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Math 479 / 568 Casualty Actuarial Mathematics

Math 479 / 568 Casualty Actuarial Mathematics. Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 15: Finance I November 4, 2014. Agenda. Basics of finance Applications of financial models to insurance. The Most Important Concepts in Finance.

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Math 479 / 568 Casualty Actuarial Mathematics

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  1. Math 479 / 568 Casualty Actuarial Mathematics Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 15: Finance I November 4, 2014

  2. Agenda • Basics of finance • Applications of financial models to insurance

  3. The Most Important Concepts in Finance • Net present value • Capital asset pricing model • Efficient capital markets • Value additivity / conservation of value • Capital structure theory • Option theory • Agency theory Source: Brealey and Myers, 1996, Principles of Corporate Finance, 5th Edition, Chapter 35

  4. Capital Asset Pricing Model E(Ri) = Rf + i [E(Rm) - Rf] where: E = expected value operator Ri = return on an asset Rf = risk free rate Rm = return on market portfolio i = Cov(Ri,Rm) / 2(Rm) = systematic risk

  5. Example • A stock has a beta of 1.5 • The risk free rate is 5 percent • The expected market risk premium is 9 percent • What is the expected return on this stock? E(Ri) = 5% + 1.50[9%] = 18.5%

  6. CAPM Assumptions 1) Investors maximize expected return and minimize risk 2) Trades do not affect prices 3) Expectations are homogeneous 4) No taxes or transaction costs 5) All borrowing and lending is at the risk free rate 6) Assets are infinitely divisible

  7. Problems with the CAPM • How can beta be measured? • What is the market portfolio? • Does the CAPM explain stock returns? • Fama and French, 1992, “The Cross-Section of Expected Stock Returns,”Journal of Finance: Size and book-to-market ratios explain stock returns better than beta over the period 1941-1990

  8. What We Do Not Know About Finance...10 Unsolved Problems 1) How are major financial decisions made? 2) What determines project risk and present value? 3) Risk and return - have we missed something? 4) Are there important exceptions to the efficient market theory? 5) Is management an off-balance sheet liability?

  9. What We Do Not Know About Finance...10 Unsolved Problems (cont.) 6) How can we explain the success of new securities and new markets? 7) How can we resolve the dividend controversy? 8) What risks should a firm take? 9) What is the value of liquidity? 10) How can we explain merger waves? Source: Brealey and Myers, 1996, Principles of Corporate Finance, 5th Edition, Ch. 35

  10. “New Facts in Finance” “New Facts in Finance,” by John H. Cochrane, Economic Perspectives, Federal Reserve Bank of Chicago, 1999 (third quarter), pp. 36-58 • Discusses the “revolution” that occurred in what we know (or think we know) about finance over the last approximately 15 years of the 20th century.

  11. “New Facts in Finance” (cont.) • Mid-1980s understanding of financial world • “The CAPM is a good measure of risk and thus a good explanation of the fact that some assets… earn higher average returns than others.” • “Returns are unpredictable, like a coin flip.” • Stock prices: random walk • Bond returns: expectations model of term structure • FX: investment rate – exchange rate “balance” • Stock volatility: relatively constant over time • “Professional managers do not reliably outperform simple indexes and passive portfolios once one corrects for risk (beta).” • Overall  asset markets are “informationally efficient”

  12. “New Facts in Finance” (cont.) • “Now (1999), we know that:” • “There are assets whose average returns can not be explained by their beta. Multifactor extensions of the CAPM dominate…” • “Returns are predictable. In particular: Variables including the dividend/price (d/p) ratio and term premium can predict substantial amounts of stock return variation. This phenomenon occurs over business cycle and longer horizons.” • Bond returns, FX also somewhat predictable. • Stock volatility: changes over time • “Some mutual funds seem to outperform simple indexes, even after controlling for risk through market betas.”

  13. Applications of FinancetoInsurance Pricing

  14. Financial Economics • Has developed tools that could be useful in insurance • Initial application in Massachusetts • confrontation-style hearings • used to depress rate levels • tended to tarnish insurers’ perception of this field • The models need to be adapted for insurance applications

  15. Caveat • Competitive markets establish appropriate profit margins, not models • A useful model can approximate appropriate profit margins and serve as an aid to management decisions • A useful model could reduce cyclicality of insurance pricing by providing advance warning of market reactions to various price levels

  16. Financial Models Applied To Pricing • Total rate of return • Capital asset pricing model • Discounted cash flow models • Myers-Cohn • NCCI • Option pricing model (to be discussed later)

  17. Total Rate of Return TRR = (IA/S)(IR) + (P/S)(UPM) TRR = Target total rate of return IA = Investable assets S = Surplus IR = Investment return P = Premium UPM = Underwriting profit margin

  18. Example An insurer with $120 million in investable assets and $40 million in surplus writes $60 million in premium for the year. The company’s investment return is 8 percent. What is the appropriate underwriting profit margin if the company has a target rate of return of 21 percent?

  19. Class Problem A small, stockholder-owned insurer that specializes in non-standard auto is using the Total Rate of Return method to establish a benchmark underwriting profit margin. The company intends to write $24 million in premium for the year, which would lead to having $48 million in investable assets. Assets are invested three-quarters in mid-maturity bonds, with a current yield of 8 percent, and one-quarter in stocks, which pay a dividend of 4 percent and have an expected capital gain of 10 percent. Last year, the net investment income yield on the company’s portfolio was 7 percent and the capital gains were negative 4 percent. The company has $8 million in surplus. The market value of the company based on its current stock price is $12 million. The company’s own stock has historically had a beta of 1.5. 1) Select and justify a reasonable Target Total Rate of Return for this company 2) Calculate the required Underwriting Profit Margin to achieve this target

  20. Capital Asset Pricing ModelApplication to Insurance Basic model without taxes: k = funds generating coefficient u = underwriting beta UPM = -kRf + u[E(Rm) - Rf]

  21. Example UPM = -0.5(8%) + 0.2[16% - 8%] = -2.4% k = .5 u= .2 Rf = 8% E(Rm) = 16%

  22. Tax Version of Insurance CAPM TA = Tax rate on investment income T = Tax rate on underwriting income S/P = Surplus to premium ratio UPM = - kRf(1 - TA)/(1 - T) + u[E(Rm) - Rf] + (S/P)Rf[TA/(1 - T)]

  23. Example Same values as before TA = 16.8% T = 30% S/P = 1.0 UPM = -1.23%

  24. Class Problem Our stockholder-owned non-standard auto insurer decides to use the tax version of the Capital Asset Pricing Model to establish a benchmark underwriting profit margin. The funds generating coefficient is 2.0 ($48 million in investable assets on $24 million annual premiums). The underwriting beta is 0.25. The company’s tax rate is 34 percent on underwriting, but only half of its investment income is taxable. The risk-free rate is 6.5 percent and the market risk premium is 9 percent. Calculate the appropriate underwriting profit margin based on a surplus value of $8 million and of $12 million.

  25. Problems Applying the CAPM To Insurance • Market risk cannot be measured for a line of business • Bankruptcy risk is ignored • Insurance risk that is not systematic with investment risk is ignored • Writing coverage at a CAPM-determined price does not add value to an insurer • CAPM may not even value stock returns appropriately

  26. Discounted Cash Flow Models • Myers-Cohn uses policyholder perspective • NCCI uses company perspective • Myers-Cohn uses NPV • NCCI uses IRR

  27. General Risk-Adjusted Discount Technique Formula PV (P) = PV (L) + PV (E) + PV (TUW) + PV (TII) where: PV = present value operator P = premiums L = losses and loss adjustment expense E = underwriting expenses TUW = taxes on underwriting profit or loss TII = taxes on investment income

  28. Example 1 • One-year policy • Premiums paid on the effective date of policy • Expenses = $20 paid on effective date of policy • Expected losses = $80 paid at the end of the year • Surplus allocated to policy = $50 • Taxes on underwriting gain or loss paid at the end of the year • Taxes on investment income paid at the end of the year • Tax rate of 34 percent applies to underwriting and investments • All cash flows discounted at 7 percent

  29. Example 1 P = + 20 + + P = 74.766 + 20 + .318 P - 6.355 - 25.421 + .667 + .022 P .66 P = 63.657 P = 96.45 80 1.07 (P - 20 - 80) (.34) 1.07 (50 + P - 20) (.07) (.34) 1.07

  30. Summary of Nominal and Discounted Values -- Example 1 Nominal ValuesDiscounted Values Losses 80 74.77 Expenses 20 20 Taxes on Underwriting -1.21 -1.13 Taxes on Investments 3.012.81 Total 101.80 96.45* *Premium = Sum of the Discounted Values of Losses, Expenses and Taxes

  31. Example 2 Risky cash flows discounted at 4 percent (P-20)(.34) 1.07 80 (.34) 1.04 (50+P-20)(.07)(.34) 1.07 80 1.04 P = + 20 + - + P = 76.923 + 20 + .318 P - 6.355 - 26.154 + .667 + .022 P P = 65.081 / 0.66 = $98.61

  32. 40 1.08 (1.04)2 40 1.08 1.04 (40 +) (.34) (40 -) (.34) (50 + P - 20) (.07) (.34) 1.07 (50(.5) + P - 20 - 40) (.07) (.34) (1.07)2 P = $98.45 Example 3 Losses paid expected to be $40 at the end of the first year and $40 at the end of the second year and the IRS discount rate is 8% 40 1.04 40 (1.04)2 (P - 20) (.34) (1.07) P = + + 20 + - - + +

  33. 40 1.08 (1.04)2 40 1.08 1.04 (40 +) (.34) (40 -) (.34) (50 + P- 20) (.07) (.34) 1.07 (50(.5) + P - 20 - 40)(.07)(.34) (1.07)2 Example 4 Expenses of $10 paid two years before the policy is effective and $10 when the policy is effective (P-(10(1.07)2)-10)(.34) (1.07) 40 1.04 40 (1.04)2 P = + + 10 (1.07)2 +10 + - - + + P = $100.00

  34. 40 1.08 (1.04)2 40 1.08 1.04 (40 +) (.34) (40 -) (.34) P (1.07)1/12 40 1.04 40 (1.04)2 (50 + P - 20) (.07) (.34) 1.07 (50(.5) + P - 20 - 40)(.07)(.34) (1.07)2 Example 5 Premiums paid (not just received by the insurer) one month after policy is effective (P-10 (1.07)2-10)(.34) (1.07) = + + 10 (1.07)2 +10 + - - + + P = $100.95

  35. Additional Adjustments Possible • Some expenses proportional to premiums • Other risk-adjusted discount rates for prepaid expenses and premium lags • Different tax rates on underwriting and investment income • Mid-period cash flows • Future renewals on book of business

  36. Problems With Risk-Adjusted Discount Technique • Determining risk-adjusted discount rate • Surplus allocation • How much? • How long?

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