Market Risk

Study guides: Exam CP-312, Exam CP-351, Exam RET-101, Exam RET-201

Market risk is the risk of loss from movements in market prices and rates: equity prices, interest rates, credit spreads, exchange rates, real estate values, and the volatilities of these. For an insurer it is measured on surplus, assets less liabilities, because many liabilities move with the same markets as the assets, such as a maturity guarantee or reserves discounted at market rates. It is one part of financial risk, beside credit risk and liquidity risk.

$\Delta S \approx \sum_k \frac{\partial S}{\partial f_k}\,\Delta f_k$

$\sigma_{\Delta S}^2 = \sum_i \sum_j \rho_{ij}\,e_ie_j\,\sigma_i\sigma_j$

$\text{VaR}_p = z_p\,\sigma_{\Delta S}$

Symbols: $S$ is surplus and $f_k$ are the market risk factors, so $\partial S / \partial f_k$ is a sensitivity, such as a dollar duration for interest rates or the equity holding for an equity shock. In the variance–covariance model, $e_i$ is the exposure to factor $i$ , $\sigma_i$ is that factor's volatility over the horizon, $\rho_{ij}$ are the correlations, and $z_p$ is the standard normal $p$ -quantile, $2.3263$ at $99\%$ . The last block assumes normal changes with a mean of zero.
Measuring it: Sensitivities such as duration, convexity and an option's delta describe small moves. VaR and TVaR from a risk measure summarise the distribution of losses, computed with the variance–covariance formula, with historical simulation, or by Monte Carlo from an economic scenario generator, and stress tests apply large specified moves. Each has a blind spot: the normal formula understates fat tails, and historical simulation sees only the moves in its data window.
Mitigating it: Asset–liability matching, such as duration matching and immunization, removes the mismatch between assets and liabilities. Hedging with futures, swaps and options offsets what remains, limits on each exposure cap it, and product design, such as charging for guarantees or using target-volatility funds, reduces how much of it new business creates.
Not the MCT margin: The market risk margin is the capital charge for this risk in Canada's Minimum Capital Test for P&C insurers, set by a regulatory formula. Market risk itself is the exposure, which an insurer measures with whatever model suits it.

Value at Risk for an Equity and Bond Portfolio An insurer holds $60 million of equities, with a one-year volatility of $18\%$ , and $40 million of bonds, with a one-year volatility of $6\%$ , and the correlation between their returns is $0.2$ . Treating the liabilities as fixed and the changes as normal with a mean of zero, find the one-year $99\%$ VaR of the portfolio, and compare it with the sum of the two holdings' VaRs.
Answer The dollar volatilities are $60(0.18) = 10.8$ and $40(0.06) = 2.4$ ($ millions).
 $\begin{align*} \sigma_{\Delta S}^2 &= 10.8^2 + 2.4^2 + 2(0.2)(10.8)(2.4) \\ &= 116.64 + 5.76 + 10.368 \\ &= 132.768 \\ \sigma_{\Delta S} &= 11.5225 \\ \text{VaR}_{0.99} &= 2.3263(11.5225) \\ &= 26.80 \end{align*}$ 
On their own the equities have a VaR of $2.3263(10.8) = 25.12$ and the bonds $2.3263(2.4) = 5.58$ , which add to $30.70$ . The portfolio's VaR is $3.90$ million lower because the two returns are far from perfectly correlated. If the correlation rose to 1 in a crisis, the benefit would vanish, which is why a stress test checks the result.

An Equity and Interest Rate Stress, Then a Fix An insurer has $800 million of bonds with modified duration 6 and $200 million of equities, against liabilities of $850 million with modified duration 9. Its surplus is $150 million. Estimate the change in surplus if equities fall $30\%$ and all yields fall by 1 percentage point. Then find the bond duration that would remove the interest rate part of the loss.
Answer
 $\begin{align*} \Delta\text{Bonds} &\approx -6(800)(-0.01) \\ &= +48 \\ \Delta\text{Equities} &= -0.30(200) \\ &= -60 \\ \Delta L &\approx -9(850)(-0.01) \\ &= +76.5 \\ \Delta S &= 48 - 60 - 76.5 \\ &= -88.5 \end{align*}$ 
Surplus falls from $150 million to $61.5 million, a loss of $59\%$ . The rate fall alone costs $28.5$ million, because the liabilities' dollar duration of $7{,}650$ exceeds the bonds' $4{,}800$ . Lengthening the bonds to a duration of $7{,}650/800 = 9.5625$ matches the two, which leaves only the $60$ million equity loss. That loss can then be capped with put options on the equity holding or limited by a smaller equity allocation.

How it appears on the exam

CP-312 asks candidates to "describe, apply, and assess" market risk models and to "evaluate the appropriateness of their usage". The work is to compute a VaR or a stressed loss with a stated model and then to say where its assumptions, such as normal returns, stable correlations or the chosen horizon, fail.

CP-351 asks candidates to "explain and implement techniques used to mitigate market risks". Its readings for that objective include ALM for Life, Annuities, and Pensions and Hardy and Saunders's chapter on risk mitigation using options and derivatives. A question gives a balance sheet and a shock, and asks for the matching or hedging trade that reduces the loss and what it leaves exposed.

Exam traps

Assets alone: Measuring market risk on the assets ignores liabilities that move with the same markets. In the second example the assets lose only $12$ million, while surplus loses $88.5$ million.

Adding VaRs: The VaRs of separate holdings add up only when their returns are perfectly correlated. Adding them overstates the portfolio's VaR, while assuming low correlation in a stress can understate it.

Scaling the horizon: Converting a one-day VaR to ten days by $\sqrt{10}$ assumes independent, identically distributed daily changes. When returns trend, the scaled figure understates the risk, and when volatility clusters it can be wrong in either direction.

Market risk and the market risk margin: The MCT's market risk margin is one regulatory measure of the risk. Using its factors as if they were the insurer's own risk model confuses a capital rule with a measurement.