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Enterprise Risk Management and Strategic Planning. Status of ERM. Started with banking regulation Banks were not quantifying risk, going broke Basic risk quantification modeling developed Expanded to insurance regulation Modeling more complex, hard to evaluate
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Status of ERM • Started with banking regulation • Banks were not quantifying risk, going broke • Basic risk quantification modeling developed • Expanded to insurance regulation • Modeling more complex, hard to evaluate • Attempts for strategic planning • Manufacturing looking at supply chains, etc. • Financial institutions model financial risk
Key Financial Issues • How much capital is needed? • Which business units have best profit after adjusting for risk? • Building models is an advancing science with many pitfalls • Difficult to quantify all the risks, even the largest problems, like competition • Today assume basic models are there and think about application
Risk-Adjusted Profit from ERM Models • ERM quantifies risk of company and each business unit • Management would like to use that information to identify units that have better and worse profitability compared to risk
Uses of Risk Adjusted Profitability • Strategic planning for insurer • Grow business units that have higher profit in relationship to risk • De-emphasize or restructure business that does not give enough profit for the risk
Typical Approach • Quantify risk by a percentile of the distribution of profit = “economic capital” • Maybe start with capital = – 1/3333 quantile • Compute – 1/100 quantile for each business unit and for company • Allocate capital by ratio of business unit quantile to company quantile • Divide unit profits by capital so allocated
Some Criticisms Historically • Not realistic capital need • Quantile is a very limited risk measure • 1/3333 quantile impossible to quantify accurately • Profit not measured relative to marginal cost of risk • Arbitrary choices required (1/100, etc.) • Not clear that growing units with higher returns will actually increase risk adjusted return or firm value
Economic Capital • Economic capital is a non-technical name for a quantile of a distribution • Not realistic capital need • Useful for regulation as a benchmark • Minimal company information becomes public – only one point on distribution
Realistic Capital Need • Shareholders want to provide as little as possible • Bondholders and customers want enough to feel secure with company guarantee • Will demand discounts if capital too low • So shareholders willing to supply enough to avoid the discounts • But practical perception issue – cannot calculate as a quantile
Relating Capital to Risk • Once actual capital is determined, return on that capital can be measured • Also riskiness of actual capital can be quantified with risk measures • Risk measures can also be applied to business units to get risk-adjusted profit by unit
Purposes of Risk Measures • Have a consistent way of comparing different risks, including asset risk, results from different businesses • Comparing profit to risk one key application • For strategic planning – which lines to grow, which to re-organize • Maybe for paying bonuses to managers • Measuring impact of risk-management • All of these work better if risk measures proportional to economic value of the risk
Relating Capital to Risk Measure • Do not have to set capital = risk measure • Useful alternative is capital as a multiple of a risk measure • Some company standards: • 3.5 to 4 times VaR @ 99%; 2.5 times TVaR @ 99% • Capital = 10 times TVaR @ 80% • Average loss in worst 20% of years is 10% of capital • Every risk worth charging for is worth measuring • Models can measure these better than 1/3333 • Includes more adverse scenarios
Which Risk Measure? • “It has been clearly demonstrated that the possibility of extreme adverse results is not the only risk driver of importance.” • Wish I knew who said it, what literature it refers to, and what other risk is important • But the idea seems sound • Losing part of capital can be a big hit to value • Even profit less than target profit can be also
Classification of Risk Measures • Moment based measures • Variance, standard deviation, semi-standard deviation • Generalized moments, like E[YecY/EY] • Tail based measures • Look only at the tail of the distribution • Transformed distribution measures • Change the probabilities then take mean or other risk measure with the transformed probabilities • Uses whole distribution but puts more weight in tails by increasing the probabilities of large losses
Variance and Standard Deviation • Do not differentiate between good and poor deviations. • Two distributions with same mean and standard deviation but Risk B has a much higher loss potential. It will produce losses in excess of 20,000 while Risk A will not. • Semi-variance does
Spectral Measures • for nonnegative function h. • gives TVaRq. • gives blurred VaR • Co-measure is • Marginal for step function or smooth h.
Tail-Based Measures • Probability of default • Value at risk • Tail value at risk • Excess tail value at risk • Expected policyholder deficit • VaR criticized for not being subadditive but not very important with co-VaR • TVaR criticized for linear treatment of large loss
Transformed Probability Measures • Risk measure is the mean (but could be TVaR, etc.) after transforming the loss probabilities to give more weight to adverse outcomes • Prices for risky instruments in practice and theory have been found to be approximated this way • Wang transform for bonds and cat bonds • Esscher transform for compound Poisson process tested for catastrophe reinsurance • Black-Scholes and CAPM are of this form as well • More potential to be proportional to the market value of the risk
Possible Transforms • G*(x) = Qk[F-1(G(x)) + l] where Qk is the t-distribution with k dof - Wang transform • l = .0453 and k [5,6] fit prices of cat bonds and various grades of commercial bonds • k can be non-integer with beta distribution • Compound Poisson martingale transform • Requires function f(x), with f(x) > – 1 for x>0 • l* = l[1+Ef(X)] • g*(x) = g(x)[1+f(x)]/[1+ Ef(X)]
Reinsurance Pricing Compared to Minimum Entropy and Least Squares • g*(y) = g(y)ecy/EY/EecY/EY • l* = lEecY/EY • Quadratic • Average
Which Risk Measures? • Useful to be proportional to value of risk being measured • Favors transformed probability measures • Tail measures are popular but ignore some of the risk
Allocation by Co-measures • Goal is additive allocation • Capital allocated separately to lines A and B will equal the capital allocated to lines A and B on a combined basis. • Start with a risk measure for the company, for example the average loss in the 1 in 10 and worse years • Then, consider only the cases where the company’s total losses exceed this threshold. In this example it is the worst 10% of possible results for the company. • For these scenarios co-measure is how much each line of business is contributing to the poor results
Definition • Denoting loss for the total company as Y, and for each line of business as Xi let: • R(Y) = E[ Y | F(Y) > a ] . Then • Co-R(Xi) = E[ Xi | F(Y) > a] • More generally: • Risk measure r(Y) defined as: E[h(Y)g(Y)| condition on Y], where h is additive, i.e., h(U+V) = h(U) + h(V) • Allocate by r(Xj) = E[h(Xj)g(Y)| condition on Y] • VaRa(Y) = E[Y|F(Y) = a], r(Xj) = E[Xj|F(Y) = a]
Marginal Decomposition of Risk Measures • Applies when allocation of capital is based on allocating a risk measure • Marginal impact of a business unit on company risk measure is decrease in overall risk measure from ceding a small increment of the line by a quota share • Marginal allocation assigns this marginal risk to every such increment in the line • Treats every increment as the last one in • If sum of all such allocations over all lines is the overall company risk measure, this is called a marginal decomposition of the risk measure • All co-measures are additive but not all are marginal
Advantage of Marginal Decomposition • You would like to have it so that: • If you increase business in a unit that has above average return relative to risk • Then the comparable return for the whole company goes up • Not all allocation does that; marginal decomposition does • Thus useful for strategic planning
Formal Definition • Marginal r(Xj) = lime0[r(Y+eXj) – r(Y)]/e . • Take derivative of numerator and denominator wrt e. • L’Hopital’s rule then gives r(Xj) = r’(Y+eXj)|0 . • Consider r(Y) = Std(Y) • r(Y+eXj) = [Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]½ so r’(Y+eXj)|0 = • [Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]-½ [Cov(Xj,Y) + eVar(Xj)]|0 • r(Xj) = Cov(Xj,Y)/Std(Y) • With h(X) = X – EX and g(Y) = (Y – EY)/Std(Y) • r(Y) =E[(Y – EY)(Y – EY)/Std(Y)] = Std(Y) • r(Xj) =E[(Xj – EXj)(Y – EY)/Std(Y)] = Cov(Xj,Y)/Std(Y) • So this co-measure gives marginal allocation
Example – Tail Value at Risk, etc. • Co-TVaR, co-Var are marginal decompositions • Increasing Xj by (1+a) increases co-measure and measure by same amount • EPDa = (1 –a)[TVaRa– VaRa] is expected insolvency cost if capital = VaRa • Co – EPD is a[co-TVaR – co-VaR] and is marginal
Some Criticisms Historically • Quantile is a very limited risk measure • 1/3333 quantile impossible to quantify accurately • Profit not measured relative to marginal cost of risk • Arbitrary choices required (1/100, etc.) • Not clear that growing units with higher returns will actually increase risk adjusted return or firm value
Improvements Round 5 – Capital Consumption Risk Adjusted Performance Without Capital Allocation
Alternative to Capital Allocation(for measuring risk-adjusted profit) • Charge each business unit for its right to access the capital of the company • Profit should exceed value of this right • Essentially an economic value added approach • Avoids arbitrary and artificial notions of allocating capital • Business unit has option to use capital when premiums plus investment income on premiums run out (company provides stop-loss reinsurance at break-even) • Company has option on profits of unit if there are any • Pricing of these options can determine economic value added
Insurance Viewpoint • Company implicitly provides stop-loss reinsurance to each business unit • Any unit losses above premium and investment income on premium are covered • Value of this reinsurance is an implicit cost of the business unit • Higher for higher risk units • Subtracting this value from profit is the value added of the unit • A form of risk adjusted profitability • Right measure of profit to compare is expected value of profit if positive times probability it is positive • Company gets the profit if it is positive • Company pays the losses otherwise • Comparing value of these options
Some Approaches to Valuing • Units that have big loss when firm overall does cost more to reinsure, so correlation is an issue • Limits on worth of stop loss • Probably worth more than expected value • Probably worth less than market value • Stop-loss pricing includes moral hazard • Company should be able to control this for unit • Or look at impact of unit loss on firm value • Need to understand relationship of risk and value
Capital Consumption Summary • Perhaps more theoretically sound than allocating capital • Does not provide return on capital by unit • Instead shows economic value of unit profits after accounting for risk • A few approaches for calculation possible • Really requires market value of risk
So … • Marginal decomposition with co-measures improves allocation exercise • Choice of risk measure can make result more meaningful • Capital consumption removes some arbitrary choices and artificial notions • Market value of risk is really what is needed
Improvements Round 6 Market Value of Risk
Market Value of Risk Transfer • Needed for right risk measure for capital allocation • Needed to value options for capital consumption • If known, could compare directly to profits, so neither of other approaches would be needed
Two Paradigms • CAPM • Arbitrage-free pricing • And their generalizations
CAPM and Insurance Risk • Insurance risk is zero beta so should get risk-free rate? • But insurance companies lose money on premiums but make it up with investment income on float • Really leveraged investment trust, high beta? • Hard to quantify • Cummins-Phillips using full information betas found required returns around 20%
Problems with CAPM • How to interpret Fama-French? • Proxies for higher co-moments? • Could co-moment generating function work? • What about pricing of jump risk? • Earthquakes, hurricanes , … • Two standard approaches to jump risk: • Assume it is priced • Assume it is not priced • Possible compromise: price co-jump risk
Arbitrage-Free Pricing • Incomplete market so which transform? • Same transform for all business units?
No Good Deals • Generalization of arbitrage-free idea • Rules out arbitrage and good deals • Good deals have some risk but so much more potential reward that anyone would take the deal • Defined by some arbitrary standard – maybe 7 flavors already – but gives more restricted pricing ranges than no arbitrage
So … • Marginal decomposition with co-measures improves allocation exercise • Choice of risk measure can make result more meaningful • Capital consumption removes some arbitrary choices and artificial notions • Market value of risk is really what is needed