Interest Rate Risk

Study guides: Exam 9, Exam CP-351, Exam ILA-101

Interest Rate Risk is the risk that a change in the level or shape of the yield curve reduces an insurer's economic value — through the market value of its bonds, the present value of its loss reserves and the value of its future business — or its earnings, through the rates at which its cash flows are reinvested. Asset–liability management (ALM) is the practice of measuring and controlling it.

$\Delta S \approx -\left(A\,D_A - L\,D_L\right)\Delta y$

$\Delta V \approx -\left(A\,D_A - L\,D_L + F\,D_F\right)\Delta y$

$A$ and $L$ are the market values of assets and liabilities, $S = A - L$ surplus, $D$ each item's modified (or effective) duration, and $\Delta y$ a parallel yield change. $F$ is franchise value — the present value of profits on future renewals — and $V = A - L + F$ the firm's total economic value. Surplus duration $D_S = (A D_A - L D_L)/S$ shows how leverage amplifies any mismatch.
Price and reinvestment risk. Rising rates cut bond values (Price Risk); falling rates cut the yield on reinvested cash flows (Reinvestment Risk). The two offset when durations are matched, which is the basis of Duration Matching and Immunization; Convexity governs how well a match survives large moves.
Beyond parallel shifts and fixed cash flows. Key-rate durations capture twists in the curve. Assets with embedded options — mortgage-backed securities (Prepayment Risk) and callable bonds — need effective duration and have negative convexity. P&C reserves are not fixed nominal amounts: claim inflation, which tends to move with rates, partly offsets the fall in their present value when rates rise.
Franchise value (Panning). ALM as usually practised manages only the balance sheet that accounting shows and ignores franchise value, which is invisible to accounting but can be a large share of market value. It has a long duration, especially when renewal premiums are priced to a fixed target return: a rate rise then shrinks both the discount factor and the premiums themselves. Hedging it with asset duration alone may require a zero or negative asset duration — infeasible, and alarming to regulators and rating agencies who see only the accounting numbers. Panning's alternative is a pricing strategy whose target return moves with interest rates, which shortens the duration of franchise value directly.
Tools. Duration and cash-flow matching, interest rate swaps and futures, and pricing policy. Interest rate risk is measured on the same economic balance sheet as the rest of the insurer's financial risk, and managed within Financial Risk Management.

Surplus Duration and the Immunizing Asset Duration Assets are $\$1{,}000$ M with duration $4.5$ ; liabilities are $\$700$ M with duration $2.5$ . Estimate the change in surplus if yields rise $1\%$ , and find the asset duration that would immunize surplus.
Answer
 $\begin{align*} D_S &= \frac{1{,}000(4.5) - 700(2.5)}{300} \\ &= \frac{2{,}750}{300} \\ &= 9.17 \\ \Delta S &\approx -9.17 \times 0.01 \times 300 \\ &= -\$27.5\text{M} \end{align*}$ 
Surplus falls about $9.2\%$ although assets fall only $4.5\%$ . Setting the dollar durations equal, $1{,}000\,D_A = 700(2.5)$ , gives $D_A = 1.75$ .

Franchise Value and Total Economic Value (Panning) Panning's simplified insurer holds assets of $126.19$ against losses with present value $71.43$ , both treated as having duration $1$ , so its current economic value is $C = 54.76$ with duration $1$ . Its franchise value is $F = 28.57$ with duration $17.62$ .

(a) Find the duration of total economic value. (b) Find it again if invested assets are moved to duration $0$ . (c) What asset duration would make it zero?
Answer (a) $V = 54.76 + 28.57 = 83.33$ .
 $\begin{align*} D_V &= \frac{54.76(1) + 28.57(17.62)}{83.33} \\ &= \frac{558.16}{83.33} \\ &= 6.70 \end{align*}$ 
Franchise value contributes a dollar duration of $28.57 \times 17.62 = 503.40$ .

(b) With zero-duration assets, the current balance sheet's dollar duration is just $-71.43$ from the liabilities:
 $\begin{align*} D_V &= \frac{-71.43 + 503.40}{83.33} \\ &= 5.18 \end{align*}$ 
(c) Zero requires $126.19\,D_A = 71.43 - 503.40$ , so $D_A = -3.42$ .

A duration-matched balance sheet leaves the firm's value with a duration of $6.7$ , and even cash-only assets leave $5.2$ . Panning shows that changing the pricing rule so that the target return moves one-for-one with the risk-free rate cuts the franchise value's duration to $7.62$ and the firm's to $3.27$ — without any asset position a regulator would question.