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Individual Risk and Return Preferences:
A Preliminary Survey

William F. Sharpe

    STANCO 25 Professor of Finance, Emeritus

    Chairman, Financial Engines, Inc.

    www.wsharpe.com

    www.financialengines.com

Topics

The goal of the survey
Economic models of investor choice over long horizons
The survey
Analysis of results
Conclusions

Topic 1

The goal of the Survey

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

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?

Topic 2

Economic Models of Investor Choice over long horizons

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

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

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^1-l / (1-l )

Marginal utility

mu(w) = w^-l

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

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

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

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 2ⁿ possible sets of one-period outcomes

each will have an ending stock value relative

stock value-relative in ending state s = W_s

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

2ⁿ states

2ⁿ prices (but not all different)

price for $1 in ending state s = P_s

Price/wealth relationships with binomial returns

Define log-linear price/wealth (LLPW) as:
- ln(P_s ) = a + b*ln(W_s )

Assume binomial IID process with
- n = number of periods
- nu_s = number of periods stock is up for ending state s
- nd_s = n-nu_s = number of periods stock is down for ending state s

If returns are binomial IID

W_s = wu^nus*wd^n-nus
P_s = pu^nus*pd^n-nus

LLPW relationships

If
- W_s = wu^nus*wd^n-nus
- P_s = pu^nus*pd^n-nus

Taking logarithms
- ln(W_s ) = nu_s*ln(wu) + (n-nu_s)*ln(wd)
- ln(P_s ) = nu_s*ln(pu) + (n-nu_s)*ln(pd)

Therefore, both ln(W_s ) and ln(P_s ) are linear functions of nu_s and

ln(P_s ) = a + b*ln(W_s )

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_s ) = k* P_s

for CRRA utility
- mu(w_s ) = w_s^-l

Thus

P_s = k*(w_s^-l )
ln(P_s) = ln(k) - l*ln(w_s)

Sufficient conditions for LLPW relationship

Stock returns follow a binomial IID process, or

Representative investor has a CRRA utility function

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

Topic 3

The Survey

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

Measure of wealth

Standard of living in retirement

Expressed as a percentage of pre-retirement standard of living

Markers

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 pattern in the rows in a playing field

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The Playing Field

Sample Rows

Actual Rows

Reserve Row
Rows from 5% to 200%

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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The Control Panel

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%
  Computing the set of state prices
  - Properties of state prices
    - ln(P) = a - b*ln(W)
    - S_s P_s= 1 / 1.02¹⁰
    - S_s P_s*W_s = 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
  Determining the cost of a distribution
  - The cost of wealth in state s
    - c_s = P_s*w_s
  - The total cost of the distribution
    - C = S_s c_s
    Obtaining a probability distribution
    - For each marker
      - determine wealth level
      - assign to a state
    - Cost of the assignment
      - C = S_s c_s
      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
        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
        Responses by Participant #1
        
        R2 values for all participants
        
        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
        R2 values for Adjusted Values
        
        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
        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
        Constant-mix Strategies for all participants
        
        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
        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
        Mean and Standard Deviation
        
        Mean and Standard Deviation for Adjusted Values
        
        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
        Preferences of the Representative Investor
        
        Summary data
        
        aggregate of all markers
        
        Consistency with models of capital markets
        
        total wealth desired in each state
        Aggregate of All Markers
        
        Desired ln(w) and ln(P)
        
        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 market strategy with 26.5% stocks
        Wealth Levels -- Representative Investor and Market Strategy
        
        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
        Cumulative Value of Differences by Market Outcome
        
        Cumulative Value of Differences by State
        
        Topic 5
        
        Conclusions
        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
        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
        FINIS