A. - value of a share of stock B. k – discount rate, required rate of return 푉0 C. g – constant growth rate D. plowback ratio – portion of earnings retained and reinvested in the firm E. ROE – return on equity F. - risk-free rate of return G. [ ] - expected market rate of return 푟푓 H. [ ] - expected equity market risk premium 퐸 푟푚 I. - Beta, a measure of systematic market risk 퐸 푟푚 − 푟푓 J. WACC – Weighted average cost of capital 훽 K. APV – Adjusted Present Value

L. BVt – Book Value at time t

M. P0 – Price per share of equity N. EPS – Earnings per share

III. The Dividend Discount Model (DDM)

A. The DDM is a very basic model, which values a company by discounting future dividend payments based on a series of assumptions 1. Overview ( ) ( ) ( ) = + + + a. ( ) ( ) ( ) (general case) 퐸 퐷푖푣1 퐸 퐷푖푣2 퐸 퐷푖푣3 2 3 푉0 1+ 푘 1+푘 1+푘 ⋯ ( ) b. = (constant growth case, also the Terminal Value) 퐸 퐷푖푣1 푉0 푘− 푔

©Actex 2013 CAS Exam 7 ICV-2 “P&C Insurance Company Valuation”

B. Terminal Value 1. The payments beyond a certain horizon can generally not be reliably forecast so it is necessary to rely on growth assumptions

( ) ( ) = + + + 2. ( ) ( ) ( ) (for forecast horizon n) 퐸 퐷푖푣1 퐸 퐷푖푣푛 푇푒푟푚푖푛푎푙 푉푎푙푢푒 푛 푛 푉0 1+푘 ⋯ 1+푘 1+푘 ( ) 3. = 퐸 퐷푖푣푛 C. Application푇푒푟푚푖푛푎푙 푉푎푙푢푒 푘−푔 1. Expected dividends during the forecast horizon a. Beyond the scope of the paper

2. Dividend growth rates beyond forecast horizon a. Can use same growth rates as forecast horizon b. Growth rate based on dividend payout ratio:

= ×

c. 푔Growth푝푙표푤푏푎푐푘 rate has 푟푎푡푖표major impact푅푂퐸 on assessed value of firm d. Considerations i. Growth rate relative to rate for entire economy ii. Growth rate relative to dividend payout rates – i.e. high growth rate and high payout unsustainable iii. High growth rate indicates more risk

3. Risk-adjusted discount rate a. Typical to use a risk-adjusted discount rate rather than a risk-adjusted cash flow b. Private versus Equilibrium Market Valuation i. Private valuation – each investor assesses risk based on their own investment portfolio ii. Equilibrium market valuation – all investors assess the risk in the same way c. Discount rate determination i. Most popular model is the Capital Asset Pricing Model (CAPM)

= + ( [ ] )

푓 푚 푓 d. Estimating푘 푟 Beta훽 퐸 푟 − 푟 i. Firm Beta – determined through linear regression of company’s returns versus industry returns ii. Industry Beta – need to be careful with industry measures due to statistical inconsistencies a) Industry means are often better b) Use firms with similar mix of business c) Effect of debt leverage is removed with all-equity beta

©Actex 2013 CAS Exam 7 “P&C Insurance Company Valuation” ICV-3

e. Estimating the risk free rate i. Risk-free rate is based on risk-free securities (government) with a time horizon similar to the average time to payment ii. Long-term yields include liquidity premium which should be removed

f. Estimating equity market risk premium i. The spread between the market return and the risk-free rate should be consistent with the duration of the risk-free rate selected ii. Arithmetic (single period preference) vs. geometric averages (multiple period preference) iii. Historical vs. implied risk premiums

IV. Discounted Cash Flow Model (DCF)

A. The DCF method is similar to the DDM in the way that it discounts future amounts to determine value B. The DCF simplifies the cash flows by assuming that all free cash flows will (eventually) be paid out 1. i.e cash that is not paid out immediately as a dividend is reinvested to earn a fair risk adjusted return. C. Free Cash Flow to the Firm (FCFF) Approach 1. Values the entire firm then subtracts market value of debt to value equity 2. Problematic when applied to insurance companies a. Policyholder liabilities vs. debt b. WACC vs. APV – policyholder liabilities make both difficult to assess. D. Free Cash Flow to Equity (FCFE) Approach 1. Similar to FCFF but reflects free cash flows after deductions for interest payments, net of tax consequences, net changes in borrowings 2. Cash flow to equity allows use of levered equity return as the discount rate, which alleviates the difficulty of determining the WACC 3. Definition of FCFE for a property and casualty insurer:

Net Income plus Non-Cash Charges - Excluding reserve changes Less Net Working Capital Investment Less Increase in Capital Required plus Net Borrowing FCFE

4. The increase in capital required results from regulatory and rating agency considerations 5. Reserve changes can be excluded since they impact both non-cash charges and capital expenditures and thus cancel out in the above formula

©Actex 2013 CAS Exam 7 ICV-4 “P&C Insurance Company Valuation”

E. Application 1. Application is very similar to DDM 2. Growth rates a. Reinvested Capital = Required Ending Capital – Beginning Capital b. FCFE = Net Income – Reinvested Capital c. Reinvestment Rate = Reinvested Capital/Net Income d. Growth Rate = Reinvestment Rate * ROE e. Similar to DDM use growth rate from last year of forecast horizon as the horizon growth rate 3. Discount rate a. Basically the same as in the DDM approach b. DDM has a larger proportion of risk coming from marketable securities than from underwriting – likely should have different discount rates, but the paper ignores this 4. Considerations a. Terminal Value – similar to DDM – can be computed as a multiple of the final forecast horizon FCFE b. Average Discount Rates – be careful if mix of business is changing c. FCFE is a net cash flow incorporating various revenues and expenses which have different risk profiles – the use of a common average discount rate means that any piece discounted in isolation is not technically correct

V. Abnormal Earnings Valuation Method (AE) A. The DCF approach cash flows often don’t resemble the forecasted values within a firm’s planning process B. The Abnormal Earnings approach relies more heavily on the accounting measure of net income. 1. Background a. Valuing equity under the AE approach is analogous to valuing a default free bond b. If the firm can earn more than the required rate of return it should be worth more than book value and vice versa. c. = [ × ] = ( ) [ ] d. = + 퐴퐸푡 푁푒푡 퐼푛푐표푚푒 − 푅푒푞푢푖푟푒푑( 퐸푞푢푖푡푦) 푅푒푡푢푟푛 퐵푉푡−1 푅푂퐸푡 − 푘 퐵푉푡−1 ∞ 퐸 퐴퐸푡 e. Similar to DDM and DCF,0 there푡=1 will be푡 a forecast horizon and a terminal value 푉푎푙푢푒 표푓 퐸푞푢푖푡푦 퐵푉 ∑ 1+푘 f. Abnormal earnings are unlikely to continue in perpetuity so terminal value is generally much smaller than in DDM or DCF

2. Accounting Distortions a. Need to be careful with the impact of accounting rules on valuation results (e.g. discounting of loss reserves) b. Distorted earnings will sometimes be offset by distorted equity values c. A more accurate accounting methodology results in a more accurate representation of book value and thus a smaller terminal value

©Actex 2013 CAS Exam 7 “P&C Insurance Company Valuation” ICV-5

3. Application a. The book value is often adjusted to the tangible book value by removing the impact of intangible assets b. Net income forecasts can be adjusted for bias – eg. Discounting of loss reserves c. CAPM can be used for the required rate of return d. Future growth should generally decline to zero after a determined period of time e. Ensure growth does not require additional capital contributions or adjustment is necessary f. Terminal values can be negative if negative growth rate determined

VI. Relative Valuation Using Multiples

A. All of the methods presented so far have the disadvantage that detailed financial analyses are necessary 1. Can be next to impossible for outside investors 2. Ratio methods appear easier to implement

B. Price-Earnings Ratio 1. Based on fundamentals

a. = , price-earnings ratio based on the DDM with constant 푃0 퐷푖푣푖푑푒푛푑 푃푎푦표푢푡 푅푎푡푖표 dividend퐸푃푆1 and growth푘−푔 assumption 2. Industry P-E ratios a. Forward leading P-E ratio is based on expected future earnings b. Trailing P-E ratio is based on prior period earnings c. When using trailing P-E ratio may need to smooth unusual events 3. Alternative uses for P-E ratios a. Validate assumptions of other models b. Valuation short-cut if industry average performance expected c. Guidance in determining terminal value

C. Price-Book Value Ratio 1. Similar to P-E ratio, can derive using the abnormal earnings approach 2. = 1 + , assuming excess returns in perpetuity 푃푟푖푐푒 푅푂퐸−푘 3. 퐵푉 = 1 + 푘−푔 1 , assuming ROE declines to cost of capital after n years 푃푟푖푐푒 푅푂퐸−푘 1+푔 푛 퐵푉 푘−푔 � − �1+푘� � D. Firm vs. Equity Multiples 1. Equity multiples are preferred for the same reason that FCFE is preferred to FCFF

©Actex 2013 CAS Exam 7 ICV-6 “P&C Insurance Company Valuation”

E. Market vs. Transaction Multiples 1. Fluctuations in ‘multiples’ can be extreme even on a daily basis 2. Transaction (IPOs, mergers and acquisitions) multiples can be used instead a. Typically determined by sophisticated parties b. Need to be cautious: i. Control premiums – M&A transaction prices inflated ii. Underpricing in IPO transactions iii. Reported financial variables may not reflect true expectations iv. Economic environment may have changed

F. Application of Relative Valuation for Multi-Line Firms 1. Comparable ‘peer’ companies can be difficult to find 2. A multi-line firm can be segmented into the various segments to make valuation easier 3. Use of pure play peers a. Collect data by segment – treat the segments of the company as separate entities b. Select comparable pure play ‘peer’ companies for each segment – they should have similar characteristics to the particular segment c. Choose multiples for the peer companies in each segment d. Apply average peer multiples to the firms financials e. Consolidate the segments to get the total firm value f. Validate the results using the multiples of other diversified companies

VII. Option Pricing Methods

A. Valuing Equity as a Call Option 1. Background a. When equity holders issue debt (D) they no longer hold all the value of the firm (V) b. Equity at time T, after issuing debt is given by:

= max ( , 0), which is similar to a call option with strike price D

c. 퐸푇The Black푉-푇Scholes− 퐷 option pricing formula can be used , with modification to value the call option

2. Application a. Difficulty in assessing a single expiration date for policyholder liabilities makes this method impractical for insurance companies

©Actex 2013 CAS Exam 7 “P&C Insurance Company Valuation” ICV-7

B. Real Value Options 1. Background – measurement of management flexibility a. Abandonment Option – project that can be stopped early and liquidated – American Put option with strike equal to liquidation value b. Expansion Option – American Call option with strike equal to cost of expansion c. Contraction Option – American Put option with strike equal to cost savings d. Option to Defer – American Call option on value of the project e. Option to Extend – European Call option on asset’s future value

2. Valuation Considerations a. Time to option maturity – uncertain in practice b. Exercise type – options can typically be exercised at any time (American)

3. Reasonableness of Real Option Values a. New information increases value of options b. Expansion options only have value if possess the ability to exercise them c. Exercise price must be fixed for option to have value

©Actex 2013 CAS Exam 7 ICV-8 “P&C Insurance Company Valuation”

©Actex 2013 CAS Exam 7 “P&C Insurance Company Valuation” ICV-9

Past CAS Examination Questions

1. Consider the following information and table of data for company Z:

i) Capital asset pricing model equity beta is .84. ii) Risk-free rate is 4.38%. iii) Equity risk premium is 5.50%. iv) Assume growth rate year 3 is 6.00%. v) GAAP equity values at the beginning of years 1–3 equal $100,000, $105,000, and $109,700 respectively. vi) Net income for years 1–3 equal $10,100, $10,600, and $11,073 respectively.

a. Calculate the equity value of company Z based on the abnormal earnings valuation method, using the constant abnormal earnings in perpetuity assumption. b. Calculate the equity value of company Z based on the abnormal earnings valuation method, using a three-year time horizon for abnormal earnings to fall to zero after the forecast horizon. c. Based on economic theory, explain why assuming that the abnormal earnings fall to zero is a more reasonable assumption than assuming constant abnormal earnings in perpetuity. Show all work. (06–8–41–2/2/.5)

2. You are given the following information about XYZ Insurance Company:

Year GAAP Equity (Beginning of Year) Net Income 2007 $100,000 $10,000 2008 105,000 11,000 2009 110,000 12,000 2010 115,000 13,000 2011 120,000 14,000

i) The risk-free rate is 3%. ii) The expected market return is 5%. iii) Beta equals 2.0. iv) Abnormal earnings will decline linearly over the six-year period after 2011.

Assume that the growth in book value does not require additional capital contributions. Calculate the total equity value of the insurance company using the abnormal earnings method. Show all work. (07–8–39–2.5)

©Actex 2013 CAS Exam 7 ICV-10 “P&C Insurance Company Valuation”

3. Suppose that a firm has one zero-coupon bond outstanding and that the bond matures at time T. You are given the following:

i) V0 is the value of the company’s assets today. ii) VT is the value of the company’s assets at time T. iii) E0 is the value of the company’s equity today. iv) ET is the value of the company’s equity at time T v) D is the debt repayment due at time T

Demonstrate how the value of the company’s equity at time T is equivalent to a call option on the value of the assets with a strike price equal to the required debt repayment at time T. (09–8–15–1)

4. Given the following information for an insurance company:

i) The U.S. GAAP equity as of January 1, 2010 is $100,000. ii) The net income during 2010 is $15,000. iii) The expected annual growth rate in net income during 2011 and 2012 is 4.5%. iv) The expected annual growth in minimum capital requirement during 2011 and 2012 is 3.5%. v) The risk-free rate is 4%. vi) The equity risk premium is 5%. vii) The stock’s beta is .85. viii) There are no noncash charges included in the net income figure other than changes in reserves. ix) There are no net working capital investments. x) There are no increases in borrowings.

a. Calculate the company’s free cash flow to equity for the 2010–12 time period. b. Calculate the company’s free cash flow growth rate beyond the forecast horizon. c. Calculate the company’s value using the free cash-flow-to-equity method. Show all work. (09–8–35–1/1/2)

5. Given the following about a company:

Year GAAP Equity Beginning of the Year Net Income 2009 $500,000 $35,000 2010 510,000 35,400 2011 520,000 35,825

i) The growth in book value does not require additional capital contributions. ii) The beta of the company is 1.5. iii) The expected market return is 5%. iv) The risk-free rate is 2%.

Assume that abnormal earnings will linearly decline to zero over a four-year period. Calculate the total equity value of the insurance company using the abnormal earnings method. (10–8–31–2.5)

©Actex 2013 CAS Exam 7

“P&C Insurance Company Valuation” ICV-11

6. Given the following information to be used in the valuation of a firm:

Firm's ROE: 15% Firm's plowback ratio: 90% Market Return: 9% Yield on long-term T-bonds: 5.2% Liquidity premium for long-term T-bonds: 1.2% Average growth rate for firms in the same industry: 4.2% Average Beta for firms in the same industry: 1.4 Beta for this firm: 2.0

• The above values are assumed to continue indefinitely into the future.

Projections for the firm’s net income after tax:

Calendar Year Net Income after Tax ($000) 2011 10,100 2012 11,464 2013 13,011 a. Demonstrate why it is reasonable for this firm’s Beta to be significantly higher than the average Beta for firms in its industry. b. Use the dividend discount model to estimate the value of this firm as of December 31, 2010. c. Discuss how a company’s growth rate may affect its dividend payout rate and risk-adjusted discount rate. (11-7-12-0.75/2/0.5)

7. Given the following information for a firm:

Return on equity (ROE): 10% Book value growth rate: 5% Discount rate: 8%

• The above values are projected to continue indefinitely. • As of December 31, 2010, the firm’s book value was $1,000,000.

a. Use the price-to-book value ratio to estimate the value of the firm as of December 31, 2010. b. Discuss whether it is reasonable to assume that the relationships among the values for the ROE, the book value growth rate and the discount rate are likely to continue indefinitely. (11-7-13-0.75/0.5)

©Actex 2013 CAS Exam 7 ICV-12 “P&C Insurance Company Valuation”

8. Given the following financial information for an insurance company (in $000,000):

Actual Projections 2011 2012 2013 2014 2015 2016 Net Income 180 190 200 211 222 234 Beginning Equity 2,000 2,108 2,222 2,342 2,468 2,602 Ending Equity 2,108 2,108 2,222 2,342 2,468 2,742

. The expected market return on equity for peer insurance companies is 10%. . The risk-free rate based on US Treasury bill rates is 2%. . The equity beta, based on peer companies, is 0.85. . The company’s business plan contemplates a 60% plowback ratio. . CAPM is an appropriate model for determining the company’s risk-adjusted discount rate.

Determine the value of this company as of December 31, 2011 based on the dividend discount model and the projections for 2012 through 2016. (12-7-12-3.5)

©Actex 2013 CAS Exam 7

“P&C Insurance Company Valuation” ICV-13

Solutions to Past CAS Examination Questions

1. a. ROE = + ( ) = 4.38% + (.84)(5.5%) = 9.0%

(Abnormal푟 푓Earnings)훽 퐸푞푢푖푡푦1 = 10,100푅푖푠푘 푃푟푒푚푖푢푚 - (100,000)(.09) = 1,100 AE2 = 10,600 - (105,000)(.09) = 1,150 AE3 = 11,073 - (109,700)(.09) = 1,200

Abnormal Earnings Terminal Value = 1,200/.09 = 13,333

PV(Abnormal Earnings) = 1,100/1.09 + 1,150/(1.09)2 + (1,200 + 13,333)/(1.09)3 = 13,199

Equity Value = Book Value + PV(AE) = 100,000 + 13,199 = 113,199, pp. 28, 30–31.

b. PV(Abnormal Earnings) = 1,100/1.09 + 1,150/(1.09)2 + 1,200/(1.09)3 + 900/(1.09)4 + 600/(1.09)5 + 300/(1.09)6 = 4,110

Equity Value = Book Value + PV(AE) = 100,000 + 4,110 = 104,110, pp. 28, 32.

c. "Abnormal earnings are less likely to continue in perpetuity and are more likely to decline to zero as new competition is attracted to businesses with positive abnormal earnings," p. 28.

2. ROE = rf + (Equity Risk Premium) = 3.0% + (2)(5.0% - 3.0%) = 7.0%

훽 (Abnormal Earnings)1 = 10,000 - (100,000)(.07) = 3,000 AE2 = 11,000 - (105,000)(.07) = 3,650 AE3 = 12,000 - (110,000)(.07) = 4,300 AE4 = 13,000 - (115,000)(.07) = 4,950 AE5 = 14,000 - (120,000)(.07) = 5,600 AE6 = 4,800 AE7 = 4,000 AE8 = 3,200 AE9 = 2,400 AE10 = 1,600 AE11 = 800

PV(Abnormal Earnings) = 3,000/1.07 + 3,300/(1.07)2 + 4,300/(1.07)3 + 4,950/(1.07)4 + 5,600/(1.07)5 + 4,800/(1.07)6 + 4,000/(1.07)7 + 3,200/(1.07)8 + 2,400/(1.07)9 + 1,600/(1.07)10 + 800/(1.07)11 = 27,322

Equity Value = Book Value + PV(AE) = 100,000 + 27,322 = 127,322, pp. 28, 32.

3. “When a firm is owned entirely by equity holders, they own all of the assets of the firm – the physical assets plus the income that these assets produce over the life of the company. If the equity holders issue debt (i.e., borrow money), then the equity holders no longer own all of the value of the firm, V. Instead, the own the excess of the value of the firm over the debt that they have to repay at time T, denoted D. In other words ET = max(VT – D, 0), which looks like a call on the value of VT with a strike price of D,” p. 45.

4. a. Since there are no noncash charges, net working capital investments, or net borrowing:

FCFE = NI – Capital Expenditures

FCFE10 = 15,000 – (100,000)(.035) = 11,500

FCFE11 = (15,000)(1.045) – [100,000][(1.035)2 – 1.035] = 12,052.50

FCFE12 = (15,000)(1.045)2 – [100,000][(1.035)3 – (1.035)2] = 12,631.09

FCFE10-12 = 11,500 + 12,052.50 + 12,631.09 = 36,183.59, pp. 21–22.

b. Use the values from 2012.

Net Income2012 – FCFE2012 Horizon Growth Rate = = GAAP Equity1/2012 (15,000)(1.045)2 – 12,631.09 = 3.5% (100,000)(1.035)2 pp. 22–23.

c. 1) Use CAPM to calculate the discount rate:

kE = rf + b(E(rp) = .04 + (.85)(.05) = .0825

2) Calculate the terminal value at 1/1/2013 and discount it and the FCFE for 2010-13:

(FCFE12)(1 + HGR) (12,631.09)(1.035) Terminal Value = = = 275,224.80 kE – HGR .0825 – .035

Company Value = 11,500/(1.0825) + 12,052.50/(1.0825)2 + (12,631.09 + 275,224.80)/(1.0825)3

CV = 247,838.70, pp. 10, 23–24.

5. ROE = rf + (Equity Risk Premium) = 2.0% + (1.5)(5.0% - 2.0%) = 6.5%

훽 (Abnormal Earnings)1 = 35,000 - (500,000)(.065) = 2,500 AE2 = 35,400 - (510,000)(.065) = 2,250 AE3 = 35,825 - (520,000)(.065) = 2,025 AE4 = 1,519 AE5 = 1,013 AE6 = 506 AE7 = 0

PV(Abnormal Earnings) = 2,500/1.065 + 2,250/(1.065)2 + 2,025/(1.065)3 + 1,519/(1.065)4 + 1,013/(1.065)5 + 506/(1.065)6 = 8,297

Equity Value = Book Value + PV(AE) = 500,000 + 8,297 = 508,297, pp. 28, 32.

“P&C Insurance Company Valuation” ICV-15

6. a. Calculate the firm’s dividend growth rate as = × = 0.9(0.15) = 13.5% Since the average growth rate for the firms 푔in the푃푙표푤푏푎푐푘 same industry푅푎푡푖표 is 4.2%푅푂퐸 there is likely 푔more risk associated with this firm and thus it should have a higher beta than the industry average. pp. 8,11

b. = 5.2% 1.2% = 4.0% = + = 4.0% + 2(9.0% 4.0%) = 14.0% 푟푓 −

푘 푟푓 훽�푟푚 − 푟푓� − ( ) = 10,100(1 0.9) = 1,010 ( ) = 11,464(1 0.9) = 1,146.40 퐸 퐷푖푣2011 − ( ) = 13,011(1 0.9) = 1,301.10 퐸 퐷푖푣2012 −

2013 퐸 퐷푖푣 ( − )( ) , . ( . ) = = = 295,350 . 퐸 퐷푖푣2013 1+푔 1 301 10 1 135 푇푒푟푚푖푛푎푙 푉푎푙푢푒 푘−푔 0 05% , , . , . , = + + + = 202,017 . . . . , pp.6-7 1 010 1 146 40 1 301 10 295 350 1 2 3 3 푉0 1 14 1 14 1 14 1 14 c. High dividend payout rates are unlikely to be sustainable and so the effects of high growth rates are likely to be offset by lower dividend amounts. The dividend payments for firms with high growth rates are likely to be riskier than those of firms with low growth rates. As a result, the effect of high growth rates are likely to be offset by discounting dividends to present value using higher risk adjusted discount rates.

7. a. Determine the price to book-value ratio of the firm: . . = 1 + = 1 + = 1.667 . . 푃 푅푂퐸−푘 0 1−0 08 퐵푉 = 푘−(푔 ) = 1.6670 08(−10,05000,000) = 1,667,000, pp. 36 푃

푉푎푙푢푒 퐵푉 퐵푉 b. Since the ROE is greater than the firm’s discount rate, the firm is earning abnormal profits. This is unlikely to continue in the long-term as new competition will be attracted to the industry. pp. 28

8. = + × = .02 + 0.85(0.1 0.02) = 0.088 = 1 = 0.4 푘 푟푓 퐵 �푟푚 − 푟푓� − 2012 2013 2014 2015 2016 퐷푖푣푖푑푒푛푑 − 푝 Dividends 76.0 80.0 84.4 88.8 93.6 PV(Dividends) 69.9 67.6 65.5 63.4 61.4 ROE 0.09 0.09 0.09 0.09 0.09

= × = 0.6(. 09) = 0.054

푔 푝 =푅푂퐸( ) + ( ) . × . = 327.72 + = 2230.96 [(. . )( . ) ] 푉푎푙푢푒 푃푉 퐷푖푣푖푑푒푛푑푠93 6 1푃푉054퐹표푟푒푐푎푠푡 ℎ표푟푖푧표푛 5 푉푎푙푢푒 088− 054 1 088

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