Stan wants to start an IRA that will have $250,000 in it when he retires in 25 years. How much should he invest semiannually in his IRA to do this if the interest is 6% compounded semiannually? Assume an Annuity Due. Round to the nearest cent.

Question

tkhunny · Answer

25 years: No Payment, but total is 250000
6 months earlier.  Payment of "P".  It's value 1/2 year later is P(1+0.03)
6 months earlier.  Payment of "P".  It's value 1 year later is P(1+0.03)^2
6 months earlier.  Payment of "P".  It's value 1½ years later is P(1+0.03)^3
6 months earlier.  Payment of "P".  It's value 2 years later is P(1+0.03)^4

We need to recognize these patterns.  Similarly, we can identify the accumulated value of all 50 payments of "P".  Starting from the last payment normally is most clear.

P(1.03) + P(1.03)^2 + P(1.03)^3 + ... + P(1.03)^50

That needs to make sense.  After that, it's an algebra problem.

P[(1.03) + (1.03)^2 + (1.03)^3 + ... + (1.03)^50]

$P\dfrac{1.03 - 1.03^{51}}{1 - 1.03} = 250000$, and you are almost done.