Financial Math FM/Immunization

Learning objectives
The Candidate will understand key concepts concerning cash flow matching and immunization, and how to perform related calculations.

Learning outcomes
The Candidate will be able to:
 * Define and recognize the definitions of the following terms: cash flow matching, immunization (including full immunization), Redington immunization.
 * Construct an investment portfolio to:
 * Redington immunize a set of liability cash flows.
 * Fully immunize a set of liability cash flows.
 * Exactly match a set of liability cash flows.

Redington immunization
Consider a fund with cash flows and  cash flows. We use the following notations: We have the following definition of immunized conditions:
 * $$P_A(i)$$: the present value of the assets at the effective interest rate $$i$$
 * $$P_L(i)$$: the present value of the liabilities at the effective interest rate $$i$$
 * $$\overline v_A(i)$$: the volatility of the asset cash flows
 * $$\overline v_L(i)$$: the volatility of the liability cash flows
 * $$\overline c_A(i)$$: the convexity of the asset cash flows
 * $$\overline c_L(i)$$: the convexity of the liability cash flows

In practice, we use some other equivalent to conditions to check that whether the fund is immunized under Redington immunization. We have such equivalent conditions as follows.

Full immunization
is an even stronger immunization technique than Redington immunization, in the sense that if a fund is fully immunized, then it is Redington immunized, but the converse may not be true. In particular, Redington immunization only works for small changes of interest rate, but full immunization works for changes of interest rate with arbitrary magnitude.

Exact matching
of cash flows is a simple immunization strategy. As suggested by the name, in this strategy, each of the liability cash outflows are exactly matched by cash inflow(s), in the sense that the amount of the cash inflow(s) equals that of the liability cash outflow, and the cash inflow(s) is (are) made at the same time as that of the liability cash outflow.

A common way for the exact matching is using suitable zero-coupon bond(s) to exactly match the liabilities. However, this is not the only way, and sometimes suitable zero-coupon bond(s) is (are) unavailable. An alternative way for the exact matching is using suitable coupon bond(s).

As a result of exact matching, the present value of the cash inflow(s) used for exact matching equals that of the liability cash outflows.