1. Exam FM Problem 013 Learning Objective: “Interest Rates”
Wisconsin Center of Actuarial Excellence Technology Enhanced Learning Project
Exam FM Problem 013
Learning Objective: “Interest Rates”
Welcome to the tutorial on exam FM. Today we are going to go over problem 13, which is part of the learning objective “Interest Rates”. Here is the view of the problem.

2. Please choose one of the answers and submit it
Please choose one of the answers and submit it. If incorrect, hit the gray ‘x’ in the top right to try again.

3. Here we are given that 100 is deposited at time 0 and X is deposited at time 3. The interest earned from time 3 to time 6 is X.
By the definition of interest, the account value at time 6 equals to the account value at time 3 plus the interest earned during this time period. So X is just the account value at time 6 minus the account value at time 3. To find X, we want to find the account value at time 6 and account value at time 3, denoted by AV6 and AV3 in this problem. We are also given the force of interest, which can be used to find the accumulation function.

4. Before we start calculating the accumulated value, let’s have a review on the force of interest. The accumulation function a(t) is the accumulated value at time t for the value of $1 at time 0. It equals to (1+i)^t or e^(δt) when the interest rate is constant. The force of interest δt is the rate of growth at a point in time, which is 1 over a(t) times the derivative of a(t). This is just the derivative of ln(a(t)). Take integral and exponential on both sides, we get a(t) equals to exponential integral 0 to t δt.
One common mistake you may try to avoid is when you calculate the accumulation function from time m to m+t. You can’t get it by using integral from 0 to t. Instead, we get the accumulation function by bring it back from time m to time 0 first, and then bring it forward to time m+t. So a(m,m+t) is a(m+t) over a(m).Plug in the formula we get above, we find this equation. Notice the integral is from time m to m+t. We will use these two formulas later in this problem.

5. Now we can solve the problem. We want to find AV6 and AV3
Now we can solve the problem. We want to find AV6 and AV3. The account value at time 3 is just X plus the value of the 100 at time 3, which is 100* a(3). Since the force of interest is given, use the formula for a(t). a(3) is just exponential integral 0 to 3 t^2 over 100 dt. This is just
Plug in a(3), we know AV3 is X

6. The account value at time 6 is the account value at time 3 times the accumulation function from time 3 to time 6. To calculate the AV6, as we already know the account value at time 3, we can use the second formula for a(m, m+t).
So a(3,6) equals to exponential integral 0 to 6 t^2 over 100 dt. We get for a(3,6), which is the accumulation function that brings the time value from time 3 to time 6.
So the account value at time 6 is just time AV3.

7. Now we know the account value at time 6 and account value at time 3 with only X unknown in the equations. Since AV6 minus AV3 equals to X, we can solve for X. Plug in AV6 and AV3 into this equation. We get , which is answer E.

8. Thanks for watching. Questions. Comments
Thanks for watching! Questions? Comments? Please us at: Funding provided by the Society of Actuaries and the Wisconsin School of Business Voice: Ting Xia Faculty Supervisor: EW(Jed) Frees
That’s the end of this tutorial. Thanks for watching!