Liability hedging and immunization

Definition Consider deterministic cash flows without default risk, valued at time $0$ by discounting with continuously compounded zero rates $z(t)$, so that a payment $c$ at time $t$ in years is worth $c\,e^{-z(t)\,t}$ in currency units. Most of this lesson uses a flat curve $z(t) = y$, under which a position paying $c_j$ at times $t_j$ is worth

Let $A$ be the value of the assets and $L$ the value of the liability, both computed with the same curve. The asset-liability surplus is

and for $L > 0$ the funding ratio is $A/L$. A liability hedge aims to control the changes of $\mathcal{S}$, not those of $A$ alone. Valuing the liability with the same curve as the assets is a modeling choice. A different discount rule for the liability changes both $L$ and its rate sensitivities.

Definition For a position with value $V(y) > 0$ under the flat continuous yield, the duration is $D = -V'(y)/V(y)$ in years. For nonnegative cash flows it is the present-value-weighted mean payment time,

and under continuous compounding Macaulay and modified duration coincide. Duration matching chooses the assets so that

which gives $\mathcal{S}(y) = 0$ and $\mathcal{S}'(y) = -A D_A + L D_L = 0$. The surplus is then zero and has no first-order exposure to a parallel shift of the yield. When $A \ne L$, matching the dollar durations $A D_A = L D_L$ removes the first-order exposure of the surplus in currency units, while matching $D_A = D_L$ removes that of the funding ratio, since

With $m$ compounding periods per year, derivatives with respect to that yield carry the factor $1/(1 + y/m)$, so the assets, the liability and the shift must all use one convention. Duration matching controls only the first-order effect of a parallel shift, and it does not match payment dates.

Definition The normalized convexity of a position with value $V(y) \ne 0$ is

in years squared, as distinct from the dollar convexity $V''(y)$. Under the flat continuous yield and for nonnegative cash flows,

so $C \ge D^2$, with equality exactly when all the value sits at one date. Under value and duration matching the surplus has

and Taylor’s theorem gives, for a small parallel shift $\delta$,

a local approximation with an error of order $\delta^3$. Under periodic compounding the convexity with respect to that yield has extra factors, as in Duration convexity and key rate sensitivities.

Definition A position is immunized against a stated family of curve changes if its surplus immediately after any change in the family is at least its surplus before. The following result covers one liability payment and parallel shifts. Let the liability be a single payment $F > 0$ at $T$, let the assets be deterministic cash flows $c_j \ge 0$ at times $t_j$, not all zero, valued with a flat continuous yield $y$, and suppose $A(y) = L(y)$ and $D_A = T$. Then for every real $\delta$

The equality holds because $A(y + \delta) = A(y) \sum_j w_j e^{-\delta t_j}$ and $L(y + \delta) = L(y)\, e^{-\delta T}$. The inequality is Jensen’s inequality for the convex exponential, since the weights are nonnegative and add to $1$, and the final exponent is $0$ because $\sum_j w_j t_j = D_A = T$. So $\mathcal{S}(y + \delta) \ge 0$ for every parallel shift, and for $\delta \ne 0$ equality holds only when all the asset value sits at $T$. The same proof works for a parallel shift of any continuously compounded zero curve, with the weights computed from that curve. For several liability payments, matching value and duration with $C_A > C_L$ makes $\delta = 0$ a strict local minimum of the surplus, so the surplus stays nonnegative for small parallel shifts, and in general the guarantee is only local. These are statements about values immediately after the shift. Funding the actual payment at $T$ also involves reinvesting early payments and selling later ones at rates that are not yet known. The result also shows that a one-period model in which the only possible change is an immediate parallel shift of a flat yield, nonzero with positive probability, cannot be free of arbitrage if five-year zero-coupon bonds trade at model prices. Buying the two bonds and selling a five-year bond of the same value costs nothing, never loses and gains after every nonzero shift.

Definition A curve shift changes the continuously compounded zero rates by $\Delta z(t)$. It is parallel when $\Delta z(t) = \delta$ for every $t$ and nonparallel otherwise, as in a steepening, a twist or a move concentrated at some maturities. For deterministic cash flows the exact change in value comes from repricing at the new rates. To first order, with asset present values $p_j = c_j e^{-y t_j}$ and one liability payment $F$ at $T$,

Duration matching makes this first-order change vanish when all the $\Delta z$ values are equal, and in general not otherwise. A duration is a derivative along the single parallel direction, and the curve has many other directions.

Definition Represent the curve by continuously compounded zero rates $z_1, \dots, z_m$ at chosen node maturities, and let a fixed interpolation rule give the zero rate at every other date. The key rate exposure of a position with value $V$ to node $i$ is $-\partial V/\partial z_i$, in currency units per unit of rate, or $10^{-4}$ times that per basis point. Collect the exposures of one unit of each of $K$ hedge instruments as the columns of an $m \times K$ matrix $H$, and those of the liability in a vector $\ell$. For holdings $\theta \in \mathbb{R}^K$ and a move $\Delta z$ of the nodes, the first-order change of the surplus is

A key rate hedge solves $H\theta = \ell$, usually together with the value condition $v^{\top}\theta = L$, where $v$ holds the instrument prices,

This system has a solution exactly when its right side lies in the column space of the matrix on the left, and the solution is unique when those columns are linearly independent. Otherwise one minimizes a weighted residual $(H\theta - \ell)^{\top} \Omega\, (H\theta - \ell)$ with $\Omega$ positive definite, and when $H$ has full column rank the minimizer is

possibly with funding and position constraints added. When the interpolation weights $\partial z(t)/\partial z_i$ add to $1$ at every date, as with linear interpolation, the key rate exposures add to the exposure to a parallel shift, so matching every key rate implies dollar duration matching but not conversely. The nodes and the interpolation rule determine $H$ and $\ell$, so two systems that both report a five-year key rate can give different exposures. The framework is linear. It removes first-order exposure to the chosen node moves, leaves second-order effects, and does not address moves the nodes cannot express or uncertain liability cash flows.