<title.htm>
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Topics
The goal of the survey
Economic models of investor choice over long horizons
The survey
Analysis of results
Conclusions
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Topic 1
The goal of the Survey
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The Survey
Designed to elicit individual choices under uncertainty
. decisions with long horizons
. focus on a single outcome
.. retirement standard of living
. provide trade-offs based on plausible model of capital market opportunities

Current results preliminary
. small number of participants
. sample may not be adequately representative of investors
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Key questions
Compatibility of responses with a standard model of investor choice
. maximization of expected utility 
. utility a power function of wealth
.. Constant Relative Risk Aversion (CRRA)

Two issues
. How many participants make CRRA compatible choices?
. In aggregate, are the responses CRRA compatible?
.. Does the "representative investor" maximize a CRRA utility function?
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Topic 2
Economic Models of Investor Choice over long horizons
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Models of Investor Behavior
Equilibrium models
. based on assumptions about the preferences of the "representative investor"
 
Financial planning advice
.  assumptions about the preferences of each individual investor
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Expected Utiity Maximization
An investor's utility is a function of consumption and/or wealth

The investor's goal is to maximize the expected value of utility
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CRRA Utility Functions
Assume outcome can be stated as a measure of future wealth (w)

Constant Relative Risk Aversion (CRRA) utility function
. u(w) = w<sup><font face="Symbol">1-l</font></sup> / (1-<font face="Symbol">l</font>&nbsp; )

Marginal utility
. mu(w) = w<sup><font face="Symbol">-l</font></sup>
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IID Processes
A return process is independent and identically distributed (IID) if
. the distribution of returns is the same each period
. regardless of the returns in prior periods
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Constant-mix Strategies
If 
. joint distribution of returns is IID, and
. an investor maximizes the expected value of a CRRA utility function

Then
. the investor should hold the same proportional values of assets every period
. a constant-mix strategy (CMS)

Therefore
. IID + CRRA ===>  CMS

Many financial planners advocate constant-mix strategies for their clients
. thus they may implicitly assume CRRA preferences for every investor
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Lognormal distributions
The value-relative (1+return) after n periods will equal the product of the n one-period value relatives

Therefore the logarithm of the n-period value relative will equal the sum of the logarithms of the n one-period value relatives

If the return process is IID
. the n-period value relative will approach lognormality as n increases
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Binomial IID Processes
Assume that there are two investments
.  a riskless bond
.  a stock (representing the stock market)

Assume that in each period there are two states of the world
. in one the stock goes up
.. value-relative = ru
. in the other the stock goes down
.. value-relative = rd

For a horizon of n periods there will be 2<sup>n</sup> possible sets of one-period outcomes
. each will have an ending stock value relative
.. stock value-relative in ending state s = W<sub>s</sub>
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State Prices
The Arrow-Debreu Price for a state
. the present cost of obtaining $1 if and only if that state occurs

One-period
. two states
. two prices
.. pu
.. pd

n-period outcomes
. 2<sup>n</sup> states
. 2<sup>n</sup> prices (but not all different)
.. price for $1 in ending state s = P<sub>s</sub>
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Price/wealth relationships with binomial returns
Define log-linear price/wealth (LLPW) as:
.   ln(P<sub>s</sub> ) = a + b*ln(W<sub>s</sub> )

Assume binomial IID process with
. n = number of periods
. nu<sub>s</sub> = number of periods stock is up for ending state s
. nd<sub>s</sub> = n-nu<sub>s</sub> = number of periods stock is down for ending state s

If returns are binomial IID
.  W<sub>s</sub> = wu<sup>nu</sup>s*wd<sup>n-nu</sup>s
.  P<sub>s</sub> = pu<sup>nu</sup>s*pd<sup>n-nu</sup>s
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LLPW relationships
If
.  W<sub>s</sub> = wu<sup>nu</sup>s*wd<sup>n-nu</sup>s
.  P<sub>s</sub> = pu<sup>nu</sup>s*pd<sup>n-nu</sup>s

Taking logarithms
. ln(W<sub>s</sub> ) = nu<sub>s</sub>*ln(wu) + (n-nu<sub>s</sub>)*ln(wd) 
. ln(P<sub>s</sub> ) = nu<sub>s</sub>*ln(pu) + (n-nu<sub>s</sub>)*ln(pd) 

Therefore, both ln(W<sub>s</sub> ) and ln(P<sub>s</sub> ) are linear functions of nu<sub>s</sub> and
.   ln(P<sub>s</sub> ) = a + b*ln(W<sub>s</sub> )
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First-order condition for expected utility maximization
For every state s
. expected marginal utility of wealth = k*price

If every state is equally probable
. mu(w<sub>s</sub> ) = k* P<sub>s</sub>

for CRRA utility
. mu(w<sub>s</sub> ) = w<sub>s</sub><sup><font face="Symbol">-l</font></sup>

Thus
. P<sub>s</sub> = k*(w<sub>s</sub><sup><font face="Symbol">-l</font></sup> )
. ln(P<sub>s</sub>) = ln(k) - <font face="Symbol">l</font>*ln(w<sub>s</sub>)
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Sufficient conditions for LLPW relationship
Stock returns follow a binomial IID process, or

Representative investor has a CRRA utility function
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Constant-mix Strategies
Constant-mix Strategy
. bond/stock mix constant in percentage terms
. in a continuous process, constant every instant
. in a binomial process, constant every period

Ending Value Relatives
. lognormally distributed
. ln(w) linearly related to ln(P)
. ln(w) linearly related to ln(w) for any other constant mix strategy
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Topic 3
The Survey
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Participants
Employees and friends from a financial services firm
. most between 20 and 40 years old
. well educated
. many with degrees in finance or engineering
. many familiar with investing for retirement 

Participation solicited for three weeks

Number of participants
. 71 total
.. 66 first-time
..  5 repeat
. 66 utilized

Median time to complete survey = 10.7 minutes
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Measure of wealth
Standard of living in retirement

Expressed as a percentage of pre-retirement standard of living
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Markers
100

Each represented by a question mark

One is the participant

The participant doesn't know which one represents him or her

Initially, all markers are in a reserve row

The ojective is to place the markers in a <strong>pattern</strong> in the rows in a playing field
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<field.htm>
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The Budget Constraint
The cost of a pattern
.  computed based on set of state prices
.  survey cannot be completed unless the cost is between 99%% and 100%% of a fixed budget

The Riskless Alternative
. all markers in the 75%% row
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<panel.htm>
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States of the World
All returns in real terms

100 States 
. Obtained from discrete approximations of lognormal distributions
..  each state equally probable
..  taken from cumulative distribution at 0.005, 0.015,.. 0.995

Properties of stock value relatives
.  10 year horizon
.  All returns in real terms
.  Riskless rate of interest
..   2%% per year
.  Stock returns independent and identically distributed (IID)
.  Lognormal distribution of stock market value relatives
.  One-year stock risk and return
..   expected return = 8%%
..   standard deviation = 18%%
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Computing the set of state prices
Properties of state prices
.  ln(P) = a - b*ln(W)
. <font face="Symbol">S</font><sub>s</sub> P<sub>s </sub>= 1 / 1.02<sup>10</sup>
. <font face="Symbol">S</font><sub>s</sub> P<sub>s </sub>*W<sub>s</sub> = 1

Solve non-linear equation to find a and b
. determines the price for each state

State prices can be used to value any set of payoffs in states
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Determining the cost of a distribution
The cost of wealth in state s
. c<sub>s</sub> = P<sub>s</sub>*w<sub>s</sub>

The total cost of the distribution
. C = <font face="Symbol">S</font><sub>s</sub> c<sub>s</sub>
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Obtaining a probability distribution
For each marker
. determine wealth level
. assign to a state

Cost of the assignment
. C = <font face="Symbol">S</font><sub>s</sub> c<sub>s</sub>
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Determining a least-cost assignment
A distribution will not be least cost if
. payoff in state s  < payoff in state s+1 and
. price for state s  < price for state s+1

To find the least-cost of a Distribution
.  order states from lowest to highest state price
.  assign payoffs to states from highest to lowest

Every distribution is assigned to states automatically using this method
.  guarantees least-cost 
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Measuring Conformance to a CRRA utility function
Given distribution, assign wealths to states
. based on least-cost

Regress ln(w) on ln(P)
. since P values are given, and participant chooses w values
. result is the best-fit CRRA utility function

Slope coefficient measures risk tolerance (rt)

R-squared (r2) measures degree of conformance to CRRA utility function
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Responses by Participant #1
<img src="part1s.jpg">
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R2 values for all participants
<img src="r2.jpg">
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Adjusted Values
Grid for responses is coarse
. 5%%, 10%%, ..., 200%%

Measured r2 value is a lower bound on r2 that might be obtained with a finer grid

To find an upper bound
.  adjust the wealth in each state towards the regression line
. maximum adjustment = 2.5%%

Fit a new regression line and measure r2
. provides an upper bound 
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R2 values for Adjusted Values
<img src="ar2.jpg">
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Conformance to CRRA utility function
A majority of participants conform reasonably well to CRRA
. for them a constant mix strategy may be appropriate

A significant minority of participants exhibit preferences that do not conform well to CRRA
. for them a dynamic strategy may be preferable
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Finding Comparable Constant Mix Strategies
Use risk tolerance from regression of log(w) on log(p)

Find combination of bonds and stocks with the same slope

Result is comparable constant mix strategy
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Constant-mix Strategies for all participants
<img src="cmstrat.jpg">
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Participant Risk Tolerances
All participants more conservative than an all-stock investor

May be representative of smaller investors
. market portfolio of traded securities approximately 60%% stocks
. retirement standard of living includes lower-risk assets
..  real estate
..  social security
..  money market funds
..  other real assets
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Mean/Variance Efficiency
A strategy is mean/variance efficient if no other strategy has
.  the same variance (standard deviation squared) and a higher mean return, or
.  the same mean return and a lower variance, or
.  a higher mean and a lower variance

Some utility-maximizing distributions will not be mean/variance efficient
. unless utility is well approximated by a quadratic function of wealth
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Mean and Standard Deviation
<img src="msd.jpg">
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Mean and Standard Deviation for Adjusted Values
<img src="amsd.jpg">
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Mean/Variance Efficiency of Participant Choices
Participant choices are not all mean/variance efficient in terms of final wealth
. especially true for higher-risk choices

But choices are mean/variance efficient in the short run
. for a binomial process, each period
. for a continuous process, each instant
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Preferences of the Representative Investor
Summary data
. aggregate of all markers

Consistency with models of capital markets
.  total wealth desired in each state
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Aggregate of All Markers
<img src="aggdist.jpg">
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Desired ln(w) and ln(P)
<img src="aggpw.jpg">
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Characteristics of the Representative Investor
Relatively conservative
. comparable constant-mix strategy = 26.5%% stocks

Conforms quite well to CRRA utility
. r2 = 0.98

Compare with <strong>market strategy</strong> with 26.5%% stocks
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Wealth Levels -- Representative Investor and Market Strategy
<img src="ww.jpg">
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Deviations from market strategy
Probably not significant in terms of preferences
.  small sample size
.  possible biases
..  reference points from prior experience (70%% as a target standard of living)

Characteristics 
. desire for downside protection in severe bear markets
. willingness to give up some return in moderate bear and bull markets
. desire for higher return in substantial bull markets
.  willingness to give up some return in extreme bull markets

Economic values of differences small
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Cumulative Value of Differences by Market Outcome
<img src="aggvw.jpg">
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Cumulative Value of Differences by State
<img src="aggvs.jpg">
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Topic 5
Conclusions
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Conclusions: Aggregate Preferences
Aggregate preferences conform well with standard assumptions
. the representative participant has preferences close to those of an investor with CRRA utility
. the economic value of the differences is less than 1%% of total value
. there is less than a 5%% probability of downside protection
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Conclusions: Individual Preferences
A majority of participants conform reasonably well to standard assumptions
.  25 of 66 had r2 values above 0.90
.  51 of 66 had upper bound r2 values above 0.90
.  for them constant mix strategies may be appropriate

A substantial minority of partipants did not conform well with standard assumptions
.  for them dynamic strategies may be preferable
.  but only if added expected utility exceeds cost

All results are highly preliminary
.  more research is needed
==============