New Video: Immunization, Part 2

It was a post that delved into the details of my previous YouTube video “Actuarial Exam 2/FM Prep: Use Redington Immunization to Find a Ratio of Assets in a Portfolio” embedded below.

Using advanced financial terminology, we can describe the problem in this video as one where the goal is to try to “match” the asset and liability cashflows, relative to a certain interest rate (assuming a flat yield curve), by equating their present values and durations at that interest rate, and by creating a certain kind of mismatch in their convexities at that interest rate.

We can be more precise by using mathematical notation: for an arbitrary effective periodic interest rate let represent the present value of the asset cashflow and let represent the present value of the liability cashflow. If the given interest rate of the problem is then: 1) to match present values entails setting and 2) to match durations entails both condition (1) to be true and also setting In the video above, these two conditions allow us to solve for unknown asset cashflows when and then form their ratio, which is the answer to the problem that is posed.

The desired mismatch in the convexities entails: 3) to make sure conditions (1) and (2) hold, in addition to setting It is not necessary to use this condition to solve the problem in the video, but it is a condition that guarantees that Redington immunization has been achieved: the assets will be in a surplus position compared to the liabilities of the interest rate changes a little bit away from It can be checked in the problem from the video that this condition holds after the unknowns have been solved for.

To see why condition (3) is used, apply the Second Derivative Test to the difference function Since it guarantees that, for small changes in the value of away from the present value of the asset cashflow will remain in a surplus position compared to the present value of the liability cashflow. That is, for all sufficiently close to (basically because the graph of will be concave up near at least when is a continuous function).

Based on the solution to the problem in the video above, the relevant graphs of these functions are shown below.

Assuming interest rates do not change much and our cashflows do not change, this should also imply that we will be in a surplus position of assets over liabilities for future values as well. If interest rates do change, then we may have to alter how we structure these cashflows to remain in a surplus position.

A Similar Problem Solved Using a Different Time-Valuation Point

In the latest video, which is the “new video” from the title of this post and is embedded below, I solve a similar problem. When solving this problem, it is more convenient to use a time other than the present to evaluate the time-value of both the asset and liability cashflows.

Instead of evaluating the present values of the asset and liability cashflows, the cashflows are time-valuated at a positive moment in time (which is in the video), where is the interest rate of valuation ( in the video).

The general method is similar. If is the time-value of the asset cashflow at time and is the time-value of the liability cashflow at time then we want: 1) 2) and 3) Once again, only the first two conditions are needed to solve the problem in this new video by using them to find the unknowns and then computing their ratio.

If we confirm that the third condition holds, this is sufficient to achieve Redington immunization of the liabilities by the assets. In fact, the problem statement from the new video calls for us to achieve “full” immunization of the liabilities by the assets. This actually means we desire more: we want for all (or at least ). In the video, we see that this occurs because, in fact, for all

We can again think of this in terms of a difference function: let Then our conditions for Redington immunization become: 1) 2) and 3) and we can apply the Second Derivative Test. If, in addition, for all then the graph of is always concave up. Thus, for all and we will actually have achieved full immunization. It does not matter how much the interest rate changes; the assets will always be in a surplus position compared to the liabilities.

Why Does This Produce the Same Answer?

Now we explore why we can time-valuate at and still get the same answer as when we time-valuate in the present

Suppose that, for the symbols and represent, respectively, the asset and liability cashflows at time Many of these values could be zero, but they are all taken to be nonnegative. Then represents the net cashflow at time which can be positive, negative, or zero.

Given an arbitrary interest rate the present value functions are and By properties of summations, we can also write the difference function as

The condition is equivalent to and

Since the condition is equivalent to and

And since the condition whether or not, is equivalent to and

Furthermore, since we can also say that when the condition is equivalent to

Now consider time-valuation of the assets and liabilities at some other time The corresponding time-valuation functions would be and By properties of summations, we can also write the difference function as

We could have guessed these relationships from the start, but it is nice to seem them work out by using properties of exponents and summations.

The first thing we see from the relationship is that if and only if

By the Product Rule, Therefore, when both and

Is the reverse statement also true? That is, is when both and Yes. Just apply the Product Rule to the equivalent (and symmetric) relationship

How about the second derivative? We compute, by linearity and the Product Rule:

Therefore, if and then as well. Once again, it works to switch around the roles of and in this last sentence because the same calculations could be performed on (it does not matter that ).

Furthermore, it can be seen that the same summation conditions as shown above for (Redington) immunization still hold. That is, 1) if and only if 2) if and only if and 3) if and only if

The graphs of both and for the new video are shown below. Note that they both are concave up at their common minimum point . It is also interesting to note that even though has a graph that is always concave up, does not. This emphasize the importance of the fact that we focused on the value rather than an arbitrary value of in the previous paragraph.

All that we have done here is also described on pages 409-413 of the 7th Edition of Mathematics of Investment and Credit, by Samuel A. Broverman.