In a previous post, I talked about a few popular measures of interest rate risk: Macaulay Duration, Modified Duration, and Convexity. However, I didn’t mention the practical implementation of these metrics or their relationship with the concept of **Immunization**. Broadly, the task of managing the interest rate risk associated with a given portfolio of financial assets comes down to minimizing the impact of a specific case of rate fluctuations: the decrease in asset value that results from an increase in interest rates. It’s hard to come up with good definition of immunization, and rather than copy and paste cookie cutter bullshit I’ll just say that a portfolio is “immunized” when its value is guarded from said interest rate fluctuations. It is not difficult to mathematically derive the conditions that are necessary for this to be the case, and once they’ve been derived they can be expressed in terms of the familiar interest risk metrics Macaulay Duration, Modified Duration, and Convexity.

First we need to generalize the concept of the Duration of a single asset to the Duration of a portfolio. In the simplest case, we have two assets, A and B. The change in the portfolio value when interest rates change is just the sum of the changes in value to assets A and B individually, **assuming that the change in the interest rate is the same for assets A and B. **If you want to be fancy, this uniform interest rate fluctuation across all assets can be called a **parallel shift in the yield curve**. Below, an expression is derived for the change in portfolio value:

This expression doesn’t say a whole lot about the underlying process. Since we assumed that the assets are affected by the same change in interest rates, we could factor that term (delta*i) out of the above expression. We could also manipulate the expression so as to express the change in price as the multiplicative product of P and some other term containing the relevant duration and price metrics for each asset. It takes some algebraic simplification and clever factoring, but you can pretty easily show that the total change in portfolio value is simply the weighted average of the changes in asset A and B, which implies that the **Modified Duration of the portfolio is the weighted average of the Modified Durations of the individual assets in the portfolio, where an asset’s weight is its proportion of total portfolio value**. The formula the duration of a portfolio consisting of m investments (X_{1} ,…, X_{m} ) is below.

It’s important to remember that we made a simplifying assumption in deriving this formula, namely that interest rate fluctuations are the same for each individual asset that comprises the portfolio. In other words, when interest rates change (in our simplified model) it’s due to a parallel shift in the yield curve. This won’t usually be the case, but the above model is still a useful approximation.

I’m convinced that a concise definition of Immunization doesn’t really exist, but I’ll explain the steps involved in “immunizing” against interest rate fluctuations in mathematical terms and then try to express the result in terms of duration and convexity to bridge the gap between the math and portfolio management. Suppose at any given time you have some assets and some liabilities. The liabilities are to be paid out at future dates, but you’ve estimated the present value of said payments given an assumption about the current interest rate. Ideally you’d like to match those liabilities with assets in order to cover them; for example, you would like to match a liability of $P at time t by purchasing an asset today for some price that yields exactly $P at time t, a situation sometimes referred to as an **exact match**. But that definition is useless because the idea that a portfolio of any realistic size could be matched exactly given the infinite number of possible combinations of assets that exist in our society is fucking retarded. Maybe it’s “possible”, but I’m using the word lightly and what I actually mean is a firm could theoretically (maybe) hire someone, make his job title “exact matcher” and it would take him an incredible amount of time to (maybe) eventually find an answer. So exact matching is conceptually only mildly retarded, but, in practice it is incredibly and wildly retarded.

So let’s stop thinking in finance buzzwords for a minute and just think about the conditions necessary for you to not get fucked by an increase in interest rates. Actually, first, let’s figure out if and why you’d get fucked if interest rates change. You have some liabilities with a present value that you’ve estimated, and we’ll assume that you’ve picked assets so as to match them at the current time. (Note – this is not “exact matching” because I’m talking about the present). It’s not unreasonable to say that at the present time assets are equal to liabilities. Suppose interest rates increase; the present value of your assets will decrease. This sucks, but won’t the present value of your liabilities decrease too? Yes. So what are we worried about, then? Immunization deals with guarding against an interest rate change that disproportionately affects the present values of assets and liabilities; specifically, the case in which an interest rate variation results in the PV of liabilities exceeding the PV of assets.

Naturally, our first condition to immunize a portfolio is that for **small **(this is important, but we’ll come back to it later)** **change in interest rates, which we’ll denote as a change from i_{0} to some nearby i, causes the PV of assets to exceed the PV of liabilities:

Inequalities suck so let’s define a new function h(i) as the difference between the PV of assets and the PV of liabilities. So the statement above is equivalent to saying h(i) is positive, and the condition mentioned earlier, that the present value of assets should equal that of liabilities at time 0, is equivalent to saying h(i_{0}) = 0, or, more generally, h(i) = 0 at i_{0}. So we have the following:

What do we know about h(i)? The first condition is pretty obvious and uninteresting; all that is said in requiring that h(i) = 0 is that PV assets = PV liabilities initially, which, has already been stated more than enough. The next condition is more interesting; regardless of the direction, any small change in the independent variable must result in a positive increase to the function h. This is obviously a local minimum, which, by calculus is a stationary point with a positive second derivative. The result is summarized below in terms of the function h and also in terms of the actual assets and liabilities.

If we wanted to, we could write the preceding expressions in terms of duration and convexity, since they are derived from the first and second derivative respectively. Deriving these expressions is computational (involves a lot of substituting and rearranging and dealing with negatives but nothing actually hard, just annoying) so I will leave that part out. If you don’t believe me you can practice your computational high school algebra skills and try it on your own.

I think there are finance-y terms for each of the three conditions above, but I can’t remember and a quick Internet search didn’t yield any helpful results. I don’t think buzzwords matter anyway, but if I had to explain them in words, I’d say something like the following (you can put a buzzword-y spin on it):

1) The present values of Assets and Liabilities are equal

2) The Durations of Assets and Liabilities are equal

3) The Convexity of Assets is greater than the Convexity of Liabilities

4) You’re Immunized!

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