---------- 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 Discrete counterpart to standard continuous time model Two investments . a riskless bond . a stock (representing the stock market) Two states of the world in each period . in one the stock goes up . in the other the stock goes down 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 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 ) If returns are binomial IID . 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 -------------- Properties of 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 If stocks are LLPW, for any constant mix strategy . 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 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 pattern in the rows in a playing field ---------- ------------ 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 ---------- ----------- 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 Assume LLPW 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 Initial budget . 75 / 1.02¹⁰ = 61.52 ------------- Implementing a probability distribution For each marker . determine wealth level . assign to a state Cost of the assignment . C = S_s c_s Express as a percent of initial budget ----------------- 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 with 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 . statistical, not necessarily economic --------- 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 the r2 value 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 with CRRA Utility Function A majority of participants conform reasonably well with CRRA . for them a constant mix strategy may be appropriate A significant minority of participants exhibit preferences that do not conform well with CRRA . for them a dynamic strategy may be preferable -------------- Finding Comparable Constant Mix Strategies Estimate risk tolerance from a regression of log(w) on
log(p) Find combination of bonds and stocks with the same risk tolerance Result is a comparable constant mix strategy --------- Constant-mix Strategies for All Participants -------------------- Participant Risk Tolerances All participants are 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 and higher mean, . the same mean and lower variance, or . higher mean and lower variance Unless utility is well approximated by a quadratic function of wealth . some utility-maximizing distributions will not be mean/variance efficient ------------ 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 with CRRA utility . r2 = 0.98 Can be compared 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 . downside protection in severe bear markets . lower return in moderate bear and bull markets . higher return in substantial bull markets . lower return in extreme bull markets Economic values of differences small . expressed as percent of initial budget ------------- 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 ----------- Conclusions: Overall All results are highly preliminary Expanded surveys planned . larger and more diverse sample size . after choice made, show comparable constant mix strategy .. ask participant to indicate %% of budget that would be spent to obtain this distribution .. would provide an economic measure of the certainty equivalent of the loss in expected utility . vary riskless alternative among participants .. would provide evidence of sensitivity of preferences to wealth More research is needed ==============