# Financial Math Problem

Q: John is to invest $100 at the end of every 3 months for the next 12 yrs. 20 years from now he will retire. Calculate the amount of accumulated money he will have when he retires if the money is invested at 3.4% compounded quarterly. My approach: Find FV after 12 years, investing$100 at the end of every 3 months using given interest rate. Let that FV = x, i still have 8 years of gap. Do i just do x(1+r)^8 where r is the annual interest rate?

I have interpreted your question to mean that the interest rate of the investment is a nominal annual rate of $$i^{(4)} = 0.034$$ compounded quarterly, thus the effective quarterly rate is $$j = i^{(4)}/4 = 0.0085$$. Since twelve years of quarterly payments is equivalent to $$48$$ payments in total, at the time of the last payment, the accumulated value is $$100 s_{\overline{48}\rceil j} = 100 \frac{(1+j)^{48} - 1}{j}.$$ At the time of retirement, which occurs $$8$$ years or equivalently $$32$$ quarters after the last payment is made, the accumulated value will be $$100(1+j)^{32} s_{\overline{48}\rceil j} = 100 (1+j)^{32} \frac{(1+j)^{48} - 1}{j}.$$ I have left the actual computation as an exercise for the reader.
It is worth noting that the cash flow at the time of retirement can be represented by the sum of each payment accumulated to retirement; i.e., $$100 (1+j)^{79} + 100 (1+j)^{78} + \cdots + 100 (1+j)^{32}.$$ You can see that this makes sense, because at retirement in $$20$$ years or $$80$$ quarters, the first payment occurring at the end of the first quarter, there will have been $$79$$ quarters for the first payment to accumulate value. The final payment, occurring at the end of the $$12^{\rm th}$$ year or $$48^{\rm th}$$ quarter, will have $$80 - 48 = 32$$ quarters to accumulate value. The total number of payments is $$79 - 32 + 1 = 48$$. Factoring this sum and noting it is a geometric progression, we can write it in the simplified form shown above.