n + a((1+r)^(n-1)-1)/r
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Here, a=50, n=30*12 months, r=0.08/12.
According to my calculation, and you should definitely check this yourself to make sure I haven't made any mistake,
a(1+r)^n = 546.7864
a((1+r)^(n-1)-1)/r = 73974.8070
so the total is T=74,521.59

The amount of her deposits is given by D=a*n, which equals 50*360=18,000.00