
Asset Pricing Simulation (work in progress) William F. Sharpe STANCO 25 Professor of Finance, Emeritus Stanford University www.wsharpe.com
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
Part 1 Asset Pricing Models
Portfolio Portfolio Asset Pricing and Portfolio Choice Position Position Investor 2 Investor 1 Preferences Preferences Predictions Predictions Market Trades Prices Outcomes
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
Asset Pricing Models:Kernel Asset Pricing • Taught in PhD courses • Used in fixed income analyses • Used in Financial Engineering
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
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
Part 2 Kernel Asset Pricing for Dummies
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
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”)
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
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”
An Individual’s Optimal Allocation * PPC * * * PPC’ * * * * * C’ Future Consumption
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
The Market Portfolio * PPC * * * * * * * * Future Consumption
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
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
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
Equilibrium Expected Returns Expected Return * * * * * * ER’ * * * C’ Future Consumption
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
Individual and Market Consumption Individual’s Consumption * * * * * Ci’ * * * * Total Consumption C’
The Empirical Question:Where is the Pricing Kernel? * PPC * * ? * * * * * * Future Consumption
Part 3 Asset Pricing Simulation
Asset Pricing Simulation • Can incorporate elements of • Mean/variance asset pricing • Kernel Asset Pricing • Arbitrage Pricing • Behavioral Finance
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
One-period models • Consumption • Now • Later • Investment • Sacrifice consumption now for consumption later • (Total) Return = consumption later / consumption now
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
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
Predictions • Agreement • Everyone agrees on the probabilities of the states • Disagreement • People have different probability assessments
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
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
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
The Four Cases Reality Theory
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?
Key Inputs • Securities • People • Preferences • Predictions • Portfolios
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
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
Part 4 A Simple Simulation Example
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
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
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)
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
A Round of Trading • Trade security 1 • Trade security 2 • ……… • Trade security N