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F. M. Redington, a British actuary, coined the term immunization in 1952. In an influential address to the Society of Actuaries, he proposed that life insurance companies manage interest rate risk by matching what he called the mean term of assets to liabilities. Fie described immunization to occur when the first derivative of the value of assets given a change in the interest rate equals the first derivative of the value of liabilities and the second derivative of the value of assets exceeds the second derivative of the value of liabilities. To me, Redington and Macaulay (along with Irving Fisher, who first decomposed nominal rates into the real rate and inflation) are the fathers of bond math.

We can translate Redington's criteria for immunization to be: (1) The modified duration of assets equals the modified duration of liabilities, and (2) the convexity of assets exceeds the convexity of liabilities. These rules are at the heart of what nowadays is called liability-driven investing (LDI). The idea is that an investment portfolio should not be structured and managed in isolation. Instead, one should look across the balance sheet to identify the sensitivity of liabilities to the main drivers of change in interest rates – that is, changes in the real rate, inflation, and credit risk. LDI is clearly applicable to pension funds, insurance companies, and university endowment funds.

Consider a defined-benefit pension plan. Its liabilities are the payments owed to employees in retirement. Typically, the obligations are tied to the employee's wage level and the number of years worked. Some plans index the retirement benefits to inflation; in others they are set in nominal terms. Obviously, measuring the plan's liabilities is a huge task requiring many assumptions, including future wage levels that depend on inflation and productivity growth, expected lifetime of the retirees, number of employees whose pensions become vested, and the interest rates used to discount the future payments back to the current date.

Suppose that the pension plan, or at least its actuarial consultant, has a big, complex model to measure its liabilities. It can raise and lower the baseline interest rate assumptions in the model to obtain estimates of MV(initial), MV(up), and MV(down). From those, it can get the effective duration of liabilities and even effective convexity, although that might be pushing the analysis a bit too far. The key point is that the modified duration of pension fund assets also can be estimated. But this, too, is difficult because most defined benefit pension plans are heavily invested in equity. The interest rate sensitivity of common stock is not nearly as clean cut as it is with a fixed-rate bond because the change in value depends on why the nominal rate changes (i.e., due to the real rate or inflation), and that relationship might not be stable over time.

Despite these modeling problems – and the model risk that results – the pension plan still can measure the duration gap between the assets and liabilities. Usually asset duration is significantly lower than liability duration. The next step is to decide to maintain or to reduce that gap, for instance, using interest rate derivatives overlays, such as receive-fixed swaps, that have positive duration. Another possibility is to estimate the real rate and inflation durations of the liabilities and then to use linkers as part of the risk reduction strategy.

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