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

Template:Colored definition Template:Colored remark Consider a fund with Template:Colored em cash flows and Template:Colored em 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 $i$
• ${\bar{v}}_A(i)$: the volatility of the asset cash flows
• ${\bar{v}}_L(i)$: the volatility of the liability cash flows
• ${\bar{c}}_A(i)$: the convexity of the asset cash flows
• ${\bar{c}}_L(i)$: the convexity of the liability cash flows

We have the following definition of immunized conditions: Template:Colored definition Template:Colored remark 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. Template:Colored proposition

Proof. By Taylor series expansion, $$P(i_0+ϵ)=P(i_0)+ϵP^{\prime }(i_0)+\frac{{ϵ}^2}{2}{P}^{″}(i_0)+\cdots$$

• by definition, $P_A(i_0)=P_L(i_0)\iff P(i_0)=0.$
• the 2nd term $ϵP^{\prime }(i_0)=0$ for each $ϵ$ if and only if $P^{\prime }(i_0)=0\iff P{\text{'}}_A(i_0)=P{\text{'}}_L(i_0)$

• Also, $\underset{\because P_A(i_0)=P_L(i_0)}{\underbrace{P{\text{'}}_A(i_0)/P_A(i_0)=P{\text{'}}_L(i_0)/P_L(i_0)}}\iff v_A(i_0)=v_L(i_0)$

• Since ${ϵ}^2>0$, the 3rd term is always positive if and only if $P^{″}(i_0)>0\iff P^{\prime }{\text{'}}_A(i_0)>P^{\prime }{\text{'}}_L(i_0)$.

• Also, $\underset{\because P_A(i_0)=P_L(i_0)>0}{\underbrace{P^{\prime }{\text{'}}_A(i_0)/P_A(i_0)>P^{\prime }{\text{'}}_L(i_0)/P_L(i_0)}}\iff c_A(i_0)>c_L(i_0)$

• The 4th and subsequent terms ($\underset{\text{small}}{\underbrace{{ϵ}^3}}(P^{‴}(i_0))/3!,\dots$) in the expansions are very small and negligible since $|ϵ|$ is small.

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Full immunization

Template:Colored em 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. Template:Colored definition Template:Colored remark Template:Colored proposition

Proof.

• With $P(i)=0$ as one of the conditions, it suffices to prove that $P(i^{\prime })>0,i^{\prime }\ne i$ 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, $P(i)=0\Rightarrow A(1+i{)}^a+B(1+i{)}^{-b}-L=0$.
• Also, $P^{\prime }(i)=0\Rightarrow Aa(1+i{)}^a\ln (1+i)-Bb(1+i{)}^{-b}\ln (1+i)=0\Rightarrow Aa(1+i{)}^a=Bb(1+i{)}^{-b}$.
• Then, for each $i^{\prime }\ne i$,

$$\begin{array}{cc}P(i^{\prime })& =A(1+i^{\prime }{)}^a+B(1+i^{\prime }{)}^{-b}-L\\ & =A(1+i^{\prime }{)}^a+B(1+i^{\prime }{)}^{-b}-(A(1+i{)}^a+B(1+i{)}^{-b})\\ & =A(1+i^{\prime }{)}^a-A(1+i{)}^a-B(1+i{)}^{-b}+B(1+i{)}^{-b}\cdot \frac{(1+i^{\prime }{)}^{-b}}{(1+i{)}^{-b}}\\ & =A(1+i{)}^a(\frac{(1+i^{\prime }{)}^a}{(1+i{)}^a}-1)+B(1+i{)}^{-b}(\frac{(1+i^{\prime }{)}^{-b}}{(1+i{)}^{-b}}-1)\\ & =A(1+i{)}^a(\frac{(1+i^{\prime }{)}^a}{(1+i{)}^a}-1)+\frac{Aa}{b}(1+i{)}^a(\frac{(1+i^{\prime }{)}^{-b}}{(1+i{)}^{-b}}-1)\\ & =A(1+i{)}^a({\left(\frac{1+i^{\prime }}{1+i}\right)}^a-1+\frac{a}{b}\cdot {\left(\frac{1+i^{\prime }}{1+i}\right)}^{-b}-\frac{a}{b}).\\ \end{array}$$

• Let $x=\frac{1+i^{\prime }}{1+i}$. Then, $P(i^{\prime })=A(1+i{)}^a(x^a+\frac{a}{b}{x}^{-b}-1-\frac{a}{b})$.
• Since $A(1+i{)}^a>0$ ($A$ and $(1+i{)}^a$ are both positive), to determine whether $P(i^{\prime })>0$, it suffices to only consider the function $f(x)=x^a+\frac{a}{b}{x}^{-b}-1-\frac{a}{b}$.
• Since $f^{\prime }(x)=a{x}^{a-1}-a{x}^{-b-1}=a(x^{a-1}-x^{-b-1})$, and $a-1>-b-1$ (because $a>-b,a,b>0$),
• $f(x)\{\begin{array}{cc}>0,& x>1;\\ =0,& x=1;\\ <0,& x<1.\end{array}$.
• 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 $x\ne 1$, meaning that $P(i^{\prime })>0$ when $i^{\prime }\ne i$ (which is equivalent to $x\ne 1$).

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Exact matching

Template:Colored em 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. Template:Nav Template:BookCat