Question

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.

Answer 1

To calculate j, we need to find the accumulated value of the retirement fund from age 25 to age 60 and then use that value to calculate the monthly annuity payment on the 65th birthday. By using the formulas for compound interest and present value of an annuity, we can solve for j. The value of j is approximately 2.35%.

Step-by-step explanation:

To calculate j, we need to find the accumulated value of the retirement fund from age 25 to age 60, and then use that value to calculate the monthly annuity payment on the 65th birthday. Here are the steps:

1. Calculate the accumulated value of the retirement fund from age 25 to age 60. This can be done using the formula for compound interest: A = P(1 + r)^n, where A is the accumulated value, P is the annual deposit, r is the annual effective rate, and n is the number of years. In this case, P = 750, r = 5%, and n = 60 - 25 = 35. Plugging in the values, we get A = 750(1 + 0.05)^35.
2. Calculate the monthly annuity payment using the accumulated value. This can be done using the present value of an annuity formula: PV = PMT((1 - (1 + r)^(-n))/r), where PV is the present value, PMT is the annuity payment, r is the annual effective rate, and n is the number of years. In this case, PMT = 580, r = 4%, and n = 25. Plugging in the values and using the accumulated value as PV, we get 750(1 + 0.05)^35 = 580((1 - (1 + j)^(-25))/j).
3. Solve the equation for j. By rearranging the equation, we can isolate j: (1 - (1 + j)^(-25))/j = (750(1 + 0.05)^35) / 580. Using numerical methods or a calculator, we find that j is approximately 2.35%.

Answer 2

9.09%

Step-by-step 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%