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1 P U B L I C A T I O N S The Experts In Actuarial Career Advancement Product Preview For More Information: Support@ActexMadRiver.com or call 1(800)

2 NOTES I have updated the manual originally written by Peter J. Murzda, Jr., FCAS, ASA and updated by Chris Van. Kooten, FCAS, FCIA to reflect the 2015 syllabus material and 2014 CAS Exam 9 with one exception. The outlines for the Bodie, Kane, Marcus book chapters other than Chapter 13 are from the ninth edition, rather than the tenth edition, which is the one currently on the syllabus. I understand the differences are minor but plan to update the ninth edition outlines in the 2016 Study Guide Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society. They are reproduced in this study manual with the permission of the CAS solely to aid students studying for CAS exams. Students may also request past exams directly from the CAS or find them on the CAS website. I am very grateful to the CAS for its cooperation and permission to use this material. The CAS is not responsible for the structure or accuracy of this manual. In some cases, questions and answers have been edited or altered to be more accurate, reflect syllabus changes, or provide a better organized manual. Students should keep in mind that there may be more than one correct way to answer a question even if only one is shown. Exam questions are identified by numbers in parentheses at the end of each question. Questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. MC indicates that a multiple choice question has been converted into a true/false question. Page numbers (p.) with solutions refer to the reading to which the question has been assigned unless otherwise noted. I have made a conscientious effort to eliminate mistakes and incorrect answers, but a few may remain. I am grateful to students who previously pointed out errors and encourage those who find others to bring them to my attention. Please check the ACTEX website for corrections subsequent to publication. Margaret Tiller Sherwood FCAS, FSA, MAAA, FCA, CPCU, ARM, ERMP, CERA January 2015 ii

3 Bodie 1 & 2 1 Bodie Kane Marcus Investments, Ninth Edition: Chapters 1 and 2 Mortgages (pp , 39-41) OUTLINE I. Changes in Housing Finance A. Mortgage backed securities were created when the Federal National Mortgage Association (FNMA or Fannie Mae) and the Federal Home Loan Mortgage Corporation (FHLMC or Freddie Mac) began securitizing bundles of mortgages (low-risk conforming mortgages) 1. Homeowner would get mortgage from originator 2. Originator would sell mortgage to Fannie Mae or Freddie Mac and receive a fee for continuing to service the mortgage 3. Fannie or Freddie sell a group of mortgages to investors but retain the credit risk by taking a fee and guaranteeing the default risk of the loans B. This changed when private firms began offering securitizations of subprime loans with higher default risks and passed that default risk on to the end investor 1. Originator of loan underwrote the risk with the intention of passing it through and retained no credit risk (no incentive to reduce risk) 2. More and more loans were created without proper underwriting C. Adjustable rate mortgages became popular and when interest rates rose homeowners couldn t afford the payments D. Rising housing prices helped to offset potential losses until When housing prices fell in 2007 many homeowners just walked away since the loan was worth more than the house 2. Defaults rose substantially and the credit crisis began E. By this time Fannie Mae and Freddie Mac had been encouraged by Congress to enter into the subprime market (to support affordable housing) 1. They guaranteed the loans against default 2. When defaults rose they had to be bailed out by the US Treasury 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

4 2 Bodie 1 & ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

5 Bodie 6 3 Bodie Kane Marcus Investments, Ninth Edition: Chapter 6 Risk Aversion and Capital Allocation of Risky Assets I. Notation A. R() r Expected return B. 2 Variance of returns C. U Utility value D. A Index of investor s risk aversion E. P Portfolio of risky assets F. r P Risky rate of return of P G. E( r P) Expected rate of return of P H. P Standard deviation of P I. F Risk-free asset J. r f Risk-free return K. B r f Rate charged to borrow OUTLINE L. E Proportion of risky portfolio in equities M. B Proportion of risky portfolio in bonds N. C Complete portfolio O. r C Rate of return of C P. y Proportion of C that is made up of the risky portfolio Q. y * Optimal allocation to the risky portfolio R. CAL Capital allocation line II. Risk and Risk Aversion A. Risk, Speculation and Gambling 1. Speculation assumption of considerable investment risk to obtain commensurate gain a. Considerable risk risk sufficient to affect decision b. Commensurate gain positive risk premium 2. Gamble bet or wager on an uncertain outcome (no positive risk premium) 3. Fair game risky investment with 0 risk premium B. Risk Aversion and Utility Values 1. History shows that investors are risk averse and require a risk premium to compensate for risk taken 2. Risk averse investors penalize expected returns for the risk involved 3. Choose among risky portfolios by assigning a utility score U E() r 1 2 A, ( E( r) written as a decimal) a. Risk averse investors A 0 b. Risk neutral investors A 0 (no penalty for risk) c. Risk lover A 0 4. Utility score can be interpreted as a certainty equivalent rate of return a. Investment is desirable only if certainty equivalent return exceeds the risk-free rate ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

6 4 Bodie 6 5. Mean-Variance Criterion: a. Portfolio A dominates B if Er ( A) Er ( B) and A B with at least one inequality strict 6. Indifference curve connects points in the mean-standard deviation plane with the same utility values a. Investor has no preference amongst portfolios on the same indifference curve C. Estimating Risk Aversion 1. Questionnaires 2. History of portfolio composition changes over time 3. Tracking of groups of individuals to determine averages III. Capital Allocation Across Risky and Risk-Free Portfolios A. Broad asset allocation between investment classes is most important part of portfolio construction 1. Treasury bills low risk 2. Long-term bonds moderate risk 3. Stocks higher risk B. First task is to determine allocation between a generic risky portfolio and the risk-free asset 1. The composition of the risky portfolio doesn t change with the amount invested in the risky portfolio, y a. Treat the risky portfolio as a single fund holding both bonds and equities in fixed proportion IV. The Risk-Free Asset A. Treasury bills are viewed as the most risk-free asset 1. Not truly risk-free in real terms, but government issued debt has default free guarantees which makes it the closest thing 2. The short-term nature makes it insensitive to interest rate fluctuations (can simply hold to maturity to lock in return) B. Money market funds are also close to risk-free due to very low default rates and short durations 1. Often comprised of the following: a. Treasury bills b. Bank certificates of deposit (CDs) c. Commercial paper (CP) 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

7 Bodie 6 5 V. Portfolios of One Risky Asset and a Risk-Free Asset A. Expectations can be used to determine the rate of return of the complete portfolio, C E( rc) rf y E( rp) rf, C y P B. The equations can be rearranged to define the return standard deviation trade-off: E( rp) rf Er ( C) rf C P 1. This is the equation of the line (Capital Allocation Line) passing through the risk-free portfolio and the risky portfolio in the return-variability plane 2. The slope of the line (reward-to-volatility ratio a.k.a. Sharpe Ratio) is given by the portion in brackets and provides the increased return for each additional unit of risk C. If the investor can borrow at the risk-free rate it is possible to have values beyond the risky portfolio P along the CAL (same slope) 1. In reality only governments can borrow at the risk-free rate and other investors must borrow at a higher rate B E( rp ) rf a. In this case the CAL kinks at P, where the slope changes to: b. Borrowing is typically done on margin with a broker i. Margin purchases cannot exceed 50% of the purchase value ii. Purchased securities must be maintained in a margin account and if the value declines below a maintenance margin a deposit is required through a margin call P VI. Risk Tolerance and Asset Allocation A. The formulas for utility (U) and expected return ( ) be combined to determine the optimal allocation y * E r and variance ( ) of the complete portfolio can 1. Utility is maximized by setting the first derivative with respect to y equal to 0 and solving: E( rp ) rf y * 2 A P 2. Optimal solution is directly proportion to the risky asset risk premium and inversely proportional to risk aversion B. Utility curve analysis can be used to validate the optimal portfolio 1. A utility curve is a curve in the return-variability plane that describes all possible return-variability combinations having the same utility value 2. Investors will always a prefer a higher indifference curve since for any given variability a higher curve corresponds to increased utility 3. The highest indifference curve that is tangent to the CAL will be tangent at y * C. Analysis thus far has considered returns to be normally distributed, which is frequently not the case C C 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

8 6 Bodie 6 VII. Passive Strategies: The Capital Market Line A. Portfolio decisions that avoid any security analysis are known as passive strategies 1. Investing in value-weighted indices is the most common passive strategy e.g. S&P Despite criticism history shows that passive strategies have performed better than most active strategies in recent years 3. Common criticisms: a. Not diversified i. Heavy concentration in largest funds, but evidence that most active strategies are also not well diversified b. Too top-heavy i. Implies the market is self is too top heavy c. Chase performance i. Equivalent of buying high and selling low since the proportion of a security held increases as the value increases d. You can do better i. With costs incorporated it is unlikely B. The CAL that uses treasury bills as the risk-free asset and a broad market index (or proxy) for the risky asset is known as the capital market line VIII. Appendix A: Risk Aversion, Expected Utility, and the St. Petersburg Paradox A. The appendix tries to show that investors are risk averse 1. The St. Petersburg Paradox considers a coin toss game that has infinite expected returns, but most would only pay a small entry fee to partake 2. What needs to be considered is the utility gained or lost from partaking in such a game a. The utility of an extra dollar varies with wealth b. Rather than looking at the expected return in a game which might be fair consider the expected utility c. If the expected utility of the outcome is less than the utility of the wager a risk averse investor will reject d. Can compute the certainty equivalent value of the wager by determining the wealth that corresponds to the expected utility of the wager 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

9 Bodie 6 7 PAST CAS EXAMINATION QUESTIONS A. Portfolios of One Risky Asset and a Risk Free Asset A1. Assume the expected market return is 15%, the Treasury bill rate is 6%, and that you can borrow at the risk-free rate. Further assume that you invest all your own money in the market portfolio, as well as half that amount of borrowed money. The expected return on your investment is: A. 14% B. 14.1% but 16.0% C. 16.1% but 18.0% D. 18.1% but 20.0% E. 20.0% (90 5B 65 1) A2. Assume the expected market return is 15% and the Treasury bill rate is 6%. Further assume that you invest 75% of your own money in the market portfolio, and lend the remaining 25% at a risk-free rate of interest. The expected risk premium on your investment is: A. < 7.0% B. 7.0% but < 8.0% C. 8.0% but < 9.0% D. 9.0% but < 10.0% E. 10.0% (91 5B 54 1) A3. You have $100 to invest and can borrow or lend money at the risk-free rate of 4%. The market portfolio offers an expected return of 12% with a standard deviation of 20%. a. Combining borrowing, lending, and/or investing in the market portfolio, how can you construct a portfolio to achieve an expected return on your investments of 18%? Show all work. b. What is the standard deviation of your portfolio? Show all work. (96F 5B 30 1/.5) A4. Given the following regarding a risky portfolio (P) and the risk-free asset, calculate the slope of the capital allocation line for < P. E(r P ) = 12% P = 10% r f = 5% A..05 B..07 C..10 D..50 E..70 ( ) A5. Company X invests all of its money in common stock portfolio A with an expected return and expected variance equal to that of the general market. Portfolio A is perfectly positively correlated with the market. The risk-free interest rate is 5% and the expected market risk premium is 8%. The standard deviation of returns of portfolio A is 20%. Calculate the expected return and standard deviation of company X s investments if the following investment strategies are implemented. Show all work. a. Invest half of the money in portfolio A and half of the money in risk-free securities. b. Borrow an amount equal to half of the company s current wealth at the risk-free rate and invest everything into portfolio A. (98F 5B 27 1/1) A6. According to Bodie et al., each investor is content to put money into two benchmark investments. a. Describe these two investments. Use a diagram to depict the investment strategy. b. Explain how an investor can vary his/her expected rate of return and commensurate risk. (99S 5B 21 1/1) A7. Given the following with respect to an optimal risky portfolio, calculate the slope of the capital allocation line (CAL) for this portfolio. Expected return 7% Risk premium 3% Variance 100 A..03 B..04 C..30 D..40 E (03 6 1) 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

10 8 Bodie 6 A1. r C = (Percentage Invested in Market)E(r m ) (Percentage Borrowed)r f r C = (1.5)(15%) (.5)(6%) = 19.5%, pp Answer: D A2. Expected Investment Risk Premium = (Percentage Invested in Market)(r M r f ) EIRP = (.75)(15% 6%) = 6.75%, p Answer: A A3. a. r C = (Percentage Invested in Market)E(r M ) (Percentage Borrowed)r f.18 = (1 + x)(.12) x(.04) x =.75 Amount Borrowed = (.75)(Original Investment) = (.75)(100) = 75, pp b. = (Percentage Invested in Market)( M ) = (1.75)(20%) = 35%, p A4. Slope = E(r P) r f = P Answer: E =.7, p A5. a. r A = E(r M ) = r f + Risk Premium = =.13 E(r C ) = [E(r A ) + r f )]/2 = ( )/2 =.09 P = (.5)( A ) = (.5)(.20) =.10, p b. E(r C ) = (Percentage Invested in A)E(r A ) (Percentage Borrowed)(r f ) E(r C ) = (1.5)(13%) (.5)(5%) = 17% C = (Percentage Invested in A)( A ) = (1.5)(20%) = 30%, pp A6. a. 1) Risk-free assets, i.e., Treasury bills 2) Portfolio of risky assets, i.e., the market portfolio Expected Return ( ) Er P P Capital Allocation Line F Risk-Free Assets Portfolio of Risky Assets P Standard Deviation b. By changing the proportions of risk-free assets and the portfolio of risky assets in his portfolio, an investor can vary his expected return and associated risk. For example, if the proportion invested in risk-free assets is increased, his complete portfolio moves down the capital allocation line, reducing his expected return and lowering risk, p A7. Slope = E(r P) r f P = Answer: C =.3, p ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

11 Bodie 6 9 A8. Consider the following information about a risky portfolio and a risk-free asset. i) The risk premium of the risky portfolio is 15%. ii) The reward-to-variability ratio of the risky portfolio is.75. iii) The expected return on the risk-free asset is 3.0%. Assume you can invest in some combination of the risky portfolio and the risk-free asset. Determine the equation for the capital allocation line (CAL) under these assumptions and graph the CAL. Label all items properly. Show all work. ( ). A9. You are given the following information: i) A risky portfolio has an expected return of 16% and a standard deviation of 25%. ii) The T-bill rate is 6%. a. Suppose you invest 60% of your funds in the risky portfolio and 40% in a T-bill money market fund. Calculate the expected value and the standard deviation of the rate of return of the portfolio. b. Determine the equation of the capital allocation line (CAL) of the risky portfolio and graph the CAL. Plot the position of the overall portfolio on the CAL graph. Label all items properly. Show all work. ( /1) A10. i) The return of a risk-free asset is 5%. ii) An investment company offers a risky asset, with a Sharpe ratio of.2. iii) An investor wants to hold a portfolio consisting of the risky asset and the risk-free asset. a. Calculate the expected return of the portfolio if the investor wants the standard deviation of the portfolio to be 15%. b. Graph the capital allocation line (CAL) associated with this portfolio. Plot the position of the overall portfolio on the CAL graph. Clearly label the axes, the CAL, the risk-free asset, and the overall portfolio. ( /.75) 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

12 10 Bodie 6 A8. E(r) = r f + (Reward-Variability Ratio) = CAL Expected Return r 3% t 18% Risk Premium = 15% p Standard Deviation 20 A9. a. E(r C ) = r f + y[e(r P ) r f ] =.06 + (.60)(.16.06) =.12 C = y P = (.60)(.25) =.15, p b. E(r C ) = r f + [E(r P ) r f ][ C / P ] =.06 + (.16.06)( C /.25) = C Expected Return 16% 12% Risk Premium 6% p. 172 Standard Deviation A10. a. E(r P ) = S P + r f = (.2)(.15) +.05 =.08. p. 172 b. Expected Return 8% Overall Portfolio Risk-Free Rate 5% p % Standard Deviation 15% 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

13 Bodie 6 11 A.11 Given the following about a risky portfolio, P: Expected return of P is 16%. Standard deviation of expected return of P is 24%. Borrowing rate is 10%. Lending rate is 6%. Graph the capital allocation line (CAL). Plot the position of P on the CAL graph. Clearly label the axes, the y-intercept of the CAL, and the borrowing and lending ranges. ( ) 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

14 12 Bodie 6 A ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

15 Bodie 6 13 A.12 The values in the table below were empirically estimated. Security Expected Return Beta Stock A 6% 0.5 Stock 8 10% 1.5 The expected rnarket return is 8%. The risk-free rate is 2%. a. On a single graph, draw and label: i. The Security Market Line (SML) implied by the expected market return. ii. The empirically estimated SML. iii. The risk-return point for both securities listed above. b. Explain whether Stock A or Stock 8 is a better buy according to the Capital Asset Pricing Model (CAPM). ( /0.5) 2015 ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

16 14 Bodie 6 A.12 a. b. Stock A has a positive α, αa = 1% = 6% [2% (8% 2%)] Stock B has a negative α, αb = 1% = 10% [2% + 1.5(8% 2%)] Therefore, Stock A is a better buy based on CAPM ACTEX Publications, Inc. CAS 9 - Volume I Portfolio Theory & Equilibrium in Capital Markets

17 Ferrari 501 J. Robert Ferrari, "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity"; Discussion by Rafal J. Balcarek, PCAS, LV, 1968, pp ; LVI, 1969, pp OUTLINE I. INTRODUCTION A. The Problem How do the investment and underwriting operations of an insurance company interact to produce a total return on equity? B. The Solution The author uses some basic equations and examples to show the impact of insurance leverage and to provide assistance in determining the optional capital structure for a firm. C. Definitions 1. Insurance exposure term ratio of premium to surplus; similar to the turnover rate 2. Insurance leverage factor unity plus the ratio of reserves to surplus; represents total investments 3. Underwriting profit on premiums earned premium minus incurred losses and incurred expenses divided by earned premiums; similar to sales margin D. Symbols 1. A - total assets 2. I - investment gain/loss (after appropriate tax charges) 3. P - earned premium 4. R - reserves and other liabilities (excluding equity in unearned premium reserves) 5. S - stockholders' equity (capital, surplus, and equity in unearned premium reserves) 6. T - total after-tax return to the insurer 7. U - underwriting profit/loss (after appropriate tax charges) E. Equations 1. T = I + U 2. S = A R 3. T/S = (I/A)(1 + R/S) + (U/P)(P/S) 4. T/S = I/A + (R/S)(I/A + U/R)

18 502 Ferrari II. AUTHOR'S ANALYSIS A. The Choice of the Investment Base 1. Arthur D. Little study opts for the use of total assets rather than net worth as a base 2. Financial leverage concept can be used to link the two bases. B. Total Return on Equity The Basic Equation 1. Insurance leverage does not result from the use of debt capital but from the deferred nature of insurance liabilities 2. Linkages in the basic equation a. Return on equity (investors' viewpoint) b. Return on assets (society's viewpoint) c. Return on sales (regulators' and actuaries' viewpoint) 3. Since the primary objective is to identify the return for investors, UW profit should be shown on an adjusted basis, reflecting the true incidence of expenses C. Reserves Viewed as Nonequity Capital 1. Reserve capital is the sum of total investable assets supplied by other than the owners 2. Reserve leverage factor applied to both investment income and UW profit/loss 3. UW losses equivalent to the interest paid for the use of reserve capital 4. Only when losses make the absolute value of a negative U/R larger than I/A is the leverage from the insurance portfolio unfavorable 5. Reserve liabilities mostly short- and intermediate-term debt D. The Impact of Insurance Leverage 1. Use the second equation to compare the impact 2. Keep invested assets and the investment return the same 3. Vary the proportion of invested assets that come from reserve liabilities (if none, the example is an unlevered investment trust) 4. Vary the insurance operating results as a percentage of reserve liabilities 5. The higher the leverage ratio, the greater the variability of the return on owners' equity 6. Greater increase in the return from an UW profit than decrease from an UW loss 7. Leverage ratio indicates the riskiness of the owner's investment

19 Ferrari 503 E. Actuarial Determination of the Optimum Capital Structure 1. How to determine the leverage (mix of liabilities and owners' equity) that maximizes the value of a firm 2. Two variables that determine this value a. Expected earning stream b. Rate of capitalization by the market 3. Reserves add to the earnings stream as long as the costs of financing are less than the returns from invested assets 4. Crucial to the question of the optimal capital structure is the effect of nonequity financing on the variance of earnings and thus on the rate of capitalization 5. Comparison of reserve capital (insurance leverage) with debt capital a. Debt capital cost is fixed, i.e., no variance b. A greater amount of debt increases the return required by investors, whereas premium and reserve growth may not since diversification may offset the disadvantage of poorer risks 6. Need also to consider investment return variability; if high, then should have a lower debt-to-equity ratio 7. Question whether current low reserve-to-surplus ratio is optimal a. If is optimal, does it reflect rating formulas and regulation or an aggressive investment portfolio b. It optimal is higher, current ratio indicates overcapitalization and explains holding company takeovers

20 504 Ferrari III. CONCLUSIONS AND REVIEW A. Ferrari's Conclusions 1. Actuaries should be concerned with investment returns and broad financial objectives 2. Management should maximize the total return to stockholders 3. Open competition may be the only way to incorporate investment considerations B. Balcarek's Review 1. Need to qualify premium expansion whenever investment return offsets any UW loss with the condition that there is no appreciable increase in the risk to the owners' equity 2. Formulas, moreover, best apply only to the static state as a change in one term leads to other changes a. If P/S increases, then U/P may decrease as growth without adequate controls would bring in poorer business (Ferrari) b. If P/S increases, then I/A tends to decrease 1) Uninvested asset percentage (cash, agents' balances) will tend to rise 2) Higher P/S means greater risk and a more conservative investment policy c. If U/P increases, then P/S will increase as a profitable situation allows a company to write more business safely d. If U/P increases, then I/A will increase as a profitable situation allows a more aggressive investment policy

21 Ferrari 505 A. Ferrari's First Formula PAST CAS EXAMINATION QUESTIONS A1. A group of large insurers has suggested that all investment income from all sources should be considered in ratemaking. This proposal includes interest, dividends, and rents, as well as realized and unrealized capital gains on all invested assets. The companies propose that the rating formula then provide for an after-tax net return of approximately 12% of net worth from underwriting and investments combined. If the proposal is adopted, what effect is the ratio of earned premiums to policyholder surplus likely to have on the required allowance for underwriting profit in the ratemaking formula? Give reasons for your answer. (71 9 5a 6) A2. Ferrari, in his paper on owners' equity, develops in one simple equation the relationship among investment returns from the investors', society's and the regulators' viewpoint. Given the following notation, derive this simple equation and identify these three different measures of return. T - total after-tax return to the insurer I - investment gain/loss (after appropriate tax charges) U - underwriting profit/loss (after appropriate tax charges) P - premium income A - total assets R - reserves and other liabilities (excluding equity in unearned premium reserves) S - stockholders' equity (capital, surplus and equity in unearned premium reserves) ( ) A3. Ferrari demonstrated that the total after-tax return to an insurer divided by surplus is a function of two types of leverage affecting two types of income. The figures below are ten-year averages showing the results of company operations for the Bill Quick Insurance Company during the years 1964 to 1973 (000,000 omitted). All are ten-year averages. Answer a., b., and c., both symbolically and numerically. Be sure to define all symbols that you use. Your numerical answers may be left in factor form, utilizing the above values. Adjusted underwriting income $2 Net investment income before tax 1,582 Statutory capital and surplus 15,340 Earned premiums 19,320 Equity in unearned premium 2,500 Voluntary (surplus-type) reserves 190 Federal income tax 210 Admitted assets 41,485 a. Identify Ferrari's two types of leverage. b. Identify the two types of income as rates of return on an appropriate base quantity. c. Calculate the rate of return produced by Ferrari's formula. ( b-d 3/3/2) A4. Ferrari, in the paper "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," presented the following formula: T/S = (I/A)(1 + R/S) + (U/P)(P/S) a. Define each symbol T, I, U, P, A, R, and S. b. Provide Ferrari's interpretation of each component term of the equation. ( a&b 2/2) A5. Which of the following is the insurance leverage factor defined by Ferrari? A. U/P B. 1 + R/S C. I/A + U/R D. I/A + U/A + R/P E. P/S ( ) A6. Which of the following is the insurance exposure defined by Ferrari? A. R/S B. U/P C. P/S D. U/S E. T/P ( )

22 506 Ferrari A1. Use Ferrari's first formula: T/S = (I/A)(1 + R/S) + (U/P)(P/S) If T/S equals 12% and the return from investments and the ratio of reserves to surplus are assumed to be constant, then the final term must also be constant. If U/P is negative (an underwriting loss) and P/S (the ratio of earned premium to policyholder surplus) increases (decreases), then the loss provision decreases (increases). If U/P is positive (an underwriting profit) and P/S increases (decreases), the profit provision decreases (increases), p A2. T/S = (I/A)(1 + R/S) + (U/P)(P/S) T/S - investor's return on equity I/A - society's return on assets U/P - underwriting profit or return on sales (the regulator's and actuary's viewpoint), p A3. a. 1) S = Statutory Surplus + Equity in Unearned Premium + Voluntary Reserves S = 15, , = 18,030 R = A S = 41,485 18,030 = 23,455 Insurance Leverage = (1 + R/S) = (1 + 23,455/18,030) = ) Insurance Exposure = P/S = 19,320/18,030 = 1.07 b. Assume all taxes apply to investment income. Calculate returns relative to equity: 1) Investment Income = (I/A)(1 + R/S) = [(1, )/41,485][2.30] = ) Underwriting Income = (U/P)(P/S) = (2/19,320)(1.07) =.0001 c. T/S = (I/A)(1 + R/S) + (U/P)(P/S) = =.0762, pp A4. a. T - total after-tax return to the insurer I - investment gain/loss (after appropriate tax charges) U - underwriting profit/loss (after appropriate tax charges) P - premium income A - total assets R - reserves and other liabilities (excluding equity in unearned premium reserves) S - stockholders' equity (capital, surplus, and equity in unearned premium reserves) b. T/S - total return on equity I/A - return on assets (1 + R/S) - insurance leverage factor U/P - underwriting profit or loss as a percentage of premium P/S - insurance exposure term, pp A5. B, p A6. C, p. 297.

23 Ferrari 507 A7. The president of XYZ Insurer has just attended a meeting that discussed Ferrari's paper, "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity." He has asked you as his consultant to explain the formula below. What would you say to the president? T/S = (I/A)(1 + R/S) + (U/P)(P/S) where T is the total after-tax return to the insurer, I is the investment gain/loss (after appropriate tax charges), U is the underwriting profit/loss (after appropriate tax charges), P is premium income, A is total assets, R is reserves and other liabilities (excluding equity in unearned premium reserves), and S is stockholders' equity (capital, surplus, and equity in unearned premium reserves). ( b 2) A8. Given the following information for a company writing only fire and general liability insurance: Fire General Liability Earned (and written) premium $100 million $400 million Loss reserves 35 million 800 million Investment income on assets corresponding to loss reserve 2.8 million (8%) 80 million (10%) Combined ratio 103% 115% a. What is the underwriting gain/loss for each line of business? b. What is the net company gain/loss for each line of business? c. What might explain the difference in interest rates between the two lines (8% vs. 10%)? d. Given an aggressively competitive climate, the company decides to lower its GL rates 10% in order to retain its current book of business. What must the rate of return on loss reserves be in order to maintain the total profit/loss calculated in b.? ( /1/1/2) A9. Define "insurance leverage." ( a 1) A10. Suppose an insurer writes business which, over time, generates a ratio of liabilities (excluding equity in unearned premium reserves) to annual earned premium of 1.3 and produces an underwriting profit margin of 1% of earned premium. If the insurer obtains an 8% after tax investment return on total assets, what is the minimum premium-to-surplus ratio required to provide at least a 20% return on stockholders' equity? Show all work. ( ) A11. You are the actuary for a company that is considering an acquisition. Your company is reviewing two possible candidates. Each is a monoline writer and they have identical level premium volume and surplus amounts. After the acquisition, your company plans to manage the investments of the newly acquired company, but will not change any other aspects of the acquired company. Your company expects to earn an investment rate of return equal to 8% of total assets. Although they are monoline writers, the two candidates write different lines of business. The following data regarding profits and reserves is also known about each company: Company A earns an underwriting profit of 2% and has a ratio of reserves to premium of 2.0. Company B has an underwriting loss of 15% and has a ratio of reserves to premium and has ratio of reserves to premium of 5.0. Using the concepts in Ferrari's paper on total return, calculate the return on equity for each of these two possible acquisitions. Show all of your work. ( ) A12. Using Ferrari's concepts from his paper, "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," and the following data about a particular company, what is the company's return on owner's equity? Invested assets $15 million Reserve liabilities $10 million Investment return 7% Underwriting results $1 million A. < 2.5% B. 2.5% but < 0% C. 0% but < 2.5% D. 2.5% but < 5.0% E. 5.0% ( )

24 508 Ferrari A7. T/S is the total return on equity combining both investment and underwriting income, representing the investor's viewpoint. (I/A)(1 + R/S) represents investment income as the product of the investment return on assets (I/A) and the insurance leverage factor (1 + R/S). (U/P)(P/S) represents underwriting income as the product of the underwriting profit/loss percentage and the insurance exposure term, p A8. a. Fire experienced an underwriting loss of 3%, GL one of 15%. b. T Fire = I + U = (.03)(100M _ ) + 2.8M _ = 2M _ T GL = (.15)(400M _ ) + 80M _ = 20M _ c. As fire losses are paid more quickly, greater liquidity is required in that line restricting the scope of investments. d. Investment income must increase to the extent of the drop in premium income: U = (400M _ )(.10) = 40M _ I = (40M _ + 80M _ )/800M _ = 15%, p A9. Insurance leverage is the factor (1 + R/S) where R equals reserves and other liabilities excluding equity in unearned premium reserves and S equals stockholders' equity (capital, surplus and equity in unearned premium reserves). It is applied to I/A (the investment return on assets) to produce the investment return relative to surplus, pp A10. P/S = T/S I/A U/P + (I/A)(R/P) = (.08)(1.3) = 1.28, p A11. T/S = [I/A][(1 + (R/P)(P/S)] + (U/P)(P/S) T/S A = [.08][1 + (2)(P/S)] + (.02)(P/S) =.08 + (.18)(P/S) T/S B = [.08][1 + (5)(P/S)] + (.15)(P/S) =.08 + (.25)(P/S), p A12. T = I + U = (.07)(15M _ ) 1M _ = 50,000 S = A R = (15 10)M _ = 5M _ T/S = 50,000/5M _ = 1%, p Answer: C

25 Ferrari 509 A13. The shareholders of XYZ Insurance Company desire a 20% pretax total return on surplus. XYZ earns 8% on invested assets, writes at a 2:1 premium-to-surplus ratio and has a book of business that generates loss reserves equal to the written premium. Based on the concepts developed by Ferrari in his paper, "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," what underwriting profit margin is needed to meet the target total return? (Assume no investment income is earned on the unearned premium reserve.) A. < 1.0% B. 1.0% but < 1.0% C. 1.0% but < 3.0% D. 3.0% but < 5.0% E. 5.0% ( ) A14. You have been asked by your management to determine the underwriting profit margin necessary to produce a total return of 15%. Using the total rate of return model described in Ferrari's "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owner's Equity," and given the following data, determine the underwriting profit margin. Show all work. Investable assets $375,000,000 Surplus $250,000,000 Investment rate of return 8% Risk-free rate of return 7% Premium $625,000,000 ( ) A15. You are given the following information: Investment rate of return 10% Investable funds (investable assets) $96,000,000 Risk-free rate of return 6% Surplus $40,000,000 Written premium $120,000,000 Expenses $48,000,000 Return on stock market 10% Underwriting profit margin.0168 Using the formula discussed by Ferrari in "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owner's Equity," determine the total after-tax return on stockholders' equity. ( b 1) A16. Based on Ferrari's "The Relationship of Underwriting, Investments, Leverage, and Exposure to Total Return on Owners' Equity," use the following information to compute the total after-tax return on stockholders' equity. Loss reserves $1,500,000 Unearned premium reserves $300,000 Equity in unearned premium reserves 60,000 Other liabilities 0 Statutory surplus 500,000 Other capital 0 Premium income 1,000,000 After-tax investment income 150,000 After-tax underwriting profit/(loss) (50,000) ( ) A17. An insurer holds a completely diversified portfolio of stocks and bonds. You expect the future investment returns to be less than historical returns on this portfolio. The risk-free rate, however, is expected to remain unchanged. For an underwriting profit that is based on the total-rate-of-return model, explain the effect of a lower expected investment return for the portfolio. ( a 2) A18. Given the following in millions and referring to Ferrari's "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owner's Equity," calculate the items below: Total assets $80 Statutory capital and surplus 25 Equity in unearned premium reserve 5 After-tax underwriting profit/(loss) (2) After-tax investment income 4 Earned premium 50 a. Insurance leverage factor b. Insurance exposure c. Total return on equity ( )

26 510 Ferrari A13. U/P = T/S (I/A)(1 + R/S) P/S =.20 (.08)(1 + 2) 2 =.02, p Answer: A A14. U/P = T/S (I/A)(1 + R/S) P/S =.15 (.08)(375M_ /250M _ ) 625M _ /250M _ =.012, p A15. T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (.10)(96M _ /40M _ ) + (.0168)(120M _ /40M _ ) T/S =.1896, p A16. S = Statutory Surplus + Equity in Unearned Premium =.5M _ +.06M _ =.56M _ T/S = I + U S =.15M_.05M _.56M _ = 17.9%, p A17. U/P = T/S (I/A)(1 + R/S) P/S All other things being equal, because investment return (I/A) is a subtracted term, a lower return will produce a higher underwriting profit. If, however, the total rate of return (T/S) also decreases, there will be an offsetting effect on the underwriting profit margin, p A18. a. R = Total Assets Statutory Capital and Surplus Equity in Unearned Premium Reserve R = = 50 S = Statutory Capital and Surplus + Equity in Unearned Premium Reserve S = = R/S = /30 = 8/3 b. P/S = 50/30 = 5/3 c. T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (4/80)(8/3) + ( 2/50)(5/3) = 6.7%, pp

27 Ferrari 511 A19. You have calculated medical malpractice rates, including an underwriting profit provision based on the total-rate-of-return model discussed by Ferrari in "The Relationship of Underwriting, Investments, Leverage, and Exposure to Total Return on Owners' Equity." Due to recent financial developments, your investment department now projects that the company's investment yield on assets will be 1.5 percentage points higher than you had assumed in your calculations. How much could you decrease your underwriting profit provision and still maintain the same target total rate of return? Premium equals 2/3 of investable assets. Show your work. ( ) A20. You are given: Investable assets $100,000 Investment rate of return 7.0% Premium $200,000 Underwriting profit margin 5.0% Based on the total-rate-of-return model described by Ferrari in "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," in what range does surplus need to fall to generate a 10% total rate of return? A. < $175,000 B. $175,000 but < $185,000 C. $185,000 but < $195,000 D. $195,000 but < $205,000 E. $205,000 ( ) A21. You are given the following information about an insurance company: i) Written premium levels are constant over time. ii) All policies are effective at the beginning of the year. iii) Losses are paid at the end of each year for four years (25% each year). iv) The loss ratio is 80.0%. v) Investment return on assets is 8.0%. vi) The premium-to-surplus ratio is Using Ferrari's "The Relationship of Underwriting, Investments, Leverage and Exposure to Total Return on Owners' Equity," calculate the expense ratio required to achieve a 20.0% rate of return, ignoring the effect of taxes. Show all work. ( ) A22. Based on the total-rate-of-return model described by Ferrari in "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," use the data below to determine the underwriting profit margin. Total assets $10,000 Investable assets $8,100 Surplus $3,000 Reserves $7,000 Premium $9,000 Investment rate of return 10% Risk free rate of return 6% Market portfolio rate of return 12% Target total rate of return 15% A. < 2% B. 2% but < 0% C. 0% but < 2% D. 2% but < 4% E. 4% ( ) A23. You are given the following information ($000,000): Underwriting profit (after taxes) 5 Earned premium 120 Capital and surplus 100 Equity in unearned premium reserve 2 Investment gain (after taxes) 10 Unearned premium reserve 50 Paid losses 30 Based on Ferrari's "The Relationship of Underwriting, Investment, Leverage and Exposure to Total Return on Owners' Equity," calculate the total return on equity. ( a 1)

28 512 Ferrari A19. (I/A)(1 + R/S) + (U/P)(P/S) = (I/A +.015)(1 + R/S) + (U/P)*(P/S) UP* UP = (1 + R/S)(.015) P/S = (S + R)(.015) P = (1.5)(.015) =.0225, p A20. S = (A)(I/A) + (P)(U/P) T/S = (100,000)(.07) + (200,000)(.05).10 = 170,000, p Answer: A A21. 1) Calculate the reserves-to-premium ratio. R/P = (ELR)(% Reserves for Latest Year + % Reserves for the Prior Year + % Reserves for the Second Prior Year) R/P = (.80)( ) = ) Calculate the reserves-to-surplus ratio: R/S = (R/P)(P/S) = (1.20)(2.0) = 2.4 3) Calculate the target underwriting profit: U/P = T/S (I/A)(1 + R/S) P/S =.20 (.08)( ) 2.0 =.036 4) Calculate the expense ratio: Expense Ratio = 1 U/P ELR = 1 (.036).80 =.236, p A22. U/P = T/S (A/S)(I/A) P/S =.15 (8,100/3,000)(.10) 9,000/3,000 =.04, p Answer: A A23. S = Statutory Surplus + Equity in Unearned Premium = 100M _ + 2M _ = 102M _ T/S = I + U S = 10M_ + 5M _ 102M _ = 14.7%, p. 296.

29 Ferrari 513 A24. Ferrari, in "The Relationship of Underwriting, Investment, Leverage and Exposure to Total Return on Owners' Equity," provides an equation showing the relationship between total return on equity, investment return on assets, and return on sales for an insurance operation. a. Using Ferrari's notation, show this equation and identify each of the three measures of return listed above. b. Using Ferrari's notation, show expressions for each of the following items: i) Insurance leverage factor ii) Insurance exposure term. ( a&b 1/.5) A25. Based on Ferrari's "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity" and the following data, calculate the total dollar return for the year for the insurance company, assuming no taxes. Reserves-to-surplus ratio 1.5 Premium-to-surplus ratio 2.0 Average annual portfolio yield 6% Average combined ratio 95% Beginning surplus $200 million A. < $20 million B. $20 million but < $40 million C. $40 million but < $60 million D. $60 million but < $80 million E. $80 million ( ) A26. A CFO of an insurance company is under pressure to raise the insurer's total return on equity. The CFO came up with the following two options: Option A involves reducing the insurer's equity by 50% without changing the investment return on assets. Option B involves investing in junk bonds and thus doubling the insurer's investment return on assets. The CFO was shocked to find that option A would have no effect on the insurer's total return on equity. Option B, however, would result in the total return on equity increasing by 10 percentage points. Assume that the CFO decides to implement both option A and option B simultaneously. Also assume that implementation of the options will affect neither the underwriting results nor the reserves. Based on Ferrari's "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," what would the insurer's total return on equity be? Show all work. ( ) A27. Ferrari, in "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owner's Equity," discusses an insurance company's total return on equity. You are given the following information for XYZ Insurance Company: Underwriting gain/loss as a % of premium 10.0% Premium-to-surplus ratio 1.0 Reserves-to-surplus ratio 2.0 Investment yield 6.7% Calculate XYZ's total return on equity. ( a 1) A28. Given the following information: Premium $4,000,000 Reserves $3,000,000 Surplus $2,000,000 Target total return on surplus 15% Return on total assets 7% What is the underwriting profit provision as a percentage of premium? A. < 2% B. 2% but < 0% C. 0% but < 2% D. 2% but < 4% E. 4% ( )

30 514 Ferrari A24. a. T/S = (I/A)(1 + R/S) + (U/P)(P/S) T/S - investor's return on equity I/A - society's return on assets U/P - underwriting profit or (the regulator's and actuary's viewpoint), p b. (1 + R/S) - insurance leverage factor P/S - insurance exposure term, pp A25. T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (.06)( ) + (.05)(2.0) =.25 T = (T/S)(S) = (.25)(200M _ ) = 50M _, p Answer: C A26. Option A produces the following: T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (I/A)(1 + 2R/S) + (U/P)(2P/S) (I/A)(R/S) = U/S Option B produces the following: T/S = (I/A)(1 + R/S) + (U/P)(P/S) +.10 = (2I/A)(1 + R/S) + (U/P)(P/S).10 = (I/A)(1 + R/S) = I/A U/S Combining options A and B produces the following: T/S = (2I/A)(1 + 2R/S) + (U/P)(2P/S) = (2)(I/A) + (4)(I/A)(R/S) + (2)(U/S) T/S = (2)(I/A U/S) = (2)(.10) =.20, p A27. T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (.067)(1 + 2) + (.10)(1.0) =.101, p A28. U/P = T/S (I/A)(1 + R/S) P/S =.15 (.07)(1 + 3M_ /2M _ ) 4M _ /2M _ =.0125, p Answer: B

31 Ferrari 515 A29. Given the following information from an insurance company's most recent Annual Statement: Premium $6,000,000 Reserves and other liabilities $3,000,000 Surplus $2,000,000 Underwriting income $180,000 Investment gain $500,000 Calculate the return on equity. ( a 1) A30. Given the following information about an insurance company: Premium $3,000 Loss reserves $2,500 Surplus $2,000 After-tax return on invested assets 6% Additionally, prior experience has shown the following underwriting results: Premium-to-Surplus Ratio Underwriting Gain as a Percent of Premium to 1 % to 1 3% to 1 5% to 1 7% The insurer reviews its reserves and determines that it needs to add $500 to reserves. Because of the charge, it is considering making one of three operational changes: 1) Shifting investment strategy toward equities. This will increase the after-tax return by 2%. 2) Raising $300 in capital. 3) Reducing premium volume by 25%. Determine which operational change maximizes the total return of the insurer. ( ) A31. An insurance company has reported the following financial data for calendar year 2006: Loss ratio 80% Expense ratio 25% Investment return on assets 7% Premium-to-surplus ratio 2 to 1 Reserves as a % of assets 75% Calculate the total return on equity for calendar year ( ) A32. An actuary has determined that the estimates below are applicable to a particular line of insurance. Adjusted underwriting income $150,000 Net investment income, before tax 1,750,000 Surplus 10,000,000 Earned premium 15,000,000 Equity in the unearned premium reserve 2,000,000 Federal income tax 850,000 Admitted assets 32,000,000 As discussed by Ferrari in "The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners' Equity," calculate each of the following: a. Insurance leverage b. Insurance exposure c. Total after-tax rate of return. ( /.5/1.5)

32 516 Ferrari A29. A = R + S = 3M _ + 2M _ = 5M _ T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (.5M _ /5M _ )(1 + 3M _ /2M _ ) + (.18M _ /6M _ )(6M _ /2M _ ) =.16, pp A30. T/S = (I/A)(1 + R/S) + (U/P)(P/S) Change 1: (.08)(1 + 3,000/1,500) + (.07)(3,000/1,500) =.100 Change 2: (.06)(1 + 3,000/1,800) + (.05)(3,000/1,800) =.077 Change 3: (.06)(1 + 3,000/1,500) + (.03)(2,250/1,500) =.135, p Change 3 optimizes the total return. A31. UP = 1 LR ER = =.05 1 R/S = 1/(R/A) 1 = 1 1/.75 1 = 3 T/S = (I/A)(1 + R/S) + (U/P)(P/S) = (.07)(1 + 3) + (.05)(2) =.18, pp A32. a. S = Unadjusted Surplus + Equity in the UPR = 10M _ + 2M _ = 12M _ R = A S = 32M _ 12M _ = 20M _ 1 + R/S = M _ /12M _ = b. P/S = 15M _ /12M _ = 1.25 c. I = Pretax Investment Income Federal Income Tax = 1.75M _.85M _ =.9M _ U/P =.15M _ /15M _ =.01 I/A =.9M _ 32M _ = T/S = (I/A)(1 + R/S) + (U/P)(P/S) = ( )(2.667) + (.01)(1.25) =.0875, pp