**"standard deviation"**

## Downside CAPM

*by Eric Bank on February 28, 2011 · No Comments · in Documenting the Search for Alpha*

We have devoted a lot of blog space in the past examining the pros and cons of the Capital Asset Pricing Model (CAPM). The model predicts the amount of excess return (return above the risk-free rate) of an arbitrary portfolio that can be ascribed to a relationship (called beta[1]) to the excess returns on the underlying market portfolio.

We have devoted a lot of blog space in the past examining the pros and cons of the **Capital Asset Pricing Model (CAPM)**. The model predicts the amount of **excess return** (return above the risk-free rate) of an arbitrary portfolio that can be ascribed to a relationship (called **beta ^{[1]}**) to the excess returns on the underlying market portfolio. It assumes that investors seek maximum utility from their portfolios, and that investors expect to be compensated for taking on additional risk. In CAPM, no portfolio can outperform a mix of the risk-free asset and the market portfolio on a risk-adjusted basis.

Risk and return are the only two variables of importance in CAPM. Risk is measured by beta, which is rooted in the variance (or more properly, its square root, standard deviation) of returns, and is an indication of volatility.

Recall that CAPM makes a number of simplifying (and criticized) assumptions to work. A couple of the assumptions that have been the subject to criticism are:

1) Portfolio returns are distributed symmetrically around a mean.

2) Portfolio returns are assumed to have no outliers (or “fat tails”).

Empirical evidence suggests otherwise. Researchers have thus sought alternatives to a variance-based beta that relaxes both of these assumptions, and one popular candidate is called **semivariance** – a measure of the dispersion of those values in a distribution that fall *below the mean* or target value of a data set. In short, semivariance-based modifications to CAPM concentrate on **downside-risk** only. Semivariance is a better statistic when dealing with asymmetric distributions, as it automatically incorporates the notion of **skewness**. Ignoring skewness, by assuming that variables are symmetrically distributed when they are not, will cause any model to *understate* the risk of variables with high skewness

One popular semivariant CAPM alternative is called **D-CAPM** (Downside-CAPM). Regular old beta is replaced by downside-beta (**β**** _{D}**). Different researchers have supplied different technical definitions for β

_{D}; here we will use the one provided by Javier Estrada:

β_{D } = downside covariance between asset and market portfolio / downside variance of market portfolio.

There are several ways to calculate β_{D}, but I will spare you the details. The important point is that empirical studies^{[2]} indicate D-CAPM gives better predictions compared to CAPM, especially for emerging markets. This may be due to the hypothesis that returns from emerging markets are less normal and more skewed than returns from developed markets.

D-CAPM is generally well-regarded because of its plausibility, supporting evidence, and the widespread use of D-CAPM. For instance, a study by Mamoghli and Daboussi^{[3]} concluded “*It comes out from these results that the D-CAPM makes it possible to overcome the drawbacks of the traditional CAPM concerning the asymmetrical nature of returns and the risk perception…*”

However, D-CAPM is not without its **critics**. Let me quote from one^{[4]}, in which downside correlation is taken to task: “*This measure ignores the ability of upside returns of one asset to hedge the downside returns of another asset in a portfolio**. **Within the scope of D-CAPM the standard equation to calculate portfolio’s downside semideviation cannot be correct, since the Estrada’s downside correlation equates upside returns to zeros and does not represent the true downside correlations. Specifically, the downside correlation cannot be measured, because the portfolio’s semideviation depends on the weights of assets, their standard deviations and correlation between them, rather than on the semivariance. This formula is specified for normally distributed and symmetrical returns and there is no formula invented to calculate the portfolio’s semideviation yet*”.

This criticism, if true, is especially relevant to hedge fund traders, who use long-short strategies as a matter of course.

Nevertheless, Professor Estrada responded^{[5]} as follows: *“**First, it’s true that the downside beta I suggest in my paper does not account for something that would be legitimately called risk, which is the market going up and the asset going down. The way I suggest to calculate downside beta only accounts for down-down states. But there is a reason for this: It’s the only way to achieve a symmetric semivariance-semicovariance matrix, necessary for a solution of the problem. For example, the only way to apply standard techniques to solve an optimization problem with downside risk is to have a symmetric matrix. Second, and importantly, my several papers on downside beta, D-CAPM, and downside risk in general have been published in several journals and therefore peer-reviewed. I believe Mr. Cheremushkin’s comment is an unpublished piece.”*

This controversy will no doubt be resolved over the fullness of time. We welcome ** your** views.

[1] Technically, beta is equal to the

**covariance**of an asset with the market portfolio divided by the

**variance**of the market portfolio.

[2] Estrada, Javier (2003). “*Mean-Semivariance Behavior (II): The D-CAPM*.” University of Navarra.

[3] Mamoghli, Chokri and Daboussi, Sami (2008). Valuation of Hedge Funds Portfolios in a Downside Risk Framework.

[4] Cheremushkin, Sergei V. (2009). “*Why D-CAPM is a big mistake? The incorrectness of the cosemivariance statistics*.” Mordovian State University. Also private communication 9-Aug-10.

[5] Private communication 9-Aug-10.

## Behavioral Portfolio Theory 1 – Safety First

*by Eric Bank on February 18, 2011 · No Comments · in Documenting the Search for Alpha*

ur survey of portfolio theories continues; we have already evaluated Modern Portfolio Theory, the Capital Asset Pricing Model and the Arbitrage Pricing Theory in earlier blogs, and now turn to a series of articles on Behavioral Portfolio Theory (BPT).

Our survey of portfolio theories continues; we have already evaluated **Modern Portfolio Theory**, the **Capital Asset Pricing Model** and the **Arbitrage Pricing Theory** in earlier blogs, and now turn to a series of articles on **Behavioral Portfolio Theory (BPT)**. Recall that as we continue to lay out a financial framework, our ultimate goal is to figure out whether hedge funds actually provide alpha returns to investors.

Whereas both CAPM and ABT assume that investors are proceeding from a *single unified motivation* – to maximize the values of their portfolios – BPT instead suggests that investors have a *variety of motivations* and structure their investments accordingly. Think of a layered pyramid, in which the base is set up to protect against losses and the top designed for high reward. This has been characterized as the “government bond and lottery ticket” portfolio because of the extreme divergence of goals.

One of the earliest attempts at BPT was **Safety-First Portfolio Theory (SFPT**, Roy 1952). The theory posits that investors want to minimize the probability of **ruin**, which is defined as when an investor’s **wealth W** falls below his **subsistence level s**:

**min Pr(W < s)**.

SFPT, like the other portfolio theories, makes a few assumptions:

- a portfolio has a return
**mean****μ**and return_{P}**standard deviation****σ**;_{P} - the
**risk-free asset R**is not available;_{F} - investors trade off assets according to the
**expected utility**of the assets and the law of diminishing returns; - the subsistence level s is
**low**; and - the distribution of portfolio returns is
**normal**(bell-shaped).

In short, an investor wants to own a portfolio for which

**(s – ****μ _{P}) / σ_{P}**

_{ }

is minimized.

Later refinements of the theory included:

- relaxing the fixed value s;
- constraining the probability of ruin such that it never exceeds a predetermined ruin probability
- expecting investors to maximize wealth within the safety-first constraint

As an early behavioral portfolio theory, SFPT has a somewhat simplified vision of investor as primarily risk-adverse. Also, it postulates that investors follow utility theory in their decision-making, an assumption widely discredited by empirical studies.

We progress next time to a theory that had its roots in SFPT: **SP/A Theory,** which deals with security, potential, and aspiration. Don’t miss it!

## Capital Asset Pricing Model, Part One – Normal Distribution

*by Eric Bank on January 7, 2011 · No Comments · in Documenting the Search for Alpha*

We are now ready to discuss asset pricing models, and we’ll begin with the Capital Asset Pricing Model (CAPM).

We left off last time showing how the **Security Characteristic Line** indicates the **beta** of an asset under Harry Markowitz’s **Modern Portfolio Theory (MPT)**. We are now ready to discuss asset pricing models, and we’ll begin by documenting the **Capital Asset Pricing Model (CAPM)**. This model was developed in the 1960’s by several independent researchers, including Sharpe, Treynor, Lintner and Mossin, building on Markowitz’s previous work.

CAPM is an equation that indicates the required **rate of return (ROR)** one should demand for holding a risky asset as part of a diversified portfolio, based on the asset’s beta. If CAPM indicates a rate of return that is different from that predicted using other criteria (such as P/E ratios or stock charts), then one should, in theory, buy or sell the asset depending on the relationship of the different estimates. For instance, if stock charting indicates that the ROR on Asset A should be 13% but CAPM estimates only a 9% ROR, one should sell or short the asset, which cumulatively should drive the price of Asset A down. Continue reading “Capital Asset Pricing Model, Part One – Normal Distribution” »

## Modern Portfolio Theory – Part One

*by Eric Bank on December 30, 2010 · No Comments · in Documenting the Search for Alpha*

Our risk/return series continues with a review of Modern Portfolio Theory (MPT). We’ve already looked at alpha, beta, efficient markets, and returns. Our ultimate goal is to evaluate the role of alpha in hedge fund profitability, how to replicate hedge fund results without needing alpha, and finally how you can start your own cutting-edge hedge fund using beta-only replication techniques.

Our risk/return series continues with a review of **Modern Portfolio Theory (MPT)**. We’ve already looked at **alpha, beta, efficient markets**, and **returns**. Our ultimate goal is to evaluate the role of alpha in hedge fund profitability, how to replicate hedge fund results without needing alpha, and finally how you can start your own cutting-edge hedge fund using **beta-only replication techniques**.

MPT suggests that a portfolio can be optimized in terms of risk and return by carefully mixing individual investments that have widely differing betas. Recall that beta is return due to correlation with an overall market (known as **systematic risk**). To take a trivial example, if you hold equal long and short positions in the S&P 500 index, your portfolio would have an overall beta of (.5 * 1 + .5 * -1) = 0. There would be no risk, but in this case, there would be no return either, except for the slow drain of commissions, fees, etc.

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