Financial Risk

Study guide: Exam 9

Financial Risk is the risk that an insurer's economic value or earnings fall because of movements in financial markets or the failure of a financial counterparty — interest rates, equity prices, credit spreads and exchange rates, defaults, and the inability to sell assets or raise cash when it is needed (liquidity). It sits beside insurance risk and operational risk in an insurer's risk taxonomy.

$\frac{\Delta S}{S} = \frac{A}{S}\cdot\frac{\Delta A}{A} - \frac{L}{S}\cdot\frac{\Delta L}{L}$

$A$ , $L$ and $S = A - L$ are the market values of assets, liabilities and surplus. The identity is why financial risk matters so much to an insurer: a small percentage move in assets or liabilities becomes a large percentage move in surplus, magnified by the leverage ratios $A/S$ and $L/S$ that its Capital Structure sets.
Components. Market risk — interest rate, equity, spread and currency risk; credit risk — bond issuers, reinsurers, agents and brokers, derivative counterparties; liquidity risk — being forced to sell assets at a loss to pay claims.
Both sides of the balance sheet. Financial risk is measured on surplus, not on assets alone. A bond portfolio that loses value when rates rise is partly offset by loss reserves whose present value also falls, and claim inflation moves the liabilities themselves.
Measuring it. A Risk Measure — standard deviation, VaR, TVaR, expected policyholder deficit — is applied to the distribution of surplus change, usually produced by an economic scenario generator run jointly with insurance risk inside the insurer's risk model. The two interact: a catastrophe can force asset sales into a falling market, and inflation raises claim costs and interest rates together.
Why it is managed. In a frictionless market shareholders could diversify financial risk themselves; Insurance Market Imperfections make it costly for the insurer to bear, which is the case for Financial Risk Management and for charging a Cost of Capital on the capital it consumes.

Leverage Turns an Asset Shock into a Surplus Shock An insurer has assets of $\$1{,}000$ M, liabilities of $\$800$ M and surplus of $\$200$ M. The assets have modified duration $5$ and the liabilities $3$ .

Find the percentage change in surplus if (a) assets fall $5\%$ in an equity and credit sell-off with liabilities unchanged, and (b) all yields rise by $1\%$ .
Answer (a) $\Delta A = -0.05 \times \$1{,}000\text{M} = -\$50\text{M}$ , so $\Delta S / S = -50/200 = -25\%$ .

(b) Using $\Delta X \approx -D \cdot \Delta y \cdot X$ on each side:
 $\begin{align*} \Delta A &\approx -(5)(0.01)(\$1{,}000\text{M}) \\ &= -\$50\text{M} \\ \Delta L &\approx -(3)(0.01)(\$800\text{M}) \\ &= -\$24\text{M} \\ \Delta S &= -50 - (-24) \\ &= -\$26\text{M} \end{align*}$ 
$\Delta S / S = -26/200 = -13\%$ . A $5\%$ fall in assets costs a quarter of surplus, and even a partly matched book loses $13\%$ on a one-point rate move because the asset dollar duration ( $5{,}000$ ) exceeds the liability dollar duration ( $2{,}400$ ).

Same VaR, Different Tails One-year losses ($$$$$M) on two investment portfolios:

X (high yield): $0$ w.p. $0.90$ ; $5$ w.p. $0.06$ ; $20$ w.p. $0.03$ ; $60$ w.p. $0.01$ .

Y (investment grade): $0$ w.p. $0.90$ ; $5$ w.p. $0.06$ ; $8$ w.p. $0.04$ .

Compare the expected loss, $\text{VaR}_{0.95}$ and $\text{TVaR}_{0.95}$ .
Answer VaR. For both, $F(0) = 0.90 < 0.95$ and $F(5) = 0.96 \geq 0.95$ , so $\text{VaR}_{0.95} = 5$ .

TVaR averages the worst $5\%$ of outcomes. The mass at $5$ extends $0.96 - 0.95 = 0.01$ beyond the $95$ th percentile, so $0.01$ of the tail sits at $5$ :
 $\begin{align*} \text{TVaR}_X &= \frac{0.01(5) + 0.03(20) + 0.01(60)}{0.05} \\ &= \frac{1.25}{0.05} \\ &= 25 \\ \text{TVaR}_Y &= \frac{0.01(5) + 0.04(8)}{0.05} \\ &= \frac{0.37}{0.05} \\ &= 7.4 \end{align*}$ 
Expected losses are $E[X] = 0.3 + 0.6 + 0.6 = 1.5$ and $E[Y] = 0.3 + 0.32 = 0.62$ .

VaR calls the two portfolios equally risky; TVaR shows X's tail is more than three times as severe. A capital model built on VaR would let the insurer swap Y for X without holding any more capital.