Notes on: J. David Cummins, Allocation of Capital in the Insurance Industry Risk Management and Insurance Review, 3, 2000, pp

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1 Notes on: J. David Cummins Allocation of Capital in the Insurance Industry Risk Management and Insurance Review pp This reading addresses the standard management problem of allocating capital to projects or lines of business in a firm. This is done in the context of management of an insurance firm but the methodologies presented are general in nature. Capital allocation has special importance for insurers for the following reasons: - Customers are principals in fact basically the only ones and providers of debt capital to an insurance firm. - Customers cannot diversify their debt portfolios away from a given insurer as they cannot hold many insurance policies for the same insurance purpose. - Insurers are regulated with the purpose of regulation being to prevent insolvency and hold capital to protect their policyholders. One might wonder why this capital allocation should even be analyzed for insurers as the capital is held mainly for insolvency protection but - Capital decision affects pricing underwriting and other key management decisions. - It is helpful to understand the interaction between the capital allocation decision and the Risk Based Capital (RBC) rules. - There are important emerging concepts related to capital allocation issue: Riskadjusted return on capital (RAROC) and Economic Value Added (EVA). It should be noted that the terminologies of these new methodologies discussed in the article are not standardized. Using Capital Allocation to Maximize Value - The standard approach in analyzing financial performance is via the GAAP performance. But the true objective should be the maximization of the firm s market value. - Capital allocation should be a part of the process should be used to facilitate economic profitability. - Transfer pricing applies to both banking and insurance: deposit accumulation similar to underwriting loan origination similar to investment and in both cases the liabilities prices are tied to asset prices. Both must manage maturity and duration characteristics of its sources of financing and the corresponding assets. For example must manage long-term liabilities differently than short-term liabilities and must make certain that profits cover cost of capital. - Some researchers (Merton and Perold) propose allocating less than 100% of total capital to product lines. Capital allocated to line i is denoted by C i total capital by C and x i = C i C. Using allocations to maximize value Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

2 Net Income - RAROC with RAROC i = where RAROC i is the risk-adjusted C i return on capital allocated to line i. Net Income is the income after taxes and interest expense with insurer s underwriting loss treated as equivalent to interest expense. If RAROC i is greater than the cost of capital resources should continue to be assigned to the line. But if return is lower than the cost of capital the firm should review pricing of insurance and/or its underwriting standards or withdraw from the line of business. - EVA with EVA i = Net Income i! r i C i where EVA i is for the business line i and r i is the cost of capital for the business line i. This EVA for the business line must be positive or at least non-negative. - Economic Value Added on Capital or EVAOC defined as Net Income EVAOVC i =! r i. C i It is basically the same as RAROC but the cost of capital is subtracted. Value is created if this quantity is positive. Determining the cost of capital - This is quite a challenge as insurers do not generally have data by business line and they do not have data of appropriate quality to implement VaR RAROC and EVA. - Pure play technique : estimate the cost of capital by finding other firms that offer only one line of business. - Full-information betas : use data on conglomerates to perform regressions that permit the estimation. Capital allocation techniques Overview of capital allocation techniques Risk-Based Capital (RBC): required amount of capital defined by regulators. RBC ratio is the ratio of actual capital held to that required. If ratio falls below 200% increasingly strong (with decreasing capital) regulatory actions result. Use of RBC for management of the firm not recommended as the standard is of questionable accuracy uses book values not economic values ignores duration and convexity risk (K.O. s comment: not exactly as there is the C3 allocation based on cash flow testing results for life insurers and that s related to the ALM position but you should say what the reading says on the exam) ignores risk of derivatives and is mostly applicable only for companies with average risk. CAPM: not the best solution but provides a useful benchmark can be used to compare to other methods. Value-at-Risk (VAR): defined as high percentile of possible losses under normal market conditions. If using daily data time varying volatility and account for autoregressive nature of data estimates improve. Debatable whether insurers really have sufficient data to implement this approach. Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

3 Marginal capital allocation: Value of policyholders claims = PV of losses minus insolvency put option. Two approaches: Merton-Perold (MP) and Myers-Read (MR). Detailed examination of the above methodologies Regulatory RBC. Inappropriate as a management tool but identifies key risks faced by insurers. Total amount of capital required calculated by multiplying factors by accounting entries and then adding them up. The charges are: - Investment charges R 1 factor is zero for Treasuries and then increases with risk. - Loss reserve R 2 provides for the risk of adverse reserve development factors vary by business lines multiplied by reserves. - Written premium R 3 accounts for the possibility of higher than expected loss ratio factors vary by product line multiplied by net premiums written. - Credit R 4 for the possibility of default by agents or reinsurers. - Off-balance sheet R 5 provides for the risk from contracts not on balance sheet e.g. loan guarantees to subsidiaries. - R 0 RBC charge for subsidiaries. The RBC formula is: R T = R 0 + R R R R R 5 2. The square root (covariance adjustment) is used instead of adding the charges to allow for diversification. This formula while used by regulators is not good for management use because it lacks theoretical foundation and has questionable accuracy (bad at predicting insolvencies ignores correlation among lines based on worst case scenario estimates not on statistical concepts). CAPM Expected return on equity is E( r E ) = r F +! E ( E( r M ) " r F ). Consider a simple case of an insurer with two lines 1 and 2. Then: I = r A! A + r 1! P 1 + r 2! P 2 where I is the net income r A is the return on assets A are the assets r 1 is the rate of return on underwriting from line 1 P 1 are premiums from line 1 r 2 is the rate of return on underwriting from line 2 and P 2 are premiums from line 2. The rate of return on equity is: = r E + L A ( + L 1 2 ) r E = I E = r! A + r! P + r! P A r! P 1 1 E E E + r! P 2 2 E where L 1 are the liabilities for line 1 and L 1 are the liabilities for line 2. We introduce liability leverage ratios k i = L i E for i = 12 and premium leverage ratios s i = P i E for i = 12 and this allows for the following decomposition of equity beta:! E =! A "( 1 + k 1 + k 2 ) +! 1 " s 1 +! 2 " s 2 Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

4 where! E is the firm s equity beta! A is the asset beta and! 1! 2 are betas for lines 1 and 2. The second and third terms in the above justify the traditional in the P/C industry use of premium to surplus ratio. The model can then be solved for the required rate of underwriting return on each line of business: r i =!k i r F + " i ( r M! r F ) for lines 1 and 2. Each line implicitly pays interest for the use of policyholders funds (in the term!k i r F ) and receives a rate of return based on the systematic risk of the line (the term! i ( r M " r F )). The implication is that it is not necessary to allocate capital to lines but rather charge each line its CAPM cost of capital. This approach does have some problems: it only pays attention to systematic risk and ignores tail risk; also estimation of beta is a big challenge. It is also possible that there may be more than one source of systematic risk (APT) and cost of capital may be driven by other economic factors besides betas. Value-at-Risk (VaR) The purpose of VaR is to point out possible sources of large losses. It is studied in terms of exceedance probability: ( ( ) + C i ) =! i Pr Loss i > E Loss i where Loss i is the (random) loss from line ii = 12 C i is the capital allocation to line ii = 12 and! i is some prescribed small level of probability of very large losses. If all! i 's are set equal capital is allocated by the equimarginal principle: Pr( Loss 1 > E( Loss 1 ) + C 1 ) =! = Pr( Loss 2 > E( Loss 2 ) + C 2 ). This can be also done in terms of ratios to expected losses:! Loss Pr 1 E( Loss 1 ) > 1+ C 1 $ # " E( Loss & 1 )% = ' = Pr! Loss 2 E( Loss 2 ) > 1+ C 2 $ # " E( Loss & 2 )%. The result is that lines with higher risk require more capital relative to expected losses. The reading illustrates this with graphs showing the relationship between exceedance probabilities and the required ratio to expected losses. This method does not allow for diversification benefits across lines and fails to indicate the amounts by which losses will exceed available resources. Also when a firm lacks sufficient capital what should be done? the firm can seek additional capital or try to increase profitability of the line not clear what the best course of action is. Insolvency put option Value of Policyholders Claim = PV (Losses) P( A Lr!" ) where P( A Lr!" )is the insolvency put option value or expected policyholder deficit L are the liabilities treated as the strike price of the option r is the risk-free interest rate! is the time to maturity and! is the volatility. Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

5 The goal of management is to make the ratios of EPD to liabilities of all lines achieve a specified target level by adjusting each line s asset-to-liability ratio (ALR). Note that ALR i = 1+ C i L i where C i is the capital allocated to line. As ALR increases EPD decreases. This method considers the amount that could be lost (so it is better than VAR in this respect) and considers the financial theory of pricing risky debt. But it has weaknesses: it does not take into account the relationship between lines (firms do not go bankrupt line by line and there are benefits from diversification). Recognizing diversification: Marginal capital allocation Overall risk is reduced when returns of various lines are not perfectly correlated. For that reason if a multi-line company were split into multiple single-line companies overall capital requirements would increase. How do we recognize this benefit? Merton-Perold (M-P) method Risk capital is the smallest amount that can be invested to insure the value of the firm s assets against a loss in value relative to risk-free investment of those net assets where net assets are the total assets minus the default-risk-free value of the firm s policyholders liabilities. This is basically the value of an option that assures that the firm s net assets will be there at the option maturity date. Risk capital is partially supplied by the liability holders if the firm has default risk. To account for diversification this method considers the correlations between the natural logarithms of the loss liabilities. The capital allocation process has the following steps: - Calculate the stand alone capital for each line as EPD. - Calculate the risk-capital required by all possible two-line combinations. - Calculate the marginal capital required when adding a third-firm (this is the marginal capital for that third line). This process separates capital needed for a three-line firm into: marginal capital for each of the three lines and the unallocated capital which is the difference between the sum of the marginal capital and the total capital needed (this is the portion of capital allocated to corporate level). Conclusion: allocations based on stand-alone capital will lead to rejections of products that add market value while the use of marginal risk capital in calculating RAROC and EVA produces higher estimates (unless all lines are perfectly correlated) and thus gives a value-maximizing decision. Myers-Read (M-R) approach This is an option-pricing approach. Here capital allocation is determined by the effect of very small changes in loss liabilities for each line vs. capital allocation. This is termed micro marginal allocation as opposed to M-P macro marginal allocation. Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

6 Start by determining the insolvency put value as a function of three liabilities lines P( L 1 L 2 L 3 ) valued at time zero. Then consider the put value per dollar of liabilities: ( ) p = P L 1 L 2 L 3 and calculate the surplus required per dollar of liabilities as: L 1 + L 2 + L 3 "p 2 s i = s! "# (# il! # L )! (# iv! # LV ) $ "p # "s where: s is the firm s surplus to liability ratio! is the firm s overall volatility parameter! il is the covariance between losses in line i and losses for the entire liability portfolio! L 2 is the variance of total losses! iv is the covariance between losses in line i and losses for the asset portfolio! LV is the covariance between firm s losses and losses for the asset portfolio. This allocation process has these characteristics: - Its objective is to equalize the marginal default values across lines - Amount allocated to a line is proportional to its covariability with the loss portfolio but inversely proportional to its covariability with the asset portfolio. - This process produces significantly different results than M-P. - This process avoids unallocated surplus problem. - The process conforms to the marginal nature of pricing and underwriting decision. - It may be appropriate to use M-P when adding an entire line while using M-R for regular operations. Issues and conclusion There are friction costs that can cause the insurer to earn less than full fair market return: - Agency and institutional costs (managers fail to maximize value adverse selection moral hazard) - Double taxation of investment income (investors could possibly earn more by investing directly) - Regulation: RBC allows seizure by regulators investment restrictions produce inefficient portfolios - Spread cost: cost over and above the cost of capital if this capital were not held in an insurance company but invested directly. Both spread and systematic market risk considered in determining the cost of capital for individual lines. Big question: if capital allocation well-designed and well-functioning who needs RBC? - For most insurers and reasonably small EPD targets no cost realized since insurer s total capital is in general in excess of RBC requirements. - Case 1: RBC requirement for some lines exceed marginal capital allocated to those lines but total capital exceeds RBC requirements overall no cost. Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

7 - Cost 2: RBC requirement overall exceeds firm s capital causing regulatory penalties. Need to use caution in designing and implementing systems. Stand-alone systems are especially bad. Conclusions - EPD generally better than VAR as VAR is just one number. - Option models incorporating diversification effect result in better decisions. - Cost of capital allocated to a line is the spread cost. - Capital allocation must consider both asset and liability risk and covariance. - Allocation of capital must consider duration and maturity of liabilities even when using transfer pricing. - Decision-making system should drive the design of the data system. - Winning firms in the 21-st century will be the ones who successfully implement capital allocation and other financial decision-making techniques. May 2006 Casualty Actuarial Society Course 8 Examination Problem No. 38 Consider the following information for an insurance company that writes two lines of business: Line Net Income Allocated Capital Cost of Capital Risk-Free Rate A $1100 $ % 5.0% B $400 $ % 5.0% Using the economic-value-added approach calculate and explain whether each line creates value for the insurance company. Solution. Economic value added (EVA) equals the net income minus the risk adjusted cost of capital. For line A EVA is 1100! 0.10 "10000 = 100 and for line B it is 400! 0.12 " 4000 =!80. Notes for CAS Exam 8 Copyright 2010 by Krzysztof Ostaszewski

SOLVENCY, CAPITAL ALLOCATION, AND FAIR RATE OF RETURN IN INSURANCE

C The Journal of Risk and Insurance, 2006, Vol. 73, No. 1, 71-96 SOLVENCY, CAPITAL ALLOCATION, AND FAIR RATE OF RETURN IN INSURANCE Michael Sherris INTRODUCTION ABSTRACT In this article, we consider the