Mean-drawdown risk behavior: drawdown risk and capital asset pricing

We develop an alternative approach based on mean-drawdown risk behavior versus the mean-variance behavior. We develop two risk measures as the maximum draw down risk and average drawdown risk to estimate two new betas and then propose two CAPM -like models. The data includes a comprehensive universe of more than 11,000 US equity-based mutual funds from first month of 2000 to third month of 2011.The evidence clearly shows superiority of the maximum and average drawdown betas and their pricing models, the maximum drawdown CAPM and the average drawdown CAPM , over the traditional beta and CAPM , respectively.


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the distribution of returns is normal. However, both the symmetry and the normality of stock returns are seriously questioned by the empirical evidence on the subject (Chunhachinda et al. 1997;Tee 2009;Dichtl, Drobetz 2011).
Against, the drawdown risk measure (hereafter, DRM) of returns, which is decomposed to two measures of maximum drawdown risk (hereafter, M-DRM) and average drawdown risk (hereafter, A-DRM), is a more acceptable measure of risk for several reasons: first, investors logically prefer downside volatility (Stevenson 2001;Galagedera 2007;Fortin, Hlouskova 2011). Second, unlike the downside risk, the DRM evaluates the loss from a local maximum to the next local minimum and is intuitively appealing for institutional investors (Hamelink;Hoesli 2003, Kim 2010Schuhmacher, Eling 2011). Third, the DRM is more beneficial than the traditional variance (standard deviation) when the dispersion of returns is asymmetric and just as beneficial when the dispersion is symmetric; accordingly the DRM is better measure of risk in comparison with the variance. And fourth, the DRM is a measure which the information is generated by three statistics of variance, semi-variance, and skewness, thus, it makes possible to utilize alternative single-factor models to estimate the expected returns.
Moreover, the DRM can be utilized to make a replacement behavioral assumption as mean-drawdown behavior (hereafter, MDB). As described in Hamelink and Hoesli (2003) and Gilli and Schumann (2009), MDB is perfectly correlated to the expected utility and can thus be defended across the same lines utilized by Levy and Markowitz (1979), Markowitz (1991), Eling and Schuhmacher (2007), and Caprin and Lisi (2009).
As main contribution of this study, we propose two alternative risk measures for diversified investors, the M-DRM and A-DRM beta, and two alternative pricing models based on these two risk measures. We also report the evidence from subclasses of US equitybased mutual funds' management styles, which support from the M-DRM and A-DRM beta over the traditional beta, and also the pricing models generated by the M-DRM (MD-CAPM) and the A-DRM (AD-CAPM) over the CAPM.
The rest of the paper is organized as follows. Section 1 describes the theoretical and conceptual framework by explaining two approaches of MVB and CAPM on one hand, and MDB and its relevant pricing models on the other hand. Section 2 discusses and reports the empirical evidence which clearly supports the M-DRM and the A-DRM risk measures, the M-DRM and A-DRM beta and their relevant pricing models. Finally, the last section reports some concluding remarks.

MVB vs. MDB Framework
We first explain the MVB framework and its relevant pricing model and then explain our proposed approach as MDB along its relevant pricing models. Then, we explain how to estimate the M-DRM and A-DRM betas. Finally, we compare our suggested pricing models with CAPM.

MVB and asset pricing
The MVB framework explains that an investor's utility (U) is determined by the mean ( P µ ) and variance ( P σ ) of portfolio returns, where ( , ) P P U U = µ σ . Thus, the risk of an asset i is assessed by the standard deviation of asset's return ( i σ ) as: where R and µ are the return of asset i and mean respectively. However, when asset i is just one out of many in portfolio, its risk is assessed by its covariance with respect to the market portfolio as: where m is the market portfolio. A more useful risk measure can be assessed by dividing this statistic by return's standard deviation of asset i and the market portfolio, thus we estimate asset i's correlation with respect to the market index as: Alternatively, the covariance between asset i and the market index can be divided by the variance of the market index, thus asset i's beta ( i β ) is calculated as: This measure is widely applied in the CAPM pircing model as: where ( ) i E R , R f , and R m denote the expected return on asset i, the risk-free rate, and the expected return on the market, respectively. Finaly, variance is a risk measure under symmetric condition.

M-DRM framework
In the MDB framework, investor's utility is maximum drawdown risk of returns on investor's portfolio. In the M-DRM framework, the risk of an asset i is measured by asset's downside standard deviation on the loss happened from a local maximum to the next local minimum plus the risk premium as: where 0 D is equal to 0. t D denotes the maximum loss suffered by an investor from 0 to t-1. Eq. (6) is a special case of the semi-deviation with respect to benchmark return ( ) BMi B ∑ as: In the M-DRM framework, the counterpart of fund i's covariance to market portfolio is computed by the M-DRM covariance as: Moreover, this co M-DRM is unbounded, but it can also be standardized by dividing it by return's M-DRM of fund i and market index, hence fund i's M-DRM correlation is obtained as: The co M-DRM is divided by the market return's M-DRM, hence M-DRM beta is obtained as: The M-DRM beta computes the covariance between the downside returns made by a combination of maximum loss and market risk premium over the investment period. This beta, which is defined as As defined in Eq. of (5) and (11), our model replaces by the M-DRM beta.

A-DRM framework
In the MDB framework, the investor's utility is notes the average drawdown risk of returns on the investor's portfolio. In the A-DRM framework, the risk of an asset i is assessed by asset's downside standard deviation on the loss happened from a local maximum to the next local minimum plus the risk premium as: where 0 A is equal to 0. t A denotes the average loss that an investor suffers from 0 to t-1. Eq. (12) can be more expressed with respect to any benchmark return ( ) BAi B ∑ as: We denote the A-DRM of fund i simply as A i ∑ . In the A-DRM framework, the counterpart of fund i's covariance to the market portfolio is resulted by its A-DRM covariance as: Moreover, it can also be standardized by dividing it by returns' A-DRM of fund i and the returns' A-DRM of market index, hence fund i's A-DRM correlation ( iA Θ ) is obtained as: The co A-DRM can be divided by the market return's A-DRM, hence A-DRM is obtained as: The A-DRM beta computes the covariance between the downside returns generated by a combination of average loss and market risk premium over the holding period. This beta, which is defined as ( / ) As defined in Eq. of (5) and (17), our model replaces the CAPM beta by the A-DRM beta.

A brief discussion on the M-DRM and A-DRM beta
The M-DRM and A-DRM betas given by Eq.
Finally, as another clear difference between the two betas, the DRM betas can be computed by regression analysis. Let , and also Ay µ and Ax µ be the mean of t y and t x for the A-DRM. If a regression model be run with t y as the dependent variable and t x as

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the independent variable (that is, 0 1 . , t t t y x = λ + λ + ε where ε is an error term and 0 λ and 1 λ are coefficients to be estimated), the estimate of 1 λ would be as: Alternatively, as defined in Eqs (10) and (16) Thus, the best method for estimating M DRM is to test a linear regression without considering a constant between the independent variable,

A brief discussion on the DRM risk framework
The DRM was extended by practitioners who did not base their work on theoretical considerations. Most of the literatures on DRM were found in journals outside of finance (Dacorogna et al. 2001), non-refereed finance journals, and finance journals geared to the investment community (Chekhlov et al. 2005). This measure was gradually used in finance literature as a new risk measure (Alexander, Baptista 2006;Eling, Schuhmacher 2007). The drawdown is the loss incurred over the investment period. It is the loss in perceptual from the prior local maximum to the next local minimum of an investment, which is decomposed into maximum DRM and average DRM (Gilli, Schumann 2009).
The DRM is the loss suffered when an asset is bought at a local maximum and sold at the next local minimum and or the worst loss that the portfolio suffers over the investment period (Alexander, Baptista 2006). It is the worst return suffered by an investor, e.g. the return of an investor who buys the fund at the highest price and sells it at the lowest price. Institutional investors often capture DRM as a risk measure to choose a portfolio. The A-DRM also is the average loss suffered over the holding period. It is relevant only if one trades the funds under loss condition (Gilli, Schumann 2009).
The concept of DRM was primarily introduced by Grossman and Zhou (1993) and Dacorogna et al. (2001). They investigated two risk-adjusted measures for investors with risk-averse preferences in the maximum drawdown framework. Hamelink and Hoesli (2004) studied the role of real estate in a mixed-asset portfolio when the maximum drawdown is used instead of the standard deviation. They showed that the maximum drawdown is one of the most natural risk measures, and such a framework can help reconcile the optimal allocations to real securities by institutional investors. Alexander and Baptista (2006), using a drawdown constraint, provided a characteristic of optimal portfolios in the MV framework. Eling and Schuhmacher (2007) used the maximum DRM and compared the Sharpe ratio with the DRM measures. Schuhmacher and Eling (2011) asserted that DRMs are as well as Sharpe measure and showed that the location and scale condition are sufficient for expected utility to imply the rankings of drawdown measure. However, literature shows that none of the studies use the DRM in the pricing models

Empirical evidence
We use the monthly data of US equity-based mutual funds' management styles. The data is extracted from the Morningstar database. The research population includes all the funds available in the database. Our sample includes the monthly returns adjusted by dividend for more than 11,000 funds from first month of 2000 to third month of 2011. The monthly return for the 90-day Treasury bills as free risk return and S&P500 as market index are extracted from the DataStream. The statistics for all the funds are reported in Table 1.

Statistical significance for the total sample
The first step of our analysis consists of calculations over the whole sample. One statistic (MR) reports the average return of each style, and other statistics report the risk measures. Average returns over the whole sample are summarized by mean monthly arithmetic returns; these estimates are reported in Table 1. The risk measures are three for the MVB (standard deviation, correlation coefficient and beta) and six for the MDB (DRM, A-DRM, their correlation coefficients, the DRM betas). An estimate of these measures is calculated over the whole sample. Moreover, since all the styles display positively skewed distributional attributes, this reinforces the use of the DRM models very well.
A correlation matrix containing the six measures and mean returns is displayed in Table  2 where the DRM risk measures (DRM, A-DRM and their betas) outperform the traditional measures (standard deviation and beta). In fact, the DRM measures and their betas outperform the standard deviation and beta.
Specifically, the relationship between return and risk can be extracted from our regression analysis. We begin by running a cross-sectional linear regression relating mean returns to each of the four surveying risk measures. More precisely: where i RM and i MR stand for risk measure and mean return, respectively. γ 0 and γ 1 are coefficients to be estimated, i u is an error term, and i denotes funds. The results of our six regression models are reported in panels A and B of Table 3. Panel A reports the result of OLS regressions, where two regressions describe the existence of heteroskedasticity. Panel B also reports the results of OLS in which statistical significance is reported by White's heteroskedasticity-consistent covariance matrix. The results in both panels are same except for the standard deviation: six risk measures are significant because of explanatory power. As reported in Table 3, the DRM measures outperform two tradi-

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tional measures in terms of their explanatory power. The DRM and A-DRM measures, in fact, outperform the traditional measures and report higher significant coefficient of 0.35 and 0.37, respectively. Similarly, the DRM and A-DRM betas explain a substantial 38% and 41% of the variability respectively in mean returns.   In other words, our suggested DRM betas outperform beta in terms of their explanatory power. The average R-squared for the two-factor regressions of "beta and DRM beta" and "beta and A-DRM beta" are 0.43 and 0.48 respectively, which implicate their better significant power. This R-squared for the four-factor regressions, as reported in Panel B of Table 4, are 0.72 and 0.76.

Statistical significance on management styles
In this section, we consider the management styles and re-assess the significance power of each measure. Note that DRM measures describe skewed distributions of returns better than the traditional measures. If all the distributions be symmetric, the DRM and the standard deviation would contain same information, and MDB would lose most of its appeal as a behavioral model.     Table 5 S459 Table 6 reports the results of multiple regressions splitting again the sample in the styles. The results confirm that none of the two traditional risk measures have a better significant explanatory power than the DRM measures. The DRM is significant when jointly considered with the standard deviation, and the DRM beta is significant when jointly considered with the beta. Finally, the DRM and its beta are significant when jointly considered with the two other traditional measures in the multi-factor models. Panel A also reports when considering multi-factor models of the beta and the DRM betas, five styles of Growth and Income, Index Fund, Long-Short, Market Neutral and Value have larger R-square than other combinational models. This implicates more significant of the DRM betas in the styles, a range from 0.26 to 0.68.

Economic significance: spreads in return and risk
To check for the robustness of our results, we divided all the styles into three equallyweighted portfolios classified by beta, and computed the spreads in mean returns between the riskiest portfolio and the least risky portfolio. Then, we repeated the process by ranking the portfolios made by DRM betas and computing again the spread between the riskiest portfolio and the least risky portfolio. By focusing on the joint sample of the styles (Panel A of Table 7), there seems a large difference in the spread of two risk measures of portfolios 1 and 3: the difference between traditional betas is 1.08 and between DRM betas is 1.5 and 1.3. Note, however, that the average beta of portfolio 1 (1.08) is larger than the average beta of portfolio 3 (0), in addition the average DRM End of Table 5 Journal of Business Economics and Management, 2013, 14(Supplement     End of Table 6 Panel B: The return spreads spanned by DRM betas are larger than those spanned by beta. Moreover, we obtain the relative spread by dividing the spread in monthly mean returns by the spread in the risk measure, which is 0.88 in the case of portfolios ranked by betas and 1.015 in the case of portfolios ranked by DRM betas; that is, mean returns are more sensitive to spreads in DRM betas than equal spreads in betas.
Panels B, C, D, E, F, G, I, J, K, and L of Table 7 also show same results so that there seems to be a considerable difference in the spread of two risk measures of portfolios 1 and 3 in the styles. In fact, the average betas of portfolio 1 are larger than the average betas of portfolio 3 and the average DRM betas of portfolio 1 are larger than the average DRM betas of portfolio 3. Moreover, Panels B, C, D, E, F, G, I, J, K, and L of Table 7 show that return spreads spanned by the DRM betas are higher than those spanned by beta. Finally, as evidenced by the relative spreads, mean returns are more sensitive to spreads in the DRM betas than equal spreads in beta. The second method of robustness check is due to the existence of dead and outlier funds, which imposes bias on the process of our analysis. Thus, we first decompose our sample into two periods of 2000 to 2006 and 2006 to 2011, and then repeat our analysis similar to our previous analysis. In addition, we consider a five-year period from 1995 to 2000 and again run the pooled model as reported in Table 8. In general, the results of these tests are similar to the In-sample period, in which our findings clearly support from the drawdown betas versus the traditional beta.

Expected returns on fund
As a conclusion at this point, our results detect that, when considering the joint sample of the styles, (1) the DRM measures outperform the traditional measures; (2) the best betas that describe the cross section of returns are the DRM betas (Table 3); (3) the only measures that significantly describe the cross section of returns, when all risk measures

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are jointly considered, are the DRM measures (Table 4); and (4) mean returns are more sensitive to variations in DRM betas than to equal variations in traditional beta (Table  7). (5) the DRM betas are the best measure that describes the cross section of returns (Table 5); (6) when the measures considered jointly, none of the two traditional measures significantly describes the cross section of returns, and only two DRM measures do (Table 6); and (7) mean returns are more sensitive to variations in DRM beta than equal variations in traditional beta.    (5), (11) and (17), a risk-free rate of 0.20 and a market risk premium of 6%.

Conclusion
The traditional beta, CAPM and their behavioral model (MVB) have been widely used but also extensively debated over the past 40 years. Most of the debates versus beta focus on comparing the ability of this coefficient rather than alternative risk measures to describe the cross section of assets' return. We found that the data on US equity-based mutual funds' management styles support from the DRM betas and their relevant pricing models, (MD-CAPM) and (AD-CAPM), rather than beta and CAPM. We generated a parallel between the traditional framework in terms of MVB, beta, and CAPM, and a replacement framework in terms of the DRM; that is, on MDB, the DRM and A-DRM betas, and their pricing models. We proposed some methods to estimate the DRM betas and to extend them into pricing models, (MD-CAPM) and (AD-CAPM). Our findings support from the DRM versus the traditional measures and show that mean returns are much more sensitive to spreads in DRM betas than equal spreads in beta. Moreover, unlike the CAPM, the drawdown CAPM plausibly generates a higher expected return.
However, we believe that our suggested measures are able to be replaced with the traditional measures, where financial markets experience a depression. It seems that our suggested measures and models provide a better framework to evaluate portfolio in asymmetric condition.