1. Asset Pricing Simulation (work in progress) William F. Sharpe STANCO 25 Professor of Finance, Emeritus Stanford University www.wsharpe.com

2. Parts • Asset Pricing Models • Kernel Asset Pricing for Dummies • Asset Pricing Simulation • A Simple Simulation Example • Mean Variance Asset Pricing • Estimating Expected Returns • Non-linear Pricing Kernels • Experimental Pricing Kernels • Extensions

3. Part 1 Asset Pricing Models

4. Portfolio Portfolio Asset Pricing and Portfolio Choice Position Position Investor 2 Investor 1 Preferences Preferences Predictions Predictions Market Trades Prices Outcomes

5. Asset Pricing Models:Mean/Variance Asset Pricing • Taught in MBA courses • The basis for most of the quantitative methods used in the investment management business • The basis for many corporate cost of capital applications

6. Asset Pricing Models:Kernel Asset Pricing • Taught in PhD courses • Used in fixed income analyses • Used in Financial Engineering

7. Asset Pricing Models:Arbitrage Pricing • Strictly speaking, can only price assets that are redundant • Value of an asset is based on the cost of obtaining the same outcomes using existing assets • Deals only with relative prices • Used in Financial Engineering

8. Behavioral Finance • Assumes individuals are not “rational economic agents” • For example, assumes that individuals do not act as if they maximize a smooth utility function • Not widely used in asset pricing at present

9. Part 2 Kernel Asset Pricing for Dummies

10. State Prices • Assume • State claims are traded • There is agreement • After equilibrium is established • There will be a set of state prices • Investors will have chosen the amounts to consume in each state • Additional trades will not be possible

11. Price per Chance (PPC) • A measure of the cost of consumption in a state • PPC = State Price / State Probability • The cost per chance that the outcome will take place • (also known as “m”)

12. Footnote on Stochastic Discount Factors • Stochastic Discount Factors P = E(mX) P = s ms Xs • Arbitrage in a complete market P = ps Xs P = s (ps / s) Xs ms = ps / s

13. Assumed Investor Behavior • Other things equal, a person will be willing to pay more for added consumption in a state in which there is less consumption • PPC is inversely related to consumption • “The more you have, the less you will pay for another unit”

14. An Individual’s Optimal Allocation * PPC * * * PPC’ * * * * * C’ Future Consumption

15. The Market Allocation • The market consumption in a state is the sum of the individuals’ levels of consumption in that state • If each individual wants more consumption in state A than state B the total desired market consumption in state A will be greater than in state B

16. The Market Portfolio * PPC * * * * * * * * Future Consumption

17. Equilibrium • Given production, the amount of market consumption in each state is given • Thus prices must adjust until the individuals’ collective demand for consumption in a state equals that available • This implies • States with the same aggregate consumption will have the same PPC • States with more aggregate consumption will have lower PPCs

18. Expected Total Returns • A state claim pays $1 • To purchase it one pays its price. Its total return is thus • 1 / price • The probability of receiving its return is given by its probability • This its’ expected total return is Probability of state State price =1 / PPC

19. Expected Returns and Consumption • If PPC is lower in states in which aggregate consumption is greater, then • Expected total return is higher in states in which aggregate consumption is greater

20. Equilibrium Expected Returns Expected Return * * * * * * ER’ * * * C’ Future Consumption

21. Portfolio Choice • For each level of market consumption there is a PPC • Higher levels of market consumption  lower PPCs • For each PPC there is a level of individual consumption • Lower PPCs  higher levels of individual consumption • Consequently: • Higher levels of market consumption  higher levels of individual consumption • Therefore: • Each individual should arrange to have consumption that is (1) related directly to market consumption and (2) related only to market consumption

22. Individual and Market Consumption Individual’s Consumption * * * * * Ci’ * * * * Total Consumption C’

23. The Empirical Question:Where is the Pricing Kernel? * PPC * * ? * * * * * * Future Consumption

24. Part 3 Asset Pricing Simulation

25. Asset Pricing Simulation • Can incorporate elements of • Mean/variance asset pricing • Kernel Asset Pricing • Arbitrage Pricing • Behavioral Finance

26. Discrete Time approaches • At each point in time there will be one and only one state of the world • One-period models • Two dates • Now • Later • Multi-period models • More than two dates

27. One-period models • Consumption • Now • Later • Investment • Sacrifice consumption now for consumption later • (Total) Return = consumption later / consumption now

28. Securities • “Standard securities” • Pay off in many states of the world • Time-state claims • Each one pays off in only one state of the world

29. Markets • Complete market • A time-state claim for every state of the world • Incomplete market • Some desired time-state claims are not available (directly or indirectly) • Sufficiently complete market • There is no demand for any unavailable time-state claim

30. Predictions • Agreement • Everyone agrees on the probabilities of the states • Disagreement • People have different probability assessments

31. Equilibrium • A situation in which no one wishes to do anything more • No more trades of existing securities • No introduction of new securities • No changes in anyone’s predictions

32. Completely costless equilibria • Assume securities can be introduced and traded without cost • Assume that information can be disseminated without cost • When equilibrium is reached: • Markets will be sufficiently complete • There will be agreement about probabilities

33. More realistic equilibria • Investors disagree about the future • Different probability assessments • Markets are incomplete • It is not possible to trade state claims for every state

34. The Four Cases Reality Theory

35. APSim: an Asset Pricing Simulator • Can analyze economies with or without • Agreement • Complete markets • Can find • Implications for the determination of asset prices • Implications for optimal portfolio choices • Can illuminate questions such as • Are market-based strategies efficient? • Is there a market risk premium? • Are security and portfolio expected returns related to beta values?

36. Key Inputs • Securities • People • Preferences • Predictions • Portfolios

37. People • Objects (“black boxes”) • Properties of a person that do not change • Name • Preferences • Predictions • Properties of a person that change • Portfolio • Consumption • Determined by portfolio and security payoffs

38. Questions that people can answer • Security bid price • Maximum amount bid to get n units of security S • Security ask price • Minimum amount asked to give up n units of security S • Consumption bid price • Maximum amount bid to get n units of consumption in state s • Consumption ask price • Minimum amount asked to give up n units of consumption in state s • Certainty equivalent • Amount for certain in each period equivalent to current consumption

39. Part 4 A Simple Simulation Example

40. Example 1 • One period, two dates • 5 states of the world • 1 now • 4 later • 2 people • 4 securities • Consumption now • Riskless • Security A • Security B

41. Example 1: Securities

42. Example 1: Initial Portfolios

43. Example 1: Initial Consumption

44. Example 1: Predictions

45. Example 1: Preferences

46. Trading • A Lot size is set • number of shares per trade • Each person submits • a sealed bid to purchase one lot of the security • a sealed bid to sell one lot of the security • The market maker then finds the maximum number of lots that can be traded • All trades are executed at a price halfway between the lowest bid and the highest ask prices for those who trade

47. Credit Checks • No one is allowed to submit a bid or ask if execution at that price would result in negative consumption in one or more states • Subject to this constraint, people can • Buy any existing security • Sell any security currently held • Sell any security not currently held • (sell short)

48. Assumed Trading Behavior • Investors submit their maximum bid and minimum ask prices • Doing otherwise can • Get the same result, or • Lose a beneficial trade, or • Or get a better price, but only if the investor would be the marginal trader in both cases • Thus the assumed behavior is generally in the investor’s best interests in this type of market

49. Initial Bids and Asks with 12 People

50. A Round of Trading • Trade security 1 • Trade security 2 • ……… • Trade security N