Here is the question:
10000 can be invested under two options:
Option 1: Deposit the 10000 into a fund earning an effective annual rate of i
Option 2: Purchase an annuity-immediate with 24 level annual payments at an effective annual rate of 10%. The payments are deposited into a fund earning an effective annual rate of 5%.
Both options produce the same accumulated value at the end of 24 years. Calculate i.
I have been working on this question for hours and still cannot get the right answer, which is i = 0.0689. Any help would be greatly appreciated.
Hi! I've been working on this problem and I finally figured it out! Bear with me b/c I do not know how to do "math symbols" on the computer.
The accumulated value in option 1 = 10,000 * (1+i)^24
The accumulated value in option 2 = X * s(24).05 --> Here the ( ) is representing the annuity symbol
X is each payment, which is deposited into a fund earning 5% over the 24 years
We need to figure out X. Let's use the information about our annuity-immediate.
10,000 = X * a(24).10
Now we can solve for X, which we find to equal 1112.997764
Now we set our two accumulated value functions equal to each other.
X * s(24).05 = 10,000 * (1+i)^24
(1112.997764) * s(24).05 = 10,000 * (1+i)^24
Just solve for i and you get .068939.
Thank you very very much!!!
Although I just had one question. I am just having a little bit of difficulty of how you got 10000 = X * a(24).10
I am new to studying annuities, but it's my understanding that you use a(n)i when discussing "present value" and s(n)i when disscussing "future value." We want to purchase an annuity-immediate that has a present value of 10,000, so we are using a(n)i. Our n = 24 = 24 one year payments. We are given that i = .10.
Thus we get the PV = 10,000 = (Our Payment Amount) * a(n)i = X * a(24).10
We used s(n) earlier when discussing the accumulated value (i.e. future value) that the fund earning 5% would reach. In this case,
FV = (Our Payment Amount) * s(n)i = X * s(24).05
I hope this answered your question -- sometimes I still get a bit confused on when to use a(n)i vs s(n)i