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Text Box: The Insider’s Guide to Varying Force of Interest Problems-The Four Types of Problems Note from Joe: This tip is different in that it involves a direct strategy for a whole class of problems. Some of the tips we post are more from a standpoint of something that might resolve some nagging issue or make some aspect clearer. This tip is something I try to make sure I always cover in my portion of the seminar. We haven’t explicitly said so, but there are some of the other tips that we always try to cover and some that are designed just to give you background or other help. As always, we hope this helps you… Annuities with a varying force of interest, δ(t), can often pose a challenge. Part of it is setting the problem up; the other part is the integration. We can help you with the integration. As it turns out, these reduce to only about four general solutions, and one of those four is too trivial to bother with. Why are there only four?: Let me illustrate the situation with an example. Lets say we assume that the force of interest, after risk-adjusted probability, is expected to increase over time, so we say δ(t) = .02+.0025t for example. Then the formula for the annuity is:  (equation increased to make it easier to read), where PV(t) is the present value of the annuity over the next t time units; p(s) is the payout at each time s between 0 and t; and δ(r) is the force of interest at time r (between 0 and s). To further digress, I’ll write , where . Now, let’s start on our example by computing S(s) for our example. . So far, so good. But when that fairly simple S function is plugged into the formula for PV(t), we get:  This is a whole different type of nut to crack. The part of this term is related to the standard normal integral, which is known to be inexpressible in a closed form. The best you could hope for would be to perform a ‘complete the square’ calculation and hope p(s) is constant so you can end up with the standard normal cumulative density function. Then, if you’re lucky, you end up with the standard normal cumulative density function, Ф(x) evaluated at a couple of points. But, let me ask you, is there a key for Ф(x) on your calculator? Is completing the square and all the associated algebra appropriate for a speed test? They certainly could ask this question, but if they did, the tip off would be all the answers would be expressed with Ф(something). And, what if you get a s³, or cos(s), or whatever for S(s)? Trust me it just gets more unlikely that the result can be expressed in a closed form (e.g., calculated for the test). So, in order to ask you a varying force of interest problem, they have to (as far as I can tell) pose the problem in one of four ways. By mastering the four (actually three) types, you should be able to solve whatever they throw you. Type 1- S is linear, S(s) = K×s, (K a constant) The equation above was difficult because of the quadratic, etc. term. If S(s) is linear, is something you have a chance to integrate, right? But, unfortunately, S is an integral and if you differentiate it to get δ(s), you get δ(s) = K, i.e., a constant force of interest. So, this type is unlikely to be asked. Type 2- S is based on a natural logarithm of something, S(s) = K×ln(g(s)), (K a constant) Then  becomes , or , which you do have a prayer of integrating. Note that one of the other tips is a special case of this, and the utility of his tip comes from how easy it is to put such a question on the test. Type 3- p(s) is based on a derivative of S, p(s) = K×S’(s) (K a constant)  becomes , or since  , which gives the solution. Type 4- p(s) contains the reciprocal of e-S(s), p(s) = g(s)×eS(s) In this case, part of p eliminates the problem e-S(s) term  becomes , and the exponential terms cancel, leaving , which you just might be able to integrate. Note: in a variant of this, a portion of p(s) cancels all but e-Kt, but you can then integrate g(s)× e-Kt. So, what’s the skinny? I’m not sure that it’s worth remembering all the formulas, but it’s worth knowing what to look for and remember what general strategies you might employ.