Investment horizon and the attractiveness of investment strategies: A behavioral approach

Journal of Banking & Finance 34 (2010) 1032–1046

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Investment horizon and the attractiveness of investment strategies: A behavioral approach Maik Dierkes, Carsten Erner, Stefan Zeisberger * Finance Center Münster, University of Münster, Universitätsstraße 14-16, 48143 Münster, Germany

a r t i c l e

i n f o

Article history: Received 25 September 2008 Accepted 4 November 2009 Available online 10 November 2009 JEL classification: D14 D81 G11 Keywords: Behavioral finance Cumulative Prospect Theory Portfolio choice Investment strategy Investment horizon

a b s t r a c t We analyze the attractiveness of investment strategies over a variety of investment horizons from the viewpoint of an investor with preferences described by Cumulative Prospect Theory (CPT), currently the most prominent descriptive theory for decision making under uncertainty. A bootstrap technique is applied using historical return data of 1926–2008. To allow for variety in investors’ preferences, we conduct several sensitivity analyses and further provide robustness checks for the results. In addition, we analyze the attractiveness of the investment strategies based on a set of experimentally elicited preference parameters. Our study reveals that strategy attractiveness substantially depends on the investment horizon. While for almost every preference parameter combination a bond strategy is preferred for the short run, stocks show an outperformance for longer horizons. Portfolio insurance turns out to be attractive for almost every investment horizon. Interestingly, we find probability weighting to be a driving factor for insurance strategies’ attractiveness. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The dependence of an investment strategy’s evaluation on the investment horizon is by no means straightforward. Analyzing return distributions building on relatively simple measures such as expected return, volatility, or shortfall risk represents an often used method. While these measures have the advantage of being easy to understand, they do not necessarily account for all features of the return distribution. Further, some compound measures, for example, the Sharpe ratio, are not appropriate to capture relevant characteristics of distributions like skewness or excess kurtosis (Farinelli et al., 2009; Zakamouline and Koekebakker, 2009). Another approach is to compare the utility that an investor obtains from different investment strategies. In his seminal work Samuelson (1963) shows that investors’ portfolio compositions are independent of the investment horizon. However, this result holds true only under specific assumptions (Ross, 1999; Barberis, 2000; de Brouwer and van de Spiegel, 2001). Furthermore, Expected Utility Theory (EUT) fails to explain the empirically observed demand for some strategies (Branger and Breuer, 2008).

* Corresponding author. Tel.: +49 251 83 22455; fax: +49 251 83 22690. E-mail addresses: [email protected] (M. Dierkes), carsten.erner@ uni-muenster.de (C. Erner), [email protected] (S. Zeisberger). 0378-4266/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2009.11.003

Shefrin and Statman (1993) suggest a behavioral explanation for the differences in attractiveness, namely, that different framing might lead to different attractiveness levels of otherwise identical strategies. Polkovnichenko (2005) explicitly calls for rank-dependent utility theories like Cumulative Prospect Theory (CPT) to investigate investors’ portfolio choices. In this study, we systematically examine how a CPT investor assesses different investment strategies. CPT is widely considered to be the most successful descriptive theory for decision making under uncertainty (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992). We analyze six investment strategies: pure stocks, pure bonds, buy-and-hold, constant mix, protective put, and constant proportion portfolio insurance (CPPI). Based on a dataset of the S&P 500 and US Treasury Securities returns from 1962 to 2008, we use a bootstrap procedure to calculate the attractiveness a CPT investor assigns to these strategies. The analyses are conducted for different investment horizons ranging from 1 to 84 months, based on the median investor as described by Tversky and Kahneman (1992). We further conduct extensive sensitivity analyses to disentangle the influences of curvature, loss aversion, and probability weighting on a CPT investor’s optimal investment strategy, depending on the investment horizon. Moreover, we integrate experimentally elicited preference parameters taken from Abdellaoui et al. (2007) to get a sense of how a real set of investors would decide.

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Our study is the first that investigates the interplay of CPT preferences and the investment horizon. Breuer and Perst (2007) examine discount certificates and reverse convertibles within a CPT framework. Driessen and Maenhout (2007) report demand for protective put strategies by CPT investors. Both studies, however, neglect the investment horizon. Barberis and Huang (2008) focus on probability weighting and find that CPT investors demand right skewed payoff profiles. Focusing on static portfolio compositions of stocks and bonds, Aït-Sahalia and Brandt (2001) find horizon effects for CPT investors, that is, optimal portfolio compositions vary with the investment horizon. The authors, essentially, identify loss aversion to be the driving factor and conclude that probability weighting only plays a minor role. Portfolio compositions of stocks and bonds, however, are not particularly suitable for generating right-skewed payoff profiles. The results of these studies thus suggest varying optimal strategies depending on the interplay of investment horizon and preference parameters. Using right-skewed payoff profiles, for example protection strategies, and at the same time allowing for probability weighting generates important insights in addition to those of Aït-Sahalia and Brandt (2001). Our main results are as follows. Stocks outperform for long investment horizons owing to their high mean returns whereas a pure bond strategy is the best choice for short horizons. Investors conducting probability weighting prefer right-skewed payoff profiles as generated by the protective put and CPPI strategy for various curvature and loss aversion combinations and for almost every time horizon. Interestingly, most previous studies ascribe the demand for insurance strategies solely to loss aversion. Therefore, our results further stress the necessity to consider insurance strategies and to investigate the impact of probability weighting carefully. Additionally to our sensitivity analyses on CPT parameters, we conduct numerous robustness checks. We control for the peso problem, that is, investors’ belief in extreme adverse events which have not yet materialized. Rietz (1988) and Barro (2006) find that the possibility of such shocks helps to explain the observed equity premium in the past. In the spirit of these studies, we account for the possibility of a low probability, exorbitant negative stock return. We also consider an extended return dataset beginning in 1926 to check for long-term variations in the equity premium. As there might be potential regime shifts in investors’ beliefs, it seems reasonable to investigate different time periods by splitting the return dataset or by using a rolling belief formation window. Furthermore, we control for autocorrelation of stock returns by using a block bootstrap method. The many robustness checks corroborate the characteristic horizon effects outlined above. In the following we will use the term ‘‘horizon effects” in general, although referring to the specific horizon effects outlined above. Throughout our analysis, we make the assumption that investors evaluate the terminal return distribution. In order to explain the equity premium puzzle (Mehra and Prescott, 1985), however, Benartzi and Thaler (1995) assume that investors act myopically, proposing the theory of myopic loss aversion. In their analysis, they show that an evaluation period of 12 months resolves the equity premium puzzle because Tversky and Kahneman’s (1992) median decision maker finds the historical bond and stock returns equally attractive for this evaluation period. Twelve months represents a natural evaluation period as, for example, most people file taxes annually, and thus evaluate their portfolios yearly. In the light of these results, our assumption, that is, evaluating terminal distributions, seems not indisputable, at least from a behavioral point of view. There are, however, some arguments in favor of our approach. The results of Benartzi and Thaler (1995) refer to the market level by using the CPT median decision maker. Therefore, the question arises of how individual investors with varying

preference parameters assess the different strategies. Furthermore, in the context of investment counseling, terminal distributions are frequently shown and explained to investors. For example, investors might be asked to construct a desired terminal distribution through software tools like the distribution builder (Sharpe et al., 2000). Moreover, when buying structured financial products that comprise an investment strategy similar to the ones analyzed in our study, issuers regularly show terminal distributions on their websites (for structured products see, for example, Stoimenov and Wilkens, 2005). Evidence that people are influenced by the framing of return distributions comes from, for example, Redelmeier and Tversky (1992), showing that subjects’ investment decisions depend on whether return distributions are presented in a segregated or an aggregated way. In addition, we focus on relatively short investment horizons of only 1 to 84 months, that is, horizons investors should be able to cope with. Thus, investors might well think in terms of terminal distributions once the investment decision is framed appropriately. The remainder of the study is structured as follows. In Section 2, we briefly present the fundamentals of CPT and describe the functional forms that are applied. We also discuss the bootstrap procedure as well as the calculation of the strategies’ utilities. In Section 3, we describe the set of investment strategies. Section 4 presents the results. We summarize our insights in Section 5. 2. Cumulative prospect theory 2.1. Overview We assume a decision maker with preferences according to CPT. CPT distinguishes between gains and losses derived relative to a reference point, that is, investors care about changes in wealth rather than absolute wealth levels. The reference point is usually set to the status quo, see, for example, Benartzi and Thaler (1995). We follow this convention and set the reference point to a strategy’s return of zero.1 Loss aversion implies that losses yield a higher (negative) utility than equally large gains. In addition, probabilities are distorted, that is, decision makers weight outcomes by decision weights that non-linearly depend on the original probabilities. Consequently, risk attitude under CPT consists of three components: basic utility (curvature), loss aversion, and probability weighting. We will approximate the terminal return distribution for each investment strategy by using a bootstrap approach. The bootstrapped return distribution can be interpreted as a prospect with n outcomes xi with probability pi: P ¼ ðx1 ; p1 ; . . . ; xn ; pn Þ. Outcomes are given as percentage returns, representing a gain or a loss from the strategy. We then sort them in ascending order to ensure monotonicity ðxn >    > xlþ1  0 > xl >    > x1 Þ, with n  l gains and l losses. Defining mðÞ as the value function and pi as the decision weights, a CPT decision maker evaluates an investment strategy by the utility

CPTðStrategyÞ ¼

l X i¼1

pi  mðxi Þ þ

n X

pi  mðxi Þ:

ð1Þ

i¼lþ1

2.2. Functional forms and parameterizations We assume the value function mðÞ to be of the two-part power utility form

1 For individual stocks, Kliger and Kudryavtsev (2009) find that company-specific events may cause investors to update their reference points. We assume this effect to be negligible in our study as we consider a broad stock index.

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Table 1 Descriptive statistics for the monthly return data of interest rates. 5-Years US Treasury Securities (%)

10-Years US Treasury Securities (%)

0.81 4.31

0.44 0.22

0.49 0.23

0.53 0.22

0.54 0.21

0.56 0.20

-0.2

0.0

0.2

0.4

0.6

-0.4

-0.2

0.0

0.4

0.6

-0.2

0.0

0.2

-0.2

0.4

0.6

0.0

0.2

0.4

0.6

frequency

cppi

0

frequency

pp

0 -0.4

-0.4

strategy return

1000 2000 3000 4000 5000 6000

cmix

0

frequency

0.2

strategy return

1000 2000 3000 4000 5000 6000

strategy return

1000 2000 3000 4000 5000 6000

-0.4

bah

0

frequency

bonds

0

frequency

stocks

1000 2000 3000 4000 5000 6000

3-Years US Treasury Securities (%)

1000 2000 3000 4000 5000 6000

1-Year US Treasury Securities (%)

0

frequency

1-Month US Treasury Securities (%)

1000 2000 3000 4000 5000 6000

Mean Standard deviation

S&P 500 (%)

-0.4

-0.2

0.0

0.2

strategy return

strategy return

0.4

0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

strategy return

Fig. 1. Return distributions for the investment strategies. This figure shows the return distributions of the investment strategies for horizon m = 12.

mðxÞ ¼



xa kðxÞb

if x  0; if x < 0;

ð2Þ

where x is an outcome, k is the loss aversion coefficient, and a and b reflect sensitivity towards gains and losses. a and b are usually less than one. In this case, the decision maker shows diminishing sensitivity for gains and losses, leading to concavity in the domain of gains and to convexity in the domain of losses. Besides this, the valuation by CPT strongly depends on the parameter set, which highlights the necessity for conducting sensitivity analyses. Using a probability weighting function wðÞ, the decision weight pi for a gain (loss) xi is calculated as

pi ¼ wðpcum Þ  wðpcum; Þ; i i

ð3Þ

where pcum is the probability of obtaining a gain (loss) at least as i is the probability of obtaining large (at most as small) as xi and pcum i a gain (loss) larger (smaller) than xi. We follow again Tversky and Kahneman (1992) and assume w to be of the form

wðpÞ ¼

pd ðpd

1

þ ð1  pÞd Þd

:

ð4Þ

The inverse-S shape of the probability weighting function puts extra weight on the tails of the return distribution. This effect is greater for lower values of d and smaller for d close to one. For d = 1, there is no probability distortion. Given the sometimes extreme outcomes of the investment strategies, we expect probability distortion to be a main driving factor for the valuation. Schmidt and Zank (2008) show that, given these parametric assumptions, second order stochastic dominance analyses as in, for example, Annaert et al. (2009) are not feasible for CPT.

2.3. Bootstrapping and calculating the CPT value For our base scenario, the dataset comprises monthly returns from January 1962 through December 2008.2 The S&P 500 index represents the stock market, with returns being calculated as total returns, that is, including reinvestment of dividends.3 For the bond market we use the yields of one-month, and one-, three-, five-, and ten-year US Treasury Securities. To be able to analyze different investment horizons, we linearly interpolate the yield curves based on these maturities. Descriptive statistics in Table 1 show, on average, an upward sloping shape for the yield curve. As the number of non-overlapping, discrete time intervals in our dataset becomes relatively small for longer investment horizons, we follow Benartzi and Thaler (1995) and conduct a resampling method (bootstrapping). With this technique, the return distributions are smoothed by randomly drawing samples out of the dataset. As we will describe later, for each investment horizon m, we need a series of m monthly stock returns, one yield for the one-month Treasury Security and one yield for the m-months Treasury Security. We randomly draw the series of stock returns with replacement and s = 100,000 simulation runs. The yields for the two Treasury Securities are drawn pair-wise, thereby capturing the term structure.

2 All data in this study is taken from Cornell (1999) and Thomson Reuters Datastream. 3 Intermediate cash flows during the investment period, that is, interest payments and dividends, would lead to a different evaluation, especially given the segregated valuation for multiple mental accounts, see for example, Thaler (1985). However, we assume direct reinvestment of such cash flows and thereby focus on the terminal return distributions.

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γ=2

-0.7

-0.6

stocks bah bonds cmix pp cppi

-0.9

0.2

0.3

0.4

Expected Utility

γ=1

0.0

-1.0

0.1

Expected Utility

0.5

stocks bah bonds cmix pp cppi

-0.8

0.6

M. Dierkes et al. / Journal of Banking & Finance 34 (2010) 1032–1046

1

12

24

36

48

60

72

84

1

12

24

48

60

72

84

-0.05

-0.40

72

84

-0.10

γ=5

-0.25

-0.50

stocks bah bonds cmix pp cppi

-0.15

-0.35

-0.30

Expected Utility

γ=3

-0.20

-0.25

stocks bah bonds cmix pp cppi

36

Investment Horizon in Months

-0.45

Expected Utility

-0.20

Investment Horizon in Months

1

12

24

36

48

60

72

84

Investment Horizon in Months

1

12

24

36

48

60

Investment Horizon in Months

Fig. 2. Horizon effects for Expected Utility Theory with constant relative risk aversion. This figure shows the expected utilities of an investor with CRRA in the form of 1c mðxÞ ¼ x1 c for different levels of risk aversion c over different horizons.

We ensure comparability by employing the same set of data for each investment strategy. The drawn returns are compounded according to the strategies’ functional forms. As a result, we receive s terminal returns for each combination of strategy and investment horizon which we then treat as prospects with n = 100,000 outcomes.4 Consequently, we calculate a CPT value for each combination of strategy and investment horizon. 3. Investment strategies Given the broad range of investment strategies, a fundamental classification distinguishes between forecast-based and forecastfree strategies. Instead of considering expectations about future asset prices, we focus on forecast-free strategies. In total, we analyze six strategies, three strategies without and three with rebalancing. The initial investment amount for each strategy is set to one. Buy-and-hold strategies do not require a rebalancing as they trace the price development of the initial portfolio composition. The pure stock strategy (stocks) embodies an investment of 100% in the S&P 500. Similarly, the pure bond strategy (bonds) represents an investment of 100% in the US Treasury Securities. As a mixture, a 50/50 buy-and-hold strategy (bah) consisting of 50% S&P 500 and 50% US Treasury Securities is also considered.5 The particular mmonths US Treasury Security for the bond component of the bonds and buy-and-hold strategies is chosen from the interpolated yield curve such that its maturity exactly matches the respective investment horizon m.

4 Strictly speaking, the number of (disjunctive) outcomes is smaller than or equal to 100,000, since some of the simulation runs might lead to identical outcomes. 5 Importantly, our results are not restricted to the case of a 50/50 composition. We also investigated a buy-and-hold strategy with an optimal composition from a CPT viewpoint. While such an optimal buy-and-hold strategy (naturally) shows a slightly higher performance, the analysis corroborates the qualitative validity of our results presented in Section 4.

For the dynamic strategies the rebalancing interval is set to one month. Consequently, the bond component of the three following strategies consists of one-month US Treasury Bills. The constant mix strategy (cmix) embodies a combination of 50% S&P 500 and 50% US Treasury Bills. We rebalance such that the 50/50-weighting is reestablished after each month. The protective put strategy (pp) protects a pre-specified level through a long put option. Usual protection levels include at-the-money or slightly out-of-the-money put options. We follow this convention and apply a strike price of 95% of the initial investment amount. Tradable options on futures and options exchanges usually exhibit only short maturities. Consequently, we replicate the option on a monthly basis through a dynamic delta hedge based on the Black/Scholes-model.6 As a second insurance approach we consider the constant proportion portfolio insurance strategy (cppi), which protects the floor level through constant rebalancing (see, for example, Black and Perold, 1992). The multiplier of the CPPI strategy depends on how much risk the investor is willing to take. We set the floor level to 95%, comparably to the protective put, and the multiplier to 2.5.7 For the rebalancing of portfolios we take transaction costs into account with a cost rate of 0.3% for stock and 0.1% for bond transactions.8 Transaction costs reduce the portfolio value. We do not apply transaction costs for the buildup of the initial portfolio. Fig. 1 depicts histograms of return distributions for an exemplary investment horizon of 12 months. Appendix A shows key descriptive

6 The historic volatility is derived from a rolling estimation period. We further apply a surcharge of one percentage point to account for the empirically observed difference between historic and implied volatility (see, for example, Christensen and Prabhala, 1998). 7 As a robustness check, we also analyzed protection levels of 90% and 99% both for the protective put and CPPI and multipliers of 1.5 and 5 for the CPPI strategy. Although parameter variations show some influence on attractiveness, the overall qualitative results and horizon effects do not change. 8 Our results are largely independent of the size of transaction costs, even for double and triple costs.

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0.4 0.0

0.2

CPT Value

0.6

0.8

stocks bah bonds cmix pp cppi

24

36

48

60

72

84

1

12

2

stocks

3

cppi

4

pp

5

bah cmix 6

Rank of Strategy (1=best, 6 =worst)

1

bonds 1

12

24

36

48

60

72

84

Investment Horizon in Months Fig. 3. CPT values and ranks of investment strategies for median decision makers. This figure shows evaluations of the investment strategies according to Tversky and Kahneman’s (1992) median CPT parameters, that is, a = b = 0.88, k = 2.25, d = 0.69, d+ = 0.61. The upper panel shows CPT values of all investment strategies for the median decision maker over horizons ranging from 1 to 84 months. The lower panel shows the ranks from 1 (best) to 6 (worst) of the different investment strategies.

statistics for the strategies for several investment horizons up to 84 months and Appendix B shows scatter plots of the strategies’ returns given different levels of stock returns. The convex return profile for the protective put and the CPPI strategy are noteworthy here. 4. Results 4.1. Evaluation under expected utility theory To obtain a benchmark for our results under CPT, we first assess the attractiveness of investment strategies under EUT. Specifically, we calculate the expected utility for a decision maker with preferences described by constant relative risk aversion (CRRA), the most frequently assumed EUT specification. While absolute attractiveness generally rises with a longer time horizon, our results reveal that the relative attractiveness is mostly independent of the investment horizon, depending primarily on the risk aversion parameter of the utility function. Hence, strategies that are preferred for the short run, also show higher evaluations for longer investment horizons. The pure stock strategy, for instance, is the most attractive strategy for lower degrees of risk aversion (c = 1), whereas it is the least attractive one for higher degrees, see Fig. 2. The only strategy for which the time horizon plays a role under EUT is the CPPI strategy, which is illustrated by crossing lines in the graphs. Here, the relative attractiveness depends on the level of risk aversion: for lower degrees, the CPPI strategy performs better for longer horizons and vice versa. In a nutshell, there are only minor horizon effects under EUT.9

9

An analysis of EUT with constant absolute risk aversion leads to similar results.

4.2. Evaluation for the median CPT decision maker As a starting point for our analysis we consider an investor with the median parameters elicited by Tversky and Kahneman (1992), that is, a = b = 0.88, k = 2.25, d = 0.69, d+ = 0.61.10 Fig. 3 shows the main results for investment horizons of 1–84 months. Evidently, not only the absolute but also – and in sharp contrast to our EUT results – the relative attractiveness of the strategies depends crucially on the investment horizon.11 For short horizons of up to 19 months the pure stock strategy has the lowest CPT value. The best performing strategies are the pure bond strategy (1–11 months), followed by the CPPI strategy. Despite the fact that standard deviations of return distributions are small for the short run, we observe a rather large difference in CPT evaluations. For medium-term horizons (m  24), these differences become slightly smaller as the formerly worse performing strategies become relatively more attractive. The two protection strategies perform best up to 51 months, the constant mix strategy is the least attractive. For longer time horizons, stocks become even better and are as attractive as the two protection strategies, for a time horizon exceeding 57 months the stock investment

10 As a robustness check, we also applied different probability weighting specifications. However, the results remain unchanged. In detail, we used: wðpÞ ¼ expððln pÞa Þ (Prelec (1998)), with a = 0.74; wðpÞ ¼ expðb  ðln pÞa Þ (Prelec (1998)) with a = 0.534 and b = 1.08 (parameters were originally only elicited for the gain domain, but are here also applied in the loss domain); as well as wðpÞ ¼ dpc =ðdpc þ ð1  pÞc Þ with d = 0.65, c = 0.6 in the gain domain and d = 0.84, c = 0.65 in the loss domain (Wu and Gonzalez, 1996). Parameterizations are taken from Bleichrodt and Pinto (2000) and Wu and Gonzalez (1996). 11 This and the following results do not depend on the term structure, that is, results are similar if we use a flat yield curve.

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M. Dierkes et al. / Journal of Banking & Finance 34 (2010) 1032–1046 − +

= 1.00, = 1.00

− +

= 0.84, = 0.76

− +

= 0.69, = 0.61

− +

= 0.54, = 0.46

1

3

6

12

bonds cppi

24 pp

36

48

bah

60 stocks

bah

bonds

72

84

Fig. 4. Sensitivity analysis over CPT preference parameters. This figure shows the best investment strategy for different (a; k) combinations and for different investment horizons. In each graph the horizontal axis represents different values for k, ranging from k = 1 at the left to k = 5 at the right. The vertical axis represents different values for a, with a = 0.5 at the top and a = 1.2 at the bottom. Probability distortions become larger (lower values for d+ and d) from left to right, while the graphs on the left side represent investors with no distortion. The rows from top to bottom correspond to investment horizons m = 1, 3, 6, 12, 24, 36, 48, 60, 72, and 84.

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M. Dierkes et al. / Journal of Banking & Finance 34 (2010) 1032–1046

base case

= 0.0005

= 0.001

= 0.002

1

3

6

12

bonds

24

cppi pp

36

48

60 stocks bah

72

84

Fig. 5. Impact of low probability adverse events. This figure shows the best investment strategy for different (a; k) combinations and for different investment horizons. In each graph the horizontal axis represents different values for k, ranging from k = 1 at the left to k = 5 at the right. The vertical axis represents different values for a, with a = 0.5 at the top and a = 1.2 at the bottom. The crash probability amounts to 1  (1  p)m+1 for time horizon m. The crash return equals 30% and occurs in two consecutive months. The rows from top to bottom correspond to investment horizons m = 1, 3, 6, 12, 24, 36, 48, 60, 72, and 84.

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becomes best. These three strategies set themselves apart from the other three as the investment horizon enlarges. For the pure bond and the constant mix strategies, we observe a clear underperformance. The pure bond strategy becomes the worst for horizons exceeding 44 months. The buy-and-hold strategy shows a medium performance with CPT values between stocks and bonds. The protection strategies perform relatively well for almost the entire set of horizons. This is an interesting result as return distributions’ median values for the pure bond and the CPPI strategies are very similar across investment horizons up to 72 months. The results demonstrate that strategy attractiveness under CPT largely depends on the investment horizon. In summary, strategies with high stock proportions show low performance in the short run but high performance in the long run. The reason for this observation is the fact that for short horizons, the high volatility of stock returns leads to high shortfall risk, especially compared with a pure bond strategy. Due to loss aversion in the CPT value function, this leads to a relatively poor evaluation for stocks. As the investment horizon increases, the positive mean return of stocks reduces the shortfall risk, leading to a more favorable evaluation of strategies with a high stock proportion. Thus, the relative

4.3. The roles of curvature, loss aversion, and probability weighting The analysis in Section 4.2 is based on an investor with median parameters. The fact that investors have different preferences, however, highlights the necessity of sensitivity analyses. As there are five CPT parameters, an analysis that varies all parameters is complex. It is, nonetheless, necessary to vary as many variables as possible simultaneously as there are substantial interaction effects. To guarantee a structured analysis, we therefore divide the sensitivity analysis into two parts. First, we consider the value function and allow for variation of curvature and loss aversion.

0.4

0.6

Pre-crash Period (01/1962–09/1987)

0.0

0.2

CPT Value

0.8

1.0

stocks bah bonds cmix pp cppi

performance of the pure stock strategy increases with longer time horizons, whereas the pure bond strategy becomes worse as returns are not sufficiently large to match the benefits of (risky) stocks. The constant mix strategy performs poorly over all horizons as the performance suffers from transaction costs and low-yielding one-month bonds. Importantly, the results also show that the trade-off between slightly curtailing gains and limiting large losses, as in the case for the insurance strategies turns out to be relatively favorable from a CPT perspective.

1

12

24

36

48

60

72

84

60

72

84

Investment Horizon in Months Post-crash Period (12/1987–08/2008)

0.6 0.4 0.0

0.2

CPT Value

0.8

1.0

stocks bah bonds cmix pp cppi

1

12

24

36

48

Investment Horizon in Months Fig. 6. Impact of belief formation period (before and after 1987 crash). This figure shows CPT evaluations of the investment strategies according to Tversky and Kahneman’s (1992) median CPT parameters. The upper panel shows the results for the case of investors evaluating returns from January 1962 to September 1987, the lower panel for returns from December 1987 to September 2008.

M. Dierkes et al. / Journal of Banking & Finance 34 (2010) 1032–1046

0.02 0.01 -0.01

0.00

CPT Value

0.01 0.00 -0.01

-0.02

-0.03

-0.02

CPT Value

0.02

0.03

1040

stocks

1926

bah

1936

bonds

1946

cmix

pp

1956

cppi

1966

stocks

bah

1962

1976

cmix

pp

cppi

1982

Rolling Window

1.0 0.8 0.4

0.2

0.6

0.4

0.6

CPT Value

0.8

1.0

1.2

Rolling Window

CPT Value

bonds

1972

stocks

1926

1936

bah

bonds

1946

cmix

1956

pp

cppi

1966

stocks

1976

Rolling Window

1962

bah

bonds

cmix

1972

pp

cppi

1982

Rolling Window

Fig. 7. Impact of belief formation period (rolling time windows). This figure shows CPT values of the median decision maker for different belief formation periods. The left panels show the results for the enlarged time horizon 1926–2008 with a rolling window of 30 years as the belief formation period, the right panels show the results for the base time horizon 1962–2008 with a rolling window of 20 years while the numbers on the time axis indicate the beginning year of the rolling window. The upper panels represent an investment horizon of m = 3, the lower panels represent an horizon of m = 60.

To keep the analysis clear we assume that a equals b. Second, we also allow for different values of the probability weighting parameters. Fig. 4 depicts the results of the sensitivity analyses for selected investment horizons of m = 1, 3, 6, 12, 24, 36, 48, 60, 72, and 84. The different grayscales represent the different strategies. For a given (a, k) parameter combination, the respective grayscale represents the best strategy for this combination, meaning the strategy with the highest CPT value. Hence, each point in the graphs represents a different potential investor. The horizontal axis represents different values for k, ranging from 1 (no loss aversion) on the left to 5 (high loss aversion) on the right, and the vertical axis represents different values for curvature a, with a = 0.5 at the top and a = 1.2 at the bottom. The four graphs in a row represent a different set of probability weighting parameters, which will be discussed later. For the beginning of our analysis we focus on the third column, that is, we assume median parameters for the probability weighting functions. For a given probability weighting, risk aversion in terms of curvature and loss aversion increases from the lower left to the upper right corner.

4.3.1. The impact of curvature and loss aversion For very short horizons, a large number of (a, k) combinations in the right area of the graphs (high loss aversion) show a preference for the pure bond strategy. Only investors with low loss aversion and a less curved12 value function (lower left area) prefer the CPPI and the protective put strategies. As the time horizon becomes

12 We use the term ‘‘low” curvature to characterize investors that lie in the lower parts of the panels, although this area also includes investors having curvature values a > 1. Investors exhibiting no curvature lie on a horizontal line at a = 1, that is, this line divides the panel into two parts with 2/7 of the area below that line.

slightly longer, the optimality of the protection strategies is mainly given for low curvature, and is, interestingly, largely independent of loss aversion. The stock investment is most attractive only for investors with low loss aversion. The buy-and-hold strategy does not play any role for these horizons, that is, it is not preferred whatever (a, k) parameter combination is assumed. As the time horizon increases, the areas located in the lower left corner of the panel become larger, replacing the pure bond investment and indicating an increasing preference for the two protection strategies. For a time horizon of 24 months, for example, only investors with substantial curvature (upper area of graph) prefer the pure bond strategy. The CPPI and the protective put strategy become dominant for the medium-term. Further, the stock strategy becomes more attractive, but only for investors with low loss aversion and curvature. For even longer time horizons (m P 36), the buy-and-hold strategy almost replaces the pure bond strategy. The pure stock investment becomes more and more attractive and fills roughly half of the panel for m = 84. The buy-and-hold strategy is only attractive for investors with strong curvature and a long investment horizon. For longer horizons, the pure stock strategy benefits from its fast increasing mean return (see Appendix A for descriptive statistics). This leads to high CPT values if curvature and loss aversion are not substantial. Hence, the pure stock strategy replaces the others from the bottom left to the top right as the horizon increases. The buy-and-hold strategy becomes optimal for investors with substantial curvature as in this case high gains of the pure stock strategy do not increase the CPT value significantly. Generally speaking, the stronger the curvature is, the more important are gain and loss likelihoods. Particularly in the case of high curvature, the high returns of the pure stock strategy do not have such a high impact on the CPT value. Therefore, the protection strategies benefit from curtailing losses while slightly lowering gains.

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4.3.2. The impact of probability distortions Comparing panels in Fig. 4 horizontally illustrates the impact of probability distortions on the attractiveness of investment strategies. Probability distortions become larger from left to right, assuming no distortion in the left column. Generally speaking, the results illustrate that stronger distortions, that is, lower values for d, favor strategies with less extreme outcomes. Higher probability distortions give a larger weight to both tails of the distribution thus weighting extreme outcomes more heavily. In combination with loss aversion, this leads to a lower attractiveness, particularly for stocks. Consequently, the stock strategy is less attractive for investors exhibiting a stronger degree of probability weighting. The pure bond strategy and particularly the two protection strategies, on the contrary, are the main beneficiaries. This insight is important as it has been generally believed that protection strategies mainly benefit from loss aversion (see, for example, Berkelaar et al., 2004). Our results show, however, that it is primarily probability weighting that drives the results. This not only holds true for investors with heavy probability distortions but also for those with median values. However, the protection strategies only become best for a large set of investors if probability distortions as heavy as for the median investor are assumed. Concerning the time horizon in general, we observe probability distortions to have only a minor effect for short horizons, but to have a strong impact for longer ones. Particularly for horizons exceeding 12 months the panels differ substantially.

= 1

1

3

6

12

bonds

24

cppi pp

4.4. Influence of low probability adverse events Investors might be afraid of low probability adverse events even though these have not been observed in the past. In this case, investors would not evaluate the actual return distribution, but would include the probability of adverse outcomes. Motivated by Rietz (1988) and Barro (2006), and to provide a robustness check for our results, we incorporate extreme (negative) outcomes with low probabilities. More precisely, we assume crashes consisting of two consecutive monthly returns of 30%. This represents a severe and abrupt crash which occurs additionally to the crashes already incorporated in the dataset as the lowest return in the dataset equals 21.5%.13 In each month of the investment horizon, this crash might occur with a probability p and will replace two consecutive returns drawn from the dataset. As we further assume that a crash can occur only once for a given horizon, the crash probability amounts to 1  (1  p)m+1 for investment horizon m.14 Distinguishing the three scenarios p = 0.0005, 0.001, and 0.002, the annual crash likelihood thus equals 0.648%, 1.292%, and 2.569%, respectively. Fig. 5 illustrates the results for the three scenarios. We observe that it takes slightly longer for stocks to become the most attractive strategy. Insurance strategies are performing even better than in the base scenario, and also the pure bond strategy benefits. In summary, a severe crash scenario does not change the overall picture of the results concerning horizon effects. Yet, as expected, the relatively safe strategies are the beneficiaries.

13 We also tested crashes limited to one month but with returns of –40, –50, and –60%, respectively. The results differ in the sense that particularly protection strategies perform worse. The reason is that these strategies cannot adjust portfolio values adequately due to the monthly rebalancing. Assuming two consecutive large negative returns obviously favors insurance strategies due to the positive autocorrelation and the fact that these strategies shift portfolio allocations to the better performing asset. Nonetheless, even severe crashes of 60% occurring in one month do not change the general horizon effects. 14 As we omit market timing aspects, we assume that a crash can also begin in the month directly preceding the start of the investment period. In this case, the return of the first month of the drawn dataset (that is the second month of the crash) is replaced. Hence, the number of possible crash events is m + 1. Similarly, if the crash occurs in the last month, only the return of the last month is replaced.

= 6

36

48

60 stocks

72

bah

84

Fig. 8. Influence of autocorrelation in asset returns (block bootstrap). This figure shows the best investment strategy for different (a; k) combinations and for different investment horizons. In each graph the horizontal axis represents different values for k, ranging from k = 1 at the left to k = 5 at the right. The vertical axis represents different values for a, with a = 0.5 at the top and a = 1.2 at the bottom. Block length b increases from left to right. The rows from top to bottom correspond to investment horizons m = 1, 3, 6, 12, 24, 36, 48, 60, 72, and 84.

= 12

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4.5. Influence of changes in investors’ beliefs

4.6. Influence of autocorrelation in asset returns

A long return data history spanning as much as 47 years might be split by structural breaks in the equity premium (Pastor and Stambaugh, 2001) and severe crashes that can have a substantial influence on investors’ subjective return expectations (Jackwerth and Rubinstein, 1996; Christensen and Prabhala, 1998). While we so far assumed investors’ return expectations to be equal to the empirical distribution for the whole time period from 1962– 2008, we now assume that investors evaluate shorter time periods. As a starting point for this analysis and motivated by Christensen and Prabhala (1998) we split the dataset into two parts: The time before and after the major stock market crash in October 1987. We then generalize the analysis to rolling time windows of 20 and 30 years so that the results are not restricted to that particular crash. Additionally, we consider a large-scale dataset with returns from 1926 to 2008 as a robustness check for our base scenario. Due to limited data availability for the long dataset we assume a flat term structure at the level of the one-month yield of the US Treasury Bills. While this will slightly penalize the bond and buy-and-hold strategies, the other four strategies are not affected. Fig. 6 shows the results for the pre- and post-crash analysis, that is, we assume that investors do not evaluate the total time period 1962–2008 but only 01/1962–09/1987 and 12/1987–08/2008, respectively. Due to the further dimension of the time period we only consider the CPT median investor. While the absolute level of attractiveness is influenced by the particular time period and the relative attractiveness of stocks is considerably higher after the crash, the overall horizon effects remain. Fig. 7 depicts the results for the rolling windows analysis. Each of the four graphs in Fig. 7 shows CPT values for different time windows determining the set of returns that are used to form the investor’s belief for the future return distribution. We compose the return sets by taking the 20 (30) years following each of the rolling starting points as separate periods. The analysis shows that the relative attractiveness of the investment strategies is largely independent of the time period, that is, the lines in the graphs mostly do not cross each other. In other words, the investment strategies that are better for an early time period are also better for later ones. Furthermore, for both investment horizons under consideration (m = 3 and m = 60) the order of the investment strategies is largely equal to our base case. This result further indicates that the horizon effects depicted in the previous sections are robust, as even major crashes as during World War II or in October 1987 do not change the results. Nonetheless, the absolute attractiveness depends on the time period, particularly for the case m = 60 and the 20-year window (lower right panel), as an example, with CPT values for the pure stock investment ranging from 0.4 to 1.25.

Stock and bond returns usually exhibit serial correlation, see, for example, Poterba and Summers (1988). A positive serial correlation generally leads to wider return distributions as compared to zero serial correlation and vice versa. The impact of autocorrelation on the CPT valuation is not straightforward and depends on the time horizon and preference parameters. In our dataset, for example, the first-order serial correlation for the S&P 500 returns in the base scenario equals 4.7% (portmanteau test p-value = 0.264). To account for the autocorrelation structures of stocks we conduct a block bootstrap. The sequences s comprise several blocks of returns, each with a block length of b months. A block consists of b consecutive monthly stock returns. Hence, scenarios with b > 1 capture the empirical autocorrelation structure. For the bond component, we assume the one-month and the horizon equivalent m-month returns as being constant over the investment horizon. Consequently, autocorrelation is not an issue for the bond component. Extending the base case b = 1 by using a block bootstrap with block lengths b = 6 and 12 reveals that the basic pattern of the CPT values remains almost the same (see Fig. 8). Depending on the preference parameters, however, the optimal strategy may vary. For longer block lengths, the relative performance of the insurance strategies improves. The investment horizon at which the pure stock strategy becomes the best increases. The reason for this phenomenon lies in moderate positive serial correlation for periods of up to one year (momentum), driven in particular by the autocorrelation of order one. This leads to fatter tails of the return distributions. The two insurance strategies assume zero autocorrelation in the way they are implemented, and thus do not account for the increasing probability mass in the loss area. As a consequence, the strategies’ prices for the protection of negative or low returns are smaller than they would be if positive autocorrelation was accounted for. The extent to which the dataset shows serial correlation patterns is, though, too low to have a substantial effect on the overall results. 4.7. Set of investors Previous sections deal with Tversky and Kahneman’s (1992) median parameters and show the results of parameter sensitivity analyses. We now allow for greater heterogeneity among investors. Specifically, we employ 31 sets of CPT parameters experimentally elicited by Abdellaoui et al. (2007). We regard these subjects as a small but real-world universe of possible investors in the following analysis. Fig. 9 presents the distribution of preferred investment strategies for the subjects based on their CPT parameters. In line with our previous results, stocks become increasingly popular among

Fig. 9. Distribution of preferred investment strategies for individual subjects. This figure shows the distribution of preferred investment strategies for an investor universe according to the CPT parameters elicited by Abdellaoui et al. (2007). The parametric form of the CPT calculus only allows to consider 31 of 48 subjects.

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the universe of these investors as the investment horizon increases. The proportion of investors favoring stocks increases from about 10% for horizons up to one year to about 60% for an investment horizon of seven years. Bonds are only attractive for relatively short horizons, with the proportion decreasing from about 75% for m 6 12 to only 15% after three years and down to 0% for horizons of more than 5.5 years. The CPPI, protective put, and buy-and-hold strategies mainly replace the pure bond strategy. Particularly the CPPI strategy shows a high attractiveness for horizons exceeding one year with proportions of about 50%. Even for a relatively long horizon of m = 84, one fourth of the investors favor the CPPI strategy. The buy-and-hold strategy becomes more and more attractive for longer investment horizons with a proportion of about 20% for m = 84. In line with our previous results, the constant mix strategy is not attractive to a single investor. In a nutshell, these results depict a very similar picture as the sensitivity analyses of Section 4.3.

Our analyses reveal strong horizon effects under CPT. While bond portfolios are preferred for the short run, a pure stock strategy shows an outperformance in the long run. Buy-and-hold strategies consisting of both stocks and bonds are only attractive for investors with high curvatures in the value function and long investment horizons. A constant mix strategy is inferior to buyand-hold strategies for all preference parameters and time horizons. We are also able to explain the high demand of insurance strategies. In particular, the CPPI strategy turns out to be attractive for a large set of different investment horizons. Importantly, we identify probability weighting, not loss aversion, to be the main driver for the attractiveness of insurance strategies. Our results are in sharp contrast to EUT and are robust with regard to the form of the probability weighting function, potential extreme adverse events, different evaluation periods, and autocorrelation in asset returns. In interpreting our results, the following caveat applies. By using CPT over terminal distributions, we make the important assumption that investors are aware of these distributions. This might well be the case in the investment counseling process. As is known from research in behavioral economics, the perception of return distributions might be distorted, for example, by framing effects. However, we provide various arguments why our assumption might well apply.

5. Conclusion In this paper we analyze how different investment strategies that are typically applied in portfolio decisions perform under a CPT framework. To do so, we bootstrap a return distribution out of a large-scale empirical dataset of S&P 500 and US Treasury Securities returns. We then analyze an investor with CPT preferences according to Tversky and Kahneman (1992) and conduct sensitivity analyses with regard to the preferences, thereby disentangling the effects of curvature, loss aversion, and probability distortions. Furthermore, we conduct extensive robustness checks, accounting for the possibility of extremely adverse events, change in investors’ beliefs and autocorrelation of asset returns. Lastly, we consider a set of potential real-world investors whose preferences were experimentally elicited by Abdellaoui et al. (2007).

Acknowledgements We are indebted to an anonymous referee, Alexander Klos, Thomas Langer, participants of the 2009 Campus for Finance conference, the 12th Swiss Society for Financial Market Research conference, and the Finance Center Münster research seminar for valuable comments and insights.

Appendix A Return distribution characteristics of investment strategies like mean, standard deviation, shortfall frequency, skewness, and 1%, 10%, 50%, 90%, and 99% quantiles for all investment strategies and investment horizons m = 1, 3, 6, 12, 24, 36, 48, 60, 72, and 84. Statistic

Strategy

Investment horizon 1

3

6

12

24

36

48

60

72

84

Mean

stocks bonds bah cmix pp cppi

0.008 0.004 0.006 0.006 0.008 0.005

0.025 0.014 0.019 0.019 0.021 0.015

0.050 0.028 0.039 0.038 0.043 0.031

0.102 0.061 0.082 0.077 0.088 0.064

0.214 0.132 0.173 0.161 0.188 0.143

0.340 0.211 0.275 0.252 0.304 0.241

0.473 0.297 0.385 0.348 0.429 0.357

0.625 0.395 0.510 0.455 0.571 0.498

0.791 0.497 0.644 0.568 0.729 0.658

0.974 0.611 0.793 0.693 0.903 0.839

Standard Deviation

stocks bonds bah cmix pp cppi

0.043 0.002 0.022 0.022 0.037 0.006

0.076 0.007 0.038 0.038 0.063 0.013

0.110 0.014 0.056 0.055 0.092 0.024

0.163 0.029 0.083 0.081 0.139 0.049

0.256 0.061 0.132 0.126 0.226 0.118

0.349 0.097 0.181 0.169 0.316 0.214

0.446 0.137 0.233 0.213 0.410 0.329

0.554 0.185 0.292 0.262 0.516 0.460

0.672 0.237 0.357 0.316 0.631 0.602

0.802 0.299 0.429 0.373 0.757 0.750

Skewness

stocks bonds bah cmix pp cppi

0.416 1.023 0.406 0.406 0.404 0.218

0.094 1.018 0.090 0.163 0.285 0.328

0.088 0.989 0.089 0.034 0.563 0.710

0.280 0.965 0.266 0.083 0.742 1.137

0.512 1.033 0.485 0.242 0.871 1.779

0.689 1.095 0.635 0.346 0.954 1.866

0.835 1.191 0.758 0.440 1.039 1.779

0.986 1.286 0.871 0.557 1.132 1.658

1.107 1.355 0.978 0.642 1.229 1.599

1.211 1.421 1.061 0.728 1.297 1.529

Shortfall

stocks bonds

0.386 0.000

0.359 0.000

0.324 0.000

0.269 0.000

0.206 0.000

0.159 0.000

0.126 0.000

0.100 0.000

0.081 0.000

0.066 0.000

(continued on next page)

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Appendix A (continued) Statistic

Strategy

Investment horizon 1

bah cmix pp cppi

3

6

12

24

36

48

60

72

84

0.348 0.348 0.384 0.178

0.292 0.291 0.384 0.117

0.238 0.238 0.362 0.082

0.159 0.166 0.312 0.052

0.082 0.095 0.242 0.030

0.041 0.056 0.187 0.022

0.022 0.037 0.148 0.019

0.010 0.023 0.117 0.015

0.006 0.016 0.094 0.014

0.003 0.010 0.075 0.013

1% Quantile

stocks bonds bah cmix pp cppi

0.108 0.001 0.051 0.051 0.091 0.010

0.167 0.002 0.077 0.079 0.117 0.016

0.208 0.005 0.091 0.095 0.131 0.017

0.250 0.012 0.098 0.108 0.134 0.016

0.294 0.032 0.091 0.113 0.135 0.014

0.311 0.055 0.067 0.101 0.133 0.012

0.318 0.090 0.041 0.085 0.132 0.011

0.309 0.141 0.000 0.056 0.128 0.007

0.305 0.171 0.034 0.035 0.125 0.007

0.296 0.223 0.079 0.003 0.120 0.005

10% Quantile

stocks bonds bah cmix pp cppi

0.045 0.002 0.020 0.020 0.038 0.002

0.070 0.006 0.029 0.029 0.056 0.001

0.089 0.014 0.031 0.032 0.067 0.002

0.101 0.030 0.022 0.025 0.073 0.010

0.097 0.064 0.012 0.003 0.075 0.025

0.072 0.106 0.059 0.043 0.067 0.040

0.044 0.149 0.112 0.087 0.052 0.054

0.001 0.204 0.174 0.138 0.026 0.070

0.045 0.250 0.238 0.193 0.012 0.086

0.098 0.308 0.313 0.255 0.061 0.105

50% Quantile

stocks bonds bah cmix pp cppi

0.011 0.004 0.007 0.007 0.010 0.005

0.026 0.013 0.020 0.020 0.018 0.014

0.048 0.026 0.038 0.038 0.034 0.028

0.095 0.057 0.078 0.076 0.070 0.057

0.192 0.122 0.162 0.156 0.155 0.117

0.302 0.192 0.257 0.242 0.255 0.185

0.415 0.270 0.358 0.333 0.361 0.258

0.542 0.365 0.471 0.431 0.482 0.350

0.685 0.442 0.593 0.539 0.617 0.473

0.831 0.546 0.726 0.652 0.760 0.635

90% Quantile

stocks bonds bah cmix pp cppi

0.056 0.007 0.031 0.031 0.049 0.012

0.118 0.022 0.066 0.066 0.103 0.032

0.189 0.045 0.109 0.107 0.166 0.062

0.314 0.099 0.189 0.181 0.278 0.128

0.550 0.208 0.346 0.324 0.498 0.294

0.798 0.337 0.512 0.472 0.729 0.512

1.060 0.487 0.692 0.626 0.981 0.806

1.357 0.652 0.897 0.800 1.262 1.137

1.671 0.813 1.112 0.982 1.563 1.478

2.021 1.019 1.354 1.181 1.895 1.846

99% Quantile

stocks bonds bah cmix pp cppi

0.114 0.012 0.061 0.061 0.102 0.022

0.206 0.036 0.110 0.108 0.184 0.050

0.319 0.074 0.175 0.168 0.288 0.100

0.517 0.154 0.292 0.272 0.472 0.216

0.907 0.316 0.526 0.476 0.839 0.548

1.331 0.509 0.783 0.691 1.243 1.014

1.774 0.723 1.059 0.919 1.670 1.511

2.312 0.962 1.376 1.175 2.176 2.053

2.879 1.246 1.719 1.458 2.718 2.664

3.497 1.549 2.100 1.755 3.315 3.305

Appendix B Dependence of the strategies’ returns on stock returns. This figure shows scatter plots that illustrate the dependence of the strategies’ returns on stock returns (m = 12).

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Appendix B (continued)

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