1. 1 Math 479 / 568 Casualty Actuarial Mathematics Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 16: Finance II November 6, 2014

2. 2 Agenda Option pricing theory Asset-liability management Modeling financial and economic variables

3. 3 Quick Review of Options C = Max [S - X, 0] C = Call option value at expiration S = Price of underlying asset X = Exercise price P = Max [X - S, 0] P = Put option value at expiration

4. 4 Option Values: Payoff Charts Call -- long position: Call -- short position: Put -- long position: Put -- short position: Payoff X STST X X X STST STST STST

5. 5 Payoff vs. Profit/Loss: Long a Call Option STST Profit/Loss Payoff X Call Premium

6. 6 Diffusion Processes Stochastic process with continuous paths Brown (1827) described and named Brownian method Bachelier (1900) applied to French stock prices Einstein (1905) developed mathematics of Brownian motion Lundberg (1909) applied Brownian motion to collective risk theory in insurance Wiener (1923) refined mathematics of Brownian motion

7. 7 Black-Scholes Option Pricing Model Assumptions: 1.European option 2.No taxes or transaction costs 3.Borrowing rate = Lending rate 4.No dividends 5.Asset price follows geometric Brownian motion 6.Markets are open continuously 7.No short sale restrictions

8. 8 Black-Scholes Option Pricing Model Variables required: 1.Underlying stock price 2.Exercise price 3.Time to expiration 4.Volatility of stock price 5.Risk-free interest rate

9. Black-Scholes Formula V C = S N(d 1 ) - X e -rt N(d 2 ) where d 1 = [ln(S/X)+(r+0.5  2 )t] /  t 0.5 d 2 = d 1 -  t 0.5 where N( ) = cumulative normal distribution, S = stock price, X = exercise price, r = continuously compounded risk-free interest rate, t = number of periods until exercise date, and  = std. dev. per period of continuously compounded rate of return on the stock

10. Call Option Example S = 100X = 110 r = 0.10T = 1.00 (year)  = 0.25 d 1 = [ln(100/110)+ (.10+(  2 /2))] / (  25  1 1/2 ) = 0.1438 d 2 =.1438 - ((  25  1 1/2 ) = -0.1062

11. Call Option Value C= SN(d 1 ) - Xe -rT N(d 2 ) C = (100 x.5572) - (110 e -.10 x 1 x.4577) C = 10.16 Implied Volatility –Using Black-Scholes and the actual price of the option, one can solve for the volatility

12. 12 Another Example Use the Black-Scholes Option Pricing Model to calculate the value of a call option with: Stock price = $18 Exercise price = $20 Time to expiration = 1 year Standard deviation of stock price =.20 Risk-free rate = 5% per year

13. 13 Answer* d 1 = (ln(18/20) + (.05+.5(.2) 2 )1)/(.2(1).5 ) = -.1768 d 2 = -.1768 -.2 (1).5 = -.3768 C = 18(N(-.1768))-20e -.05(1) (N(-.3768)) = 18(.4298)-20(.9512)(.3532) = $1.02

14. 14 Applying The Option Pricing Model To Insurance* Use option pricing to determine the value of each claim on an insurer ’ s assets Policyholders ’ Claim = H Government ’ s Tax Claim = T Owners ’ Claim = V * Neil Doherty and James Garven, 1986, “ Price Regulation in Property- Liability Insurance: A Contingent Claims Approach, ” Journal of Finance, December

15. 15 Option Pricing Model Applied to Insurance Stockholder Value Terminal Asset Value 0 Taxes Liabilities Beg. Assets

16. 16 S 0 =Initial equity P=Premiums (net of expenses) Y 0 =Initial assets = S 0 + P R=Investment rate k=Funds generating coefficient Y 1 =Ending assets = S 0 + P + (S 0 + kP)R L=Losses t=Tax rate i=Portion of investment income that is taxable Let:

17. 17 Value Of Various Claims At The End Of The Period Policyholders ’ claim H 1 = MAX{MIN[L,Y 1 ],0} Government ’ s tax claim T 1 = MAX{t[i(Y 1 -Y 0 )+P-L],0} Owners ’ claim V e = Y 1 - H 1 - T 1

18. 18 Determine The Value Of These Claims At The Beginning Of The Period V(Y 1 ) =Market value of asset portfolio C[A;B] =Value of call option with exercise price of B on asset with value of A E(L) = Expected losses H 0 =V(Y 1 ) - C[Y 0 ;E(L)] T 0 =tC[i(Y 1 - Y 0 ) + P 0 ;E(L)] V e =V(Y 1 ) - H 0 - T 0 =C[Y 0 ;E(L)] - tC[i(Y 1 - Y 0 ) + P 0 ;E(L)]

19. 19 Example Initial equity100 Premiums written200 Expenses 40 Net premiums160 Expected losses150  of investment returns0.5  of losses0.0 Risk-free interest rate4.0% k (FGC)1.0 i1.0 t.34 Y 1 = 100+160+(100+1.0(160)).04 = 270.4

20. 20 Calculation Of Values Owners ’ Value Without Taxes C[Y 0 ;E(L)]= C[100+200-40;150] = C[260;150] d 1 = ln( ) + (.04 +.5 (.5) 2 )1 260 150.5 (1).5 d 2 = 1.43 -.5(1).5 d 1 = 1.43 d 2 =.93

21. 21 Calculation Of Values (cont.) Owners ’ Value Without Taxes (cont.) C = 260 N(1.43) - 150e -.04(1) N(.93) C = 260(.9236) - 150 (.9608) (.8238) C [Y 0 ;E(L)] = 121.41

22. 22 Calculation of Values (cont.) Government ’ s Claim T 0 = tC[i(Y 1 - Y 0 ) + P 0 ;E(L)] =.34 C[1(270.4 - 260) + 160;150] =.34 C[170.4;150] d 1 = ln( ) + (.04 +.5 (.5) 2 )1 170.4 150.5(1).5 d 1 =.5850 d 2 =.5850 -.5(1).5 d 2 =.0850

23. 23 Calculation of Values (cont.) Government ’ s Claim (cont.) C = 170.4N(.585) - 150e -.04(1) N(.085) C = 170.4(.7207) - 150 (.9608) (.5339) C[i(Y 1 - Y 0 ) + P 0 ;E(L)] = 45.86 T 0 =.34 C[i(Y 1 - Y 0 ) + P 0 ;E(L)] = 15.59

24. 24 Valuing Owners ’ Claim This firm has an initial equity of $100, but increases the firm value to $105.82 by writing this coverage. V e = V(Y 1 ) - H 0 - T 0 = C[Y 0 ;E(L)] - tC[i(Y 1 - Y 0 ) + P 0 ;E(L)] V e = 121.41 - 15.59 = 105.82

25. 25 Asset-Liability Management (ALM) Changes in assets and liabilities may have leveraged effects on net worth (surplus) ALM can help meet company to fulfill its objectives by protecting against intermediation risk -- e.g., –Interest rate –Currency –Credit –Liquidity ALM can also help enhance returns

26. 26 ALM for Insurers Insurer ALM tends to focus on “ matching ” the interest rate sensitivities (i.e., durations) of assets and liabilities If this can be accomplished, it is claimed that the surplus of the insurer will be unaffected in the event of interest rate changes Other sources of risk also need to be considered

27. Duration of Surplus Sensitivity of an insurer ’ s surplus to changes in interest rates D S S = D A A - D L L D S = (D A - D L )(A/S) + D L where D = duration S = surplus A = assets L = liabilities

28. Surplus Duration and Asset-Liability Management To “ immunize ” surplus from interest rate risk, set D S = 0 Then, asset duration should be: D A = D L L / A Thus, an accurate estimate of the duration of liabilities is critical for ALM

29. 29 Economic Series Project CAS/SOA Request for Proposals on “ Modeling of Economic Series Coordinated with Interest Rate Scenarios ” –A key aspect of dynamic financial analysis –Also important for regulatory, rating agency, and internal management tests – e.g., cash flow testing Goal: to provide actuaries with a model for projecting economic and financial indices, with realistic interdependencies among the variables. –Provides a floor or foundation for future efforts

30. 30 Scope of Project Literature review –From finance, economics, and actuarial science Financial scenario model –Generate scenarios over a 50-year time horizon Document and facilitate use of model –Report includes sections on data & approach, results of simulations, user ’ s guide –To be posted on CAS & SOA websites –Writing of papers for journal publication

31. 31 Economic Series Modeled Inflation Real interest rates Nominal interest rates Equity returns –Large stocks –Small stocks Equity dividend yields Real estate returns Unemployment

32. 32 Inflation(q) Modeled as an Ornstein-Uhlenbeck process –One-factor, mean-reverting dq t =  q (  q – q t ) dt +  q dB q –In discrete format, an autoregressive process Parametrization –Annual regressions on AR process –Two time periods: (i) since 1913; (ii) since 1946 –Base case Speed of reversion:  q = 0.40 Mean reversion level:  q = 4.8% Volatility:  q = 0.04

33. 33 Real Interest Rates Two-factor Vasicek term structure model Short-term rate (r) and long-term mean (l) are both stochastic variables dr t =  r (l t – r t ) dt +  r dB r dl t =  l (  l – r t ) dt +  l dB l

34. 34 Nominal Interest Rates Combines inflation and real interest rates i = {(1+q) x (1+r)} - 1 where i = nominal interest rate q = inflation r = real interest rate

35. 35 Equity Returns Model equity returns (s t ) as an excess return over the nominal interest rate s t = q t + r t + x t Empirical “ fat tails ” issue regarding equity returns distribution Thus, modeled using a “ regime switching model ” –Low volatility regime –High volatility regime

36. 36 Large Stocks (1871-2002)Small Stocks (1926-1999) Low Volatility Regime High Volatility Regime Low Volatility Regime High Volatility Regime Mean 0.8%-1.1%1.0%0.3% Variance 3.9%11.3%5.2%16.6% Probability of Switching 1.1%5.9%2.4%10.0% Equities: Excess Monthly Return Parameters