Starting on your 25th birthday, and continuing through your 60th birthday, you deposit 750 each year on your birthday into a retirement fund earning an annual effective rate of 5%. Immediately after the last deposit, the accumulated value of the fund is transferred into a fund earning an annual effective rate of j. On your 65th birthday, you purchase a 25-year annuity-due paying 580 each month with the balance of the account. The purchase price of the annuity was determined using an annual effective rate of 4%. Calculate j.

9.09%

Explanation:

With the payment for first term with interest rate for 5%. we choose to set up problem as ordinary annuity, then we should use 36 rent periods because term would start at one period before first deposit.

We have formula with resulting equation to find out future value of first annuity, that gives a value of an annuity on his 60th birthday:

Formula is as under

S = R ((1 + i) ^n - 1) / i

putting values we get

= $750 ((1 + 0.05) ^36 - 1) / 0.05

S = $71,887.24

Because value of S is located Fred's 65th birthday, now you can use such value as present value of fund compounded for Five years. Future value of these fund, will later be equated to present value of annuity-due, is given by following equation:

S = P (1 + j) ^n where i=j and n=5 so ...

S = $718,772.42 (1 + j) ^5

Now you calculate present value of annuity-due & equate it to equation just give. For annuity-due, went as rent payments of $5,800 each with effective interest rate of 4%. Because this payments occur each month & annuity-due lasts for 25 years, you have (25*12) periods = 300 periods. Further, You must calculate new interest rate, given by following equation:

(1 +.04) ^1 = (1 + i (12) / 12) ^12 Therefore ... i (12) / 12 = 0.00327

Now calculate present value of annuity-due:

P = R (1 + i) (1 - (1 + i) ^-n)

P = $5800 (1 +.00327) (1 - (1 +.00327) ^300) /.00327

P = $1,111,979.

Finally, equate earlier equation with the new present value:

$1,111,979.84 = $718,772.42 (1 + j) ^5

Therefore j = 9.09%