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Equilibrium Simulation
William F. SharpeStanford University
www.wsharpe.com
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Why Analyze Equilibrium?
An investor needs to have a view of the ways inwhich asset prices are determined.
Risk and return forecasts should reflect these views.
Asset prices are set by investors operating in markets.
Equilibrium prices are those at which no investors arewilling to make further trades.
Asset prices will tend towards equilibrium untilconditions change.
Good asset pricing theory is a key ingredient forgood investment practice .
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Asset Pricing Theories
Mean/Variance
State/Preference
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Mean/Variance Analysis
All returns are jointly normally distributed, or
Investors care only about mean and variance ofportfolio return
Markowitz Portfolio AnalysisNormative theoryMaximize portfolio expected return for given riskPortfolio Optimization
The Capital Asset Pricing ModelPositive theoryExpected Returns related to Beta ValuesIndex Funds
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The Market Risk/Reward TheoremOnly market risk is rewarded with higher expected return
The Market Risk/Reward CorollaryDon’t take non-market risk
Key Implications of the CAPM
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State/Preference Analysis
Investors care about the entire distribution ofportfolio return and returns need not be jointlynormally distributed
Arrow/Debreu economies
Financial EngineeringNormative theory
Pricing KernelsPositive theory
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State/Preference AssetPricing Theory for Dummies
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States of the World
There are alternative future states of the world
One and only one will occur
For each state there is a probability
It is possible to buy and sell state claimsa claim for state s pays $1 if and only if state s occurs(similar to an insurance policy)
The markets are complete (every state claim can be traded)
When equilibrium is established, there will be a set of stateprices, one for each state of the world.
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Price Per Chance
PPC:State Price
Probability of State
With any insurance policy, one should compare the price withthe likelihood of cashing in on a claim
The higher the PPC, the less attractive is an investment
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Rational Investment Allocations
Take more of something when it costs less
PPC is a measure of cost
Allocate current wealth to obtain more future wealth in stateswith lower PPCs
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An Individual’s Optimal Allocation
PPC
Future Wealth
PPC’
W’
*
*
**
*
** *
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The Market Allocation
The market wealth in a state is the sum of the individuals’levels of wealth in that state
If each individual wants more wealth in state A than state B,the total desired market wealth in state A will be greaterthan in state B
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The Market Portfolio
PPC
Future Wealth
*
*
**
*
** *
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Equilibrium Conditions
Given production, the amount of market wealth in each stateis given.
Thus prices must adjust until the individuals’collectivedemand for wealth in a state equals that available.
This implies thatStates with the same wealth will have the same PPC, andStates with more wealth will have lower PPCs
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Equilibrium Prices: the Pricing Kernel
PPC
Future Wealth
PPC’
W’
*
*
**
*
** *
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Individual and Market Allocations
For each level of market wealth there is a PPCHigher levels of market wealth lower PPCs
For each PPC there is a level of individual wealthLower PPCs higher levels of individual wealth
Thus each individual should arrange to have wealth that isrelated directly to market wealthHigher levels of market wealth higher levels of
individual wealth
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Individual and Market Wealth
Individual’sWealth
Total WealthW’
w’
**
**
**
**
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Risk and Expected Return
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Risk and Expected Return (1)
The pricing kernel
A kernel beta
The kernel beta equation
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Risk and Expected Return (2)
If m is a decreasing function of RM
Then
And
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Risk and Expected Return (3)
If m is a linear function of RM
Conventionally, let
Then
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Equilibrium Simulation
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To what extent do the implications of the CAPM and/orState/Preference Asset Pricing theory hold when markets areincomplete and investors:
do not have mean/variance preferences,
have sources of income outside the capital market,
make different predictions,
act in accordance with findings of behavioral research,
etc..
The Key Question to be Addressed
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The Vehicle: APSIM,Asset Price and Portfolio Choice Simulator
Formulation:Discrete outcomesDiscrete time
Process:Simulate trading to reach market equilibriumAnalyze characteristics of the resulting equilibrium
The approach:Proceeds from first principlesUses simple mathematicsAllows for complex economies and preferencesCan analyze mean/variance as a special case
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References
William F. Sharpe,
Investors and Markets: Portfolio Choices, Asset Prices andInvestment Advice,
Princeton University Press, 2007
APSIM program, cases and manualwww.wsharpe.com
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Equilibrium Simulation: Key Steps
Specify investors’initial conditions
Operate markets until no further trades are possible
Examine equilibrium properties
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Position
Preferences
Predictions
Position
Preferences
Predictions
Investor 1 Investor 2
MarketTrades
FinalPortfolio
FinalPortfolio
Prices
InitialPortfolio
InitialPortfolio
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Case 1Non-Mean/Variance Preferences
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Case 1Currency: fish
2 traders (Mario and Hue)
Preferences not mean/varianceConstant relative risk aversion
4 future states of the worldTotal number of fishFavored locations
AgreementAll predictions = actual probabilities
Incomplete marketOnly assets traded
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InputsCase 1
Securities: Consume Bond MFC HFCNow 1 0 0 0BadS 0 1 5 3BadN 0 1 3 5GoodS 0 1 8 4GoodN 0 1 4 8
Portfolios: Consume Bond MFC HFCMario 49 0 10 0Hue 49 0 0 10
Probabilities: Now BadS BadN GoodS GoodNProbability 1 0.15 0.25 0.25 0.35
Preferences: Time RiskAversionMario 0.96 1.5Hue 0.96 2.5
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Trading
Do a round of trades:
For each security from 2 through n
Find investors’reservation prices
Select a trade price
Obtain bid and offered quantities
Make trades for the smaller of bids and offers
If any trades were made in the round, repeat
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Equilibrium Portfolios and ConsumptionsCase 1
Portfolios: Consume Bond MFC HFCMARKET 98.00 0.00 10.00 10.00Mario 48.77 -12.16 6.24 6.24Hue 49.23 12.16 3.76 3.76
Consumptions: Now BadS BadN GoodS GoodNTOTAL 98.0 80.0 80.0 120.0 120.0Mario 48.8 37.8 37.8 62.7 62.7Hue 49.2 42.2 42.2 57.3 57.3
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Equilibrium PricesCase 1
Security Prices: Consume Bond MFC HFCMARKET 1.00 0.96 4.35 4.89Mario 1.00 0.96 4.35 4.89Hue 1.00 0.96 4.35 4.89
State Prices: Now BadS BadN GoodS GoodNMARKET 1.00 0.21 0.35 0.16 0.23Mario 1.00 0.21 0.35 0.16 0.23Hue 1.00 0.21 0.35 0.16 0.23
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Price Per ChanceCase 1
State Prices: Now BadS BadN GoodS GoodNMARKET 1.00 0.21 0.35 0.16 0.23
Probabilities: Now BadS BadN GoodS GoodNProbability 1 0.15 0.25 0.25 0.35
PPCs: Now BadS BadN GoodS GoodNPPC 1.00 1.41 1.41 0.66 0.66
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The Pricing KernelCase 1
Pricing Kernel & Consumption
0.62
0.72
0.82
0.92
1.02
1.12
1.22
1.32
1.42
78.00 83.00 88.00 93.00 98.00 103.00 108.00 113.00 118.00
Tot a l Consumpt i on
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Investor and Market ReturnsCase 1
Returns
0.79
0.89
0.99
1.09
1.19
1.29
0.79 0.89 0.99 1.09 1.19 1.29
M a r k e t Re t ur n
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The Security Market LineCase 1
Security Market Line
1.039
1.059
1.079
1.099
1.119
1.139
-0.063 0.137 0.337 0.537 0.737 0.937 1.137
Be t a
Securit ies
Port f olios
SML
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The Capital Market LineCase 1
Capital Market Line
1.035
1.055
1.075
1.095
1.115
1.135
1.155
1.175
1.195
1.215
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450
S t a nda r d De v i a t i on
Securit ies
Port f olios
CML
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Case 10Outside Positions
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Case 10
Same investors, states of the world and securities asin Case 1
Agreement
Investors have outside positions (salary income)
Total consumption is different in each state
Incomplete market
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SalariesCase 10
Salaries: Now BadS BadN GoodS GoodNMario 0 30 15 45 20Hue 0 15 25 20 40
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The Pricing Kernel & ConsumptionCase 10
Pricing Kernel & Consumption
0.57
0.67
0.77
0.87
0.97
1.07
1.17
1.27
1.37
77.75 82.75 87.75 92.75 97.75 102.75 107.75 112.75 117.75 122.75
Tot a l Consumpt i on
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The Pricing Kernel & Market ReturnCase 10
Pricing Kernel & Market Return
0.57
0.67
0.77
0.87
0.97
1.07
1.17
1.27
1.37
0.87 0.92 0.97 1.02 1.07 1.12 1.17 1.22 1.27 1.32
M a r k e t Re t ur n
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Investor and Market ReturnsCase 10
Returns
0.42
0.62
0.82
1.02
1.22
1.42
1.62
1.82
0.42 0.62 0.82 1.02 1.22 1.42 1.62 1.82
M a r k e t Re t ur n
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The Security Market LineCase 10
Security Market Line
1.078
1.098
1.118
1.138
1.158
1.178
-0.074 0.126 0.326 0.526 0.726 0.926 1.126 1.326 1.526
Be t a
Securit ies
Port f olios
SML
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The Capital Market LineCase 10
Capital Market Line
1.073
1.123
1.173
1.223
1.273
0.000 0.100 0.200 0.300 0.400 0.500
S t a nda r d D e v i a t i on
Securit ies
Port f olios
CML
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Case 15Diverse Predictions
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“Vox Populi”Francis Galton, 1907
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0
10
20
30
40
50
60
70
80
90
100
1050 1100 1150 1200 1250 1300
Weight
Per
cen
tG
reat
er
Actual Median
Estimates of Weight of Ox
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The Index Fund Premise
None of us is as smart as all of us
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Variation 1
Few of us are as smart as all of us
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Variation 2
Few of us are as smart as all of us,and it is hard to identify such people in advance
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Variation 3
Few of us are as smart as all of us,it is hard to identify them in advance,
and they may charge more than they are worth
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Case 15
10 investors5 like Mario5 like Hue
Disagreement
Predictions unbiased but subject to errorBased on independent samples from trueprobability distribution
Incomplete market
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The Pricing KernelCase 15
Pricing Kernel & Consumption
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
390.00 440.00 490.00 540.00 590.00
Tot a l Consumpt i on
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Investor and Market ReturnsCase 15
Returns
0.51
0.71
0.91
1.11
1.31
1.51
1.71
0.51 0.71 0.91 1.11 1.31 1.51 1.71
M a r k e t Re t ur n
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The Security Market LineCase 15
Security Market Line
1.037
1.057
1.077
1.097
1.117
1.137
1.157
1.177
1.197
-0.093 0.407 0.907 1.407 1.907
Be t a
Securit ies
Port f olios
SML
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The Capital Market LineCase 15
Capital Market Line
1.036
1.056
1.076
1.096
1.116
1.136
1.156
1.176
1.196
1.216
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450
S t a nda r d De v i a t i on
Securit ies
Port f olios
CML
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Cases 21 and 23Behavioral Preferences
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Case 21
17 investors:16 “standard”(constant relative risk aversion)
1 (Kevin) “behavioral”with a reference range(kinked marginal utility function)
Agreement
Complete market
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A Kinked Marginal Utility Function
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The Pricing KernelCase 21
Pricing Kernel & Consumption
0.45
0.65
0.85
1.05
1.25
1.45
1.65
1.85
1320.90 1420.90 1520.90 1620.90 1720.90 1820.90 1920.90 2020.90 2120.90
Tot a l Consumpt i on
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Investor and Market ReturnsCase 21
Returns
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50
M a r k e t Re t ur n
Kevin
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The Security Market LineCase 21
Security Market Line
1.073
1.078
1.083
1.088
1.093
1.098
1.103
1.108
1.113
1.118
-0.080 0.120 0.320 0.520 0.720 0.920 1.120 1.320 1.520
Be t a
Secur it ies
Port f olios
SML
Kevin
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The Capital Market LineCase 21
Capital Market Line
1.073
1.078
1.083
1.088
1.093
1.098
1.103
1.108
1.113
1.118
0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160
S t a nda r d De v i a t i on
Securit ies
Port f olios
CML
Kevin
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Case 23
17 investorsEach “behavioral”Reference ranges differ
Agreement
Complete market
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The Pricing KernelCase 23
Pricing Kernel & Consumption
0.29
0.79
1.29
1.79
2.29
1320.90 1420.90 1520.90 1620.90 1720.90 1820.90 1920.90 2020.90 2120.90
Tot a l Consumpt ion
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Investor and Market ReturnsCase 23
Returns
0.77
0.87
0.97
1.07
1.17
1.27
1.37
1.47
0.77 0.87 0.97 1.07 1.17 1.27 1.37 1.47
M a r k e t Re t ur n
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The Security Market LineCase 23
Security Market Line
1.072
1.082
1.092
1.102
1.112
1.122
-0.066 0.134 0.334 0.534 0.734 0.934 1.134 1.334
Be t a
Securit ies
Port f olios
SML
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The Capital Market LineCase 23
Capital Market Line
1.072
1.082
1.092
1.102
1.112
1.122
0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140
S t a nda r d De v i a t i on
Securit ies
Port f olios
CML
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Conclusions
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General Observations
The MRRT version of the Market Risk/Reward Theorem holdsrelatively well in most cases
Equivalently, asset prices are consistent with a pricingkernel that is a decreasing function of market return
The Market Risk/Reward Corollary fails in many cases
Investors do hold portfolios with non-market riskand in at least some cases they should do so
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Sound Personal Investment Advice
Diversifyto avoid unrewarded risk
Economizeto avoid unnecessary costs
Personalizeto take into account one’s situation
Contextualizeto take into account the determinants of asset prices
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Requirements for Good Investment Practice
A well thought-out view of the ways in which asset prices aredetermined:
an equilibrium model and/or simulation
A procedure for making forecasts of possible future returns thattake into account the current market values of assets
such values reflect the opinions of investorsworldwide concerning assets’future prospects
Without both ingredients it will be difficult or impossible to evenknow whether you are betting against the market and if so, inwhat manner.