Risk Preferences and Efficiency of Household Portfolios
RRisk Preferences and Efficiency of Household Portfolios
Agostino Capponi ∗ Zhaoyu Zhang † October 28, 2020
Abstract
We propose a novel approach to infer investors’ risk preferences from their portfoliochoices, and then use the implied risk preferences to measure the efficiency of invest-ment portfolios. We analyze a dataset spanning a period of six years, consisting of endof month stock trading records, along with investors’ demographic information andself-assessed financial knowledge. Unlike estimates of risk aversion based on the shareof risky assets, our statistical analysis suggests that the implied risk aversion coefficientof an investor increases with her wealth and financial literacy. Portfolio diversification,Sharpe ratio, and expected portfolio returns correlate positively with the efficiency ofthe portfolio, whereas a higher standard deviation reduces the efficiency of the port-folio. We find that affluent and financially educated investors as well as those holdingretirement related accounts hold more efficient portfolios.
Keywords:
Risk Aversion, Capital Market Line, Portfolio Efficiency, Investors’ Demograph-ics
JEL classification : G11, D81, R20 ∗ Department of Industrial Engineering and Operations Research, Columbia University, New York, USA,10027, e-mail: [email protected]. † Department of Mathematics, University of Southern California, Los Angeles, California, USA, 90089,e-mail: [email protected]. a r X i v : . [ q -f i n . P M ] O c t Introduction
Attitude towards risk is a crucial factor in the investors’ decision making process. In thiswork, we first propose a novel approach to infer investors’ risk preferences from portfoliochoices. We then leverage a unique data set of household portfolio positions and correspond-ing demographics, spanning a six-year time horizon, to shed light on the main determinantsof risk preferences and the efficiency of portfolio allocations.The concept of portfolio efficiency dates back to the pioneering work of Markowitz (1952)on mean-variance analysis. In his work, Markowitz assumes that investors are risk averse,and trade off risk and return of their portfolios. Investors act rationally and, for a givenlevel of risk, they choose the portfolio which yields the highest expected return. Plottingthese portfolios in the risk and expected return space traces the efficient frontier (or “theMarkowitz bullet”) in the absence of a risk-free asset. If a risk-free asset is included, theline which is tangent to the efficient frontier at the portfolio with the highest Sharpe ratiois called the capital market line (CML).According to theory, any rational investor would choose a portfolio on the capital mar-ket line depending on her risk preferences. However, using our data set we observe thatthe majority of investors’ portfolios lie under the capital market line (see Figure 1 for anillustration). This provides supportive evidence that the vast majority of investors do notoutperform, and rather underperform, the market. Existing literature has identified severalreasons why this may happen: excessive trading (e.g. Odean (1998) and Kumar (2009)),overconfidence (e.g. Barber and Odean (2001)), and underdiversification (e.g. Kelly (1995)and Gaudecker (2015)).A large body of literature has investigated the process of inferring risk preferences fromdata in various environments. Sahm (2012) used panel data on hypothetical gambles overlifetime income to quantify changes in risk tolerance over time. Chiappori, Salanie, Salanie,and Gandhi (2019) identified the distribution of risk preferences among bettors in parimutuelhorse races; Kimball, Sahm, and Shapiro (2008) developed a quantitative proxy for risk toler-2nce based on hypothetical income survey. Schubert, Brown, Gysler, and Brachinger (1999)investigated the effect of gender on risk attitudes in financial decision-making via a designedexperiment. Bucciol and Miniaci (2011) estimated risk preferences from household surveyportfolio data, which distinguish between risk free assets, bonds, stocks, human capital, andreal estate. Unlike the above mentioned studies, we focus on the stock market, and considertwo types of data sets. The first are investor specific data, which consist of portfolio positionsand demographics of households (Barber and Odean, 2000) . This is a rich dataset, whichincludes end of month trading records, for a period of six years ranging from 1991 through1996. The second dataset consists of market data used to construct expected values andstandard deviations of asset returns based on Fama French factors and industry portfolios.We next describe our approach for inferring the risk preferences of an investor. In ourwork, we project the expected return and standard deviation of an investor’s portfolio ontothe capital market line. We define the investor’s risk aversion coefficient to be that associatedwith the projection of the portfolio on the capital market line. Such a risk aversion coefficient θ can be computed explicitly as θ = (1 + λ mkt ) /σ obs λ obs + 1 /λ mkt , and is determined by the market Sharpe ratio ( λ mkt ), the observed portfolio Sharpe ratio( λ obs ), and the observed portfolio standard deviation ( σ obs ). Consistent with intuition, ourformula implies that investors become less risk averse as (i) the portfolio standard deviationincreases, and (ii) the portfolio Sharpe ratio increases.Our proposed approach differs from those used in existing literature, which either usethe proportion of risky assets over total wealth (see Riley Jr. and Chow (1992)), or insteadutilize the portfolio standard deviation (see Bucciol and Miniaci (2011)) to imply the riskaversion coefficients.We use the formula developed above to assess the relationship between risk attitudes We are grateful to Prof. Terrence Odean for agreeing to share the data set with us.
In this section, we present the methodology used to infer risk preferences from investorportfolio choices, and define the measure of portfolio efficiency used.5 .1 Implied Risk Preferences
Our formulas for implying investors’ risk aversion and measuring portfolio efficiency builds onthe foundational work of Markowitz (1952). According to Markowitz’s portfolio criterion, fora given level of risk investors choose the portfolio which yields the highest expected return.Such optimal portfolios, when plotted in the standard deviation and expected return space,lie on the so-called efficient frontier. Tobin (1958) extended Markowtiz’s work by addinga risk-free asset. His “separation theorem” leads a new efficient frontier, called the capitalmarket line. This line is a graphical representation of all efficient portfolios obtained froma linear combination of the risk-free asset, with risk-free rate r f , and the market portfolio,with return µ mkt and standard deviation σ mkt . The return µ p and standard deviation σ p ofany portfolio on the capital market line satisfies: µ p := r f + µ mkt − r f σ mkt σ p = r f + λ mkt σ p , (1)where λ mkt := µ mkt − r f σ mkt is called the Sharpe ratio of the market. In Figure 1, we plot the capital market lineand observed investors’ portfolios for the month of January 1992. The data set used toconstruct the market portfolio comes from the Fama-French 49-industry portfolio. Eachrational investor chooses a portfolio along the capital market line based on her risk preference.Such a portfolio maximizes the investor’s returns for a given level of risk. The existence ofa one-to-one correspondence between the risk aversion coefficient and the portfolio on thecapital market line can be seen from the mean-variance criterion. Suppose an investor hasrisk aversion coefficient θ . Then, he would invest a proportion w ∗ of her wealth in the marketportfolio and the remaining proportion of (1 − w ∗ ) in the risk-free asset. The weight w ∗ which6aximizes the mean-variance utility is given by: w ∗ = arg max w { wµ mkt + (1 − w ) r f − θ w σ mkt } . From the first order condition, we obtain w ∗ = µ mkt − r f σ mkt θ = λ mkt θσ mkt . (2)Denote by ( µ obs , σ obs ) the expected return and standard deviation of observed portfolio forFigure 1: Capital market line and observed investor portfolios using the data set for themonth of January 1992. The data used to construct the market portfolio comes from theFama-French 49-industry portfolio. The expected return and standard deviation for in-vestors’ portfolios and the Fama-French 49-industry portfolios are estimated using a Fama-French five-factor model. The expected return and standard deviation of the market portfoliocan be computed explicitly as shown in Section 4. We take the risk-free rate from the Fama-French five-factor dataset.an investor. Because of the one-to-one correspondence between risk preferences and portfoliochoices in (2), we can use the capital market line to infer the investor’s risk aversion. Thisis accomplished by projecting the investor’s portfolio onto the capital market line. We thendefine the implied risk aversion coefficient of the portfolio to be that of the projected portfolio7 µ ⊥ obs , σ ⊥ obs ). A straightforward calculation shows that the expected return and the standarddeviation of the projected portfolio is given by( µ ⊥ obs , σ ⊥ obs ) = (cid:18) λ mkt µ obs + λ mkt σ obs + r f λ mkt , λ mkt µ obs + σ obs − λ mkt r f λ mkt (cid:19) . The risk aversion coefficient associated with the portfolio with ( µ ⊥ p , σ ⊥ p ) is obtained by match-ing the volatility of the portfolio with risk aversion θ , given by w ∗ σ mkt , with the volatility ofthe projected portfolio, λ mkt µ obs + σ obs − λ mkt r f λ mkt . Formally, w ∗ σ mkt = λ mkt µ obs + σ obs − λ mkt r f λ mkt , and it then follows that the implied risk aversion for an observed portfolio with expectedreturn and standard deviation ( µ obs , σ obs ) is given explicitly by θ = (1 + λ mkt ) /σ obs λ obs + 1 /λ mkt . (3)It is clear from our formula that the investor’s risk aversion coefficient depends bothon the portfolio mean and the portfolio standard deviation. Moreover, the formula impliesthat, given two portfolios with the same expected return, the investor who holds the onewith the higher standard deviation is less risk averse. This is illustrated in Figure 2, whereinvestor a with risk aversion coefficient equal to 39 holds a portfolio with expected returnand standard deviation of (0 . , . c with risk aversion coefficient of 27holds a portfolio with expected return and standard deviation of (0 . , . σ a , µ a ) to ( σ b , µ b )) would yield an increase ofthe risk aversion coefficient by 10. 8igure 2: Comparison of investors’ ( a, b, c ) implied risk aversion coefficients θ and portfolioefficiency E based on the expected return and standard deviation of investors’ portfolios. In our data, the vast majority of investors’ portfolios are not sufficiently close to the capitalmarket line, i.e., they are inefficient. Existing research has provided various explanationsfor the origins of these inefficiencies. For example, Barber and Odean (2000) find thathouseholds’ overconfidence leads to excessive trading, and households’ portfolios are largelyunder-diversified. As a result, the average household net annual return is much lower thanthe market returns.We next define the measure of efficiency that we use to evaluate household portfolios.
Definition 2.1
A portfolio is efficient if its Sharpe ratio is greater than the market’s Sharperatio. If, instead, the reverse inequality holds, we say that the portfolio is inefficient.
We measure the efficiency of the portfolio by the Euclidean distance between the observedportfolio and the projected portfolio on the capital market line.9ur definition of inefficiency is driven by the following considerations. If the portfolio isinefficient, the projected portfolio is the closest “optimal portfolio” (in the sense of lying onthe efficient frontier) a risk-averse investor can attain by eliminating the excess volatility andincreasing the expected return, while maintaining the same risk preference. For example, inFigure 2 an investor with risk aversion coefficient equal to 39 can achieve a “better” portfolio,if she moves from ( σ a , µ a ) to ( σ ⊥ a , µ ⊥ a ). The new portfolio has a standard deviation which islower by 0.1, and an expected return which is higher by 0.05. Mathematically, we denotethe portfolio efficiency by E , and quantify it as E = (cid:112) ( µ obs − µ ⊥ obs ) + ( σ obs − σ ⊥ obs ) if µ obs − r f σ obs > µ ⊥ obs − r f σ ⊥ obs , − (cid:112) ( µ obs − µ ⊥ obs ) + ( σ obs − σ ⊥ obs ) if µ obs − r f σ obs < µ ⊥ obs − r f σ ⊥ obs . (4) We use three data sets in our analysis: investor portfolios and demographics, Fama Frenchindustry portfolios, and Fama French factors.
Our data set of investors’ portfolio positions and corresponding investor demographics comefrom Barber and Odean (2000). This dataset is provided by a large discount brokerage firm,and it includes the end of month trading records, from January 1991 through December1996, for the accounts opened by the households as well as the corresponding demographicinformation, account type, and self-assessed suitability. We filter out households for whichthere is missing information (such as missing net worth and income information), or suchthat their portfolios consist of assets whose prices are not available in the Center for Researchin Security Prices (CRSP) database. We are then left with a dataset of 37,108 households10ncluding 54,141 accounts. Investors’ account, demographics, and suitability information are all recorded at the timeinvestors opened their accounts. They include:(1)
Net worth proxy : The net worth of each household is assigned to one of the followingsix categories based on the dollar amount: 1–24,999, 25,000–49,999, 50,000–74,999,75,000–99,999, 100,000–249,999, and > Income proxy : The income proxy is recorded in one of five categories, depending onthe dollar amount of the households’ annual income: 1–24,999, 25,000–49,999, 50,000–74999, 75,000–99,999, > Knowledge : This variable records self assessed financial knowledge. 3103 accounts haveextensive knowledge, 11188 accounts have good knowledge, 7373 accounts have limitedknowledge, 2087 accounts have zero knowledge, and the remaining accounts did notreport their knowledge information.(4)
Age : The age variable records the oldest person’s age in a household. It belongs toone of seven groups: 18–24, 25–34, 35–44, 45–54, 55–64, 65–74, >
75. These groupsconsist of 52, 1843, 8822, 10289, 6746, 5521, and 3835 households respectively.(5)
Number of children : A integer from 0 to 6. 26154 households have no child, 6236households have 1 child, 3525 households have 2 children, 966 households have 3 chil-dren, 208 households have 4 children, 17 households have 5 children, and 2 householdshave 6 children.(6)
Marital status : The marital status categories include married, single, inferred married,inferred single, and unknown. They consist of 23834, 6165, 1146, 1792 and 4171households, respectively. Our original dataset of positions consists of 128829 accounts. The demographic dataset includes 55432households. The suitability dataset, which contains self-reported net worth, income, and financial knowledgeinformation, includes 77995 households. The base dataset (containing account information) has 158006accounts opened by 77995 households.
Length of residence : An integer ranging from 0 to 15. Its 25, 50 and 75 percentilesare 3, 7 and 14 respectively. A household has a value of 0 if it has been in residencefor less than one year, and a value of 15 if it has been in residence for fifteen years ormore. The values of 1 through 14 refer to the number of years the household has beenin residence. The mean of this variable is equal to 7.76. (8) Number of cars : The data set shows that 10778 households have 1 car, 5748 householdshave 2 cars, 2383 households have 3 cars, and the remaining households do not own acar.(9)
Number of credit cards : This is an integer from 0 to 6. We observe from that the dataset that 1193 households, 3006 households, 6240 households, 11693 households, 12983households, 1958 households and 35 households have 0, 1, 2, 3, 4, 5, 6 credit cardsrespectively.(10)
Account types : Cash (12618 accounts), Individual Retirement (18734 accounts), Keogh(478 accounts), Margin (5462 accounts), Schwab One (16849 accounts). • The
Cash Account is a type of brokerage account, in which the investor must paythe full amount for securities purchased, and he is not allowed to borrow fundsfrom his broker to pay for transactions in the account. • The
Individual Retirement Account (IRA) is an investment account for retirementsavings. • The
Keogh Account is a tax-deferred pension plan available to self-employed in-dividuals or unincorporated businesses for retirement purposes. • The
Margin Account is a brokerage account, in which the broker lends the cus-tomer cash to purchase stocks or other financial products.(11)
Client segments : Households can be grouped into three categories: active trader (6450accounts), affluent household (10325 accounts), general brokerage (37366 accounts): The raw data indicate that number of households in each of those categories are respectively, 1715, 3512,3272, 2816, 2423, 2470, 2095, 2100, 1909, 1688, 1203, 984, 868, 633, 766, 8654. Affluent household : Households which hold more than $ • Active trader : Households which make more than 48 trades in any year. If ahousehold qualifies as either active trader or affluent, it is assigned the activetrader label. • General brokerage : All other households are labeled as general.
We use data from Fama-French 49 industry portfolios to characterize the capital marketline. The portfolio consists of all NYSE, AMEX, and NASDAQ stocks classified basedon their Standard Industrial Classification (SIC) codes. We use the value-weighted dailyreturns of the industry portfolio. In addition, we obtain the historical returns of stocks inthe households’ portfolios from the CRSP dataset. The historical returns of the 49 industryportfolios and investors’ portfolios span a period of three years.
We use a Fama-French five-factor dataset to estimate the expected return and covariancematrix of the portfolios described above. We estimate the coefficients ( µ, Σ), both for the49-industry portfolio and investors’ portfolios.Fama and French (1993) developed an asset pricing model consisting of three factors.Their model is able to capture anomalies that are not explained by the capital asset pricingmodel developed by Sharpe (1964). Roughly speaking, the excess return ( r i,t − r f,t ) of anasset i at time t linearly depends on the excess return of the market ( r mkt,t − r f,t ), thedifference between the returns of a portfolio consisting of small cap and large cap stocks,denoted by SM B t at time t , and the difference between the returns of a portfolio composed of The SIC is a system used to classify industries based on a four-digit code. Established in the UnitedStates in 1937, it is used by government agencies to classify industry areas.
HM L t , at time t . In a subsequentstudy, Fama and French (2014) added the profitability and investment factors into the three-factor model, leading to a five-factor model. In addition to the aforementioned three factors, RM W t denotes the return spread of the most profitable over the least profitable firms, and CM A t denotes the return spread of firms that invest conservatively over those that investaggressively. The methodology discussed in Section 2 allows inferring investors’ risk aversions and portfolioefficiencies from (i) the capital market line, and (ii) the mean and standard deviation ofreturns of each portfolio in each account.First, we estimate the mean and covariance matrix of portfolio returns in each accountfor each household, as well as those of the 49-industry portfolio. The Fama-French five-factormodel states that the excess return of asset i over the risk free rate, ( r i,t − r f,t ), in a universeof p assets satisfies: r i,t − r f,t = α i + β i ( r mkt,t − r f,t ) + β i SM B t + β i HM L t + β i RM W t + β i CM A t + (cid:15) i,t , (5)where β ij , i = 1 , . . . , p and j = 1 , . . . , (cid:15) , · , . . . , (cid:15) p, · are p idiosyncratic error terms. These errors have zero mean, and are uncorrelated with thespecified factors f , where we set f := [( r mkt − r f ) , SM B, HM L, RM W, CM A ] T . Given the realizations of observed factors at t = 1 , ..., T , the factor betas for each asset areestimated from multivariate regression. Thus, the estimated expected return vector (ˆ µ T )14nd the estimated covariance matrix ( ˆΣ) areˆ µ = r f + ˆ α + ˆ β ¯ f , ˆΣ = ˆ β (cid:100) Cov( f ) ˆ β T + diag (Var( (cid:15) ) , . . . , Var( (cid:15) p )) , where ˆ β is the matrix of estimated regression coefficients, and (cid:100) Cov( f ) is the sample covariancematrix of the factors f .Next, the capital market line specifies the optimal combination of a risk-free asset andthe market portfolio, and it is tangent to the efficient frontier. We characterize the mar-ket portfolio from Fama-French 49-industry portfolio, which assign all stocks from NYSE,AMEX, and NASDAQ to 49 categories based on their SIC codes. Such a market portfoliocan be computed explicitly, as we show in the next proposition. Proposition 4.1
Consider a market consists of p risky assets, and let r f be the risk-freerate. Let µ = [ µ , µ , . . . , µ p ] T be a column vector of mean returns of p risky assets, and Σ the covariance matrix of returns. Then the expected return and standard deviation of themarket portfolio admits the explicit expression given by µ mkt = A − Br f B − Cr f ,σ mkt = (cid:113) A − Br f + Cr f B − Cr f , (6) where A = µ T Σ − µ,B = µ T Σ − e = e T Σ − µ,C = e T Σ − e. (7)Using the above expressions for the expected return and volatility of the market port-folio, we can immediately obtain the capital market line, which connects the risk-free ratewith the market portfolio. Hence, we can obtain the mean return and volatility for each15ortfolio, at any time, from the Fama-French five-factor model. The capital market line isalso explicitly given from the above computation. The risk aversion coefficients are readilyobtained by projecting the observed portfolios on the capital market line according to themethod described in Section 2. We imply the risk aversion coefficient for each account of each household. This yields atotal of 54,141 accounts, together with 1,981,277 risk aversion coefficients, one for each ofthe 72 monthly periods between 1991 and 1996 over which the household invests. The 25th,50th, and 75th percentiles of such risk aversion coefficients are 40.97, 67.49, and 108.74,respectively.Figure 3 presents scatter plots of risk aversion coefficients versus portfolios’ Sharpe ratiosand standard deviations. The data indicate that an investor becomes less risk averse as herportfolio Sharpe ratio increases, and that she becomes more risk averse if she experiences aloss and her portfolio Sharpe ratio becomes negative. In summary, we find that investorswho achieve higher portfolio Sharpe ratio in the stock market are willing to take more risk.To the best of our knowledge, this finding has not been highlighted in earlier literature.Additionally, such finding is consistent with results in the behavioral finance literature. Forinstance, Thaler and Johnson (1990) find that prior gains can increase subjects’ willingnessto accept gambles, while prior losses decrease their willingness to take risks. Moreover, andconsistent with intuition, investors who are willing to take more risk, i.e., less risk averse,tend to hold more volatile portfolios. These findings are consistent with earlier studies,such as Cohn, Lewellen, Lease, and Schlarbaum (1974), who elicits investors’ risk preferenceinformation via questionnaires, Riley Jr. and Chow (1992) who uses financial data for alarge random sample of U.S. households, and Bucciol and Miniaci (2011) who uses a survey16f consumer finances dataset.Figure 3: Left: Scatter plot of risk aversion versus portfolio Sharpe ratio. Right: Scatterplot of risk aversion versus portfolio standard deviation.Next, we analyze the dependence of risk preferences on investors’ demographics, financialliteracy and wealth characteristics. Figure 4 presents histograms of the time average ofinvestors’ risk aversion coefficients, obtained from the procedures discussed in Sections 2and 4, versus investors’ account types, net worth, knowledge, segments, ages, and numberof children in the household.The first plot in Figure 4 presents the distribution of risk preferences across all accounttypes. The data show that more than 60% of the margin accounts have risk aversion smallerthan 80, which is the highest percentage across all account types. We also observe that morethan 50% of investors with Keogh account have risk aversion greater than 80, which is thehighest risk aversion among account types. The other types of accounts all lie somewherein the middle. These observed statistics can be understood from the purposes of theseaccounts: the margin account allows customers to borrow money from the broker, while theKeogh Account is for retirement purposes only and thus associated with a high level of riskaversion.The second and the third graph in Figure 4 show that more wealth and financially moreliterate investors tend to be more risk averse. This can be understood by the fact thatfinancially more knowledgeable and wealth investors are able to diversify their risks better17y holding more securities.The fourth plot of Figure 4 shows that general households are the least risk averse, whileaffluent households are the most risk averse. This can be explained by the fact that affluenthouseholds hold more diversified portfolios, which imply a higher risk aversion. Furthermore,the fifth plot of Figure 4 indicates that, investors become more risk averse as they get older.This finding is consistent with a large body of literature (see, for instance, Sahm (2012),Bucciol and Miniaci (2011), Palsson (1996), and Morin and Suarez (1983)). Lastly, investorstend to be less risk averse as the number of children grows from 0 to 2, but investors withmore than 3 children tend to be more risk averse. The “number of children” factor mighthave little impact on investors’ risk aversion, since the distribution of risk aversion acrossthe number of children is not very obvious.We conclude with an investigation of how investors’ portfolio efficiency depends on de-mographics and account information. The 25%, 50% and 75% percentiles of the portfolioefficiencies are -0.073, -0.05, and -0.035 respectively. Figure 5 reports histograms of investors’portfolio efficiency versus investors’ account types, net worth, knowledge, segments, ages, andthe number of children in their households. The top left histogram shows that portfolios inmargin and cash accounts are less efficient than those in the retirement related accounts.As for the the relationship between portfolio efficiency and net worth, from the second plotin Figure 5, we see that investors with more net worth hold more efficient portfolios whenthe net worth exceeds $ In this section, we examine the determinants of risk aversion coefficients and portfolio effi-ciencies using a linear panel regression. We also compare our results to existing literature.
We perform a linear regression of implied risk preferences on investors’ demographic infor-mation, account information, portfolio’s characteristics, and market conditions. The corre-sponding results are reported in Table 1.We first remove outliers, i.e., implied risk aversions that are above or below 1.5 times theinterquartile range. After removing 119,315 outliers out of 1,861,962 points, the adjusted25%, 50% and 75% percentiles of the implied risk aversions are 39.61, 64.14, and 99.01.The demographic information includes net worth proxy, income proxy, financial literacy,ages, number of children, marital status, length of residence, number of cars, and numberof credit cards. The account information includes investors’ account types, client segments,and investors’ portfolio information include the number of stocks in their portfolio, portfolioSharpe ratio, portfolio expected return, and portfolio volatility. Lastly, we use the VIX indexas a proxy for market volatility. The VIX typically rises during times of financial stress andmarket selloffs, and falls as investors become complacent.Table 1 reports the results of the linear panel regression model. We account for timefixed effects in Column (1) and Column (2) of Table 1, where we control for variables thatare constant across entities but vary over time, such as market volatility.Column (1) shows the existence of a statistically significant positive relation betweeninvestors’ implied risk aversion and their net worth. Investors’ financial literacy is also20igure 5: Histograms of portfolio efficiency based on account types, net worth, financialknowledge, client segments, ages, and number of children of households.21 statistically significant predictor. Investors who are financially less literate, i.e., whoseknowledge moves from extensive to limited, are less risk averse. This confirms statisticallythe visual findings from Figure 4. It is worth remarking that existing literature (e.g. RileyJr. and Chow (1992), Guiso and Paiella (2008), and Rooij et al. (2011)) has measured theindividual risk preferences of an investor from its proportion of total wealth allocated torisky assets. They argue that, with an increase in wealth and education level, the investors’willingness to accept more risk would increase. This means that investors are willing toallocate a larger proportion of total wealth to the risky assets. We would like to remark thatour results are not contradicting theirs. In the market of stocks we are considering, wealthierand financially more literate investors are more likely to hold a more diversified portfolio,relative to less wealthy and financially less savy investors. This in turn reduces the risk theyare taking, and as a result their implied risk aversion coefficient is higher.Interestingly, our regression shows a statistically significant and negative linear depen-dence between risk aversion and investors’ income, for large incomes (greater than $ After controlling for the demographic information and account information, in Column(2) of Table 1, we confirm statistically that investors holding more securities portfolios are We remark that the data set in Guiso and Paiella (2008) consists of only 2377 observations, and hefinds a R coefficient of about 12%. Our data set consists of a significantly large number of observations,1 , , R . Column (3) of Table 1 shows that market variables impact riskpreferences, and the R is equal to 0.039. Specifically, investors become more risk aversewhen the aggregate risk in the market is high. Malmendier and Nagel (2011) come to asimilar conclusion by measuring risk attitudes using four measures (survey of willingness totake risk, stock market participation, bond market participation, and the fraction of liquidassets invested in stocks) and data from Survey of Consumer Finances from 1960 to 2007.If we additionally control for portfolio characteristics, such as number of stocks, portfolioSharpe ratio, portfolio mean and volatility beyond the VIX index, the proportion of explainedvariation increases to nearly 10%. Despite investors’ net worth are significant, there seems to be no obvious linear relationshipbetween net worth and their portfolio efficiency (see Column (1) of Table 2). Portfolios aremore efficient for investors whose net worth is larger than $ In addition, This is obtained by removing the time-invariant characteristics, such as demographic and account infor-mation. The comparison of the coefficients of the categorical variables are based on the reference group. Thereference group in this regression consists of investors whose age is between 18 and 24, and who are inferredmarried, and affluent traders invested in Cash account with 0-24999 net worth, 0-24999 annual income, and R = 0 . extensive financial knowledge. Moreover, any rational investor would prefer to maximize return for a speci-fied level of risk, or equivalently, to minimize risk for a given level of return. This explainswhy expected return impacts positively, while volatility impacts negatively, the portfolio’sefficiency. Controlling for demographics, account, and portfolio characteristics altogetherexplain all variation in the data, i.e., the R reaches 99.4%.We next turn to analyze the impact of market conditions on portfolio efficiency, account-ing for fixed entity effects. Column (4) of Table 2 shows that the VIX index has a negativeimpact on portfolio efficiency. However, there appears to be no evidence of any correlationbetween these two variables ( R is almost 0, and adjusted R is negative). Accounting forportfolio characteristics, in addition to market variables, the R coefficient reaches 91 .
5% (seeColumn (4) of Table 2). Hence, we can conclude that the portfolio efficiency is primarilydriven by portfolio characteristics and very little by market variables.Lastly, to further investigate which portfolio variables contribute the most to portfolioefficiency, we perform two way fixed effect regressions on portfolio variables in Table 3. Wethen conclude that portfolio volatility is a driving factor of portfolio efficiency. This suggeststhat, as investors’ portfolios become more volatile, their portfolios also become less efficient,and portfolio under-diversification could be one of the reasons for this decrease in efficiency.
In this study, we have analyzed portfolio stock holdings of investors and related the impliedrisk aversion profiles to their demographics. Our study shows that the majority of investors’portfolios are inefficient in the sense that they lie below the capital market line. As thecapital market line traces the optimal portfolios associated with any possible risk aversion Gibbons et al. (1989) proposed a test of efficiency based on the based on the Sharpe ratios differencebetween the observed and the efficient portfolio. able 1: Linear Panel Regression Models of Risk Aversion
Dependent variable:
Risk Aversion(1) (2) (3) (4)Net Worth 25,000-49,999 − ∗ ∗∗∗ ∗∗∗ Net Worth 75,000-99,999 − ∗ − ∗∗∗ ∗∗∗ Net Worth > ∗∗∗ ∗∗∗ Income 25,000-49,999 2.165e-03 0.177Income 50,000-74,999 0.061 − − ∗∗∗ − ∗∗∗ Income > − ∗∗∗ − ∗∗∗ Good Knowledge − ∗∗∗ − ∗∗∗ Limited Knowledge − ∗∗∗ − ∗∗∗ None Knowledge − ∗∗∗ − ∗∗∗ Unknown Knowledge − ∗∗∗ − ∗∗∗ Ages 25-34 − − ∗ − ∗ Ages 55-64 2.796 ∗∗∗ ∗∗∗
Ages 67-74 8.402 ∗∗∗ ∗∗∗
Ages >
75 11.168 ∗∗∗ ∗∗∗
Num. of Children − − ∗∗∗ − ∗∗∗ Married − ∗∗ − ∗∗∗ Single − − ∗∗∗ Unknown Marital − ∗∗∗ − ∗∗∗ ength of Residence 0.062 ∗∗∗ − ∗∗∗ ∗∗∗ Num. of Credit Cards − ∗∗∗ − ∗∗∗ IRA Account 2.681 ∗∗∗ ∗∗∗
Keogh Account 4.428 ∗∗∗ ∗∗∗
Margin Account − ∗∗∗ − ∗∗∗ Schwab Account 4.431 ∗∗∗ − − ∗∗∗ − ∗∗∗ Active Trader − ∗∗∗ − ∗∗∗ Num. of Stocks 4.134 ∗∗∗ ∗∗∗
Portfolio Sharpe Ratio − ∗∗∗ − ∗∗∗ Portfolio Expected Return − ∗∗∗ − ∗∗∗ Portfolio Std. Deviation − ∗∗∗ − ∗∗∗ General Brokerage x Num. of Stocks 0.68 ∗∗∗ ∗∗∗
Active Trader x Num. of Stocks − ∗∗∗ − ∗∗∗ VIX Index 2.572 ∗∗∗ ∗∗∗
Observations 1861962 1861962 1861962 1861962R ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ (df = 33; 1861858) (df = 39; 1861852) (df = 1; 1807948) Note: ∗ p < ∗∗ p < ∗∗∗ p < Table 2: Panel Regression Models of Portfolio Efficiency
Dependent variable:
Portfolio Efficiency(1) (2) (3) (4) (5)Net Worth 25,000-49,999 − ∗∗ − ∗∗∗ − ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ > ∗∗∗ ∗∗∗ − ∗∗∗ − ∗∗∗ − − ∗∗∗ − ∗∗∗ − ∗ (1.533e-04) (1.520e-04) (1.187e-05)Income 75,000-99,999 − ∗∗∗ − ∗∗∗ − ∗∗ (1.525e-04) (1.512e-04) (1.181e-05)Income > − ∗∗∗ − ∗∗∗ − − ∗∗∗ − ∗∗∗ − ∗∗ (1.077e-04) (1.070e-04) (8.360e-06)Limited Knowledge − ∗∗∗ − ∗∗∗ − None Knowledge − ∗∗∗ − ∗∗∗ − − ∗∗∗ − ∗∗∗ − ∗∗∗ ∗∗∗ − ∗ ∗ − ∗∗∗ ∗∗∗ − ∗∗∗ ∗∗∗ − ∗∗∗ ∗∗∗ − >
75 0.012 ∗∗∗ ∗∗∗ − − ∗∗∗ − ∗∗∗ − − ∗∗∗ − ∗∗∗ − − − ∗∗ − − ∗∗ − − ∗∗∗ − ∗∗∗ − − ∗∗∗ (4.460e-06) (4.420e-06) (3.500e-07)Num. of Cars 1.764e-05 2.536e-04 ∗∗∗ ∗∗ (2.371e-05) (2.355e-05) (1.840e-06)Num. of Credit Cards − ∗∗∗ − ∗∗∗ − ∗∗∗ (1.998e-05) (1.982e-05) (1.550e-06)IRA Account 4.294e-04 ∗∗∗ ∗∗∗ − ∗∗∗ − ∗∗∗ − ∗∗∗ − ∗∗∗ (5.810e-05) (4.580e-06)Active Trader − ∗∗∗ − ∗∗∗ (8.123e-05) (6.400e-06)Num. of Stocks 1.378e-05 ∗∗∗ ∗∗∗ (5.700e-07) (2.580e-06)Portfolio Sharpe Ratio 1.000e-08 2.700e-07(3.000e-07) (5.700e-07)Portfolio Expected Return 0.523 ∗∗∗ ∗∗∗ (1.086e-04) (3.010e-04)Portfolio Std. Deviation − ∗∗∗ − ∗∗∗ (5.242e-05) (2.002e-04)VIX Index − ∗∗∗ − ∗∗∗ (4.380e-06) (1.290e-06) Observations 1841177 1841177 1841177 1841177 1841177R − ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ (df = 27; 1841079) (df = 33; 1841073) (df = 37; 1841069) (df = 1; 1788885) (df = 5; 1788881) Note: ∗ p < ∗∗ p < ∗∗∗ p < Table 3: Panel Regression on Efficiency with Time Fixed Effects
Dependent variable:
Portfolio Efficiency(1) (2) (3) (4)Num. of Stocks 3.589e-03 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ (7.840e-06) (7.840e-06) (7.730e-06) (1.320e-06)Portfolio Sharpe Ratio 1.829e-05 ∗∗∗ ∗∗∗ − ∗∗∗ ∗∗∗ (9.429e-04) (1.561e-04)Portfolio Std. Deviation − ∗∗∗ (1.019e-04)Observations 1841177 1841177 1841177 1841177R ∗∗∗ (df = 1; 1788815) 1.049e+05 ∗∗∗ (df = 2; 1788814) 9.048e+04 ∗∗∗ (df = 3; 1788813) 1.939e+07 ∗∗∗ (df = 4; 1788812) Note: ∗ p < ∗∗ p < ∗∗∗ p < Proof of Proposition 4.1
The portfolios on the efficient frontier are weighted linear combinations of the p risky assetswith weights w = [ w , . . . , w p ] T , which minimize risk for a given level of return. We denotethe return level by a , and use e to denote a column vector whose entries are all equal toone. The weights on the efficient frontier are recovered as the solution to the followingoptimization problem: min w w T Σ w s.t. w T µ = a and w T e = 1 . By introducing the vector of Lagrange multipliers ( λ, ν ), the Lagrangian L can be ex-pressed as L = w T Σ w + λ ( w T µ − a ) + ν ( w T e − . Using the first order condition, we obtain that the solution to the above optimization problemis equivalent to the solution of the system of equations:2Σ w + λµ + νe = 0 , w T µ = a, w T e = 1 . Such a system can be easily solved, and its solution is given by λ = 2 − aC + BAC − B ,ν = 2 − A + aBAC − B ,w = 1 AC − B Σ − (( Cµ − Be ) a + ( Ae − Bµ )) , where we have introduced A, B, C for notational convenience: A = µ T Σ − µ, B = µ T Σ − e = e T Σ − µ, C = e T Σ − e. σ p , µ p ) on the efficient frontier satisfies σ p = w T Σ w = 1 AC − B ( Cµ p − Bµ p + A ) . Next, we use the fact that the capital market line is tangent to the efficient frontier, andthe market portfolio is on the efficient frontier as well as on the capital market line. We firsttake the derivative of σ p with respect to µ p , and obtain ∂σ p ∂µ p = 12 (cid:18) AC − B ( Cµ p − Bµ p + A ) (cid:19) − / (cid:18) AC − B (2 Cµ p − B ) (cid:19) = Cµ p − B ( AC − B ) / ( Cµ p − Bµ p + A ) / . Then, we evaluate this derivative at ( µ mkt , σ mkt ), and set it equal to the inverse slope of thecapital market line, i.e., Cµ mkt − B ( AC − B ) / ( Cµ mkt − Bµ mkt + A ) / = (cid:0) AC − B ( Cµ mkt − Bµ mkt + A ) (cid:1) / µ mkt − r f . After solving the quadratic equation, we obtain µ mkt , and σ mkt from (A), i.e., µ mkt = A − Br f B − Cr f , σ mkt = (cid:113) A − Br f + Cr f B − Cr f . Using the above expressions for the expected return and volatility of the market portfolio,we can immediately obtain the capital market line, which simply connects the risk-free ratewith the market portfolio. 35 eferences
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