Welcome to the World of Immunisation!

Welcome! If you have ever wondered how insurance companies and pension funds make sure they can still pay their future claims even when interest rates are jumping around, you are in the right place. In this chapter, we explore Redington’s Theory of Immunisation.

Think of immunisation as a financial shield. It is a strategy used by actuaries to protect a portfolio's surplus from the "infection" of changing interest rates. By the end of these notes, you will understand how to set up this shield using three specific rules known as Redington's Conditions.


The Goal: Protecting the Surplus

Before we dive into the math, let's look at the basic "ingredients":

1. Assets (\(P_A\)): The money the company has invested (bonds, cash, etc.).
2. Liabilities (\(P_L\)): The money the company owes in the future (payouts to policyholders).
3. Surplus (\(S\)): The difference between the two: \(S(i) = P_A(i) - P_L(i)\).

The value of both assets and liabilities depends on the interest rate \(i\). If \(i\) changes, the values of \(P_A\) and \(P_L\) change. Our goal is to ensure that if \(i\) changes by a small amount, the Surplus does not decrease. In fact, we want it to ideally increase or stay the same!


Redington’s Three Conditions

F.M. Redington, a famous actuary, proposed three conditions to "immunise" a portfolio against small, parallel shifts in interest rates. For a given interest rate \(i\), these are:

Condition 1: Matching Present Values

\(P_A(i) = P_L(i)\)

In plain English: The current value of your assets must exactly equal the current value of your liabilities. You need to have enough money today to cover what you owe.

Condition 2: Matching Durations

\(P'_A(i) = P'_L(i)\)

In plain English: The "sensitivity" of the assets to interest rate changes must match the sensitivity of the liabilities. If the interest rate moves, we want both sides of the balance sheet to move by the same amount. Since Duration is related to the first derivative of the price, this condition effectively means Duration of Assets = Duration of Liabilities.

Condition 3: Greater Asset Convexity

\(P''_A(i) > P''_L(i)\)

In plain English: This is the "safety net." If interest rates move away from our starting point, we want the assets to gain more value (or lose less value) than the liabilities. Mathematically, this ensures our surplus function \(S(i)\) has a local minimum at the current interest rate. Imagine a "U-shaped" curve (a smiley face!)—no matter which way the interest rate moves, the surplus goes up!

Quick Review: Match the values, match the slopes, and make sure the asset "curvature" is higher.


Why does this work? (The Taylor Series Shortcut)

Don't worry if the calculus looks intimidating! Redington used a mathematical tool called a Taylor Series Expansion to show how the surplus \(S(i + \epsilon)\) changes when interest rates change by a tiny amount \(\epsilon\):

\(S(i + \epsilon) \approx S(i) + \epsilon S'(i) + \frac{\epsilon^2}{2} S''(i)\)

Let's see what happens when we apply Redington's conditions:

  • From Condition 1: \(S(i) = P_A(i) - P_L(i) = 0\). (The starting surplus is zero).
  • From Condition 2: \(S'(i) = P'_A(i) - P'_L(i) = 0\). (The first term disappears).
  • From Condition 3: \(S''(i) = P''_A(i) - P''_L(i) > 0\). (The second term is positive!).

Because \(\epsilon^2\) is always positive, the whole expression becomes positive. This means any change in interest rates (\(+\epsilon\) or \(-\epsilon\)) results in a surplus that is greater than zero!


Step-by-Step: Solving Immunisation Problems

When you face an exam question asking you to "determine if the portfolio is immunised" or "find the amount of assets needed," follow these steps:

Step 1: Calculate \(P_L(i)\), \(P'_L(i)\), and \(P''_L(i)\)
Find the present value of the liabilities and its first two derivatives. Use the force of interest \(\delta\) if it makes the derivatives easier (remember \(v^t = e^{-\delta t}\)).

Step 2: Set up the Asset equations
Usually, you are given two or three assets (like zero-coupon bonds at different times). Let \(A_1\) and \(A_2\) be the amounts invested in these assets.

Step 3: Solve for Assets using Conditions 1 and 2
Use \(P_A(i) = P_L(i)\) and \(P'_A(i) = P'_L(i)\) as a system of simultaneous equations to find the required amounts of each asset.

Step 4: Test Condition 3
Plug your asset values into the second derivative formula. If \(P''_A(i) > P''_L(i)\), you have successfully immunised the portfolio!


Important Limitations (The "Catch")

Redington’s theory is brilliant, but it isn't perfect. In the real world, you must remember these three limitations for your exam discussions:

1. Small Changes Only: The Taylor Series approximation only works for very small shifts in interest rates. If rates jump by 5%, the "shield" might break.
2. Parallel Shifts Only: It assumes the entire term structure of interest rates moves up or down by the same amount. In reality, short-term rates might go up while long-term rates go down (a "twist" in the yield curve).
3. Continuous Rebalancing: As time passes and interest rates move, the durations and convexities change. To stay immunised, the actuary must constantly buy and sell assets to reset the conditions. This costs money (transaction costs).


Quick Summary Checklist

  • Condition 1: \(P_A = P_L\) (Initial values match)
  • Condition 2: \(P'_A = P'_L\) (Volatility/Duration matches)
  • Condition 3: \(P''_A > P''_L\) (Assets are more "curved" than liabilities)
  • Goal: To ensure \(S(i)\) is at a local minimum.
  • Constraint: Only applies to small, parallel shifts in the yield curve.

Pro-tip for CM1B (Excel): You can use the "Solver" or "Goal Seek" tool in Excel to find the asset amounts that make the difference in durations zero!

Did you know? F.M. Redington described immunisation as "investing in such a way that the existing business is immune to a change in the rate of interest." It revolutionized how life insurance companies manage their money!