# 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

Definition. (Immunization) Immunization is a technique to reduce or even eliminate the impact of interest rate movements.
Remark. There are different ways of immunization. We will discuss several immunizations in this chapter, and will discuss Redington immunization in this section.
Consider a fund with asset cash flows and liability cash flows. We use the following notations:
* $P_{A}(i)$: 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
We have the following definition of immunized conditions:
Definition. (Redington immunization conditions) At an interest rate $i_{0}$, the fund is immunized against small movements in interest rate $\epsilon$ if and only if $P_{A}(i_{0})=P_{L}(i_{0})\quad {\text{and}}\quad P_{A}(i_{0}+\epsilon )\geq P_{L}(i_{0}+\epsilon ).$
Remark.
* i.e. small changes in interest rate ($\epsilon$) in either direction ($\epsilon >0$ or $\epsilon <0$) will increase $P$
* in practice, there are difficulties in implementing this immunization strategy, since, e.g., continuous rebalancing of portfolios to keep the asset and liability volatilities equal is needed, which is infeasible
   * although there are some problems, immunization can still improve investment strategies
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.
Proposition. (Alternative conditions for Redington immunization) Let $P(i)$ be the surplus $P_{A}(i)-P_{L}(i)$. At a interest rate $i_{0}$, the fund is immunized against small movements in interest rate $\epsilon$ if and only if the following three conditions are met:
* (correct amount of assets to support liabilities) $P(i_{0})=0$ or $P_{A}(i_{0})=P_{L}(i_{0})$
* (same interest sensitivity for assets and liabilities) $P'(i_{0})=0$, $P'_{A}(i_{0})=P'_{L}(i_{0})$(, or ${\overline {v}}_{A}(i_{0})={\overline {v}}_{L}(i_{0})$ (result from satisfying both $P_{A}(i_{0})=P_{L}(i_{0})$ and $P'_{A}(i_{0})=P'_{L}(i_{0})$))
* $P''(i_{0})>0$, $P''_{A}(i_{0})>P''_{L}(i_{0})$(, or ${\overline {c}}_{A}(i_{0})>{\overline {c}}_{L}(i_{0})$ (result from satisfying both $P_{A}(i_{0})=P_{L}(i_{0})$ and $P''_{A}(i_{0})>P''_{L}(i_{0})$))
Proof. By Taylor series expansion, $P(i_{0}+\epsilon )=P(i_{0})+\epsilon P'(i_{0})+{\frac {\epsilon ^{2}}{2}}P''(i_{0})+\cdots$
* by definition, $P_{A}(i_{0})=P_{L}(i_{0})\Leftrightarrow P(i_{0})=0.$
* the 2nd term $\epsilon P'(i_{0})=0$ for each $\epsilon$ if and only if $P'(i_{0})=0\Leftrightarrow P'_{A}(i_{0})=P'_{L}(i_{0})$
   * Also, $\underbrace {P'_{A}(i_{0})/P_{A}(i_{0})=P'_{L}(i_{0})/P_{L}(i_{0})} _{\because P_{A}(i_{0})=P_{L}(i_{0})}\Leftrightarrow v_{A}(i_{0})=v_{L}(i_{0})$
* Since $\epsilon ^{2}>0$, the 3rd term is always positive if and only if $P''(i_{0})>0\Leftrightarrow P''_{A}(i_{0})>P''_{L}(i_{0})$.
   * Also, $\underbrace {P''_{A}(i_{0})/P_{A}(i_{0})>P''_{L}(i_{0})/P_{L}(i_{0})} _{\because P_{A}(i_{0})=P_{L}(i_{0})>0}\Leftrightarrow c_{A}(i_{0})>c_{L}(i_{0})$
* The 4th and subsequent terms ($\underbrace {\epsilon ^{3}} _{\text{small}}\left(P'''(i_{0})\right)/3!,\ldots$) in the expansions are very small and negligible since $|\epsilon |$ is small.
$\Box$
Remark.
* the 3rd condition assures that a $\downarrow (\uparrow )$ in interest rate will cause asset values to $\uparrow (\downarrow )$ by more (less) than the $\uparrow (\downarrow )$ in liability values (can be observed from Taylor series expansion)

## Full immunization

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.
Definition. (Full immunization) Let $P(i)$ be the present value of cash flows from a fund. The fund is fully immunized if $P(i)>0,\quad P(i_{0})=0{\text{ and }}i\neq i_{0}$.
Remark.
* That is, except the interest rate $i_{0}$ at which the present value is zero the present value is positive for each other interest rate. This means that the present value is always nonnegative.
Proposition. (Conditions for full immunization) Let $P(i)$ be the present value of cash flows from a fund. The fund is fully immunized if the following conditions are satisfied:
* for each liability cash outflow, there are two corresponding assets providing cash inflows, one of which is made before the liability cash flow, and another one is made after the liability cash flow;
* $P(i)=0$;
* $P'(i)=0$.
Proof.
* With $P(i)=0$ as one of the conditions, it suffices to prove that $P(i')>0,\quad i'\neq i$ from the remaining two conditions.
* First, consider one of the liability cash outflows with amount $L$ , and suppose a cash inflow of $A$ is made at $a$ units of time before the liability cash outflow, and another cash inflow of $B$ is made at $b$ units after the liability cash outflow.
* Then, $P(i)=0\Rightarrow {\color {blue}A(1+i)^{a}+B(1+i)^{-b}-L=0}$.
* Also, $P'(i)=0\Rightarrow Aa(1+i)^{a}\ln(1+i)-Bb(1+i)^{-b}\ln(1+i)=0\Rightarrow Aa(1+i)^{a}=Bb(1+i)^{-b}$.
* Then, for each $i'\neq i$,
${\begin{aligned}P(i')&=A(1+i')^{a}+B(1+i')^{-b}-{\color {blue}L}\\&=A(1+i')^{a}+B(1+i')^{-b}-({\color {blue}A(1+i)^{a}+B(1+i)^{-b}})\\&=A(1+i')^{a}-A(1+i)^{a}-B(1+i)^{-b}+B(1+i)^{-b}\cdot {\frac {(1+i')^{-b}}{(1+i)^{-b}}}\\&=A(1+i)^{a}\left({\frac {(1+i')^{a}}{(1+i)^{a}}}-1\right)+B(1+i)^{-b}\left({\frac {(1+i')^{-b}}{(1+i)^{-b}}}-1\right)\\&=A(1+i)^{a}\left({\frac {(1+i')^{a}}{(1+i)^{a}}}-1\right)+{\frac {Aa}{b}}(1+i)^{a}\left({\frac {(1+i')^{-b}}{(1+i)^{-b}}}-1\right)\\&=A(1+i)^{a}\left(\left({\frac {1+i'}{1+i}}\right)^{a}-1+{\frac {a}{b}}\cdot \left({\frac {1+i'}{1+i}}\right)^{-b}-{\frac {a}{b}}\right).\\\end{aligned}}$
* Let $x={\frac {1+i'}{1+i}}$. Then, $P(i')=A(1+i)^{a}\left(x^{a}+{\frac {a}{b}}x^{-b}-1-{\frac {a}{b}}\right)$.
* Since $A(1+i)^{a}>0$ ($A$ and $(1+i)^{a}$ are both positive), to determine whether $P(i')>0$, it suffices to only consider the function $f(x)=x^{a}+{\frac {a}{b}}x^{-b}-1-{\frac {a}{b}}$.
* Since $f'(x)=ax^{a-1}-ax^{-b-1}=a(x^{a-1}-x^{-b-1})$, and $a-1>-b-1$ (because $a>-b,\quad a,b>0$),
* $f(x){\begin{cases}>0,&x>1;\\=0,&x=1;\\<0,&x<1.\end{cases}}$.
* This is because $f(y)=x^{y}$ is strictly increasing when $x>1$, always equals one when $x=1$, and strictly decreasing when $x<1$.
* This shows that $f(x)$ has the global minimum at $x=1$ with zero value (by first derivative test), and thus $f(x)>0$ when $x\neq 1$, meaning that $P(i')>0$ when $i'\neq i$ (which is equivalent to $x\neq 1$).
$\Box$

## Exact matching

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.
* ↑ In particular, $x^{a-1}>x^{-b-1}$ since $a-1>-b-1$, and thus $f(x)>0$. The results for other cases have similar reasoning.