1. Casualty Actuarial Society Experienced Practitioner Pathway Seminar Lecture 5 – Advanced Quantitative Analysis Stephen P. D’Arcy, FCAS, MAAA, Ph.D. Robitaille Chair of Risk and Insurance California State University – Fullerton D’Arcy Risk Consulting, Inc.

2. Overview 2EPP Lecture 5: Advanced Quantitative Analysis What you need to know to be part of the conversation –Stochastic processes –Interest Rate Models –Regime switching and transition matrices –Copulas –Extreme value theory –Option pricing models

3. Stochastic Processes A stochastic process is an elaborate term for a random variable Future values of the process is unknown We want to model some stochastic process –Future interest rates can be viewed as a stochastic process Basic stochastic processes: –Random walk –Brownian motion –Wiener process –Ornstein-Uhlenbeck process 3EPP Lecture 5: Advanced Quantitative Analysis

4. Features of a Random Walk Example – someone moving one step at a time north or south, but the direction of each step is random Memory loss –History reveals no information about the future Expected change in value is zero –Over any length of time, the best predictor of future position is the current position –This feature is termed a martingale Variance increases with time –As more time passes, there is potential for being farther from the initial position 4EPP Lecture 5: Advanced Quantitative Analysis

5. Brownian Motion A Brownian Motion is the limit of the discrete case random walk –This is a continuous time process The simplest form of Brownian Motion is a Wiener process (dz) 5EPP Lecture 5: Advanced Quantitative Analysis

6. Generalizing Pure Brownian Motion Let’s interpret the following expression: First, recall that we are modeling the stochastic process x –Think of x as a stock price, a level of interest rates and an inflation rate The equation states that the change in variable x is composed of two parts: –A drift term which is non-random –A stochastic or random term that has variance σ 2 –Both terms are proportional to the time interval 6EPP Lecture 5: Advanced Quantitative Analysis

7. Ornstein-Uhlenbeck Process A constant drift term does not make economic sense Most models assume mean reversion –For interest rates and inflation rates there is a long-run average value 7EPP Lecture 5: Advanced Quantitative Analysis

8. Classifications of Interest Rate Models Discrete vs. Continuous Single Factor vs. Multiple Factors General Equilibrium vs. Arbitrage Free 8EPP Lecture 5: Advanced Quantitative Analysis

9. Discrete Models Discrete models have interest rates change only at specified intervals Typical interval is monthly Daily, quarterly or annually also feasible Discrete models can be illustrated by a lattice approach 9EPP Lecture 5: Advanced Quantitative Analysis

10. Continuous Models Interest rates change continuously and smoothly (no jumps or discontinuities) Mathematically tractable Accumulated value = e rt Example $1 million invested for 1 year at r = 5% Accumulated value = 1 million x e.05 = 1,051,271 10EPP Lecture 5: Advanced Quantitative Analysis

11. Single Factor Models Single factor is the short term interest rate for discrete models Single factor is the instantaneous short term rate for continuous time models Entire term structure is based on the short term rate For every short term interest rate there is one, and only one, corresponding term structure 11EPP Lecture 5: Advanced Quantitative Analysis

12. Multiple Factor Models Variety of alternative choices for additional factors Short term real interest rate and inflation (CIR) Short term rate and long term rate (Brennan-Schwartz) Short term rate and volatility parameter (Longstaff-Schwartz) Short term rate and mean reverting drift (Hull-White) 12EPP Lecture 5: Advanced Quantitative Analysis

13. General Equilibrium Models Start with assumptions about economic variables Derive a process for the short term interest rate Based on expectations of investors in the economy Term structure of interest rates is an output of model Does not generate the current term structure Limited usefulness for pricing interest rate contingent securities More useful for capturing time series variation in interest rates Often provides closed form solutions for interest rate movements and prices of securities 13EPP Lecture 5: Advanced Quantitative Analysis

14. Arbitrage Free Models Designed to be exactly consistent with current term structure of interest rates Current term structure is an input Useful for valuing interest rate contingent securities Requires frequent recalibration to use model over any length of time Difficult to use for time series modeling 14EPP Lecture 5: Advanced Quantitative Analysis

15. Which Type of Model is Best? There is no single ideal term structure model useful for all purposes Single factor models are simpler to use, but may not be as accurate as multiple factor models General equilibrium models are useful for modeling term structure behavior over time Arbitrage free models are useful for pricing interest rate contingent securities How the model will be used determines which interest rate model would be most appropriate 15EPP Lecture 5: Advanced Quantitative Analysis

16. Regime Switching Models Many real life processes are too dispersed to reflect a single regime model One way to deal with this dispersion is to allow different regimes Data have too many outliers (leptokurtic) to fit common distributions If volatility is set high enough to generate the outliers, there are not enough mid-range observations Solution is to use a regime switching model with a transition matrix Examples –Stock returns – Mary Hardy –Inflation – Ahlgrim and D’Arcy 16EPP Lecture 5: Advanced Quantitative Analysis

17. Copulas Copulas are conditional joint distribution functions Early applications –Joint life estimations by life actuaries Mortality is generally independent, but the likelihood of a spouse dying increases if their partner has died recently –Common event –Adverse health reaction to stress of losing spouse –Stock and bond prices for a single company Prices generally move independently If stock price declines significantly, bond prices decline as well to reflect increased risk of bankruptcy Copulas now widely used in quantitative risk models 17EPP Lecture 5: Advanced Quantitative Analysis

18. Commonly used copulas Archimedean copulas –Joint probability distribution can be expressed in a closed form –They are defined using only one or two parameters –Examples One parameter –Gumbel (upper tail dependence only) –Frank (neither upper nor lower tail dependence) –Clayton (if α >0 lower tail dependence only, otherwise none) Two parameters –Generalized Clayton (upper and lower tail dependence) Gaussian (or normal) copula –Linear correlation (Pearson’s Rho) –Every pair of variables can have different correlation –No tail dependence 18EPP Lecture 5: Advanced Quantitative Analysis

19. Dependency Structure Comparison - 1 19EPP Lecture 5: Advanced Quantitative Analysis Normal Copula Gumbel Copula Correlation =.50 Gumbel parameter 1.5

20. Dependency Structure Comparison - 2 20EPP Lecture 5: Advanced Quantitative Analysis Frank Copula Clayton Copula Frank parameter = 3.7 Clayton parameter 1.0

21. Drawbacks of Copulas Commonly used copulas may be mathematically elegant, but not appropriate for specific application Asymmetry issues –The relationship between two variables may not be the same for large increases as for large decreases Need to consider tail dependency and select appropriate copula –One variable may depend on another, but the relationship may not be reciprocal –Each of two variables could depend on a third variable, but not on each other Try to determine and model key variable and relate other variables to it 21EPP Lecture 5: Advanced Quantitative Analysis

22. Extreme Value Theory In many cases the extreme scenarios are what is important Extreme value theory focuses on the maximum value from a set of independent observations (eg. the largest hurricane each year) Regardless of the distributions generating the observations, the maximum values will follow a known distribution, the Generalized Extreme Value (GEV) 22EPP Lecture 5: Advanced Quantitative Analysis

23. Generalized Extreme Value Distribution Advantage –By determining the GEV parameters, the shape of the tail is determined Power law Exponential Finite endpoint Disadvantages –Ignores most of the observations –Extreme value is not necessarily the important value Extreme value might be too large to be relevant All values above a particular level (solvency) might be more important 23EPP Lecture 5: Advanced Quantitative Analysis

24. Generalized Pareto Distribution Considers all observations above a chosen hurdle level As the hurdle increases, the conditional loss distribution converges to a Generalized Pareto distribution, regardless of the underlying distributions Key to select the right hurdle (or threshold) –Too high and not enough observations to calculate parameters –Too low and it does not focus on the tail 24EPP Lecture 5: Advanced Quantitative Analysis

25. What is an Option Contract? Options provide the right, but not the obligation, to buy or sell an asset at a fixed price –Call option is right to buy –Put option is right to sell –Only option sellers (writers) are required to perform under the contract (if exercised) –After paying the premium, option owner has no duties under the contract 25EPP Lecture 5: Advanced Quantitative Analysis

26. Some Option Terminology The exercise or strike price is the agreed on fixed price at which the option holder can buy or sell the underlying asset Exercising the option means to force the writer to perform –Make option writer sell if a call, or force writer to buy if a put Expiration date is the date at which the option ceases to exist American option can be exercised anytime prior to expiration European option can only be exercised on the expiration date 26EPP Lecture 5: Advanced Quantitative Analysis

27. Option Valuation Basics Two components of option value –Intrinsic value –Time value Intrinsic value is based on the difference between the exercise price and the current asset value (from the owner’s point of view) –For calls, max(S-X,0) X= exercise price –For puts, max(X-S,0) S=current asset value Time value reflects the possibility that the intrinsic value may increase over time –Longer time to maturity, the higher the time value 27EPP Lecture 5: Advanced Quantitative Analysis

28. In-the-Moneyness If the intrinsic value is greater than zero, the option is called “in-the- money” –It is better to exercise than to let expire If the asset value is near the exercise price, it is called “near-the- money” or “at-the-money” If the exercise price is unfavorable to the option owner, it is “out-of-the- money” 28EPP Lecture 5: Advanced Quantitative Analysis

29. Determinants of Call Value Value must be positive Increasing maturity increases value Increasing exercise price, decreases value American call value must be at least the value of European call Value must be at least intrinsic value For non-dividend paying stock, value exceeds S-PV(X) –Can be seen by assuming European style call 29EPP Lecture 5: Advanced Quantitative Analysis

30. Determinants of Call Value (p.2) As interest rates increase, call value increases –This is true even if there are dividends As the volatility of the price of the underlying asset increases, the probability that the option ends up in-the-money increases 30EPP Lecture 5: Advanced Quantitative Analysis

31. Put-Call Parity Consider two portfolios –One European call option plus cash of PV(X) –One share of stock plus a European put Note that at maturity, these portfolios are equivalent regardless of value of S Since the options are European, these portfolios always have the same value –If not, there is an arbitrage opportunity (Why?) 31EPP Lecture 5: Advanced Quantitative Analysis

32. Fisher Black and Myron Scholes Developed a model to value European options on stock Assumptions –No dividends –No taxes or transaction costs –One constant interest rate for borrowing or lending –Unlimited short selling allowed –Continuous markets –Distribution of terminal stock returns is lognormal Based on arbitrage portfolio containing stock and call options Required continuous rebalancing 32EPP Lecture 5: Advanced Quantitative Analysis

33. Black-Scholes Option Pricing Model C= Price of a call option S= Current price of the asset X= Exercise price r= Risk free interest rate t= Time to expiration of the option σ= Volatility of the stock price N= Normal distribution function 33EPP Lecture 5: Advanced Quantitative Analysis

34. Using the Black-Scholes Model Only variables required –Underlying stock price –Exercise price –Time to expiration –Volatility of stock price –Risk-free interest rate 34EPP Lecture 5: Advanced Quantitative Analysis

35. Use of Options Options give users the ability to hedge downside risk but still allow them to keep upside potential This is done by combining the underlying asset with the option strategies Net position puts a floor on asset values or a ceiling on expenses 35EPP Lecture 5: Advanced Quantitative Analysis

36. Common Uses of Options Interest rate risk Currency risk Equity risk –Market risk –Individual securities Catastrophe risk 36EPP Lecture 5: Advanced Quantitative Analysis

37. Summary 37EPP Lecture 5: Advanced Quantitative Analysis If you don’t understand terminology or techniques being used in a model, then ask for an explanation You should have a basic understanding of the key terms to grasp the most important issues if they are explained well Basic topics covered –Stochastic processes –Interest Rate Models –Regime switching and transition matrices –Copulas –Extreme value theory –Option pricing models

38. References Copulas and extreme value theory –Paul Sweeting, Financial Enterprise Risk Management Regime switching – Ahlgrim and D’Arcy, The Effect of High Inflation or Deflation on the Insurance Industry –Hardy, A Regime Switching Model for Long-Term Stock Returns, North American Actuarial Journal Interest rate models –D’Arcy and Gorvett, Hacking a Path through the Thickets, Global Reinsurance, 2000. 38EPP Lecture 5: Advanced Quantitative Analysis