How loss averse are investors in financial markets?

Journal of Banking & Finance 34 (2010) 2425–2438

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How loss averse are investors in financial markets? Soosung Hwang a,*, Steve E. Satchell b a b

School of Economics, Sungkyunkwan University, 53 Myeongnyun-Dong 3-Ga, Jongno-Gu, Seoul 110-745, Republic of Korea Trinity College, Cambridge University, Cambridge CB2 1TQ, UK

a r t i c l e

i n f o

Article history: Received 7 August 2009 Accepted 29 March 2010 Available online 1 April 2010 JEL classification: G11

a b s t r a c t We investigate loss aversion in financial markets using a typical asset allocation problem. Our theoretical and empirical results show that investors in financial markets are more loss averse than assumed in the literature. Moreover, loss aversion changes depending on market conditions; investors become far more loss averse during bull markets than during bear markets, indicating their more profound disutility for losses when others enjoy gains. Contrary to most previous results, we find that investors are more sensitive to changes in losses than changes in gains. Ó 2010 Elsevier B.V. All rights reserved.

Keywords: Loss aversion utility Asset allocation Pension funds

1. Introduction According to prospect theory, proposed by Kahneman and Tversky (1979), individuals maximize a weighted sum of a ‘value’ function. Decisions are made in terms of gains or losses rather than final wealth, and the ‘value’ of a loss is compensated for by two to three times the ‘value’ of a gain equivalent to that loss; hence, the notion of loss aversion. Loss aversion ‘power’ utility is a functional form of prospect theory devised by Kahneman and Tversky (1992); this utilizes power utility and is designed to satisfy the properties of prospect theory. Unlike power utility that involves one parameter, the loss aversion ‘power’ utility (henceforth LA utility) involves three parameters: two curvature parameters explain the sensitivity of utility to losses and gains, and a coefficient of loss aversion measures the relative disutility of losses against gains. Understanding these parameters is essential in the same way as the risk aversion coefficient of power utility is crucial in expected utility theory.1 Kahneman and Tversky (1992) suggest a set of parameter values for LA utility using experiments. Benartzi and Thaler (1995), Barberis et al. (2001), Ang et al. (2005), and Barberis and Xiong (2009) use similar values in their studies. Other studies such

* Corresponding author. Tel.: +82 (0)2 760 0489; fax: +82 (0)2 744 5717. E-mail addresses: [email protected], [email protected] (S. Hwang), ses11@ econ.cam.ac.uk (S.E. Satchell). 1 For instance, for the coefficient of the CRRA, many studies have theoretically suggested that the admissible range lies between one and two. However, debate regarding the appropriate ranges of the CRRA coefficient is far from over. Mehra and Prescott (1985) suggest that the equity premium puzzle can be solved when the coefficient is of the order of 30. 0378-4266/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2010.03.018

as Wu and Gonzalez (1996) estimate that the values of the two curvature parameters are identical but significantly smaller than those of Kahneman and Tversky (1992). However, Burnes and Neilson (2002) show that these values cannot simultaneously explain gambles on unlike gains and the Allais paradox. In general, decision makers are loss averse and there is more utility curvature for gains than for losses. See Abdellaoui et al. (2007) and Wakker et al. (2007) for further discussion of the shape of LA utility. Despite the importance of loss aversion, the robustness and appropriateness of loss aversion utility function in financial markets have not been addressed. Although considerable field data has been accumulated since Kahneman and Tversky (1979), the majority of the studies have been conducted in laboratory experiments with students in the fields of decision theory or psychology. Differences may exist in the way these decision makers behave in experiments and in real financial markets (Levitt and List, 2007), because it is difficult to design experiments that include other important components in practice, such as the probability density function of asset returns or decision making with a large dollar amount of investment in financial markets. In our study, we attempt to determine the appropriate ranges of LA parameters in financial markets using a typical asset allocation problem for investors with LA utility.2 By analyzing asset allocation 2 Our approach is a partial equilibrium portfolio choice problem. Unlike an expected utility framework, we do not estimate a reasonable range of parameter values of LA utility function through a set of equilibrium relations (through Euler conditions) linking expected asset returns to covariances with consumption growth. Our study is different from that of Berkelaar and Kouwenberg (2009) who investigate the effects of loss aversion on asset prices.

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decisions of investors who maximize their LA utility, we propose several theoretical results for the LA utility function and the optimal asset allocation, and then show that these analytical results are empirically supported in the US and UK markets. Our main results can be summarized as follows. First, we find that investors are more loss averse than Kahneman and Tversky (1992) suggest; the loss aversion coefficient of 2.25 that has been widely used in the finance literature is lower than what we calculate, in particular, in the US market. This may be interpreted that when investors choose risky prospects that involve large amounts, they become more loss averse than the subjects (students) in laboratory experiments who are asked to choose risky prospects that are smaller in dollar amount or who are asked to choose risky prospects of large amounts in hypothetical situations. Second, loss aversion changes depending on market conditions. In particular, we find that the loss aversion coefficient should be larger during boom periods (bull markets) than during recessions (bear markets), indicating investors’ more profound disutility for losses when other investors enjoy their gains. The time variation of loss aversion adds a new dimension to the loss aversion literature, which is already hindered by many different versions of loss aversion (see, for example, Abdellaoui et al., 2007). Third, the curvature on losses should be larger than that of gains, and thus investors are more sensitive to the changes in losses than to the equivalent changes in gains. More utility curvature for losses than for gains is surprising because mostly the opposite is found; see Note 1, Wakker et al. (2007) for a survey. Finally, when UK and US investors are compared, the equity proportion in the typical US pension funds is lower than that in similar UK funds. If we make the simplistic assumption that the loss aversion coefficient is the sole source of the different equity proportions in pension funds, the lower investment proportion in the US pension funds could be interpreted that US investors have a larger loss aversion coefficient than UK investors. However, this does not necessarily suggest that US investors are more loss averse than UK investors, as the loss aversion coefficient could be affected by the differences other than loss aversion, e.g., difference in asset distributions, social/economic welfare, or characters of pension funds. What parameter values should be used for the LA utility in financial markets? We propose loss aversion coefficients of 3.25 and 2.75 for the US and UK investors, respectively, which should be increased and reduced by 1.5 during bull and bear markets, respectively; and the difference between the two curvature parameters should be 0.2 and 0.25 for the US and UK markets, respectively. Therefore, for investors who are risk averse for gains and risk loving for losses, we suggest that the curvature parameters of the UK investors should be 0.7 and 0.95 for gains and losses, respectively, while those of the US investors should be 0.7 and 0.9 for gains and losses, respectively. This paper is structured as follows: in Section 2, we present the details of the LA utility and our principal analytic results. In Section 3, we use UK and US asset allocation problems to empirically evaluate the values of LA parameters. Conclusions are provided in Section 4. 2. Admissible analytical ranges of the LA parameters 2.1. Loss aversion utility Several different definitions of loss aversion have been proposed in the literature; for instance, see Abdellaoui et al. (2007). In an effort to investigate the properties of the function, we use the standard LA function proposed by Kahneman and Tversky (1992) (henceforth KT) with minor modifications. Let W and B represent final wealth and some appropriate benchmark, respectively. The LA utility in our study is defined as

uðXÞ ¼

8 v < Xv 1 ;

if X  0;

1

: k

ðXÞv 2

v2

;

if X < 0;

ð1Þ

where X = W  B determines gains or losses, and the three parameters (i.e., v1, v2, and k) are assumed to be positive. As X v 1 and ðXÞv 2 are divided by v1 and v2, respectively, the value of k is not directly comparable with that of KT, whereas the two curvature parameters, v1 and v2, are not influenced by the modification. To observe this, when Eq. (1) is multiplied by v1, we have the LA function of KT:

uðXÞKT ¼

 v1 X ;

if X  0;

kKT ðXÞv 2 ;

if X < 0;

ð2Þ

where kKT ¼ k vv 12 . Therefore, depending on the values of v1 and v2, k is larger or smaller than kKT , and when v1 = v2, our LA utility function is the same as the original LA function of KT. The properties of LA utility are dependent on the selection of different parameter values, but there appear to be few theoretical results that suggest appropriate values for v1, v2, and k.3 Previous choices for these values are generally based on surveys or experiments such as those reported in the studies of Fishburn and Kochenberger (1979) and KT. When v1 > 1 and v2 > 1, the investor is risk loving with regard to gains since u00 ðXÞ ¼ ðv 1  1ÞX v 1 2 > 0, while she is risk averse with regard to losses since u00 ðXÞ ¼ ðv 2  1ÞðXÞv 2 2 < 0. This is similar to the reversed S-shape utility function of Markowitz (1952) and Post et al. (2008). Other two cases are also possible, i.e., ‘risk loving for gains and losses’ and ‘risk averse for gains and losses’. Balzer (2001) assumes that both the upside and downside are concave, such that the investor is risk averse in both directions. A common assumption regarding investors’ behavior in academic studies is the case that u() is ‘risk averse’ with regard to gains but ‘risk loving’ with regard to losses, which is what Kahneman and Tversky (1979) determine with a series of experiments. KT propose that v1 = v2 = 0.88 and k ¼ 2:25, and Burnes and Neilson (2002), Barberis et al. (2001), and Ang et al. (2005) have followed these suggested values.4 Which of these alternatives is the more plausible is not apparent. One simple method to avoid this parameter choice problem is to assume v1 = v2 = 1, such that u() would be risk neutral with regard to gains or losses, which has been used academically and commercially, see Benartzi and Thaler (1995). Several risk-control strategies, prevalent in the market, capture certain aspects of these alternative cases. For example, a stop-loss strategy controls downside risk and is presumably consistent with v2 > 1. A take-profit strategy controls upside risk and may be consistent with v1 < 1. In this study, we focus on the case that u() is risk averse with regard to gains but risk loving with regard to losses, following Kahneman and Tversky (1979, 1992). Using a parameter-free method, Abdellaoui et al. (2007) show strong evidence that supports prospect theory; decision makers are loss averse to various definitions, risk averse with respect to gains, and risk loving with respect to losses both at the aggregate and at the individual level. 3 In the loss aversion utility function in Eqs. (1) or (2), it is possible that marginal utility declines as wealth approaches zero. For the case when W is less than B, and W approaches zero (i.e., X < 0), the sign of the second derivative of the utility function would be positive if v2 < 1, and negative if v2 > 1. Thus, the utility function can be consistent with microeconomic theory (loss aversion) but also allows for possible loss tolerance. 4 Fishburn and Kochenberger (1979) put forward two-piece utility functions as an example of conventional expected utility theory, and present some survey evidence suggesting that k > 1 and that 0 < v1 < 1 and 0 < v2 < 1. They also refer to other papers that present experimental survey studies supporting these assumptions, namely that investors are risk averse for gains and risk loving for losses.

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2.2. Asset allocation problem in loss aversion world

Solving the above equation with the first-order condition in (6) gives

A simple two-asset case is used to demonstrate how the asset allocation decision is made in a loss aversion world, and then the admissible ranges of LA parameters are evaluated in the light of the optimal asset allocation decision. In order to investigate the LA parameters we introduce two hypothetical portfolios: a benchmark portfolio and a performance portfolio. The benchmark portfolio (reference point) could be any portfolio that an investor has reason to expect; i.e., the current state of wealth or riskfree asset.5 On the other hand, the performance portfolio is the one that aims for better performance by taking the risk of possible losses. These two portfolios need not consist of different asset classes; they may consist of the same classes of assets, but with different weights placed on the classes of assets. The investor must decide how much should be invested in these portfolios. Intuitively, when she is more loss averse, her investment proportion in the benchmark portfolio increases. On the other hand, the investment proportion in the performance portfolio increases when she is willing to take possible losses (less loss averse) to achieve higher performance. Denoting h as the proportion held in the performance portfolio, the final wealth of the investor is as follows:

v 2 > v 1:

W ¼ W 0 ð1  hÞð1 þ r b Þ þ W 0 hð1 þ rp Þ ¼ B þ hW 0 y;

ð3Þ

where rb and rp represent the returns of the benchmark and performance portfolios, respectively, B = W0(1 + rb), and y = rp  rb is the excess return of the performance portfolio. Therefore, gains or losses are denoted by hW0y and the asset allocation problem is to obtain the value of h that maximizes the loss aversion utility function in (1), i.e., u(hW0y). In order to obtain the optimal value of h, we let uþ ¼ Eðyv 1 jy > 0Þ; u ¼ EððyÞv 2 jy < 0Þ; p ¼ probðy > 0Þ, where the expectation is taken with a subjective measure (subjective probability density function) as suggested by Kahneman and Tversky (1979). Since X in Eq. (1) is equivalent to hW0y, the expected LA utility, ULA, is given by

U LA ¼

1

v1

ðhW 0 Þv 1 puþ 

k

v2

ðhW 0 Þv 2 ð1  pÞu :

ð4Þ

From Eq. (4) the first-order condition for maximizing ULA is

U 0LA ¼ W v0 1 uþ phv 1 1  kW v0 2 u ð1  pÞhv 2 1 ¼ 0: We first consider the case of tion becomes

ð5Þ

v1 = v2 = v. Then, the above equa-

U 0LA ¼ W v0 hv 1 ðuþ p  ku ð1  pÞÞ ¼ 0: When v = 0.88 < 1 as suggested by KT, the optimal solution for h is 1. On the other hand, when v > 1, the optimal solution for h is h = 0. Ang et al. (2005) come to similar conclusions when v1 = v2. Additionally, when v = 1 or uþ p ¼ ku ð1  pÞ, h is indeterminate. The results suggest that the assumption regarding these parameters in earlier papers, i.e., v1 = v2, is not appropriate in asset allocation. Barring the above cases, we need v1 – v2 to calculate an optimal investment proportion that matches historical experience. Solving for the case v1 – v2, we note that

 h¼

uþ p  pÞ

ku ð1

v

1 2 v 1

;

ð6Þ

where W0 = 1. The second order condition is

U 00LA ¼ ðv 1  1Þuþ phv 1 2  kðv 2  1Þu ð1  pÞhv 2 2 < 0:

5 See Kahneman and Tversky (1979) for further explanation of the appropriate choice of the benchmark.

ð7Þ

Proposition 1. The necessary and sufficient conditions for h to be optimal for the maximization of the expected LA utility in (4) are

h¼

 v 1 v 2 1 1 uþ p ;  W 0 ku ð1  pÞ

ð8Þ

v 2 > v 1:

ð9Þ

Proof. See the explanation above. h Eq. (8) demonstrates how much we should invest in the performance portfolio for given values of loss aversion parameters. The investment proportion is a non-linear function of u+, u, and p, the details of which will be discussed in the following section. Finally, by substituting (8) into (4), we can obtain the optimal LA utility, v2

U LA ¼ ðpuþ Þv 2 v 1



v2  v1 v 1v2



v

ðku ð1  pÞÞ

v1 2 v 1

;

ð10Þ

from which we can obtain the sensitivities of the optimal LA utility with respect to u+, u, and p around the optimal LA utility. Proposition 2. For the optimal LA utility in (10), the elasticities of U LA with respect to u+ and u are

@ ln U LA v2 ; ¼ @ðln uþ Þ v 2  v 1 @ ln U LA v1 ; ¼ @ðln u Þ v2  v1 and the semi-elasticity of U LA with respect to p is

@ ln U LA v 2 ð1  pÞ þ v 1 p ¼ : ðv 2  v 1 Þpð1  pÞ @p Proof. By taking logs of U LA and differentiating the equation with respect to u+, u and p, we have the results. h As expected, the elasticities of the optimal LA utility with respect to u+ and u are positive and negative, respectively, under the assumption that v2  v1 > 0. When v2 is close to v1, the elasticities increase. For example, when v2 is close to 0.9 and v2  v1 is 0.2, which we suggest later in our study, a percent change in u+ increases U LA by 4.5 percent. Note that the elasticity of U LA with respect to u+ is always larger than that with respect to u as v2 > v1, and therefore, a percent increase in u+ increases U LA further than a percent decrease in u. The semi-elasticity of U LA with respect to p is large. For example, with p = 0.5, v1 = 0.7, and v2 = 0.9, U LA increases by 16% when p increases by 0.01. The large semi-elasticity is driven by two effects; firstly, p is quadratic in the denominator, giving us a factor about 4 (i.e., p(1  p)), and secondly, v2  v1 is small (0.2), giving us multiplicative effects. In intuitive terms, an increase in p increases gains and at the same time decreases losses, which make the difference between additional utility from gains and disutility from losses large in the context of loss aversion utility. 2.3. Admissible ranges of loss aversion parameters As a part of our efforts to reduce the possible combinations of parameters, we use the results in the previous subsection in order to investigate the relationship between the three LA parameters and the investment proportion in the performance portfolio. We propose the lower bound of the loss aversion coefficient, and

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how the optimal investment proportion in the performance portfolio is influenced by the three loss aversion parameters. Proposition 3. If v1 – v2 and the proportion of wealth held in the performance portfolio is an increasing function of the probability that the performance portfolio outperforms the benchmark portfolio, then we have v2  v1 > 0. Proof. See Appendix. h Proposition 3 is a sensible constraint since it implies that, ceteris paribus, investors will increase their investment in the performance portfolio as the probability of gains (or positive excess returns, y > 0) rises. The proposition yields a theoretical motivation for v2 > v1 which satisfies the second order condition. This simple but strong result is useful in the derivation of other results in our study. We note that the proposition, which is also supported by our empirical results in the next section, is in sharp contrast with most previous studies that find more utility curvature for gains than for losses. According to a survey by Wakker et al. (2007), only a few studies such as Fishburn and Kochenberger (1979) have similar results as ours. Proposition 4. For the LA utility in (1) with the final wealth in (3), when W0 is set to 1 and 1  h  0, then any positive v2  v1 would þ p imply k  uuð1pÞ .

on the performance portfolio (h) increases with v2  v1, but decreases as k increases.

Proof. See Appendix. h The above proposition demonstrates that as relative curvature of losses to gains increases, the investment proportion on the performance portfolio will increase. Moreover, investors with a large loss coefficient must reduce the investment proportion on the performance portfolio, as the possibility of losses increases with increasing investment proportion on the performance portfolio. Our analytic results eliminate a few cases with respect to the two curvature parameters. For example, if v1 > 1 and v2 < 1 (i.e., risk loving with regard to gains and losses), then we would contravene the second order condition and Proposition 3. Obviously, the case of risk neutrality for losses and risk loving for gains is also excluded. Therefore, we have a function similar to that elucidated by Kahneman and Tversky (1979, 1992) (i.e., 1 > v2 > v1), the one proposed by Markowitz (1952) (i.e., v2 > v1 > 1), or the case of risk aversion with regard to gains and losses (i.e., v2 > 1 > v1). Considering the recent evidence of Abdellaoui et al. (2007), we focus on the case that is risk averse with regard to gains but risk loving with regard to losses, but report the results of the other two cases for comparison purposes. 3. Empirical investigation

Proof. See Appendix. h Proposition 4 suggests that k should be sufficiently large to ren-

Remark 1. When u+, u, and p change over time, the lower bound of k in Proposition 4 changes. The lower bound during bull markets, where we expect u+ to be relatively larger than u and p is expected to be larger than 0.5, is larger than that observed during bear markets in which u+ is relatively smaller than u and p is expected to be less than 0.5. This differs from regret or house money effects, both of which can be explained conditional on prior gains or losses. Barberis et al. (2001) model the regret and house money effects in LA utility by making k a function of the previous gains and losses. However, the changes in k in Remark 1 explain the contemporaneous relationship; if k varies with the changes of the lower bound, investors may experience more disutility for losses during bull markets, whereas they experience less disutility for the same amount of losses during bear markets. The intuition about Remark 1 is that an investor suffers relatively more deprivation from losses when other investors enjoy gains, whereas she experiences relatively less disutility from the same amount of losses when most other investors lose money. In other words, Remark 1 suggests that relative performance matters. Therefore, loss aversion utility can be generalized further by making k a function of current market performance for the relative effect, and of past performance for regret and house money effects, respectively. Finally, the investment proportion on the performance portfolio has the following relationship with the LA utility.

In this section, we investigate what values of v1, v2 and k are appropriate to explain the empirical value of h for the UK and US financial markets. The asset allocation problem in the LA utility is a typical single period decision making problem as in many other asset allocation studies; monthly data are used over a certain time period in order to optimize the investment weights. However, we face a few additional empirical difficulties in the optimization problem of (4). Ideally, a non-linear regression could be used for the estimation  of v1, v2, and k from Eq. (6) with the time series uþ t , ut , and ht (the subscript t represents that these variables are time varying and observed at time t). However, the performance and benchmark port folios required for the calculation of uþ t and ut cannot be readily identified. The characteristics of the performance and benchmark portfolios are not expected to be the same for different styles of funds, e.g., hedge funds and pension funds. In our study, we investigate the LA utility in the asset allocation problem for long-term investors (pension funds). Pension funds are widely used as a representative agent for asset allocation problems. See Campbell and Viceira (2002) for example. Moreover, by using pension funds, we can create performance and benchmark portfolios from the recommendations by professional investment advisors as in Canner et al. (1997). Using other portfolios such as hedge funds or derivatives does not seem to be appropriate for our purpose, as they are based on non-arbitrage trading (or risk neutral). Investors in these funds are likely to have quite different utility from those in pension funds.6 Second, the non-linear regression does not appear to be appropriate to our objectives. For example, we use 41 years of annual UK pension funds’ asset allocation data from 1963 to 2003 for various combinations of asset classes, but fail to obtain the estimates of the LA parameters. There are several reasons for this convergence error. First of all, ht is highly persistent and follows a unit root process (empirically). This is not unexpected as asset allocation in pension funds is not active. Changes in ht are highly positively correlated with the stock returns, thereby suggesting that a significant pro-

Proposition 5. For the LA utility in (1) with the final wealth in (3), when W0 = 1, 1 P h P 0, and v2  v1 > 0, the investment proportion

6 For studies on the LA parameters in derivatives markets, see Gurevich et al. (2009).

der ku ð1  pÞ  uþ p non-negative regardless of h, and thus

uþ p u ð1pÞ

can be interpreted as the lower bound of k. If the excess returns (rp  rb) are symmetrical with a mean zero and v1 is close to v2, then

uþ p u ð1pÞ

 1, which is similar to Fishburn and Kochenberger

(1979). However, in general, the expected excess returns are positive (i.e., u+ > u) and p is larger than 0.5; thus k should be larger than 1. Proposition 4 indicates that the lower bound of k may change over bull and bear markets as in the following remark.

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2429

Fig. 1. Smoothed probabilities of bull and bear market states. The bull and bear markets are identified with the regime switching model in equation (11) that allows different expected returns and volatilities. The smoothed probabilities show what regime the market was in at time t based on observations obtained through a later date T.

portion of ht is endogenous in the stock returns. This also follows from pension funds naively increasing the allocation in the asset class that has recently performed well. Therefore, the non-linear equation in (6) may not be well specified in general. Another reason is that the three LA parameters are linked strongly, and it is difficult to differentiate between them. The analytical results in the previous section indicate that h, v1  v2, and k are not independent of each other, which is also shown in the empirical results later. Third, it is worth noting that, as in Barberis and Xiong (2009), we use an objective probability density function rather than a subjective weight function to calculate the conditional expected returns. This is because, unlike in other areas, the objective probability density function and its properties are well known to investors in financial markets. Moreover, the investors who use their subjective weight functions that differ significantly from the objective probability density function would mis-price assets and thus be arbitraged out. Therefore, in an effort to further investigate the admissible ranges of v1, v2, and k within a more general framework, we calibrate Eq. (8) with various sets of v1, v2, and k for various investment proportions in the performance portfolio. Calibration imposes restrictions on certain parameter values in order to estimate others by replicating a dataset as a model solution. The estimated parameter values obtained via calibration allow us to

determine whether or not the theoretical arguments in the previous section are empirically supported. In addition, we can also evaluate how sensitive these parameters are to changes in other parameters, and how these parameters change for different investors or over different periods. We use the parameter values proposed by KT as our guidance in the empirical study. Although the results from laboratory experiments may not be directly applicable to financial decisions, Levitt and List’s (2007) study indicates that these parameters are still useful for qualitative insights.

3.1. Benchmark and performance portfolios The benchmark and performance portfolios can be constructed in many different ways.7 Following Canner et al. (1997) and Campbell and Viceira (2002), we construct these portfolios using cash, bond, and equity. There are many issues that we do not consider in our study. For example, we assume no tax and no transaction costs, and also that short sales are not allowed. Other asset classes, such as real estate or human capital, are not considered. Therefore, the empirical analysis in our study may not capture all of the aspects 7 We also used stocks and cash as proxies for the performance and benchmark portfolios. However, the results are very similar to those reported in this study.

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Table 1 Properties of US and UK market returns, riskfree rates, and excess market returns. Asset classes Bond

Cash

(A) US market returns, riskfree rates, and excess market returns Entire sample period (January Mean 0.008 0.003 1989–December 2008) Standard 0.025 0.002 deviation Skewness 0.156 0.043 Kurtosis 5.333 2.600 Jarque– 55.4 1.7 Bera Bull period (177 nonths)

Bear period (63 months)

Mean Standard deviation Skewness Kurtosis Jarque– Bera Mean Standard deviation Skewness Kurtosis Jarque– Bera

Bear period (76 months)

Mean Standard deviation Skewness Kurtosis Jarque– Bera Mean Standard deviation Skewness Kurtosis Jarque– Bera

Investment horizon

Two-fund separation

Performance portfolio

Performance portfolio

Performance portfolio

Benchmark portfolio

Benchmark portfolio

Benchmark portfolio

0.008 0.042

0.008 0.032

0.006 0.016

0.007 0.032

0.007 0.019

0.008 0.029

0.003 0.002

0.666 4.527 41.0

0.586 4.579 38.6

0.315 4.509 26.7

0.600 4.573 39.1

0.304 4.478 25.5

0.492 4.594 35.1

0.043 2.600 1.7

0.007 0.024

0.004 0.002

0.016 0.032

0.014 0.026

0.008 0.014

0.014 0.025

0.009 0.017

0.013 0.024

0.004 0.002

0.357 4.218 14.7

0.045 3.142 0.2

0.101 3.040 0.3

0.104 3.342 1.2

0.029 3.778 4.5

0.107 3.296 1.0

0.051 3.799 4.8

0.065 3.569 2.5

0.056 3.145 0.2

0.012 0.029

0.003 0.002

0.016 0.056

0.010 0.041

0.000 0.018

0.010 0.041

0.000 0.021

0.006 0.035

0.003 0.002

0.807 5.662 25.4

0.413 1.909 4.9

0.175 2.956 0.3

0.298 3.258 1.1

0.443 4.593 8.7

0.278 3.201 0.9

0.426 4.629 8.9

0.411 3.680 3.0

0.417 1.922 4.9

0.008 0.043

0.008 0.033

0.007 0.017

0.007 0.033

0.007 0.020

0.008 0.030

0.005 0.002

0.458 3.958 17.6

0.356 3.854 12.4

0.020 3.739 5.5

0.371 3.878 13.2

0.010 3.670 4.5

0.243 3.744 7.9

1.499 4.316 107.3

(B) UK market returns, riskfree rates, and excess market returns Entire sample period (January Mean 0.008 0.005 1989–December 2008) Standard 0.023 0.002 deviation Skewness 0.126 1.499 Kurtosis 3.184 4.316 Jarque– 1.0 107.3 Bera Bull period (164 months)

Stock

Investors characteristics

0.008 0.021

0.005 0.002

0.017 0.031

0.014 0.025

0.010 0.014

0.014 0.024

0.010 0.016

0.014 0.023

0.005 0.002

0.168 3.242 1.2

1.843 5.985 153.7

0.159 3.974 7.2

0.167 3.944 6.8

0.096 3.795 4.6

0.165 3.955 7.0

0.107 3.774 4.4

0.160 3.884 6.0

1.843 5.986 153.8

0.007 0.027

0.006 0.003

0.012 0.057

0.007 0.044

0.001 0.021

0.007 0.044

0.000 0.025

0.005 0.039

0.006 0.003

0.441 2.792 2.6

0.775 2.041 10.5

0.194 2.795 0.6

0.281 2.909 1.0

0.587 3.449 5.0

0.267 2.896 0.9

0.579 3.387 4.7

0.372 3.030 1.8

0.775 2.041 10.5

Notes: The investor characteristics-based method uses 20%, 5% and 75% for bonds, cash, and stocks, respectively, for the performance portfolio, whereas it uses 35%, 35% and 30% for bonds, cash, and stocks, respectively, for the benchmark portfolio. The investment horizon-based method uses 16.67%, 8.33% and 75% for bonds, cash, and stocks, respectively, for the performance portfolio, whereas it uses 43.33%, 21.67%, and 35% for bonds, cash, and stocks, respectively, for the benchmark portfolio. Finally, the twofund separation-based method uses 35.71%, 0% and 64.29% for bonds, cash, and stocks, respectively, for the performance portfolio, whereas it uses 100% for cash for the benchmark portfolio. The bull and bear markets are identified with the regime switching model in Eq. (11) that allows different expected returns and volatilities. The state variable is assumed to follow a first-order Markov chain. We estimate the regime switching model using the Bayesian Markov chain Monte Carlo Gibbs sampling estimation. We use standard conjugate Gaussian distributions for the expected returns and the inverted gamma distribution for the volatilities. The transition probabilities are estimated using conjugate beta priors, but in order to avoid regimes changing too frequently, weak priors are used for the probabilities. All results are generated with 10,000 iterations after at least 10,000 burn-in iterations.

of the LA utility but may provide partial aspects of LA utility in financial markets. Despite the restrictions, using the three most popular asset classes is anticipated to shed light on loss aversion in financial markets. The first method to construct the performance and benchmark portfolios is to use the investment proportions recommended by professional investment advisors as in Canner et al. (1997). We use the investment proportions of conservative and aggressive investors in order to construct the benchmark and performance portfolios, respectively. The portfolios for the conservative and aggressive investors are calculated by taking middle investment

proportions from the four recommendations of the following advisors; Fidelity, Merrill Lynch, Jane Bryant Quinn, and New York Times in Canner et al. (1997). For the benchmark portfolio, the weights for bonds, cash, and stocks are 35%, 35%, and 30%, respectively, while for the performance portfolio they are 20%, 5%, and 75%, respectively. These portfolios are based on investor characteristics (IC). The second method is to consider the investment horizon (IH). We use a widely known rule-of-thumb that suggests the stock allocation as 100 minus the investor’s age. The analysis should provide us with LA parameters from a quite different point of view, and

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S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438 Table 2 The values of

v1 n h

v2 for given sets of v1 and investment proportion in performance portfolio for the entire sample period when k = 2.25.

US market 0.10

0.20

UK market 0.50

0.60

0.70

0.80

0.90

1.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

(A) Investors characteristics-based portfolios 0.3 0.38 0.39 0.40 0.40 0.41 0.5 0.59 0.60 0.61 0.61 0.62 0.7 0.79 0.81 0.81 0.82 0.83 0.9 1.00 1.02 1.02 1.03 1.04 1.1 1.21 1.23 1.24 1.25 1.25 1.3 1.42 1.44 1.45 1.46 1.47 1.5 1.64 1.65 1.67 1.68 1.69 1.7 1.85 1.87 1.89 1.90 1.91 1.9 2.06 2.09 2.10 2.12 2.13

0.30

0.40

0.41 0.62 0.83 1.04 1.26 1.48 1.70 1.92 2.14

0.42 0.63 0.84 1.05 1.27 1.48 1.71 1.93 2.15

0.42 0.63 0.84 1.06 1.27 1.49 1.71 1.94 2.16

0.42 0.63 0.85 1.06 1.28 1.50 1.72 1.94 2.17

0.43 0.64 0.85 1.06 1.28 1.50 1.73 1.95 2.18

0.38 0.59 0.80 1.01 1.23 1.44 1.65 1.86 2.08

0.39 0.60 0.82 1.03 1.24 1.46 1.67 1.89 2.10

0.40 0.61 0.82 1.04 1.26 1.47 1.69 1.90 2.12

0.40 0.62 0.83 1.05 1.27 1.48 1.70 1.92 2.14

0.41 0.62 0.84 1.06 1.27 1.49 1.71 1.93 2.15

0.41 0.63 0.84 1.06 1.28 1.50 1.72 1.94 2.16

0.42 0.63 0.85 1.07 1.29 1.51 1.73 1.95 2.17

0.42 0.64 0.85 1.07 1.29 1.52 1.74 1.96 2.18

0.42 0.64 0.86 1.08 1.30 1.52 1.75 1.97 2.19

0.43 0.64 0.86 1.08 1.31 1.53 1.75 1.98 2.20

(B) Investment horizon-based portfolios 0.3 0.39 0.40 0.41 0.42 0.5 0.60 0.61 0.62 0.63 0.7 0.81 0.82 0.83 0.84 0.9 1.02 1.03 1.04 1.05 1.1 1.23 1.24 1.26 1.27 1.3 1.44 1.46 1.47 1.48 1.5 1.65 1.67 1.69 1.70 1.7 1.87 1.89 1.91 1.92 1.9 2.09 2.11 2.13 2.15

0.42 0.63 0.85 1.06 1.27 1.49 1.71 1.94 2.16

0.43 0.64 0.85 1.07 1.28 1.50 1.72 1.95 2.17

0.43 0.64 0.86 1.07 1.29 1.51 1.73 1.96 2.18

0.44 0.65 0.86 1.08 1.30 1.52 1.74 1.97 2.19

0.44 0.65 0.87 1.08 1.30 1.52 1.75 1.97 2.20

0.44 0.66 0.87 1.09 1.31 1.53 1.76 1.98 2.21

0.39 0.60 0.81 1.02 1.23 1.44 1.66 1.87 2.08

0.40 0.61 0.82 1.04 1.25 1.46 1.68 1.89 2.11

0.41 0.62 0.83 1.05 1.26 1.48 1.69 1.91 2.12

0.41 0.63 0.84 1.06 1.27 1.49 1.70 1.92 2.14

0.42 0.63 0.85 1.06 1.28 1.50 1.72 1.93 2.15

0.42 0.64 0.85 1.07 1.29 1.51 1.73 1.94 2.16

0.43 0.64 0.86 1.08 1.29 1.51 1.73 1.95 2.17

0.43 0.65 0.86 1.08 1.30 1.52 1.74 1.96 2.18

0.44 0.65 0.87 1.09 1.31 1.53 1.75 1.97 2.19

0.44 0.66 0.87 1.09 1.31 1.54 1.76 1.98 2.20

(C) Two-fund separation-based portfolios 0.3 0.36 0.37 0.37 0.38 0.38 0.5 0.56 0.57 0.58 0.58 0.58 0.7 0.77 0.77 0.78 0.79 0.79 0.9 0.97 0.98 0.99 0.99 1.00 1.1 1.18 1.19 1.19 1.20 1.21 1.3 1.38 1.39 1.40 1.41 1.42 1.5 1.59 1.60 1.61 1.62 1.63 1.7 1.80 1.81 1.82 1.83 1.84 1.9 2.01 2.02 2.03 2.04 2.05

0.38 0.59 0.79 1.00 1.21 1.42 1.63 1.85 2.06

0.39 0.59 0.80 1.01 1.22 1.43 1.64 1.85 2.07

0.39 0.59 0.80 1.01 1.22 1.43 1.64 1.86 2.08

0.39 0.60 0.80 1.01 1.22 1.44 1.65 1.87 2.08

0.40 0.60 0.81 1.02 1.23 1.44 1.65 1.87 2.09

0.38 0.59 0.80 1.01 1.22 1.43 1.63 1.84 2.05

0.39 0.60 0.81 1.02 1.23 1.44 1.65 1.86 2.07

0.39 0.61 0.82 1.03 1.24 1.46 1.67 1.88 2.09

0.40 0.61 0.83 1.04 1.25 1.47 1.68 1.89 2.11

0.41 0.62 0.83 1.05 1.26 1.48 1.69 1.91 2.12

0.41 0.62 0.84 1.06 1.27 1.49 1.70 1.92 2.13

0.41 0.63 0.85 1.06 1.28 1.50 1.71 1.93 2.14

0.42 0.63 0.85 1.07 1.29 1.50 1.72 1.94 2.15

0.42 0.64 0.86 1.07 1.29 1.51 1.73 1.94 2.16

0.43 0.64 0.86 1.08 1.30 1.52 1.74 1.95 2.17

Notes: The values of v2 are calculated numerically with the optimization problem in (12). The performance and benchmark portfolios for the three cases are constructed as in Table 1.

should help us to answer the question as to whether or not our loss aversion parameters change with age. Here, we change the proportion of stocks in the portfolio, and the proportion of bonds to cash is kept constant. Consider young and old investors whose ages are 25 and 65, respectively. Applying the rule-of-thumb gives us 75% and 35% investment proportions to stocks. The remainder is divided into bonds and cash with a fixed 2-to-1 proportion in accordance with the reported proportion of the moderate investors in Canner et al. (1997). Finally, in the simplified world where the two-fund separation (TFS) works, there are only two asset classes, cash and the market portfolio. The market portfolio, which consists of stocks and bonds, is used to increase the performance of a fund, whereas cash is used as the benchmark portfolio. Using the proportions for the moderate investors in the study of Canner et al. (1997), we fix the proportion of stocks to bonds to 1.8 . 3.2. Statistical properties of portfolio returns A total of 240 monthly returns from January 1989 to December 2008 is used for the UK and US markets. The difference between the two countries is expected to show the difference in the LA parameters between the US and the UK. For cash and bonds, we use the 3-month Treasury bill and the 10-year bond total returns from Citigroup. For stock returns, the FTSE All-share index and the S&P500 index total returns are used in the UK and US markets, respectively. In order to test whether LA parameters change over time, we need to identify market states. Motivated by the regime switching literature and the modelling of business cycles (e.g., Hamilton, 1989; Henry, 2009), we identify bull and bear markets using the following regime switching model:

r mt ¼ l1 S1t þ l2 S2t þ rt et ;

rt ¼ r1 S1t þ r2 S2t ;

ð11Þ

where rmt is the market return, li and ri are the expected market return and volatility of regime i, respectively, and the dummy (state) variable, Sit, is one when regime i is selected, and zero otherwise. Following Hamilton (1989), we allow the state variable to be governed by a first-order Markov chain, and estimate the regime switching model using the Bayesian Markov chain Monte Carlo Gibbs sampling estimation.8 Once we identify the two states, we label them according to the characteristics of the expected market return and volatility. Fig. 1 reports the smoothed probabilities of the two market states. Although there is slight difference between the countries, bull states are identified during the periods from 1993 to 1998 and from 2003 to 2007 whereas bear states are identified in the early 1990s, 1998, from 2000 to 2003, and from 2007 to the end of the sample period. The statistical properties of the returns for the entire sample period, bull and bear periods are reported in Table 1. The average return of stocks is not higher than that of bonds for the 20 years, mainly due to the sharp plummet of share prices in 2008. The bear period appears to be less than a third of the entire sample period. As expected, stock returns during this period are characterized by a low average return with higher volatility. The difference between the US and UK does not appear to be significant, although cash shows fat-tails 8 We use standard conjugate Gaussian distributions for li and the inverted gamma distribution for ri. The transition probabilities are estimated using conjugate beta priors, but in order to avoid regimes changing too frequently, we use weak priors for the transition probabilities. We allow for a large number of burn-in iterations to guarantee convergence. All results are generated with 10,000 iterations after at least 10,000 burn-in iterations. For detailed explanations, see Kim and Nelson (1999).

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S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438

Fig. 2. Difference between the two curvature parameters (v2  v1).

and positive skewness in the UK market whereas it does not in the US market. The properties of the portfolios based on the IC are close to those based on the IH. The investment pattern of young investors does not differ significantly from that of the ‘aggressive’ investors recommended by professional investment advisors. In both cases, owing to diversification effects, the performance and benchmark portfolios have smaller volatilities and are closer to normality than stocks. However, the performance and benchmark portfolios show large differences in volatility or higher moments, which could be relevant to asset allocation problems when the non-linear LA utility function is used. On the other hand, the benchmark portfolio in the TFS case is equivalent to cash, and is thus significantly different from the other two benchmark portfolios. 3.3. Empirical properties of the LA utility To determine what might be appropriate sets of v1, v2, and k in asset allocation decisions, we first investigate whether the relationship v2 > v1 holds for various values of h. We then analyze what

the magnitude of v2  v1 is and if the estimated magnitude of v2  v1 is robust for various loss aversion coefficients or for different market conditions. The loss aversion coefficient and its lower bound are estimated in a similar fashion. Finally, we suggest LA parameter values for the US and UK markets. The values of the two curvature parameters in the two countries can be identified using the empirical investment proportions in the performance portfolio, h, which is not known. We extract h from the investment proportions in stocks. The investment proportions in stocks for large pension funds in the US and UK were approximately 50% and 78% in 1993, respectively.9 These proportions in stocks correspond to a 30% (IC case) and 50% (IH case) investment in the performance portfolio in the US, whereas they are equivalent to a 100% (both IC and IH cases) investment in the performance portfolio in the UK.

9 See Elton (1999) for example. Ex post returns are not necessarily anticipated to converge to the ex ante returns in finite samples.

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S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438 Table 3 The values of

v1 n h

v2 for given sets of v1 and investment proportion in performance portfolio for the entire sample period when k = 1.5 and 3.

k = 1.5

k=3

US market 0.10

0.30

UK market 0.50

0.70

US market

UK market

0.90

0.10

0.30

0.50

0.70

0.90

0.10

0.30

0.50

0.70

0.90

0.10

0.30

0.50

0.70

0.90

(A) Investors characteristics-based portfolios 0.3 0.32 0.32 0.33 0.33 0.33 0.5 0.52 0.53 0.53 0.53 0.54 0.7 0.73 0.74 0.74 0.74 0.74 0.9 0.94 0.94 0.95 0.95 0.96 1.1 1.14 1.15 1.16 1.17 1.17 1.3 1.35 1.37 1.37 1.38 1.39 1.5 1.57 1.58 1.59 1.60 1.61 1.7 1.78 1.80 1.81 1.82 1.83 1.9 1.99 2.01 2.03 2.04 2.05

0.32 0.53 0.74 0.95 1.16 1.37 1.58 1.79 2.01

0.32 0.53 0.74 0.96 1.17 1.39 1.60 1.82 2.03

0.33 0.54 0.75 0.96 1.18 1.40 1.61 1.83 2.05

0.33 0.54 0.75 0.97 1.19 1.40 1.62 1.84 2.06

0.33 0.54 0.76 0.97 1.19 1.41 1.63 1.85 2.07

0.43 0.63 0.84 1.05 1.26 1.47 1.69 1.90 2.12

0.45 0.66 0.87 1.08 1.30 1.51 1.73 1.95 2.17

0.47 0.68 0.89 1.10 1.32 1.54 1.76 1.98 2.20

0.48 0.69 0.90 1.12 1.34 1.56 1.78 2.01 2.23

0.49 0.70 0.92 1.13 1.35 1.58 1.80 2.03 2.26

0.43 0.64 0.85 1.06 1.28 1.49 1.70 1.92 2.13

0.45 0.67 0.88 1.10 1.31 1.53 1.75 1.97 2.18

0.47 0.68 0.90 1.12 1.34 1.56 1.78 2.00 2.22

0.48 0.70 0.92 1.14 1.36 1.58 1.81 2.03 2.25

0.49 0.71 0.93 1.16 1.38 1.60 1.83 2.05 2.27

(B) Investment horizon-based portfolios 0.3 0.33 0.34 0.34 0.34 0.5 0.54 0.54 0.55 0.55 0.7 0.74 0.75 0.76 0.76 0.9 0.95 0.96 0.97 0.97 1.1 1.16 1.17 1.18 1.19 1.3 1.37 1.39 1.40 1.41 1.5 1.58 1.60 1.62 1.63 1.7 1.80 1.82 1.84 1.85 1.9 2.01 2.04 2.06 2.07

0.35 0.56 0.77 0.98 1.20 1.41 1.64 1.86 2.09

0.33 0.54 0.75 0.96 1.17 1.38 1.59 1.80 2.01

0.34 0.54 0.76 0.97 1.18 1.39 1.61 1.82 2.04

0.34 0.55 0.76 0.97 1.19 1.40 1.62 1.84 2.05

0.34 0.55 0.77 0.98 1.20 1.41 1.63 1.85 2.07

0.34 0.56 0.77 0.99 1.20 1.42 1.64 1.86 2.08

0.44 0.64 0.85 1.06 1.28 1.49 1.70 1.92 2.14

0.46 0.67 0.89 1.10 1.32 1.53 1.75 1.97 2.19

0.48 0.69 0.91 1.12 1.34 1.56 1.78 2.01 2.23

0.49 0.71 0.92 1.14 1.36 1.58 1.81 2.03 2.26

0.51 0.72 0.94 1.16 1.38 1.60 1.83 2.06 2.29

0.43 0.64 0.86 1.07 1.28 1.49 1.71 1.92 2.13

0.46 0.67 0.89 1.10 1.32 1.54 1.75 1.97 2.19

0.48 0.69 0.91 1.13 1.35 1.56 1.78 2.00 2.22

0.49 0.71 0.93 1.15 1.37 1.59 1.81 2.03 2.25

0.50 0.72 0.94 1.16 1.38 1.61 1.83 2.05 2.28

(C) Two-fund separation-based portfolios 0.3 0.30 0.29 0.29 0.29 0.29 0.5 0.49 0.49 0.49 0.49 0.49 0.7 0.70 0.70 0.69 0.69 0.69 0.9 0.90 0.90 0.90 0.90 0.90 1.1 1.10 1.10 1.10 1.11 1.11 1.3 1.31 1.31 1.31 1.31 1.31 1.5 1.52 1.52 1.52 1.52 1.53 1.7 1.72 1.73 1.73 1.73 1.74 1.9 1.93 1.94 1.94 1.95 1.95

0.31 0.52 0.73 0.93 1.14 1.35 1.56 1.77 1.97

0.31 0.52 0.73 0.94 1.15 1.36 1.57 1.78 1.99

0.31 0.52 0.73 0.95 1.16 1.37 1.58 1.80 2.01

0.31 0.53 0.74 0.95 1.17 1.38 1.59 1.81 2.02

0.32 0.53 0.74 0.96 1.17 1.39 1.60 1.81 2.03

0.41 0.61 0.81 1.02 1.23 1.43 1.64 1.85 2.06

0.43 0.63 0.84 1.05 1.26 1.47 1.68 1.89 2.10

0.44 0.65 0.86 1.07 1.28 1.49 1.70 1.92 2.13

0.46 0.66 0.87 1.08 1.29 1.51 1.72 1.94 2.16

0.47 0.67 0.88 1.10 1.31 1.52 1.74 1.96 2.18

0.43 0.64 0.85 1.06 1.27 1.48 1.69 1.90 2.11

0.45 0.67 0.88 1.10 1.31 1.52 1.74 1.95 2.16

0.47 0.69 0.90 1.12 1.34 1.55 1.77 1.98 2.20

0.49 0.70 0.92 1.14 1.36 1.58 1.80 2.01 2.23

0.50 0.72 0.94 1.16 1.38 1.60 1.82 2.04 2.25

Notes: The values of v2 are calculated numerically with the optimization problem in (12). The results are obtained using 240 monthly returns from January 1989 to December 2008. Bold numbers represent a violation of Proposition 3, i.e., v2 > v1.

3.3.1. The values of v2 and v1 for given investment proportions Appropriate values of v2 are optimized for given values of v1 and h via the Newton–Raphson algorithm as follows:

" min h  v2



v 1 v #2 2 1 uþ p ; ku ð1  pÞ

ð12Þ

where v1 is set to values from 0.3 to 2, h is set to values from 0.1 to 1, k is set to 2.25, and W0 = 1 following the asset allocation literature. The two components, u+p and u(1  p), are estimated using

uþ p ¼

T 1X yv 1 Iy 0 ; T t¼1 t t

u ð1  pÞ ¼

T 1X ðyt Þv 2 ð1  Iyt 0 Þ; T t¼1

where Iyt 0 is the indicator variable, which is one when yt P 0 and zero otherwise.10 In our study, we set the maximum value of v1 to 2 since both u+ and u rapidly approach zero as v1 and v2 increase, respectively. Table 2 reports the estimated values of v2 for the given values of v1 and h, and Fig. 2 shows the values of v2  v1 for the given values of v1 and h. The results support Proposition 3 in that we have v2  v1 > 0 for all cases, thereby suggesting that the experimental 10 In a previous version of the paper, we used two parametric distribution functions for the estimation of u+ and u as in Hwang and Pedersen (2004); the normal distribution and the scale Gamma distribution. However, the results obtained using these probability density functions do not differ significantly from those obtained using the nonparametric method we report in this study.

evidence of v1 = v2 = 0.88 by KT and others may be too restrictive in asset allocation decisions. The difference between v2 and v1 is not negligible. When the tenets of prospect theory – namely, risk aversion for gains and risk loving for losses – are assumed, the difference between v1 and v2 is up to 0.19 in the US and UK markets. The values of v2  v1 are slightly smaller for TFS portfolios than for the other two portfolios. As in Markowitz (1952), if v2 > v1 > 1, the value of v2  v1 increases up to 0.3 (IH). Investors with the reversed S-shape utility become more sensitive to changes in losses than in gains as the values of v1 and v2 increase. A close look at the table and figure reveals several interesting results. First, there is little difference in v2  v1 between the US and UK. However, UK investors seem to have larger values of v2  v1 as their investment proportion in the performance portfolio is close to 1. For example, for h = 1 and v1 = 0.7, the values of v2  v1 are close to 0.16. On the other hand, in the US market, the values of v2  v1 are approximately 0.12 for h = 0.3 and v1 = 0.7. These results show that, ceteris paribus, UK investors are more sensitive to the changes in losses or less sensitive to gains than are US investors. Second, the values of v2  v1 increase with v1 (or v2) and h, thus supporting Proposition 5, but this increase is not large. Finally, the values of v2  v1 are similar for the cases of IC and IH. In terms of the two curvature parameters, young investors are not particularly different from aggressive investors; old investors, whose investment horizons are shorter, can generally be regarded as conservative investors. This can be understood by the results shown in Table 1, which report little difference in the portfolio returns between the two cases. In order to evaluate if the sensitivity of the curvature parameters to different values of k, we test two different values of k, i.e., 1.5 and 3. Table 3 shows that when k increases or decreases by 0.75, v2  v1

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S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438

Table 4 The values of

v1 n h

v2 for given sets of v1 and investment proportion in the performance portfolio for the bull and bear periods: investors characteristics-based portfolios.

Bull market

Bear market

US market 0.10

0.30

UK market 0.50

0.70

0.90

0.10

0.30

US market

UK market

0.50

0.70

0.90

0.10

0.30

0.50

0.70

0.90

0.10

0.30

0.50

0.70

0.90

(A) The estimated values 0.3 0.25 0.24 0.5 0.44 0.43 0.7 0.63 0.62 0.9 0.83 0.81 1.1 1.02 1.00 1.3 1.21 1.19 1.5 1.40 1.38 1.7 1.59 1.57 1.9 1.78 1.75

of v2 for 0.23 0.42 0.61 0.80 0.99 1.18 1.37 1.55 1.74

given sets of v1 0.23 0.23 0.42 0.41 0.61 0.60 0.80 0.79 0.98 0.98 1.17 1.16 1.36 1.35 1.54 1.53 1.73 1.72

and h with k = 1.5 0.24 0.23 0.22 0.43 0.42 0.41 0.63 0.61 0.60 0.82 0.81 0.80 1.02 1.00 0.99 1.21 1.19 1.18 1.41 1.39 1.38 1.61 1.59 1.57 1.80 1.78 1.77

0.22 0.41 0.60 0.79 0.98 1.17 1.37 1.56 1.76

0.21 0.40 0.59 0.78 0.97 1.17 1.36 1.56 1.75

0.50 0.72 0.93 1.14 1.35 1.56 1.77 1.98 2.19

0.55 0.77 0.98 1.19 1.41 1.62 1.84 2.05 2.27

0.58 0.80 1.01 1.23 1.45 1.66 1.88 2.10 2.32

0.60 0.82 1.04 1.26 1.48 1.70 1.92 2.14 2.36

0.62 0.85 1.07 1.29 1.51 1.73 1.95 2.17 2.39

0.47 0.69 0.90 1.11 1.32 1.53 1.74 1.95 2.15

0.51 0.73 0.94 1.16 1.37 1.59 1.80 2.01 2.22

0.53 0.76 0.97 1.19 1.41 1.62 1.83 2.05 2.26

0.55 0.78 1.00 1.22 1.44 1.65 1.87 2.08 2.29

0.57 0.80 1.02 1.24 1.46 1.68 1.89 2.11 2.32

(B) The estimated values 0.3 0.31 0.31 0.5 0.50 0.50 0.7 0.69 0.69 0.9 0.89 0.88 1.1 1.08 1.07 1.3 1.27 1.26 1.5 1.46 1.45 1.7 1.65 1.64 1.9 1.84 1.83

of v2 for 0.31 0.50 0.69 0.88 1.07 1.26 1.45 1.64 1.82

given sets of v1 0.31 0.31 0.50 0.50 0.69 0.69 0.88 0.88 1.07 1.07 1.26 1.25 1.44 1.44 1.63 1.63 1.82 1.81

and h with k = 2.25 0.30 0.30 0.29 0.49 0.49 0.49 0.69 0.68 0.68 0.88 0.88 0.88 1.08 1.07 1.07 1.27 1.27 1.27 1.47 1.47 1.46 1.67 1.66 1.66 1.87 1.86 1.86

0.29 0.49 0.68 0.87 1.07 1.26 1.46 1.66 1.85

0.29 0.49 0.68 0.87 1.07 1.26 1.46 1.65 1.85

0.57 0.79 1.00 1.21 1.42 1.63 1.84 2.06 2.27

0.63 0.85 1.07 1.28 1.50 1.71 1.93 2.15 2.36

0.67 0.89 1.11 1.33 1.55 1.77 1.99 2.21 2.42

0.70 0.93 1.15 1.37 1.59 1.81 2.03 2.25 2.48

0.73 0.96 1.18 1.41 1.63 1.85 2.07 2.30 2.52

0.54 0.75 0.97 1.18 1.39 1.60 1.81 2.02 2.23

0.59 0.81 1.03 1.25 1.46 1.67 1.89 2.10 2.31

0.63 0.85 1.07 1.29 1.51 1.72 1.94 2.15 2.36

0.65 0.88 1.10 1.33 1.54 1.76 1.98 2.19 2.41

0.68 0.91 1.13 1.36 1.58 1.80 2.01 2.23 2.44

(C) The estimated values 0.3 0.35 0.36 0.5 0.54 0.55 0.7 0.74 0.74 0.9 0.93 0.93 1.1 1.12 1.13 1.3 1.31 1.32 1.5 1.50 1.51 1.7 1.70 1.69 1.9 1.89 1.88

of v2 for 0.36 0.56 0.75 0.94 1.13 1.32 1.51 1.69 1.88

given sets of v1 0.37 0.37 0.56 0.56 0.75 0.75 0.94 0.94 1.13 1.13 1.32 1.32 1.51 1.51 1.69 1.69 1.88 1.88

and h with k = 3 0.34 0.34 0.53 0.54 0.73 0.73 0.92 0.93 1.12 1.13 1.32 1.32 1.52 1.52 1.72 1.72 1.91 1.92

0.35 0.54 0.74 0.94 1.13 1.33 1.53 1.72 1.92

0.35 0.55 0.74 0.94 1.13 1.33 1.53 1.72 1.92

0.62 0.83 1.05 1.26 1.47 1.68 1.90 2.11 2.32

0.69 0.91 1.13 1.35 1.56 1.78 2.00 2.21 2.43

0.74 0.96 1.18 1.40 1.62 1.84 2.06 2.28 2.50

0.78 1.00 1.23 1.45 1.67 1.89 2.12 2.34 2.56

0.81 1.04 1.27 1.49 1.72 1.94 2.16 2.39 2.61

0.59 0.80 1.02 1.23 1.44 1.65 1.86 2.07 2.28

0.65 0.87 1.09 1.31 1.52 1.74 1.95 2.16 2.37

0.69 0.92 1.14 1.36 1.58 1.79 2.01 2.22 2.44

0.73 0.95 1.18 1.40 1.62 1.84 2.06 2.27 2.49

0.76 0.99 1.21 1.44 1.66 1.88 2.10 2.32 2.53

0.35 0.54 0.74 0.93 1.13 1.33 1.52 1.72 1.92

Notes: The values of v2 are calculated with the optimization problem in (12). The bull and bear periods are identified with the regime switching model in (11). Bold numbers represent violation of Proposition 3, i.e., v2 > v1.

increases or decreases, and thus, when k ¼ 1:5, we have cases of v2 6 v1 which violate Proposition 3. As the proposition should hold regardless of k as far as investors would increase the proportion of the performance portfolio with the probability that the performance portfolio outperforms the benchmark portfolio, we conclude that k ¼ 1:5 is too low. On the other hand, when k ¼ 3, the condition v2  v1 > 0 is always satisfied. In this case, the values of v2  v1 increase such that they are approximately 0.18 (for h = 0.3 and v1 = 0.7) for the US and 0.24 (for h = 1 and v1 = 0.7) for the UK, respectively, and we have kKT ¼ k vv 12 similar to 2.25. The results also support our arguments (Proposition 5) that for a given value of v2  v1, investors reduce the investment proportion on the (risky) performance portfolio when they are more loss averse. We can also find some suggestions as to why estimating the three parameters in a non-linear regression is difficult. Tables 2 and 3 clearly demonstrate that there are many different combinations of v1, v2, and k, which explain the US investment proportion in stocks. For example, we could select values of v1 = 0.5 and v2 = 0.61, v1 = 0.7 and v2 = 0.8, or v1 = 0.9 and v2 = 1.02 for 50% investment in the performance portfolio in the US (see panel A, Table 2). Even if ht changes over time, as far as ht is highly autocorrelated (or non-stationary) or does not change frequently, it becomes difficult to estimate the three LA parameters. This is essentially an issue of parameter identification; the least squares criterion function is not empirically identified. The three parameters, v1, v2, and k, are also connected together; k increases with v2  v1, which again increases with h. KT estimated v2 and v1 separately using experimental data, and were able to avoid the problems associated with the estimation of the parameters.

3.3.2. Loss aversion in bull and bear markets We conduct the same optimization procedure in (12) for the bull and bear markets identified with the regime switching model. Table 4 indicates that the value of k needs to be higher than 2.25 in both countries during the bull markets, as Proposition 3 is violated in many cases. On the contrary, during the bear markets, all cases satisfy this condition. In order to investigate the changes of the loss aversion coefficient under different market conditions, we calculate the lower bound of k suggested in Proposition 4 for different values of v2  v1 and report the results in Table 5. The results in panels A and B in Table 5 demonstrate that the lower bound in the bull markets is approximately six times larger than that in the bear market, supporting that k is a function of contemporaneous market performance, thus supporting Remark 1. Although the values of the lower bounds do not directly show what the value of k should be, the large variation in the lower bounds during bull and bear markets indicates that Kahneman and   Tversky’s (1992) fixed value i:e:; kKT ¼ k vv 12 ¼ 2:25 may not be reflective of the disutility for losses, particularly during the bull markets. Comparing the two countries, the lower bounds of the US and the UK are similar; for example, for both the US (v2  v1 = 0.2, v1 = 0.7) and the UK (v2  v1 = 0.25, v1 = 0.7), the lower bounds are approximately 7.35 during the bull markets, but they become approximately 1.1 during the bear markets. However, these results do not suggest that the two countries have similar loss aversion, as the lower bounds do not reflect the difference in the investment

2435

S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438 Table 5 Lower bound of the loss aversion coefficient with investors characteristics-based portfolios.

v1

v2

(A) US 0.5 0.5 0.5 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 1.5 1.5 1.5

market 0.65 0.70 0.75 0.85 0.90 0.95 0.95 1.00 1.05 1.05 1.10 1.15 1.65 1.70 1.75

Lower bound of k

Lower bound of kv1/v2

Entire sample period 2.377 2.905 3.543 2.257 2.739 3.320 2.194 2.655 3.207 2.130 2.569 3.095 1.729 2.055 2.440

1.829 2.075 2.362 1.858 2.131 2.446 1.848 2.124 2.444 1.826 2.102 2.422 1.572 1.814 2.091

Entire sample period (B) UK market 0.5 0.70 0.5 0.75 0.5 0.80 0.7 0.90 0.7 0.95 0.7 1.00 0.8 1.00 0.8 1.05 0.8 1.10 0.9 1.10 0.9 1.15 0.9 1.20 1.5 1.70 1.5 1.75 1.5 1.80

2.813 3.422 4.155 2.587 3.127 3.774 2.481 2.991 3.600 2.381 2.862 3.437 1.873 2.226 2.643

Lower bound of k

Lower bound of kv1/v2

Lower bound of kv1/v2

Bull period (177 months)

Bear period (63 months)

5.309 7.062 9.384 5.502 7.291 9.653 5.608 7.419 9.807 5.722 7.558 9.975 6.578 8.625 11.304

0.860 1.094 1.390 0.814 1.030 1.303 0.793 1.001 1.263 0.773 0.974 1.226 0.659 0.822 1.025

4.084 5.044 6.256 4.531 5.671 7.113 4.723 5.935 7.472 4.904 6.184 7.807 5.980 7.611 9.689

Bull period (164 months) 2.009 2.281 2.597 2.012 2.304 2.642 1.985 2.279 2.618 1.948 2.240 2.578 1.653 1.908 2.203

Lower bound of k

5.962 7.448 9.289 5.978 7.423 9.203 5.985 7.411 9.165 5.991 7.400 9.129 6.007 7.330 8.937

4.258 4.965 5.806 4.650 5.469 6.442 4.788 5.647 6.665 4.902 5.791 6.847 5.300 6.283 7.447

0.661 0.782 0.927 0.670 0.801 0.960 0.668 0.801 0.962 0.662 0.797 0.959 0.599 0.726 0.879

hv 2 þv 1

1.20 1.27 1.35 1.20 1.27 1.35 1.20 1.27 1.35 1.20 1.27 1.35 1.20 1.27 1.35

Bear period (76 months) 1.027 1.231 1.473 0.952 1.137 1.355 0.921 1.097 1.305 0.892 1.061 1.260 0.762 0.900 1.061

0.733 0.820 0.921 0.741 0.838 0.949 0.737 0.836 0.949 0.730 0.830 0.945 0.673 0.771 0.884

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Notes: The lower bounds are calculated using the equation in Proposition 4. The values in the last column, hv 2 þv 1 , represent an additional factor for investment proportions in the performance portfolio that differ between the US and UK. Note that we use h = 0.3 for the US and h = 1 for the UK.

proportion on the performance portfolio11; as we noted earlier, the US investment proportion on the performance portfolio (30%) is less than that of the UK (100%). If we make the simplistic assumption that the loss aversion coefficient is the sole source of the difference in performance proportion, the results in Proposition 5 indicate that the investment proportion on the performance portfolio decreases as loss aversion increases. From Eq. (6) and Proposition 4, we calculate a factor, hv 2 þv 1 , that can account for the difference. The last column of Table 5 reports that the factor is always one for the UK (h = 1) whereas it ranges from 1.2 and 1.35 for the US (h = 0.3), indicating that, ceteris paribus, the US lower bounds need to be increased by 20% to 35% for the lower investment proportion on the performance portfolio. This increase in the US lower bounds suggests that the loss aversion coefficient of US investors could be larger than that of UK investors. The larger US loss aversion coefficient, however, should be interpreted with care. It does not necessarily suggest that US investors are more loss averse than UK investors. A direct comparison between the two countries may not be possible in our asset allocation problem as the differences in asset distributions, social/economic welfare, or characters of pension funds may contribute the difference in the loss aversion coefficient. We leave more rigorous international comparisons for future study. 3.4. Summary and discussions The lower bounds of k for the entire sample period lie between 2.75 and 3 in the US and UK markets, respectively, unless v1 and v2 11

For details, see the proof of Proposition 4 in Appendix A.

are large, i.e., over 1. For the US, the lower bound needs to be increased to 3.5 due to the adjustment factor of 1.27 (hv 2 þv 1 , where h = 0.3 and v2  v1 = 0.2) for the low investment proportion in the performance portfolio. However, these are ‘lower bounds’. Due to the large variations in the lower bounds and uncertainty over the estimated v1 and v2 values, loss aversion coefficients should be larger than these lower bounds. We suggest 4 and 3.5 for the US and UK markets, respectively, by adding 0.5 to the lower bounds. From the results in Table 3 we have v2  v1 = 0.18 (US) and 0.24 (UK) when k ¼ 3 and v2 < 1. As v2  v1 increases with k, we propose v2  v1 = 0.2 for the US and 0.25 for the UK. For the reversed Sshape utility, v2  v1 needs to be increased by an additional 0.1 for the two markets. How do these values compare with those proposed by KT? KT’s   loss aversion coefficients k vv 12 are 3.1 for the US (v1 = 0.7 and

v2  v1 = 0.2) and 2.6 for the UK (v1 = 0.7 and v2  v1 = 0.25). Therefore, we propose that KT’s loss aversion coefficient should be 3.25 and 2.75 for the US and UK markets, respectively. Although direct comparison may not be possible due to the difference in the definitions of various loss aversion measures, the results of Abdellaoui et al. (2007) suggest that our loss aversion coefficient is in general larger than those of other previous studies (including KT) which are based on experiments. The increase in loss aversion in empirical asset allocation reflects the difference between laboratory experiments and practice. There may be various explanations for the higher level of loss aversion, but we propose an explanation that the high level of loss aversion may reflect risky prospects with larger dollar amounts in financial markets. In typical laboratory experiments, subjects

2436 Table 6 The values of

S. Hwang, S.E. Satchell / Journal of Banking & Finance 34 (2010) 2425–2438

v2 for given sets of v1 and investment proportion in the performance portfolio: annual data from 1985 to 2003.

Performance portfolio Benchmark portfolio Investment proportion on the performance portfolio

v1 n h

Stocks and properties Bonds and cash Average investment proportion: 0.79

Stocks, bonds and properties Cash Average investment proportion: 0.95

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

0.10 0.26 0.46 0.67 0.88 1.09 1.31 1.52 1.74 1.95

0.30 0.25 0.45 0.66 0.87 1.09 1.31 1.53 1.75 1.97

0.50 0.24 0.44 0.65 0.87 1.09 1.31 1.54 1.76 1.99

0.70 0.23 0.43 0.65 0.87 1.09 1.32 1.54 1.77 2.01

0.90 0.22 0.42 0.64 0.86 1.09 1.32 1.55 1.79 2.02

0.10 0.22 0.42 0.63 0.84 1.05 1.26 1.48 1.69 1.91

0.30 0.19 0.40 0.60 0.82 1.03 1.25 1.47 1.69 1.91

0.50 0.17 0.38 0.59 0.80 1.02 1.24 1.46 1.68 1.91

0.70 0.16 0.36 0.57 0.78 1.00 1.23 1.45 1.68 1.91

0.90 0.14 0.34 0.55 0.77 0.99 1.21 1.44 1.68 1.92

k = 2.25

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

0.35 0.56 0.77 0.98 1.20 1.41 1.63 1.84 2.06

0.36 0.58 0.79 1.01 1.23 1.45 1.68 1.90 2.12

0.38 0.59 0.81 1.03 1.26 1.49 1.72 1.95 2.17

0.39 0.60 0.83 1.05 1.29 1.52 1.75 1.99 2.23

0.40 0.62 0.84 1.08 1.31 1.55 1.79 2.04 2.28

0.31 0.52 0.72 0.93 1.15 1.36 1.58 1.79 2.01

0.31 0.52 0.73 0.95 1.16 1.38 1.61 1.83 2.05

0.32 0.52 0.74 0.96 1.18 1.40 1.63 1.86 2.09

0.32 0.53 0.74 0.97 1.19 1.42 1.65 1.88 2.12

0.32 0.53 0.75 0.97 1.20 1.44 1.67 1.91 2.16

k=3

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

0.41 0.62 0.84 1.05 1.27 1.48 1.70 1.92 2.13

0.45 0.67 0.89 1.11 1.33 1.56 1.78 2.01 2.23

0.48 0.70 0.92 1.15 1.38 1.61 1.84 2.08 2.31

0.50 0.73 0.96 1.19 1.43 1.67 1.91 2.15 2.38

0.53 0.76 0.99 1.23 1.48 1.72 1.97 2.22 2.46

0.37 0.58 0.79 1.00 1.22 1.43 1.65 1.87 2.08

0.40 0.61 0.82 1.04 1.26 1.48 1.71 1.93 2.16

0.42 0.63 0.85 1.07 1.30 1.52 1.75 1.98 2.22

0.43 0.65 0.87 1.10 1.33 1.56 1.79 2.03 2.27

0.45 0.67 0.89 1.12 1.36 1.60 1.84 2.08 2.33

k = 1.5

Notes: The values of

v2 are calculated with the optimization problem in (12). Bold numbers represent violation of Proposition 3, i.e., v2 > v1.

(students) are asked to choose risky prospects that are small in dollar amount (i.e. up to hundreds of dollars for each subject), or they may be asked to choose risky prospects of large amounts in hypothetical situations (i.e. subjects do not bet real money). In reality when investors decide their lifetime investments, they could be more loss averse since their decision could have more significant and long-term effects on their life.12 We need to be careful when proposing an appropriate value for the loss aversion coefficient during the bull and bear market periods. Table 5 shows that the lower bounds of k vv 12 during bull markets are approximately six times larger than those of bear markets, confirming our analytical results that investors in bull markets become more loss averse than in bear markets. However, it is not likely that loss aversion changes in such a dramatic way. The lower bounds in Table 5 are calculated with ex post returns, and investors’ ex ante expected returns are not expected to be as volatile as the ex post returns.13 Therefore, we propose a modest increase and decrease of 1.5 for bull and bear periods, respectively. 3.5. Robustness tests with annual data In this section, we conduct the same analysis as in Table 4 using 23 years of annual UK pension funds’ asset allocation data from 1985 to 2003.14 The data include the investment proportions in seven asset classes; UK equities, overseas equities, properties, UK bonds, overseas bonds, index-linked guilts, and cash. The annual 12 We do not suggest that the level of loss aversion increases as dollar amounts in the risky prospects increase. We would rather point out the differences in decision making in risky bets between lab and reality. 13 See Elton (1999) for example. Ex post returns are not necessarily anticipated to converge to the ex ante returns in finite samples. 14 We also would like to thank David Blake for the UK asset allocation data.

data may reflect investment decisions better than monthly data. For example, see Benartzi and Thaler (1995). As in Table 5 we optimize appropriate values of v2 for given values of v1, h, and k, and report the results in Table 6.15 The seven asset classes are divided into two portfolios; (case 1) the performance portfolio (UK and overseas equities and properties) and the benchmark portfolio (UK bonds, overseas bonds, index-linked guilts, and cash) and (case 2) the performance portfolio (UK and overseas equities, properties, UK bonds and overseas bonds, and index-linked guilts) and the benchmark portfolio (cash). The returns on these portfolios are calculated with historical investment proportions on these asset classes. We take the average investment proportion in the performance portfolio over these sample periods and report it as the h value. The values of v2 in case 1 are close to those in Tables 2 and 3, and the difference between the annual and monthly returns is not large. Case 2 also reports similar difference between v1 and v2 for the given investment proportion on the performance portfolio, which does not differ substantially from the results we obtain with the TFS case in Table 2. The results with the annual data also confirm that k ¼ 1:5 is too small. Summarizing the results, our findings with monthly data do not differ substantially from those observed with the annual data: k ¼ 1:5 appears too small and v2 is larger than v1, but the difference is approximately 0.25.

15 As explained previously, we have also used 41 years of annual UK pension funds’ asset allocation data (from 1963 to 2003) in the nonlinear regression in (6), but failed to obtain estimates of the LA parameters. When the calibration method in (12) is used, the difference between v2 and v1 is smaller for this longer sample period. However, using an excessively long period (41 years) may not be appropriate when we consider the increasing investment in overseas equities since the 1980s and general changes in the institutional and legal framework influencing investment decisions.

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4. Conclusions In this study, we investigate the LA utility function in financial markets using the asset allocation problem. Despite the criticism raised by Köbberling and Wakker (2005), the two-side power function is used extensively to analyze loss aversion. We first develop several theoretical results that provide conditions that should be satisfied by the parameters of the LA utility function. We demonstrate that the curvature for the losses (v2) should be larger than that for the gains (v1). We also show that the optimal investment proportion on the performance portfolio increases with v2  v1, but decreases with k in the loss aversion world. Finally, a lower bound for the loss aversion coefficient is suggested and is demonstrated to change significantly over bull and bear markets. This may reflect changing composition of the representative agents in different markets. These analytical results are supported by the results of empirical tests. Our results demonstrate that v2 is larger than v1 by 0.25 for the UK market and by 0.2 for the US market, and the conventional loss aversion coefficient, i.e., 2.25, falls within the admissible ranges but is close to the lower bound. For Kahneman and Tversky’s (1992) loss aversion coefficient, we suggest 3.25 and 2.75 in the US and UK markets, respectively, which requires adjustment to explain the relative effect. There are a few issues that we would like to address. The parameter values estimated from observed macro aggregates are not necessarily comparable to those calculated through the analysis of experimental data. This is because, in addition to the difference between experiments and practice discussed above, the experimental tests often have repeated game structure, whereas the investor in this paper faces a static optimization. However, our main results should still hold for the dynamic loss aversion function proposed by Barberis et al. (2001). Allowing loss aversion to depend on past gains and losses should not change our main results that loss aversion increases during bull markets. Intuitively, the regret and house money effects from past losses and gains differ from the relative effects that investors feel by comparing others’ performance. The multi-period analysis, being more complex, would not be amenable to the same approach we employ in this paper. This is because gains and losses are path-dependent in the multi-period analysis and cannot be simply reduced to analysis of one period. A full extension to a dynamic LA utility function could be envisaged, which we leave for future study. Finally, in our study, we assume, as is common in asset allocation problems, that there are two hypothetical asset classes: one performance portfolio, and the other benchmark portfolio. Three different methods have been proposed for the construction of these portfolios, and then tested. It appears probable that the selection of a different benchmark will yield different attitudes to risk; under-performing cash is qualitatively different from an under-performing peer-group. Extensions to the case of multiple asset classes would require more complex methods and analytic solutions might no longer be available without more specific assumptions regarding the form of multi-asset utility. Acknowledgements We would like to thank participants of the CEMFI, FMA European and APFA/PACAP/FMA conferences for their helpful comments. We also would like to thank Inquire for their financial support, John Tomi for his research assistance, and David Blake for the UK asset allocation data.

Appendix Proof of Proposition 3. As the proportion of wealth in the performance portfolio rises with the probability of a gain, the partial derivative of ln h with respect to ln p should be positive, i.e., @ ln h > 0. Thus, from Eq. (6), @ ln p

@ ln h 1 ¼ > 0: @ ln p ðv 2  v 1 Þð1  pÞ The condition that satisfies the above equation is

v2  v1 > 0.

h

Proof of Proposition 4. When W0 is set to 1 and 1 P h P 0 in (8)

1



uþ p ku ð1  pÞ

Since

v

1 2 v 1

 0:

v2  v1 > 0,

0  logðuþ pÞ  logðku ð1  pÞÞ; and thus

k

uþ p :  pÞ

u ð1



Proof of Proposition 5. From Eq. (6), we have

  @ ln h 1 uþ p  0; ln ¼ @ðv 2  v 1 Þ ku ð1  pÞ ðv 2  v 1 Þ2 þ

p since 1  kuuð1pÞ > 0 from Proposition 4. Likewise we have

@ ln h 1 ¼ < 0: @ ln k v2  v1



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