Exam FM Problem 021 Learning Objective: “Interest Rates” Wisconsin Center of Actuarial Excellence Technology Enhanced Learning Project Exam FM Problem 021 Learning Objective: “Interest Rates” Welcome to the tutorial on exam FM. Today we are going to go over problem 21, which is part of the learning objective “Interest Rates”. Here is the view of the problem.

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. (A) 111 (B) 116 (C) 121 (D) 126 (E) 131

Here we are given the function of payment at time t, which we write PMT(t) in this problem. The graph shows the relationship between the payment at time t and time t. For any given time point t between 0 and t, the payment amount is (8+t)*k. We are also given the force of interest δt, which is also changing with time t. The last thing we know is the Accumulated value of the account at time 10, which we use symbol AV10 in this problem. To find k, we have to use the given number for AV10. So we want an equation for AV10 with only k and constants, and then we can get k. We already know the payments and force of interest. We can find the accumulation function from force of interest and then find accumulated value.

Let’s start with a single time point n Let’s start with a single time point n. We calculate the accumulated value at 10 by bring the payment at n to 10. This is just payment at time n times the accumulation function from n to 10.

Here is some helpful notes on the accumulation function and force of interest. The accumulation function from t to t+m, s(t,t+m), represents the accumulated value at time t+m of $1 at time t. When the interest rate or force of interest is constant, this is just (1+i)^m of e^δt, which you are probably quite familiar with. Since the i does not change, the value only depends on the length of the time period m and is not affected by the staring point t. however, when the force of interest changes, you need to be careful. The accumulation function equals to the exponential integral t to t+m δt dt. It does depend on the stating time t. Notice t can also be 0, and the formula will then changes to something we sees more often. If you don’t want to memorize both formula, this one is most general and you can derive other equations from this formula. Now let’s go back to find the accumulated value for a single time point.

The payment function is given but we need to find the accumulation function. We know the force of interest at time t is 1 over 8+t. By the formula for s(t,t+m), s(n,100) equals to exponential integral n to 10 δt. So s(n,10) equals to 18 over 8+t. Now we calculate the AV10, which is PMT(n) times s(n,10). We find this is just 18*k.

We find the Accumulated value for a single payment at any time point n is 18k. From the graph we can see the payment is continuous from time 0 to 10. So to get the accumulated value for the payment streams, we just take the integral of the single accumulated value from 0 to n. As given in the problem, the account value at time 10 is 20,000. Equate this to the integral, we get k equals to 111, which is answer.

Discrete Continuous Constant Non-constant PMT i Although not related to this problem, this table summarizes the steps you may take to get the future value, under different situations for interest and payment.

Thanks for watching. Questions. Comments Thanks for watching! Questions? Comments? Please email us at: TELFEEDBACK@bus.wisc.edu Funding provided by the Society of Actuaries and the Wisconsin School of Business http://instruction.bus.wisc.edu/jfrees/UWCAELearn/default.aspx Voice: Ting Xia Faculty Supervisor: EW(Jed) Frees That’s the end of this tutorial. Thanks for watching!