Portfolio Choice and Trading Volume with Loss‐Averse Investors*

Francisco J. Gomes  

London Business School

We present a model of portfolio choice and stock trading volume with loss‐averse investors. The demand function for risky assets is discontinuous and nonmonotonic: As wealth rises beyond a threshold, investors follow a generalized portfolio insurance strategy, which is consistent with the disposition effect. In addition, loss‐averse investors hold no stocks unless the equity premium is quite high. The elasticity of the aggregate demand curve changes substantially, depending on the distribution of wealth across investors. In an equilibrium setting, the model generates positive correlation between trading volume and stock return volatility but suggests that this relationship is nonlinear.

I. Introduction

 

“Value should be treated as a function in two arguments: the asset position that serves as a reference point, and the magnitude of the change (positive or negative) from that reference point” (Kahneman and Tversky 1979).

This paper solves a model of portfolio choice and trading volume with loss‐averse investors. Loss aversion specifies that individuals value wealth relative to a given reference point, that they are (much) more sensitive to losses than to gains (both measured relative to the reference point), and that they are risk averse in the domain of gains and risk loving in the domain of (moderate) losses. The first property is summarized in the preceding quote from Kahneman and Tversky (1979). The second property corresponds to the notion of first‐order risk aversion as discussed by Epstein and Zin (1990). It implies that agents exhibit significant risk aversion even for very small gambles. The last property states that following losses the investor is more willing to take additional risks (to go back to the breakeven point), while following gains he or she is more conservative.

If investors exhibit first‐order risk aversion and their attitudes toward risk are a function of the past performance of their investments, then this has important implications for the demand for risky assets and, in equilibrium, the conditional distribution of stock returns and for trading volume.

We start by studying the optimal portfolio allocation behavior of a loss‐averse investor. This behavior depends crucially on the level of surplus wealth (current wealth relative to the reference point) and how the investor's reference point reacts to changes in the current stock price. As surplus wealth reaches a certain threshold, the investor sells a significant part of his or her stock holdings and follows a (generalized) portfolio insurance rule, to protect against losses (relative to a personal reference point). Intuitively, as the stock price goes up, the investor faces a trade‐off between the potential benefit from insuring against losses and the cost of doing so: selling a large share of the portfolio and giving up the equity premium. As the stock price rises further and surplus wealth keeps increasing, the cost of a switch to the portfolio insurance rule becomes smaller: The investor need not sell as many stocks. Therefore, as the price rises enough, he or she eventually switches. This generates a behavior consistent with the disposition effect: Investors have a larger tendency to sell their winners and hold on to their losers (see Shefrin and Statman 1985 and Odean 1998, among others). In addition, this provides a rational motivation for portfolio insurance strategies and identifies the conditions under which investors are more or less likely to follow these strategies. Finally, loss‐averse investors abstain from holding equities unless they expect the equity premium to be quite large, and therefore these preferences can help to explain the low stock market participation rates observed in the data.

Heterogeneity in surplus wealth across investors generates trading volume, even in a perfect information setting. We consider a general equilibrium model with two types of investors: one type with power utility and another type exhibiting loss aversion. This corresponds to a symmetric information version of the model in He and Wang (1995), but with constant relative risk aversion (CRRA) and loss‐averse investors instead of constant absolute risk aversion (CARA) investors. Alternatively, it also corresponds to a discrete‐time version of the models in Grossman and Zhou (1996) or Basak (1995), but where the demand for portfolio insurance is endogenous and time varying. Basak (2002) also develops a model in which the demand for portfolio insurance is endogenously generated by the investor's preferences. Our models differ because we consider a different preference specification and he studies the implications for stock return volatility and risk premia, while we are concerned with trading volume and with characterizing the portfolio rules specifically generated by the loss‐aversion preferences.

The equilibrium model is solved numerically and yields two main results. First, loss‐averse investors can generate a significant degree of trading volume, even if they have homogeneous preferences and even if they are a small fraction of the population of investors. Second, when the loss‐averse investors are following the generalized portfolio insurance strategy, trading volume is positively correlated with stock return volatility. Intuitively, when the demand for portfolio insurance is stronger, the aggregate demand for stocks becomes more elastic, thus increasing both the volatility of returns and trading volume. This is the same mechanism as in Grossman and Zhou (1996), and it is consistent with the empirical evidence (see Andersen 1996; Jones, Kaul, and Lipson 1994; or Gallant, Rossi, and Tauchen 1992). However, in our model, this relationship is not always present since neither is the demand for portfolio insurance. When loss‐averse investors switch strategies, either from or to the portfolio insurance rule, the relationship between volume and volatility reverses. Consider the case in which the investor switches to the portfolio insurance rule. The optimal amount of trading is now a negative function of his or her surplus wealth, since the higher the level of surplus wealth, the smaller is the amount of stocks that the investor must sell to obtain insurance. The same logic applies in the reverse case, when the investor switches away from the portfolio insurance rule. This suggest a nonlinear relation between the two variables, volume and volatility.1

Recent economic literature has studied some of the implications of loss aversion (see Shiller 1998 or Shleifer 1999 for detailed surveys). Benartzi and Thaler (1995) provide an explanation for the portfolio allocation puzzle (the flip side of the equity premium puzzle) assuming that investors are loss averse and evaluate their portfolios only infrequently, a combination defined as myopic loss aversion. Shumway (1997) uses the same setup to explain the cross‐sectional distribution of expected returns. Epstein and Zin (1990) introduce first‐order risk aversion in the consumption Capital Asset Pricing Model (CAPM),2 while Lien (2001) studies the implications of loss aversion for futures hedging. Finally, Berkelaar and Kouwenberg (2001) derive closed‐form solutions for the optimal portfolio choice of a loss‐averse investor, assuming a complete markets setting, while Barberis, Huang, and Santos (2001) extend the consumption CAPM by assuming that investors derive utility not only from consumption but also from changes in the value of their risky asset holdings. A combination of loss aversion and influence from prior outcomes (as suggested by the evidence from Thaler and Johnson 1990) determines the preferences over this second component. We differ from these models by considering a setup with heterogeneity and studying the trading volume implications.3 Additionally, we consider a pure loss‐aversion model, in which investors are risk averse in the domain of gains and risk loving in the domain of losses. In the model of Barberis et al. (2001), following prior losses, investors actually become even more risk averse. As we show, one consequence of this distinction is that the model in this paper rationalizes a behavior consistent with the disposition effect, while the model in Barberis et al. (2001) generates the opposite pattern.

Section II describes loss aversion, derives the portfolio allocation behavior of a loss‐averse investor, and establishes some partial equilibrium results. Section III studies the equilibrium implications of a model with these investors, with a special focus on trading volume. Section IV concludes and suggests some future work.

II. Optimal Portfolio Choice with Loss Aversion

 

This section studies the optimal portfolio allocation of an investor who exhibits loss aversion. The benchmark used for comparison purposes will be the CRRA case, standard in the intertemporal portfolio choice literature.

A. Characteristics of Loss Aversion

Loss aversion is defined by three properties. First, wealth is measured relative to a given reference point. Second, the decrease in utility implied by a marginal loss (relative to the reference point) is always larger (in absolute value) than the increase in utility resulting from a marginal gain.4 Third, although agents are risk averse in the domain of gains, they are risk loving in the domain of losses. A typical utility function is where Γ denotes the reference point of the investors, and λ is a positive number, greater than 1, that determines the degree of first‐order risk aversion.

A limitation of this specification is that it implies that marginal utility is decreasing as wealth approaches zero. In an extended framework that avoids this problem, the utility function is given by where W identifies the level of wealth beyond which the utility function becomes concave (again).5 This extended setup allows for the fact that, for big enough losses (Wf<W), the decreasing marginal utility (of consumption) eventually dominates the psychological effect of the loss. This puts a limit on the amount of risk the investor is willing to take, whenever in a losing position. This is the specification used in our analysis, and it is shown in Figure 1. The vertical axis crosses the horizontal axis at the level of wealth that corresponds to the reference point.6

Fig. 1.— Value function for the loss‐averse investor

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In this specification V is continuous and everywhere differentiable except at W and at the reference point (Γ). The nondifferentiability at the reference point is a crucial property of loss aversion, while the nondifferentiability at W is a feature of the specification considered here and affects only the technical conditions for some of the results. Marginal utility is always positive, but it is increasing in the range of moderate size losses ([W, Γ]). Consistent with Tversky and Kahneman (1992), we consider γ∈[0, 1], implying that V(Γ, Γ) = 0. The level of wealth W cannot be calibrated from the available empirical evidence, but for most of this paper, that will not be required.

B. Optimal Portfolio Allocation

The analysis in this section considers a static portfolio choice problem that provides intuition for the results that follow.

In date one, the investor chooses how to allocate a given financial wealth, between two assets, one risky and the other one riskless. In date two, he or she liquidates the investment and derives utility from the terminal wealth.

The full static problem (SP) is specified by such that where α is the share of wealth invested in the risky asset, R2 is the return on the risky asset, and Rf is the return on the safe asset.

For simplicity we start by considering a binomial model:7 with R+ > Rf > R− and so that the expected excess return on the risky asset is positive.8

The dynamics of reference point (Γ) are given by: with θ ∈ [0, 1), so that the reference point is a nondecreasing function of the investor's current wealth. The parameter θ determines the speed of adjustment.9 The reference point is adjusted by the risk‐free rate, because even if the stock price remains unchanged for a given period of time, it is plausible that the investor will start considering this as a loss since he or she could have earned a riskless return instead. The notations Γt(Ri) and is used to define, respectively, the value of the reference point and the wealth level in state i (when the return on the risky asset is Ri).

1. Portfolio Allocation with CRRA Utility It is instructive to compare the results for the loss‐aversion case with the ones obtained for the CRRA case. The CRRA preferences are given by The first proposition characterizes the solution of the problem (SP) when the investor has CRRA utility, Changes in the current stock price (P1) lead to changes in the returns in each state of nature. For the purpose of this section, those effects are not important. Therefore, we study price change accompanied by changes in the expectations of future dividends, such that the distribution of the return process remains unchanged.10

For a given portfolio composition with current positive stock holdings, a change in P1 implies a change in current wealth. In the case of CRRA utility, this does not affect the optimal portfolio allocation, so, holding expectations of future returns constant, the share invested in stocks is independent of the current stock price.

Proposition 1.If the investor's preferences are given by (7), then the optimal portfolio allocation in problem P is independent of W and given by where Let P1 denote the price of the risky asset in period 1, then For the proof, see appendix A.

Since we have a positive expected equity premium (from eq. [5]), then K < 1 and α1 > 0. As R–converges to Rf we have K → 0 and therefore → .

It is important to clarify the distinction between the demand curve studied here and the one considered in the empirical literature on the slope of demand curve for stocks (see Shleifer 1986). This literature looks at market demand curves for individual stocks and is concerned with the degree of substitutability between alternative assets; namely, how stock prices react when the relative supply of these assets changes (even if no new information is released). The demand curve implicit in proposition 1 (and others to follow) is an individual demand curve for risky assets as a whole, and the investor's wealth is being changed as the stock price changes.

2. Portfolio Allocation with Loss Aversion and Zero Surplus Wealth The following proposition characterizes the portfolio allocation rule of a loss‐averse investor with zero surplus wealth. In particular, this is the situation of an investor that is out of the market and currently contemplating whether to invest some portion of wealth in stocks.

When initial surplus wealth is zero, the investor will hold stocks only if the financial gain obtained in the good state is sufficiently larger than the financial loss obtained in the bad state, since the marginal utility for losses exceeds the marginal utility for gains. Equation (11) gives a necessary and sufficient condition for this to be true. This participation constraint is more binding as the equity premium decreases or the degree of loss aversion increases. Since marginal utility decreases with the size of the loss, the investor who is willing to accept a small loss also is willing to accept a big one. This logic is valid until the loss is sufficiently large and W−<W, when eventually an optimum is reached.

Proposition 2.Assume that the investor's preferences are given by V, with W1 = Γ1. Then, the global optimum for problem SP, (W1, Γ1, P1, W) is equal to 0 unless in which case, (W1, Γ1, P1, W) is implicitly defined by For the proof, see appendix A.

The result in proposition 2 suggests that, even if the expected equity premium is positive, investors might not be willing to hold stocks, depending on the specific parameters of the utility function. Since λ>1, condition (11) defines a strictly positive lower bound on the expected equity premium. This is a direct implication of first‐order risk aversion, and it can help explain why the majority of households in the population do not invest in equities. In fact, if we take Rf = 2% and assume a binomial model for stock prices with expected return equal to 8% and standard deviation equal to 15%, then condition (11) is satisfied only if we assume λ<2.25, which is exactly the value suggested by the experimental evidence from Tversky and Kahneman (1992). In other words, with λ<2.25 this model implies that households should not invest in equities.

3. Portfolio Allocation with Loss Aversion and Negative Surplus Wealth The next proposition still considers a loss‐averse investor but studies the case in which surplus wealth is nonpositive, although still above W. Condition (13) imposes a lower bound on W1 to rule out cases in which the initial wealth is very close to W.11

Proposition 3.Assume that the investor's preferences are given by V, with ( ) < W1 ≤, Γ1,12 where ( ) is defined by with then there exits a global optimum for problem SP, (W1, Γ1, P1, W) implicitly defined by equation (12). For the proof, see appendix A.

The optimal portfolio rule is the one identified in proposition 2. Once in a losing position, the investor becomes risk loving and therefore is always willing to invest in stocks (since they have a higher expected return and higher risk than the safe asset). For levels of wealth below W, marginal utility is again decreasing, and this eventually imposes a limit on the amount of risk the investor is willing to take. The sign of , in general, is undetermined, as will become clear later.

4. Portfolio Allocation with Loss Aversion and Positive Surplus Wealth The following propositions characterize the optimal portfolio share invested in stocks when the investor exhibits loss aversion and surplus wealth is strictly positive. Proposition 4 identifies and characterizes a local optimum for this problem. This solution fully protects the investor from losses, as he or she keeps the portfolio allocation to stocks sufficiently low such that, even in the worst state of nature, wealth is still above the reference point. The conditions under which this strategy is also a global optimum are discussed in proposition 5.

Proposition 4.If the investor's preferences are given by V and W1 > Γ1, then there exits a (local) optimum for problem SP, (W1, Γ1, P1), such that where, as before, K is given by (9).

Let denote the price of the risky asset in period 1, then where → 0 as θ → 1. For the proof, see appendix A.

This expression reduces to the one obtained in the CRRA case when the reference point is equal to zero and converges to it as surplus wealth rises. Changes in the current stock price affect the investor's optimal portfolio allocation, even for given expectations of future returns, because they change the investor's surplus wealth and, therefore, his or her risk aversion. As wealth increases (for a given reference point), the investor becomes less risk averse and therefore increases his or her risk exposure. Conversely, as the reference point rises, for a given level of wealth, risk aversion increases and the optimal portfolio allocation to stocks is reduced. As surplus wealth falls toward zero, the optimal allocation to stocks also converges to zero, as the investor becomes locally (almost) infinitely risk averse.

Naturally the magnitude of is going to depend on θ. If, as the stock price increases, the reference point remains constant, then surplus wealth changes only because current wealth has changed; this leads to a higher value of α: The portfolio share invested in the risky asset is a positive function of the stock price. Consider now the limit case in which θ → 1, and therefore ∂Γt/∂Pt → ∂Wt/∂Pt. In this case, an increase in the stock price leaves surplus wealth unchanged, and therefore α1 is independent of P1, as in the CRRA case. In general, the larger is θ, the smaller is.

This solution is consistent with a portfolio insurance strategy: The investor tries to prevent current wealth from falling below the reference point. Benninga and Blume (1985) found that, in a complete markets setting, “the end‐of‐period utility function of an investor who insured his or her portfolio at some level would have to exhibit an unbounded coefficient of relative risk aversion below the insurance level and decreasing relative risk aversion above that level.” These conditions are almost perfectly satisfied by a loss‐averse investor with positive surplus wealth. That investor's marginal utility converges to infinity as surplus wealth falls toward zero. However, marginal utility also decreases rapidly as wealth falls below the reference point, and this suggests that a portfolio‐insurance‐type strategy might not be globally optimal. This issue is reconsidered later, when studying the (global) optimality of solution identified in proposition 4. In what follows, the solution from proposition 4 is referred to as a generalized‐portfolio‐insurance (GPI) rule (for the reasons discussed previously and following the terminology of Leland 1980).

The next proposition presents a necessary and sufficient condition under which the optimum from proposition 4 is a global optimum and identifies the correct global solution for the case in which this condition fails. The intuition behind this result is the following. Proposition 4 identifies the optimal portfolio allocation for an investor with positive surplus wealth, subject to the constraint that his orher terminal wealth always exceeds the reference point. This portfolio allocation was shown to be a local optimum for problem SP. However, if the investor is willing to tolerate a positive probability of a loss, then the alternative candidate for an optimum is given by proposition 5, since the same reasoning discussed then still applies: The investor who is willing to accept a small loss is also willing to accept a larger one. Condition (16) merely compares the value of the two alternative optima to determine which is the global solution.

Proposition 5.Consider that the investor's preferences are given by V and W1 > Γ1, and consider the following condition where is the optimum defined in proposition 2 by equation (12, and is the optimum defined in proposition 4 by equation (15). Then, the optimal portfolio rule for problem SP is given by A higher (smaller) W1 makes condition (16) more (less) likely to hold. For the proof, see appendix A.

Part 2 of this proposition states that, as the individual's surplus wealth rises, he or she is more likely to choose the optimum from proposition 4. So, for low levels of surplus wealth, we expect the investor not to follow the generalized portfolio insurance rule, as he or she must when first investing in stocks. The reason for this behavior is simple: This rule has a very high cost when surplus wealth is small. However, as surplus wealth increases and such cost is reduced, the investor eventually switches so as to guarantee positive surplus wealth in all states of nature.

Naturally, the “cutoff” between the two strategies depends on both the investor's preferences and the expected equity premium. So, it is possible to calibrate it, either by changing the parameters for the return process or by varying W.

5. Demand Curve for Stocks for a Loss‐Averse Investor Figure 2 plots the demand curve for the loss‐averse investor, for different values of θ. As before, the vertical axis crosses the horizontal axis at the price level for which surplus wealth equals zero.13

Fig. 2.— Demand curve for stocks for the loss‐averse inventor

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Consider first the price range on the right‐hand side of the discontinuity point. Over this range, the investor's surplus wealth is nonnegative and, as shown in proposition 4, he or she follows what was defined previously as a generalized portfolio insurance (GPI) rule. However, as shown in proposition 5, the investor must pay a pric for avoiding losses: the foregone expected return, implied by the lower portfolio share allocated to the risky asset. Eventually, if this cost is very high, the investor will choose to accept a positive probability of a loss to be able to benefit from the high expected return. The cost becomes quite high as the investor's surplus wealth converges to zero, since the GPI strategy implies that the wealth share invested in stocks should also go to zero in this case. This generates the discontinuity in the demand function, as the investor eventually switches strategies. It is important to note that, in the domain of gains, the loss‐averse investor exhibits decreasing relative risk aversion, and therefore this demand curve is not inconsistent with the observation that wealthier individuals invest a larger fraction of their wealth in risky assets.

As shown in proposition 4, an increase in θ reduces the slope of the demand for stocks under the GPI strategy. However, to the left of the discontinuity point, changing θ actually affects the level of the demand curve and not just the slope. Even when surplus wealth is zero, the two demand curves do not coincide. This occurs because the level of surplus wealth obtained in each state depends on the speed at which the reference point adjusts. For a higher θ values, a given (positive) α generates both a smaller gain and a smaller loss, respectively, in the good and bad states. Therefore, to keep the same risk exposure, the share invested in stocks must rise.

6. The Disposition Effect This demand function is consistent with the disposition effect: Investors tend to sell winners and hold on to losers (Shefrin and Statman 1985).14 However, the evidence in favor of the disposition effect is at the individual stock level, while the predictions of this model hold for risky asset holdings as a whole. Deriving the disposition effect in this framework requires one additional assumption, mental accounting for the holdings of each individual stock.

The results presented here also have important implications for the identification of the determinants and of the evolution of reference points over time. Odean (1998) and Heath, Huddart, and Lang (1999) try to identify the implied reference points of the investors by determining the cut‐off price, beyond which the probability of selling jumps. According to our results, these estimates of reference points are biased upward as the discontinuity in the probability of selling occurs when surplus wealth is already positive and not when it is zero. Investors cannot optimally sell their stocks when surplus wealth becomes marginally positive, because then they would have had no incentive to buy them in the first place.

III. Equilibrium and Trading Volume

 

This section considers a two‐period equilibrium version of the previous problem and studies the implied trading volume patterns. Solving a rational expectations equilibrium with heterogeneous investors is quite problematic, since the expectations of future prices depend on the distribution of all the relevant state variables, which in our case are the stock holdings, levels of wealth, and surplus wealth for all investors, in every possible state of nature in the future. These, naturally, depend on the current decisions and allocations, which in turn depend on these expectations. This section deals with the problem by considering a two‐period binomial model, so that it becomes feasible to solve it numerically. Appendix B discusses a model with T periods, which requires additional assumptions: myopic portfolio behavior (investors choose their portfolio allocations assuming that these are buy‐and‐hold strategies) and no adjustment of the reference points. In this case, the investors do not care about the distribution of future prices (except for the distribution of the terminal price, which is exogenous) and the problem can also be solved numerically. The main results of the two models are the same.

The structure is derived from He and Wang (1995) but in a symmetric information context. Since in this model there is a group of investors with a time‐varying demand for portfolio insurance, it can also be linked to Grossman and Zhou (1996) or Basak (1995).15 However, in the loss‐aversion model, the demand for portfolio insurance is not constant, becoming a function of the relevant state variables.

A. Setup

There are two types of investors: loss‐averse investors, and CRRA investors. The loss‐averse investors solve the following problem (DP), subject to with ωH > ωL and where St and Bt denote, respectively, risky asset holdings and riskless asset holdings at time t. The short‐selling constraint on the portfolio allocation (eq. [21]) limits the amount of risk taking in the domain of losses and is motivated by the desire to make the results independent of the choice of W, which cannot be calibrated from the data.

The CRRA investors solve the same problem but with U(W2) replacing V(W2, Γ2). The policy rules for the second period have already been derived, in proposition 1 for the CRRA investors, and in propositions 2 through 5 for the loss‐averse investors. The first‐period policy rules are obtained numerically as the problem is solved backwards.16 The market clearing condition is where represents the total (exogenous) supply of stocks, Pt is the stock price at time t, and the superscripts LA and C are now used to identify, respectively, the loss‐averse investors and the CRRA investors. As in He and Wang (1995), the supply of the riskless asset is perfectly elastic at the given risk‐free rate.

Note that, if stocks are to have a positive expected return as of date 0, then it must be the case that additionally, so that stocks do not fully dominate bonds at date 0, we must also have

B. Equilibrium

In this subsection, we discuss the implications of this model for trading volume. There are two main results. First, the presence of the loss‐averse investors can generate a significant degree of trading volume, even if they have homogeneous preferences and even if they are a small fraction of the population of investors. Second, when the loss‐averse investors follow the GPI strategy, trading volume is positively correlated with stock return volatility. However, when they switch between strategies, the sign of this correlation reverses. This suggests a nonlinear relation between the two variables.

1. Loss‐Averse Investors with Low Initial Surplus Wealth This is the case in which the initial surplus wealth for the loss‐averse investors is negative, zero, or only marginally positive, such that the optimal portfolio rule is not given by the GPI strategy. In this case, the short‐selling constraint is binding and inhvestors are fully invested in stocks. If ω1 = ωL, then the stock price falls, the loss‐averse investor remains fully invested in stocks and there is no trading volume. If ω1 = ωH, then the stock price rises and, given our calibration, the loss‐averse investor switches to the GPI strategy.17

The benchmark results are shown for the following preference parameters: λ = 1.5, γ = 0.5 for the loss‐averse investors and γ = 5 for the CRRA investors, and θ = 0.5. Results for different values of θ are presented later. We considered values of λ ranging from 1.25 to 1.75. As was shown previously, even small values of λ require large risk premia to keep the loss‐averse investors in the market. As for γ, we considered values going from 0.2 to 0.8. In all cases, the results were found to be robust.

Panel b of figure 3 plots the stock return in the high state as a function of ωH, for different values of the ratio . As argued previously, the model was calibrated so that the returns in this state are high enough to induce the loss‐averse investors to switch to the GPI strategy. As expected, a higher value of ωH is associated with a higher return, since its impact on is higher than its impact on P0. As we increase , the stock return falls. This occurs because, in this region, the aggregate demand curve is actually positively sloped. Once the loss‐averse investors follow the GPI strategy, a higher stock price actually increases their demand for stocks, since it increases their surplus wealth and therefore reduces their risk aversion. As the loss‐averse investors become more negligible, the stock price does not have to increase so much to generate sufficient demand. In the limit, we obtain the volatility of a model with only CRRA investors.

Fig. 3.— (a) Demand curve for the loss‐averse investors in the two‐period model of section III. This figure represents the case in which the initial surplus wealth of the loss‐averse investors is relatively “low,” therefore, they do not follow the GPI strategy. The solid line plots the first‐period demand curve and the initial allocation corresponds to point 0. The other two curves plot the second‐period demand curves and allocations for the different possible shocks, positive (1b) or negative (1a).

(b) Gross stock return in the good state, when the loss-averse investor's initial surplus wealth was “low” (from 0 to 1b in panel a). Results are shown for different values of the news shock (ωH) and different values of the ratio of wealth between the CRRA investors and the loss-averse investors (WC/WLA).(c) Turnover ratio in the good state, when the loss-averse investor's initial surplus was “low” (from 0 to 1b in panel a). Results are shown for different values of the news shock (ωH) and different values of the ratio of wealth between the CRRA investors and the loss-averse investors (WC/WLA). (d) Gross stock return in the good state, when the loss-averse investor's initial surplus wealth was “low” (from 0 to 1b in panel a). Results are shown for different values of the news shock (ωH) and different rates of adjustment for the reference point.

(e) Turnover ratio in the good state, when the loss-averse investor's initial surplus wealth was “low” (from 0 to 1b in panel a). Results are shown for different values of the news shock (ωH) and different rates of adjustment for the reference point.

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Panel c of figure 3 plots trading volume as a function of ωH and for different values of the ratio In all cases we find that trading volume is negatively correlated with ωH Since, from panel b, we know that a higher value of ωH corresponds to a higher return in the good state, and changing ωH has a much smaller impact on than on , trading volume is negatively correlated with stock return volatility. Remember that trading volume occurs because the loss‐averse investors partially liquidate their positions, and switch to the GPI strategy. Consequently, the larger is the price change, the smaller the amount of stocks that these investors have to liquidate.

A less intuitive result is that, as we increase , trading volume also increases. A higher value of generates two effects. On the one hand, the loss‐averse investors hold less shares and therefore should generate less trading volume. But, as we saw in panel b, this also reduces the stock return and therefore reduces surplus wealth and the optimal stock holdings of the loss‐averse investors. Naturally, as the share of loss‐averse investors becomes very small, the first effect should dominate and we should observe a nonlinear relationship between and trading volume. However, as we increase the initial weight of the CRRA investors, the risk premium in the economy falls and the loss‐averse investors are eventually excluded from the market. In our model, this occurs before the reversal actually takes place. In other words, the reversal occurs in a discontinuous way. This last result (the discontinuity) is a specific feature of our model, as we have no heterogeneity among loss‐averse investors, therefore, all trades must take place with CRRA investors. However, the main result is particularly interesting, as it suggests that an economy with loss‐averse investors can generate trading volume, even if these investors are not a large fraction of the relevant population.

In panels d and e, we study the impact of changing the adjustment rate for the reference point (θ). We find that, as we increase the value of θ, the stock return rises, but not too much. At any given level of wealth, a higher θ implies lower surplus wealth for the loss‐averse investor and therefore a smaller level of aggregate demand. Since the demand is positively sloped in this region, we have an increase in the stock price and therefore a higher return. Trading volume also rises with θ, since a higher value of θ is associated with a lower value of surplus wealth, therefore, the loss‐averse investor wants to sell a larger share of his or her stock holdings. The effect of having a higher stock return is clearly second order, since it occurs only to the extent that there is less demand from the loss‐averse investors.

2. Loss‐Averse Investors with High Initial Surplus Wealth Now, we consider the case in which initial surplus wealth for the loss‐averse investor is sufficiently positive, such that the optimal portfolio rule in period 0 is given by the GPI strategy. This is shown in figure 4, panel a. Under our calibration, if ω1 = ωL, then, as the stock price falls, the loss‐averse investor gives up the GPI strategy and invests fully in stocks. Note that, unlike the previous case, there is trading volume in this state as well. If ω1 = ωH, then the stock price rises and, given our calibration, the loss‐averse investor keeps following the GPI strategy, although the demand for portfolio insurance weakens as surplus wealth has increased.

Fig. 4.— (a) Demand curve for the loss‐averse investors in the two‐period model of section III. This figure represents the case in which the initial surplus wealth of the loss‐averse investors is relatively “high,” therefore, they follow the GPI strategy. The solid line plots the first‐period demand curve and the initial allocation corresponds to point 0. The other two curves plot the second‐period demand curves and allocations for the different possible shocks, positive (1b) or negative (1a).

(b) Turnover ratio in the good state, when the loss-averse investor's initial surplus wealth is “high” (from 0 to 1b in panel 4a). Results are shown for different values of the news shock (ωH) and different values of initial surplus wealth as fraction of total wealth for the loss-averse investors (SW).

(c) Turnover ratio in the bad state, when the loss-averse investor's initial surplus wealth is “high” (from 0 to 1b in panel 4a). Results are shown for different values of the news shock (ω1) and different values of initial surplus wealth as fraction of total wealth for the loss-averse investors (SW).

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Panel b shows trading volume in the high state as function of ωH (i.e., as a function of the stock return) and for different levels of initial surplus wealth. Now, we find that trading volume is positively correlated with stock returns. This occurs because the investor follows the GPI strategy in both periods. Therefore, larger returns generate larger fluctuations in surplus wealth and therefore more trading volume. This is consistent with the findings of Grossman and Zhou (1996): When investors following portfolio insurance strategies, trading volume and stock returns are positively correlated. Figure 4 also shows that the value of initial surplus wealth and trading volume are negatively correlated. A higher value of initial surplus wealth reduces the demand for portfolio insurance and, therefore, the elasticity of the demand curve.

The level of trading volume in panel b is much smaller than the one reported in panel c. Trading volume is at its maximum level when the loss‐averse investor switches strategies. Consistent with this, we also obtain a very high turnover ratio in the bad state (ω1 = ωL), when the investor switches away from the GPI strategy. This is shown in panel c, and it is just the reverse of the preceding case. The lower is the value of initial surplus wealth, the lower the initial stock holdings of the loss‐averse investor and, therefore, the higher the turnover ratio, as he or she is now fully invested in stocks. This result is not a consequence of the short‐selling constraint. Remember that the portfolio allocation in the domain of losses is roughly independent of the level of surplus wealth, while the GPI rule depends strongly on that level. As before, we find that, when the investor switches strategies, trading volume is negatively correlated with stock returns. Therefore, the results suggest that the relation between trading volume and stock return volatility should be nonlinear.

IV. Conclusion and Directions for Future Work

 

This paper studies the optimal portfolio allocation behavior of loss‐averse investors and its implications for trading volume. The demand function for risky assets is discontinuous and nonmonotonic. As surplus wealth reaches a certain threshold, investors sell a large part of their stock holdings and follow a (generalized) portfolio insurance rule, protecting themselves against losses (relative to their reference point). In addition, this provides a rational motivation for portfolio insurance strategies and identifies the conditions under which investors are more or less likely to follow those strategies.

Since the value function exhibits first‐order risk aversion, this implies that loss‐averse investors abstain from holding stocks unless they expect the equity premium to be quite high. Simulation results show that this model is able to rationalize the small participation rates observed in the data.

A dynamic model, in the spirit of Grossman and Zhou (1996), Basak (1995, 2000), and He and Wang (1995), (typically) yields positive correlation between stock return volatility and trading volume. When the demand for portfolio insurance increases, the aggregate demand for stocks becomes more elastic and, at the same time, trading volume increases. This generates the positive correlation between both series. However, in our model, this result is not globally valid, since the demand for portfolio insurance is not always present. When loss‐averse investors switch strategies, the relationship between volume and volatility reverses. This is also the moment in which trading volume is at its peak. Note, however, that, in a model with heterogeneous loss‐averse investors, the time at which some investors switch strategies is when others are more likely to be close to switching as well. But this implies that the demand for portfolio insurance for this second group should be quite high, contributing to a higher correlation between volume and volatility. This suggests that we should not expect a discontinuity, as obtained in the model with homogeneous (loss‐averse) investors, but rather a smooth transition. In any case, a definite prediction for the relationship between trading volume and stock return volatility is quite hard to make until such a model is actually solved.

Shiller (1981) and LeRoy and Porter (1981) show that stock prices are too volatile to be explained by realistically calibrated shocks to future cash‐flows or moderate changes in discount rates. If investors exhibit loss aversion, then moderate changes in wealth can lead to large changes in discount rates that might help explain this puzzle.

Other generalizations would be quite interesting. Allowing for asymmetric information is very important to developing a more‐complete model of trading volume, and it also helps determine the quality of market prices as signals of fundamentals. As shown by Grossman and Zhou (1996), models with portfolio insurance have important implications for option pricing. By generating a time‐varying demand for portfolio insurance, the loss‐aversion model should generate some very interesting option pricing dynamics. Deriving these dynamics and testing the model along this additional dimension is another promising direction for future research.

AppendixA
Proofs

 

Proof of Proposition 1.

 

The first‐order condition for this problem is (using the specification for U): Rearranging this expression, we obtain Defining K as and solving for α*: From expression (31), we can compute the derivative of α with respect to W1, and check that ∂α/∂W1 = 0. By definition, we have that When holding R+ and R− constant, the first two terms equal zero and we obtain and from eq. (31), we easily get the result.

Proof of Proposition 2.

 

For W1 = Γ1, the marginal utility from an infinitesimal increase in α (from α = 0) is given by where and are, respectively, the right‐hand‐side derivative of VG at 0, and the left‐hand‐side derivative of VL at 0. From their definitions, we have that : therefore, dividing both terms by 0.5W1, a necessary and sufficient condition for α* > 0 is that This condition does not depend on α and corresponds to condition (11).

If the function VL were defined over the whole domain of losses, then under condition (11), there would be no maximum, as the optimal value of α would tend to infinity. However, for values of α high enough that W−falls below W, the marginal utility loss is given by –0.5VBL(W−)’(R− – Rf)W1. As α increases, [W+ – Γ2(R+)] converges to zero, while VBL(W−)’(Rf –R−)converges to infinity, so some α must exist such that18 and this equation implicitly defines the optimum portfolio allocation.

Proof of Proposition 3.

 

This proof follows closely the proof of proposition 2, part 2. Since < W1 < Γ1, then the investor's expected utility, for a small value of α, is given by19 Since VL is a strictly convex function (the agent is risk loving in the domain of moderate size losses), we know that α ≤ 0 is not an optimum.

As we consider increasing the value of α, from α = 0, the marginal utility gain is given by Again, by the convexity of VL, we know that this always positive. When α is large enough, we will have W ≥ Γ(R+), and the marginal utility gain is now given by20 And the rest of the proof follows directly from the proof of proposition 2, part 2.

Proof of Proposition 4.

 

Using the law of motion for the reference point, we can write surplus wealth in period 2, W2 − Γ2, as And, if we have W2 = RfW1 (α = 0), this becomes As a result, if W1 > Γ1 and α = 0, then W2 > Γ2 and therefore For α small enough, it will still be true that We define from the following condition: This implies that, for any portfolio allocation in the set [o, α], all payoffs will occur in the domain of gains.

Therefore, we know that is equivalent to Let α* denote the solution to this problem. Since VG is a concave function, we know that α* must satisfy with Substituting the expressions for Γ2(R+) and Γ2(R−), we can rewrite the first‐order condition as Solving for α*, where, as before, K is given by eq. (30). Finally, since α* is a local optimum for problem SP.

Using the definition of Γ1, we can rewrite eq. (46) as By definition, we know that so since θ ∈ [0, 1).

Proof of Proposition 5.

 

1. Proposition 4 identifies a local optimum for problem SP, for an investor with positive surplus wealth. This optimum is given by eq. (46), and we denote it by . This was shown to be the optimal solution when we added the constraint W−>Γ1 to problem SP. Now, we need to consider whether larger values of α (therefore, yielding W−<Γ1) can generate higher utility than .

If the investor is willing to tolerate a positive probability of a loss then, from the proof of proposition 2, part 2, the alternative candidate for an optimum is given by eq. (12). We denote this alternative optimum by . Condition (16) merely compares the utility level given by α and to determine the global optimum.

2. The utility obtained by choosing (the optimum from proposition 4) is While the utility derived from choosing (the optimum defined in condition [12]) is Using the envelope theorem and defining , From the first order conditions for and , we know that we can define Using the definitions of and , it is possible to rewrite eqq. (50) and (51) as Now, note that which does not depend on α. Also, since > , then therefore, combining eqq. (57), (52), and (53), we have > . So, defining and we have since both H1 and H2 are positive.

AppendixB
Dynamic Model

 

This is essentially a T‐period version of the model in section III. The structure is derived from He and Wang (1995), but in a symmetric information context.

Set‐up of the Model

 

As before, the supply of the risky asset is exogenously fixed, while the supply of the risk‐free asset is perfectly elastic at a given risk‐free rate. There are two types of investors, investors with CRRA preferences and loss‐averse investors.

The relevant information set and state variable follows a (logarithmic) random walk, and the reference point is assumed to be constant during the T periods for reasons discussed later.21

The full dynamic problem (DTP) is specified by such that where Γt is the investor's reference point at time t.

In general, this problem cannot be solved, since the pricing function depends on an infinite number of state variables (the infinite regress problem discussed in Townsend 1983). This motivates the assumption of a fixed reference point and requires one additional assumption, the investor chooses αt assuming an inability to rebalance the portfolio in the future (myopic portfolio allocation).

The other constraint on αt (eq. [65]) limits risk‐taking behavior in the domain of losses, making the calibration of W and ρ virtually irrelevant. This is particularly helpful, since these parameters cannot be rigorously calibrated from existing evidence.

The market‐clearing dynamics assumed here are the following: in response to excess demand (supply), the market maker increases (decreases) the stock price.

Numerical Solution

 

Since the investor solves the problem under the assumption of no rebalancing, the optimal solution exhibits the properties derived in section II. This is true, even in the presence of the short‐selling constraint. The distribution for the state variable is approximated using Gaussian quadrature. The state‐space is made discrete along all the other dimensions, using equally spaced grids with nonbinding upper and lower bounds. The distance between any two grid points is determined by an upper bound of 2.5 percentage points for implied change in the optimal portfolio rule (share invested in the risky asset).22

The solution given by eq. (15) can be computed directly, while the solution from eq. (12) is obtained using a simple fixed‐point algorithm. The market clearing condition is solved using a recursive algorithm based on the market‐clearing rule specified previously. In each period, the algorithm is started by computing the excess demand at last period's stock price, given the new information set of each group of investors.

After solving the model, we simulate 5000 different time series and generate different cross sections from them.23

Data

 

The predictions of this model are compared against empirical evidence derived from high‐frequency data; namely, daily stock returns and daily trading volume.

The return data correspond to the value‐weighed stock return on the NYSE, the AMEX, and the NASDAQ, from July 1962 to December 1996,24 taken from the Center for Research in Security Prices (CRSP). The volume data is also taken from CRSP. The measure of volume chosen is the turnover ratio, constructed by aggregating trading volume for individual stocks, measured in dollar units, and dividing it by total market capitalization. Trading volume for each stock and its corresponding price were taken from the CRSP daily stock files. Only ordinary common shares and certificates traded on the NYSE, the AMEX, or the NASDAQ are considered (this implies dropping 7.2% of the original sample). The volume variable reports the number of shares sold on a given day, rounded to the nearest hundred. The price variable corresponds either to the closing price or the average of the bid and ask prices, on that day (the files also provide information on which one is actually being reported). For roughly 99% of observations in which trading volume is positive, the closing price is the one reported. Missing observations can be distinguished from observations with zero trading volume and are dropped from the sample (they correspond to 0.1% of the remaining sample size). Following Campbell, Grossman, and Wang (1993), we consider the natural logarithm of the turnover ratio and low‐frequency patterns are removed by subtracting a 1‐year backward moving average filter.

Results and Empirical Evidence

 

The results with simulated data correspond to the average across 20 cross sections, each containing 5000 observations. Unlike what one might think, the model has few free parameters. Results are presented for different values for λ and γ for the percentage of initial wealth of the CRRA investors (wC), the only relevant free parameters.

Table 1 reports the correlations between volume and turnover, volume and lagged volume, and turnover and lagged turnover, for both the simulated data and the CRSP data.25 It is not purpose of this section to match specific moments as this model is too simplified for that. Instead the objective is to show that the model can generate the correct qualitative predictions and that the magnitudes are economically meaningful. For all combinations of parameter values, the correlations are strongly positive and significant, consistent with the evidence from CRSP.

Table 1
Table 1 Sample Correlations from Simulated Data and CRSP Data

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When the demand for portfolio insurance is stronger than the aggregate, demand for stocks becomes more elastic and trading volume increases. This generates the positive correlation between volume and volatility, just like in Grossman and Zhou (1996). Consistent with the data, the loss‐aversion model also generates persistence in volatility and trading volume. This persistence occurs because the demand for portfolio insurance rules is itself a persistent process, as it is motivated by the level of surplus wealth.

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  • * This paper has benefited from the comments and suggestions made by one anonymous referee, Leandro Arozamena, Nicholas Barberis, Estelle Cantillon, John Campbell, Laurent Calvet, João, Cocco, Wayne Ferson, Benjamin Friedman, João Gomes, David Laibson, Andrew Metrick, Sendhil Mullainathan, Andrei Shleifer, Raman Uppal, Haiying Wang, and seminar participants at Boston College, C.E.M.F.I., Duke, Harvard, L.B.S., New University of Lisbon, M.I.T., Universtiy of Cyprus, U.S.C., Washington, Yale and at the EFA 2000. Previous versions of this paper have circulated with the title “Loss Aversion and the Demand for Risky Assets”. Financial support from the Sloan Foundation is gratefully appreciated. The usual disclaimer applies. Contact the author at fgomes@london.edu.

  • 1. Wang (1993, 1994) and He and Wang (1995) generate positive contemporaneous correlation between stock return volatility and stock turnover with models of differential and asymmetric information. The same result is obtained by Shalen (1993) in a model with dispersion of beliefs, by Brock and LeBaron (1996) in a model with learning, and by Grossman and Zhou (1996) in a model with (exogenous) portfolio insurance.

  • 2. More recent models with first‐order risk aversion include Bekaert, Hodrick, and Marshall (1997); Ang, Bekaert and Liu (2000); and Lien and Wang (2003).

  • 3. Berkelaar and Kouwenberg (2001) do not study the equilibrium implications of their model, while Barberis et al. (2001) use a representative agent setup.

  • 4. This property is defined as first‐order risk aversion (Epstein and Zin 1990) and differs from “normal” risk aversion because it holds for infinitesimal gains and losses.

  • 5. For levels of wealth below W, the properties of the utility function are given by . The two extra terms are just a constant, required to make V a continuous function at W. W is modeled as constant. Alternatively, we could specify it as a function wealth (which we do with the reference point), but this would only add to the algebra and notation in the paper, without changing its results.

  • 6. Since W is a constant, the value function depends on two variables: W, Γ. Therefore, when we wish to specify its arguments, we write V(W, Γ). Likewise, we can write VG(W, Γ) or VL(W, Γ). However, since W and Γ enter linearly in VG and VL, we often use the notation or , as it is typically more revealing.

  • 7. In the two‐state case, it is possible to characterize the solution analytically and, in certain cases, derive it in closed form, while otherwise it must be obtained numerically. The results obtained for the two‐state case are also valid for more general versions of the problem: .

  • 8. The notation Ri is used to define the risky asset's return in state i (Ri = R+ or Ri = R−).

  • 9. Note that we must have θ < 1, since for θ = 1, we always have Γt = Wt and, therefore, from eq. (2) we get V = 0.

  • 10. The results in this section, namely, the shapes of the demand curves, have also been derived assuming that only the current stock price changes, while everything else (including expectations about the future) remains unchanged. This, however, only adds to the algebra, making the crucial effects less clear.

  • 11. This condition guarantees that, for any W1 above this lower bound, if , then , i.e., if wealth in the bad state equals W, then wealth in the good state exceeds the reference point.

  • 12. Condition (13) implies that W(?) > W.

  • 13. The formulation of problem (SP) assumes that the distribution for R2 has continuous support. For tractability, all the propositions are stated and proven for the simpler binomial model. These results are also valid in the more general case, as can be seen numerically. This can be done by approximating distribution for ln(R2) using Gaussian quadrature and obtaining the portfolio rule using a standard grid‐search algorithm to deal with the nonconvexity of the objective function.

  • 14. See Odean (1998); Heath, Huddart, and Lang (1999); Locke and Mann (1999); Grinblatt and Keloharju (2000, 2001); or Ranguelova (2001) for recent empirical evidence on the disposition effect.

  • 15. The model in Basak (1995) allows for intermediate consumption, unlike the one in Grossman and Zhou (1996) or the one in this paper.

  • 16. The details on the numerical solution are given in appendix B.

  • 17. Considering values of ωH such that this switch does not occur would be of very limited interest, since the model would not generate any trading volume.

  • 18. Note that is equal to zero, while is always strictly positive; and since both are continuous monotonic functions, they must eventually cross.

  • 19. Note that, by definition, > W: and we have φ > 0 and Γ1 > W.

  • 20. From the definition of , for W+ = Γ(R+), we still have W− > W. In other words, we have assumed that W1 is “closer to the reference point” than to W.

  • 21. T is set equal to 252, the average number of trading days in a year. Therefore, this assumption implies that reference points are constant over 1‐year periods, just as in the models of Barberis, Haung, and Santos (2001) and Benartzi and Thaler (1995).

  • 22. Increasing the number of grid points (for given upper and lower bounds) does not produce any meaningful change in the results.

  • 23. To minimize both the effects of the initial conditions and horizon effects, the cross sections were taken for data points t* such that 0 < t* < T, where T is set at 252, the average number of trading days in a year. The results are robust to changes in the value of t*.

  • 24. The volume data starts in July 1962.

  • 25. The signs and economic significance of these correlations survive more‐detailed empirical studies, as shown by Gallant, Rossi, and Tauchen (1992); Bollerslev, Chou, and Kroner (1992); and several others.

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