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General form of return on capital (ROC)


ROC = income / capital


Issue with the general form of return on capital (ROC)


fails to recognize varying degrees of risk


Measures of income to use in ROC (4)


1. GAAP net income
2. statutory net income
3. IASB fair value basis net income
4. economic profit


Methods to reflect risk in ROC (2)


1. implicitly by using premium-to-surplus or reserve-to-surplus ratios
2. explicitly by adjusting income, capital, or both


Measures of capital to use in ROC that are not risk-adjusted (2)


1. actual committed capital
2. market value of equity


Measures of risk-adjusted capital to use in ROC (4)


1. regulatory required capital
2. rating agency required capital
3. economic capital
4. risk capital


Difference between IASB fair value income vs. GAAP/statutory income


IASB fair value removes biases in country-specific accounting standards (e.g. discounts loss reserves and includes a risk margin)


Economic profit


economic profit = (premium - expenses) * (1 + investment return %) - premium * LR discounted to t=1


Limitations of using economic profit as an income measure (3)


1. does not consider changes in firm value from future profits (franchise value)
2. difficult to reconcile to GAAP
3. decision rationale may not be clear to external parties if decision-making is based on economic profit but reporting is based on GAAP accounting


Actual committed capital


actual committed capital = contributed capital + retained earnings


Market value of equity


considers franchise value, so generally > committed capital


Economic capital


capital required to ensure probability of achieving a given objective


Possible objectives used for economic capital requirements (2)


1. solvency - hold enough capital that firm can meet future obligations to PH
2. capital adequacy - hold enough capital to reach other objectives such as paying dividends, gaining premium, or maximizing franchise value


Risk capital


capital contributed by shareholders to absorb the risk that liabilities will exceed premium & loss reserves


Risk-adjusted return on capital (RAROC)


RAROC = economic profit / risk-adjusted capital


Issue with RAROC


different capital allocation methods can lead to different conclusions


Common risk measures (4)


1. probability of ruin
2. percentile risk measure (VaR)
3. conditional tail expectation (CTE, aka TVaR)
4. expected policyholder deficit (EPD) ratio


Probability of ruin risk measure


estimated probability that a ruin scenario will occur

ex: insurer default or ratings downgrade


Percentile risk measure (VaR)


amount of capital required to achieve a specific probability of ruin target (= loss amount exceeded X% of the time)


Conditional tail expectation (CTE, aka TVaR) risk measure


average loss out of losses that exceed a given percentile


Approx. default probability using TVaR


approx. default probability = (1 - TVaR %) / 2


Expected policyholder deficit (EPD) ratio risk measure


avg value of shortfall b/w assets and liabilities relative to expected liabilities

(denominator is total # of scenarios/simulations, not just those generating a deficit)


Methods for selecting risk measure thresholds (3)


1. bond default probabilities for credit rating level
2. management's/shareholder's risk preferences
3. arbitrary default probability, percentile, or EPD ratio


Bond default probabilities for credit rating level risk measure threshold


set the threshold such that the probability of default = probability of default of bonds with the desired credit rating (e.g. AA-rated)


Weakness of the bond default probability method for selecting a risk measure threshold


does not address which rating firm's should target


Considerations for selecting bond default probabilities for selecting a risk measure threshold (3)


1. historical (more stable) vs. current (more responsive) default rates
2. can get different estimates from different rating agencies - should use the most reflective of current estimates
3. default probabilities should be adjusted to reflect the time horizon


Weakness of using management's risk preferences to determine risk measure threshold


management may have difficulty agreeing or preferences that clash with board/shareholders


Risk sources (aka risk distributions, 4)


1. market risk
2. credit risk
3. insurance UW risk
4. other risk sources: operational & strategic


Market risk


potential loss in value of current investments from
- changes in equity indices
- interest rates
- FX rates
- and other market variables


Credit risk


potential loss in value due to credit events such as
- counterparty default
- changes in counterparty credit rating
- changes in credit-rating specific yield spreads


Credit risk exposures (3)


1. market securities, derivative & swap positions
2. insured's contingent premiums and deductibles
3. reinsurance recoveries


Unique challenges of credit risk associated with reinsurance recoveries (3)


1. may require a broader definition of default (to include reinsurer rating downgrade)
2. substantial contingent exposure
3. reinsurance credit risk is correlated with other insurance risks


Sources of insurance UW risk (3)


1. loss reserves on prior policy years (adverse development from prior yr exposures)
2. UW risk for current policy year (risk premium charged is insufficient to cover losses & expenses)
3. property CAT risk


Loss reserve risk and components (3)


risk actual losses <> expected losses due to:

1. process risk (random variation)
2. parameter risk (inaccurate parameter estimates)
3. model risk (wrong model)


Methods for measuring current period UW risk (2)


1. LR distribution models
2. frequency and severity models


LR distribution models for measuring current period UW risk and reliances (2)


combines a distribution of losses with estimates of WP

relies on:
1. source of model parameters (historical vs. industry)
2. choice of distribution (e.g. normal, lognormal, gamma)


Frequency and severity models for measuring current period UW risk


combines separate frequency and severity distributions to generate an aggregate loss distribution


Advantages of frequency and severity models for measuring current UW risk (3)


1. more easily accounts for exposure growth
2. more accurate reflection of inflation
3. reflects changes in limit/deductible profiles


Measuring property CAT risk


use meteorological, seismological, and engineering data to produce a probability distribution of CAT losses (CAT model)


Methods to determine measures of correlation/dependency (3)


1. empirical analysis of historical data
2. subjective estimates
3. explicit factor models


Advantage of empirical analysis of historical data to measure correlation/dependency


intuitive appeal


Disadvantages of empirical analysis of historical data to measure correlation/dependency (3)


1. data may not exist
2. may contain measurement errors,
3. may fail to capture tail events


Advantages of subjective estimates to measure correlation/dependency (2)


1. reflects dependency during tail events
2. reflects user's intuition


Disadvantages of subjective estimates to measure correlation/dependency (2)


1. # of required correlations increases as # of risk sources/LOB increase
2. difficult to ensure internal consistency in estimates


Explicit factor models for measuring correlation/dependency


uses models to link variability of assets to sensitivity of common factors


Methods for finding aggregate distributions (3)


1. closed-form solutions
2. approximation methods
3. simulation methods (most common)


Approaches for reflecting dependencies in simulation methods (3)


1. Iman-Conover method
2. copula method
3. square root rule


Iman-Conover method for reflecting dependencies


simulates each random variable separately and reshuffles stand-alone results to produce a specific rank correlation


Copula method for reflecting dependencies


simulates correlated percentiles to achieve a certain level of correlation in specific parts of the distribution


Square root rule for aggregating stand-alone risk measures


Capital = sqrt(sum across all risks(capital(i)^2) + double sum rho(i,j) * capital(i) * capital(j))


Conditions for the square root rule to produce an exact approximation (2)


1. risk measure is proportional to standard deviation
2. all risk distributions are normal


Capital allocation methods (4)


1. proportional allocation
2. incremental allocation
3. marginal allocation
4. co-measures


Proportional capital allocation


calculates stand-alone risk measures for each risk source and allocates total risk capital proportionally


Incremental capital allocation


calculate a risk measure for the aggregate risk, then recalculate the same risk measure after removing one of the business units

difference in required capital = incremental capital requirement for that business unit

total risk capital is allocated in proportion to incremental capital requirements


Marginal capital allocation


measures the change in capital requirement as a small change in the exposure of the risk source and allocates total risk capital in proportion to the marginal amounts


Capital allocation using co-measures


determines the contribution each risk source has to the aggregate risk measure

sum of co-measures = total risk capital as long as the same risk measure is used (if not, allocate proportionally to the co-measures)


Advantage of the Myers-Read marginal approach to capital allocation


sum of the individual BU capital = total capital when the same risk-measure is used


Challenges with the Myers-Read marginal approach to capital allocation (2)


1. difficult to calculate the value of the default put option
2. assumes BU risk exposure can be changed without impacting the shape of the loss distribution (which is not true of insurance in general)


Applications for risk-adjusted performance metrics (5)


1. assessing capital adequacy
2. setting risk management priorities
3. evaluating alternative risk management strategies
4. risk-adjusted performance measurement
5. insurance policy pricing


Evaluating alternative risk management strategies using RAROC


calculate RAROC before and after a risk mitigation strategy and evaluate the impact


Using RAROC for insurance policy pricing


set premiums such that expected RAROC is above a specified target rate


Target RAROC for insurance policy pricing


RAROC(target) = [(premium + pi - expenses) * (1 + investment return %) - premium * LR discounted to t=1] / allocated risk capital

pi = additional risk margin
assumes risk margin does not contribute to expenses


Simplifications when using target RAROC for insurance policy pricing (3)


1. ignores investment income on allocated risk capital (means target RAROC is an excess return over investment return)
2. ignores reality of multiperiod capital commitment
3. cost of risk capital used (usually CAPM) is not directly comparable to RAROC


Adjusted target RAROC and when to use it


Adjusted target RAROC = target rate * total PV(BOY capital discounted to t=1) / initial capital

assumes 100% of capital is required in 1st period and releases capital according to a payment schedule in subsequent years

use to reflect risk over multiple periods (multi-period capital commitment)


Reasons cost of capital from CAPM is inconsistent with RAROC (2)


1. CAPM measures systematic risk whereas RAROC is focused on tail risk
2. RAROC is leveraged because it uses risk-adjusted capital (vs. actual committed capital or market value of capital)


Ways to address CAPM cost of capital and RAROC inconsistencies (3)


1. adjust CAPM return upward by ratio of total capital / risk capital
2. use total capital in RAROC instead of risk capital
3. reframe the cost of capital as frictional costs of holding capital


Practical considerations ignored by RAROC methods (4)


1. sources of risk have different time horizons
2. significant differences b/w RAROC and GAAP or statutory income measures
3. risk-based allocation relies on tail-based risk measures vs. more normal events that could materially impact firm value
4. does not consider diversification adjustments


Stranded capital


stranded capital = held capital - risk-based capital


Types of events more important to shareholders compared to tail-events (3)


events impacting
1. credit rating
2. financial strength
3. ability to continue operations


Accounting for risks measured over different time horizons (3)


1. use 1-yr horizon for all risks = change in value
2. multiperiod DFA models (lifetime)
3. ignore inconsistencies


Advantage of using 1-yr time horizon for all risks


mathematically appealing


Issues with using a 1-yr time horizon for all risks (3)


1. no reliable method for recognizing adverse reserve development
2. change in value may not fully capture risk
3. ignores large portion of risk in loss reserves b/c majority of risk will not be seen over a 1-yr horizon


Advantage of using multiperiod DFA models to address different time horizons


consistent risk exposure horizon for all risks


Disadvantage of using multiperiod DFA models to address different time horizons


complex to model long-term market and credit risk exposure


Advantage of ignoring inconsistencies to address different time horizons


simplifies required modeling


Disadvantage of ignoring inconsistencies to address different time horizons


difficult to interpret aggregate risk models