New Video: Immunization, Part 5 - Infinity is Really Big

Redington immunization is a concept that can be defined in terms of derivatives of present value functions.

Given asset cashflows $A_{0},A_{1},A_{2},\ldots,A_{n}$ and liability cashflows $L_{0},L_{1},L_{2},\ldots,L_{n},$ each set of flows occurring at times $t=0,1,2,\ldots,n,$ the expressions $P_{A}(i)=\displaystyle\sum_{t=0}^{n}A_{t}(1+i)^{-t}$ and $P_{L}(i)=\displaystyle\sum_{t=0}^{n}L_{t}(1+i)^{-t}$ represent the present values of these cashflows, as functions of an arbitrary periodic interest rate $i.$

If we let $h(i)=P_{A}(i)-P_{L}(i)=\displaystyle\sum_{t=0}^{n}C_{t}(1+i)^{-t},$ where $C_{t}=A_{t}-L_{t},$ then we say the liabilities are Redington immunized by the assets at periodic interest rate $i=i_{0}$ if: 1) $h(i_{0})=0,$ 2) $h'(i_{0})=0,$ and 3) $h''(i_{0})>0.$ By the Second Derivative Test, these conditions imply that $h(i)$ as a local (relative) minimum value of zero at $i=i_{0}$ so that $h(i)>0$ when $i$ is sufficiently close to $i_{0}.$ This means that $P_{A}(i)>P_{L}(i)$ when $i$ is sufficiently close to $i_{0},$ implying that small changes in the interest rate will put the assets in a surplus position with respect to the liabilities.

In my most recent video, “Actuarial Exam 2/FM Prep: Redington Immunization Equivalent Conditions“, embedded below, I look at alternative ways to check for Redington immunization of a liability cashflow by an asset cashflow. This is accomplished by solving a modified version of Exercise 7.2.3 from the 7th Edition of “Mathematics of Investment and Credit”, by Samuel Broverman.

Financial Math for Actuarial Exam 2 (FM), Video #173. Exercise #7.2.3 (modified) from “The Mathematics of Investment and Credit”, 7th Edition, by Samuel A. Broverman.

Many of the ideas and calculations in this video were also discussed in my post “New Video: Immunization, Part 2“. However, in that post, I also emphasized the fact that the time-values of the cashflows could be evaluated at any moment in time, not just the present.

Main Results and Their Purpose

The main results of the newest video above are expressed in the statement of Exercise 7.2.3 from Broverman’s text: letting $v_{i}=(1+i)^{-1}$ and assuming, in everything that follows, that $P_{A}(i_{0})=P_{L}(i_{0})$ (in other words, $h(i_{0})=0$ ), we can conclude that (a) $P_{A}'(i_{0})=P_{L}'(i_{0})$ ( $h'(i_{0})=0$ ) is equivalent to $\displaystyle\sum_{t=0}^{n}tA_{t}v_{i_{0}}^{t}=\displaystyle\sum_{t=0}^{n}tL_{t}v_{i_{0}}^{t}$ and (b) if $P_{A}'(i_{0})=P_{L}'(i_{0})$ , then the following three statements are equivalent (i) $P_{A}''(i_{0})>P_{L}''(i_{0})$ ( $h''(i_{0})>0$ ), (ii) $\displaystyle\sum_{t=0}^{n}t^{2}A_{t}v_{i_{0}}^{t}>\displaystyle\sum_{t=0}^{n}t^{2}L_{t}v_{i_{0}}^{t},$ and (iii) $\displaystyle\sum_{t=0}^{n}(t-D(i_{0}))^{2}A_{t}v_{i_{0}}^{t}>\displaystyle\sum_{t=0}^{n}(t-D(i_{0}))^{2}L_{t}v_{i_{0}}^{t}.$ In statement (iii), $D(i_{0})$ represents either the Macaulay duration or the modified duration of either cashflow (their durations are equal).

Side Note: in actuality, $D(i_{0})$ can be any quantity that does not depend on $t.$

What is the purpose, or value, of these results? For one thing, after matching present values, they allow you to match durations and also achieve Redington immunization without resorting to taking derivatives. They allow you, in fact, to use a spreadsheet to do so.

Why? Because once you know the cashflows and the interest rate, then the spreadsheet can easily be used to compute the various summations above (we might also need to use the form of the summations to solve for the cashflows). Details of this will be the topic of both the next video that I make and the next blog post on immunization.

Are their any other benefits of these results? Yes. There will be benefits to thinking of these results in a “probabilistic” or “statistical” way.

What do I mean by this? If you know some probability theory, the summations above should have a familiar feel to them. They should seem similar to the formulas for the mean (expected value, or “first moment“), second moment, and variance of a discrete random variable. I think this perspective can be brought into play to bring new insights into this subject. I plan to explore this in future videos and blog posts as well.