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

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Parts

1. Asset Pricing Models
2. Kernel Asset Pricing for Dummies
3. Asset Pricing Simulation
4. A Simple Simulation Example
5. Mean Variance Asset Pricing
6. Estimating Expected Returns
7. Non-linear Pricing Kernels
8. Experimental Pricing Kernels
9. Extensions

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Part 1

• Asset Pricing Models

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Asset Pricing and Portfolio Choice
Position
Position
Investor 2
Investor 1
Preferences
Preferences
Predictions
Predictions
Market Trades
Prices
Outcomes
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Asset Pricing ModelsMean/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

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Asset Pricing ModelsKernel Asset Pricing

• Taught in PhD courses
• Used in fixed income analyses
• Used in Financial Engineering

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Asset Pricing ModelsArbitrage 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

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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

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Part 2

• Kernel Asset Pricing for Dummies

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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

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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)

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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

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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

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An Individuals Optimal Allocation

PPC



PPC





Future Consumption
C
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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

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The Market Portfolio

PPC








Future Consumption
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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

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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

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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

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Equilibrium Expected Returns
Expected Return






ER



Future Consumption
C
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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

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Individual and Market Consumption
Individuals Consumption





Ci




Total Consumption
C
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The Empirical QuestionWhere is the Pricing
Kernel?

PPC


?






Future Consumption
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Part 3

• Asset Pricing Simulation

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Asset Pricing Simulation

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

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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

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One-period models

• Consumption
• Now
• Later
• Investment
• Sacrifice consumption now for consumption later
• (Total) Return
• consumption later / consumption now

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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

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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

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Predictions

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

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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 anyones predictions

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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

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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

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The Four Cases

Agreement Complete Markets
no no
no yes
yes no
yes yes
Reality
Theory
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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?

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Key Inputs

• Securities
• People
• Preferences
• Predictions
• Portfolios

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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

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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

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Part 4

• A Simple Simulation Example

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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

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Example 1 Securities
Security 1 Security 2 Security 3 Security 4
State 1 1 0 0 0
State 2 0 1 5 3
State 3 0 1 3 5
State 4 0 1 8 4
State5 0 1 4 8
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Example 1 Initial Portfolios
Security 1 Security 2 Security 3 Security 4
Ryan 53 0 10 0
Trinity 53 0 0 10
Total 106 0 10 10
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Example 1 Initial Consumption
State 1 State 2 State 3 State 4 State 5
Ryan 53 50 30 80 40
Trinity 53 30 50 40 80
Total 106 80 80 120 120
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Example 1 Predictions

State 1 State 2 State 3 State 4 State 5
Ryan 1.00 .20 .20 .30 .30
Trinity 1.00 .20 .20 .30 .30
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Example 1 Preferences
Family Time Preference Risk Tolerance
Ryan M 0.96 80
Trinity M 0.96 120
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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

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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)

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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
  investors best interests in this type of market

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Initial Bids and Asks with 12 People
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A Round of Trading

• Trade security 1
• Trade security 2
• Trade security N

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Trading Until Equilibrium is Obtained

• Trade one round
• If any trades were made in any security, repeat
• Continue until no trades are possible in any
  security
• Calculate the equilibrium prices

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Final Bids and Asks with 12 People
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Equilibrium Prices for Traded Securities

• The maximum bid across all investors
• A measure of the amount for which an additional
  unit could be sold
• Will be lower than the minimum ask price but only
  slightly so in a market with many investors

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Example 1 Final Portfolios
Security 1 Security 2 Security 3 Security 4
Ryan 51.67 18.56 3.125 3.125
Trinity 54.33 -18.56 6.875 6.875
Total 106 0 10 10
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Example 1 Final Consumption
State 1 State 2 State 3 State 4 State 5
Ryan 51.67 43.56 43.56 56.06 56.06
Trinity 54.33 36.44 36.44 63.94 63.94
Total 103 80 80 120 120
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Example 1 Certainty Equivalents
Initial Final Percent Improvement
Ryan 49.34 51.05 3.47
Trinity 51.16 52.98 3.56
Total 100.50 104.03 3.52
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Expected Returns and Risks
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Expected Returns and Beta Values
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Returns versus Market Returns
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Example 1 Alternative Markets

Securities Percent Improvement in Total CE Type of Market
State Claims 3. 52 Complete
A, B, Riskless 3. 52 Sufficiently Complete
A, B 3.34 Incomplete
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The Pricing Kernel when State Claims are Traded
2 states
2 states
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Part 5

• Mean Variance Asset Pricing

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Example 2

• One period, two dates
• 11 states of the world
• 1 now
• 10 later
• 16 people
• All members of the M family
• 8 securities
• Consumption now
• Riskless
• 6 regular securities

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Total Consumption by State
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Case 2a

• Agreement
• State Claims Traded

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PPC versus Consumption
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The Pricing Kernel
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Returns versus Market Returns
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Expected Returns and Standard Deviations
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Expected Returns and Beta Values
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Case 2a Results

• M Family characteristics
• PPC linearly related to consumption
• Consistent with maximization of a quadratic
  utility function
• Will choose a portfolio that is on a
  mean/variance efficient frontier
• Pricing Kernel
• Linear
• Efficient portfolios
• Market plus borrowing or lending
• The CAPM holds

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Case 2b

• Agreement
• State Claims not traded
• Results
• The same as in Case 2a
• Market is sufficiently complete since investors
  only want to hold the market portfolio plus
  borrowing and lending
• The CAPM holds

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Case 2c

• Disagreement
• State Claims traded

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PPC Values with Disagreementand Traded State
Claims

• Personal PPC
• Market state price
• Personal probability of state
• Market Predictions defined as
• Wealth-weighted average of investor
  probabilities, using equilibrium levels of wealth
• Market PPC
• Market state price
• Market probability of state

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PPC versus Consumption
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Returns versus Market Returns
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The Pricing Kernel
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Expected Returns and Standard Deviations with
Market Probabilities
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Expected Returns and Beta Values with Market
Probabilities
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Case 2c Results

• Using Market probabilities
• Pricing Kernel
• Approximately linear
• Efficient portfolios
• Market plus borrowing or lending
• The CAPM holds approximately

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Case 2d

• Disagreement
• State Claims not traded

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PPC Values with Disagreementand No Traded State
Claims

• Personal State Price
• Maximum amount bid for an additional unit of
  consumption
• Personal PPC
• Personal state price
• Personal probability of state
• Market State Price defined as
• Wealth-weighted average of personal state prices
• Market PPC
• Market state price
• Market probability of state

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Case 2d Results

• Smaller gains through trade than in 2c
• With state claims, CE increased by 1.50
• Without state claims CE increased by 1.06
• Magnitudes change
• Expected returns, Riskless rate
• Relationships similar to those of Case 2c
• CAPM holds approximately using market predictions

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The Efficiency of Market-based Strategies

• Index Fund Premise
• None of us is as smart as all of us
• Gabrielle M
• Predictions differ from market
• Portfolio is not market-based

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Risk and Return using Gabrielles predictions
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Risk and Return using Market Predictions
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Part 6

• Estimating Expected Returns

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Estimating Asset Expected Returns

• Which is better?
• 1 Historic average returns
• 2 Historic beta values
• Assume the conditions of example 2d hold in every
  period
• Generate simulations to determine the better
  procedure

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Simulation ProcedureOne Case

• Generate historic outcomes for 100 historic
  periods
• 1 Compute security average returns
• 2 Compute security historic beta values
• Compute correlation between
• Historic average returns and true expected
  returns
• Historic beta values and true expected returns

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Simulation ProcedureOverall

• Generate 1,000 cases
• Summarize the ranges of correlations between true
  expected returns and
• 1 Historic average returns
• 2 Historic beta values

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Cumulative Distributions of Correlations with
Expected Returns
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Part 7

• Non-Linear Pricing Kernels

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Example 3

• The same as Example 2 except all 16 people are
  members of the P family
• 11 states of the world
• 1 now
• 10 later
• 16 people
• All members of the P family
• 8 securities
• Consumption now
• Riskless
• 6 regular securities

Source Case 5p
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Members of the P Family

• PPC versus consumption is proportional
• A percentage difference in consumption is
  associated with a given percentage difference in
  PPC
• Log(PPC) is linearly related to log(consumption)
• Consistent with maximization of a CRRA utility
  function

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Case 3a

• Agreement
• State Claims Traded

Source Case 5p(sa)
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PPC versus Consumption
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The Pricing Kernel
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Returns versus Market Returns
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Expected Returns and Standard Deviations
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Expected Returns and Beta Values
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Kernel-based Beta Values

• With complete markets and agreement
• Ei-r cov(Ei-r,m)
• EM-r cov(EM-r, m)
• The pricing kernel
• m f(RM)
• Define
• cov(Ei-r, f(RM))
• cov(EM-r, f(RM))
• Then
• Ei-r Bki ( EM-r)

Bki
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Expected Returns and Kernel Beta Values
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Case 3a Results

• Pricing Kernel
• Non-linear
• Efficient portfolios
• Market-based
• Close to market plus borrowing or lending
• The CAPM holds
• Approximately with standard beta values
• Exactly with kernel beta values

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Case 3d

• Disagreement
• State Claims not traded

Source Case 5p()
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PPC versus Consumption
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Returns versus Market Returns
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The Pricing Kernel
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Expected Returns and Standard Deviations
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Expected Returns and Kernel Beta Values
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Cumulative Distributions of Correlations with
Expected Returns Kernel Beta Values
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Cumulative Distributions of Correlations with
Expected Returns Standard Beta Values
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Case 3d Results

• Compare with case 3a (agreement and state claims
  traded)
• Magnitudes change
• Expected returns, Riskless rate
• Market relationships similar
• CAPM holds approximately using market predictions
  and kernel betas

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Part 8

• Experimental Pricing Kernels

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The Distribution Builder

• Joint work with
• Dan Goldstein
• Columbia University
• Eric Johnson
• Columbia University

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Setting

• 100 People, one of which is you
• Place people in retirement rows
• E.g. row 75 retire at 75 of final salary
• Budget meter indicates percent of budget used
• Final pattern must use gt99 of budget
• Subject to constraint, select a preferred pattern

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Source http//vlab.cebiz.org/dggoldst/db/sample.
php
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Framing

• 75 row is considered standard advice
• Stated and shown on the interface
• Riskless outcome (wealth)
• For half of participants 60
• For the other half 75
• Not stated or shown explicitly

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State Prices

• 100 equally probable states
• State prices
• Derived from discrete approximation to log-normal
  distribution
• Based on 10 years investment
• Market plus riskless security
• IID
• One-year returns Sharpe Ratio 1/3

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Least-cost Investment

• Rank 100 Desired outcomes from highest to lowest
• Assign to states from cheapest to most expensive
• Provides the desired distribution at the lowest
  possible cost
• Budget shown is based on these computations

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Conditions

• Agreement
• Every state has a probability of 1/100
• Complete Market
• Least-cost calculation is based on the use of
  state claims

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Pricing Kernels

• Can infer each persons personal pricing kernel
• Can infer the collective pricing kernel
• Results subject to granularity
• All outcomes in increments of 5

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Average DistributionParticpants with Wealth 75
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The Pricing Kernel, Participants with Wealth 75
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The Pricing Kernel in logs, Wealth 75
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Average DistributionWealth 60
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The Pricing Kernel, Wealth 60
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The Pricing Kernel in logs, Wealth 60
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Average Distribution,, All Levels of Wealth
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The Pricing Kernel, All Levels of Wealth
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The Pricing Kernel in logs, All Levels of Wealth
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Experimental Pricing Kernels

• Individuals show preferences for wealth at
  reference points
• standard outcome
• Riskless outcome
• Consistent with kinked utility curves
• Behavioral Finance
• But these occur at different points for different
  people
• Aggregate pricing kernel is smooth
• Consistent with traditional asset pricing theory

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Part 9

• Extensions

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Desirable Extensions

• Multiple periods
• More than two dates
• More types of families
• Behavioral, kinked functions, etc.
• Prior Positions
• Security holdings that cannot be changed
• Houses, jobs, etc..
• Production
• Trades with nature securities at fixed prices
• Financial Institutions
• Combine securities and issue claims on securities
• Different trading procedures