Actuarial Outpost FM exam question - annuity with varying interest rates
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#1
12-03-2018, 09:47 PM
 dws098 Non-Actuary Join Date: Nov 2018 Posts: 3
FM exam question - annuity with varying interest rates

Question: Five deposits of 100 are made into a fund at two-year intervals with the first deposit at the beginning of the first year.
The fund earns interest at an annual effective rate of 4% during the first six years and at an annual effective rate of 5% thereafter.

Calculate the annual effective yield rate earned over the investment period ending at the end of the tenth year.

____________________________

Approach: I set up an equation to calculate the present value:

PV = 100 + 100(1.04)^(-2) + 100(1.04)^(-4) + 100(1.04)^(-6) + 100(1.04)^(-6) * (1.05)^(-2).

I get that the present value is 428.65, that is 428.65 = (a_5) * (1+i), where i is the two-year effective interest rate. Solving with linear interpolation, I get 8.35% for the two-year effective interest rate meaning the effective annual interest rate is 4.09%. But the solutions manual gives 4.58% as the correct answer, where they set up the accumulated value as follows:

AV = 100(1.04)^6 * (1.05)^4 + 100(1.04)^4 * (1.05)^4 + 100(1.04)^2 * (1.05)^4 + 100(1.05)^4 + 100(1.05)^2. This gives 659.27 as the accumulated value (and the annual effective interest rate would indeed be 4.58%).

Both my solution and the one in the answer key should make sense, yet they give different values for the effective interest rate. What am I doing wrong?
#2
12-03-2018, 10:13 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,096

You need to work with accumulated values, because the question asks about the 10 year period.

Your method would give the same answer if the interest rate in years 9 and 10 were 5% (as it is) or if the interest rate in years 9 and 10 were 20% (or 0%, or 4%, or anything)

Your method (slightly modified) would work if you made a total withdrawal of the accumulated value at the end of 10 years, and then solved for the interest rate that would make the PV of all the deposits and the withdrawal 0.
#3
12-03-2018, 10:51 PM
 dws098 Non-Actuary Join Date: Nov 2018 Posts: 3

Interesting, thanks.