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Asset Pricing Simulation (work in progress)

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## Asset Pricing Simulation (work in progress)

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**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