28 Asset-Liability Matching and Immunization
28.1 Simple Cash Flow Matching
28.1.1 Exact Matching
Exact matching is about maintaining \(A_t = L_t\) for all \(t\). Then, interest rates wouldn’t matter at all if the asset cash inflows were exactly matched to the liability cash outflows. There would be no need to sell assets at depressed prices at any time, or to reinvest maturities of assets at lower interest rates at any time. As we receive each asset cash inflow at time \(t\), we would immediately use it to pay an equal amount of liability cash outflow.
In reality, however, this is not easy to achieve (bonds are subject to callable risks, etc. and liabilities are often only estimates).
Example: A company expects to have liability cash outflows in one, two, three and four years of \(\$200\), \(\$400\), \(\$600\) and \(\$500\), respectively. The only investments available are the following bonds, all with annual coupons and all redeemable at par:
| Term of Bond | Annual Coupon Rate |
|---|---|
| 1 year | \(7\%\) |
| 2 years | \(4\%\) |
| 3 years | \(5\%\) |
| 4 years | \(6\%\) |
How much of each bond should the company buy in order to exactly match the liability cash outflows?
Solution: We should work backward, starting with the 4-year bond, then the 3-year bond, etc.
The face value of the 4-year bond that we wish to match with the \(\$500\) liability is: \[\frac{500}{1.06} = 471.70\] The bond has coupons \(471.70 \times 6\% = 28.30\).
To do this systematically, we set up a table as follows, fill the table from top down and match the liabilities from the furthest year to the nearest:
| Asset | t=1 | t=2 | t=3 | t=4 |
|---|---|---|---|---|
| 4-year bond | \(28.30\) | \(28.30\) | \(28.30\) | \(28.30+471.70\) |
| 3-year bond | \(27.22\) | \(27.22\) | \(27.22+544.48\) | |
| 2-year bond | \(13.25\) | \(13.25+331.23\) | ||
| 1-year bond | \(8.58+112.64\) | |||
| Total | \(200\) | \(400\) | \(600\) | \(500\) |
28.1.2 Duration Matching
The duration matching strategy involves constructing a portfolio of assets such that the following conditions hold: \[\begin{align}P_A(i) &\ge P_L(i) \\ D_\text{Assets}(i) &= D_\text{Liabilities}(i)\end{align}\] Under duration matching, when there is an immediate one-time small shift in interest rate, the surplus \(S(i) = P_A(i) - P_L(i)\) is preserved (i.e., non-negative).
To improve the strategy, we may take into account the convexity of the asset and liability portfolios. Using second-order approximation, we reconsider the rate of change of the asset-liability ratio in equation to obtain Redington immunization as below.
28.2 Immunization
Suppose we purchase a portfolio of bonds that we will use to pay liabilities, such as benefits that have been guaranteed to customers. At the start, we make sure that everything is “in balance”, i.e., the PV of the cash inflows from the bonds is equal to the PV of the liability cash outflows at a specified interest rate \(i = i_0\).
However, the interest rate that we used to balance the cash flows in the beginning is subject to changes. The process of protecting a financial enterprise from changes in interest rates is known as immunization.
28.2.1 Redington Immunization
The surplus is defined to be the net present value of assets minus liabilities: \(S(i) = P_{\text{Assets}}(i)- P_{\text{Liabilities}}(i)\). Frank Mitchell Redington derived three conditions for a surplus to be immunized against small changes in \(i\). Using the Taylor expansion: \[S(i) = S(i_0) \bigg( 1 - (i-i_0)\times D_{\text{mod, S}}(i_0) +\frac{(i-i_0)^2}{2}\times C_{\text{mod, S}}(i_0)\bigg) + O(i^3)\] These conditions must hold for the interest rate \(i = i_0\) at which we want to immunize the enterprise:
\[\begin{align}P_{\text{Assets}}(i_0) &= P_{\text{Liabilities}}(i_0) \\ D_{\text{Assets}}(i_0) = D_{\text{Liabilities}}(i_0) &\Leftrightarrow P'_{\text{Assets}}(i_0) = P'_{\text{Liabilities}}(i_0)\\ C_{\text{Assets}}(i_0) > C_{\text{Liabilities}}(i_0) &\Leftrightarrow P''_{\text{Assets}}(i_0) > P''_{\text{Liabilities}}(i_0)\end{align}\] For a change \(i = i_0 \pm h\), we want to keep \(S(i) \ge S(i_0)\). The three sufficient conditions for what has come to be known as Redington immunization.
It can be proved that we can use either modified or Macaulay duration and convexity for Redington immunization.
Example: A company must make payments of \(\$10\) annually in the form of a 10-year annuity-immediate. It plans to buy two zero coupon bonds to fund these payments. The first bond matures in 2 years and the second bond matures in 9 years, and both are purchased to yield \(10\%\) effective. Using Redington immunization, determin the face amount of each bond should the company buy in order to be immunized from small changes in the interest rate.
Solution: Let
- \(X = \text{face amount of the 2-year bond}\)
- \(Y = \text{face amount of the 9-year bond}\)
The conditions for Redington immunization can be rewritten by: \[\begin{align}v^2 X + v^9 Y &= a_{\overline{10|}}\\ -2v^3X -9v^{10} Y &= \frac{d}{di}10(v + v^2 + ... + v^{10}) = -10v(Ia)_{\overline{10|}}\\ \end{align}\] Solving for \(X\) and \(Y\), we have \(X = 45.40\) and \(Y = 56.41\). For good measure, we can substitute in calculate the second derivative to check if the convexity of assets is higher than the convexity of liabilities. Having \(P''_{\text{Assets}} = 1,965.49\) and \(P''_\text{Liabilities} = 1,774.32\), we can conclude that the annuity can be immunized by purchasing the said assets.
28.2.2 Full Immunization
Full immunization means that the company is protected against any change in the interest rate, no matter how large. It turns out that it’s very simple to fully immunize a single liability cash outflow. The condition for full immunization can be written as:
\[S(i) = P_{\text{Assets}}(i) - P_{\text{Liabilities}}(i) \ge 0 \qquad \forall\ i > 0 \] Full immunization of a single outflow liability can be achieved by having two cash inflows, one before and one after the payment of the liability. Then, the below conditions must still hold: \[\begin{align}P_{\text{Assets}}(i_0) &= P_{\text{Liabilities}}(i_0) \\ D_{\text{Assets}}(i_0) &= D_{\text{Liabilities}}(i_0) \end{align}\] (Both Macaulay and modified duration can be used).
Proof: Let the cash flow \(C_1\) be the inflow at time \(T_1\), \(C_2\) be the inflow at time \(T_2\) and \(L\) be the outflow at time \(T_L\) such that \(T_1 \le T_L \le T_2\). Let \(\Delta_1 = T_L - T_1\) and \(\Delta_2 = T_2 - T_L\).
The two conditions above can be rewritten as:
\[\begin{align} C_1 (1+i_0)^{-T_1} + C_2 (1+i_0)^{-T_2} &= L (1+i_0)^{-T_L} \\ T_1 \times C_1 (1+i_0)^{-T_1} + T_2\times C_2 (1+i_0)^{-T_2} &= T_L \times L (1+i_0)^{-T_L}\end{align}\] Multiplying \(T_L\) to the first equation and substituting the second equation into the first, we obtain \[C_1 = \bigg(\frac{\Delta_2}{\Delta_1}\bigg) C_2 \times (1 + i_0 )^{-(\Delta_1 + \Delta_2)} \] Substitute back, we have \[L = C_2 \times (1 + i_0 )^{-\Delta_2} \bigg(\frac{\Delta_2}{\Delta_1} + 1\bigg) \] Let \(S(i) = P_{\text{Assets}}(i) - P_\text{Liabilities}(i)\), we have
\[\begin{align} S(i) &= \bigg(\frac{\Delta_2}{\Delta_1}\bigg) C_2 \times (1 + i_0 )^{-(\Delta_1 + \Delta_2)}(1+i)^{-T_1} + C_2 (1+i)^{-T_2} \\ &\hskip6em - C_2 \times (1 + i_0 )^{-\Delta_2} \bigg(\frac{\Delta_2}{\Delta_1} + 1\bigg) (1+i)^{-T_L} \\ &= \bigg[C_2 (1+i_0)^{-\Delta_2}(1+i)^{-T_L} \bigg]\times \bigg[\bigg(\frac{\Delta_2}{\Delta_1}\bigg)\bigg(\frac{1+i}{1+i_0}\bigg)^{\Delta_1} +\bigg(\frac{1+i}{1+i_0}\bigg)^{-\Delta_2} + \bigg(\frac{\Delta_2}{\Delta_1} + 1\bigg) \bigg] \\ &= C(i)\times \eta(i) \end{align}\] where \(C(i)\) is a positive function and
\[\eta(i) = \bigg[\bigg(\frac{\Delta_2}{\Delta_1}\bigg)\bigg(\frac{1+i}{1+i_0}\bigg)^{\Delta_1} +\bigg(\frac{1+i}{1+i_0}\bigg)^{-\Delta_2} + \bigg(\frac{\Delta_2}{\Delta_1} + 1\bigg) \bigg]\] Taking the derivative, we have \[\eta'(i) = \frac{d\eta(i)}{di} = \frac{\Delta_2}{1+i}\bigg[\bigg(\frac{1+i}{1+i_0}\bigg)^{\Delta_1} + \bigg(\frac{1+i}{1+i_0}\bigg)^{-\Delta_2}\bigg]\] When \(0 < i < i_0\), \(\eta(i)\) decreases when \(i\) increases; and vice versa, when \(i > i_0 > 0\), \(\eta(i)\) increases when \(i\) increases. It can be shown that \(\eta(i)\) achieves absolute minimum at \(i=i_0\). Thus, \(S(i) \ge 0\) for all \(i \ge 0\).
28.2.3 Rebalancing
Immunization requires that the average durations of assets and liabilities be kept equal at all times. This makes it necessary to rebalance the portfolio investments regularly, because the years remaining in the planning period grow shorter with each passing year. Coupon income, reinvestment income, proceeds from maturities and sales proceeds must be reinvested in securities that will keep the portfolio’s duration equal to the remaining years in the planning period.