**"market portfolio"**

## Intertemporal CAPM (I-CAPM)

*by Eric Bank on March 2, 2011 · No Comments · in Documenting the Search for Alpha*

I-CAPM was first introduced in 1973 by Merton. It is an extension of CAPM which recognizes not only the familiar time-independent CAPM beta relationship, but also additional factors that change over time (hence “intertemporal”).

The **Capital Asset Pricing Model (CAPM),** which estimates the return on an asset by its co-varying relationship (its **beta**) with the market portfolio, has spawned a whole field of study devoted to improving upon the original model. We have already explored **D-CAPM**, which modified beta to measure downside risk rather than total risk. Our attention now turns to another alternative theory, the **Intertemporal CAPM (I-CAPM)**. As in previous discussions, we omit the complicated mathematics.

I-CAPM was first introduced in 1973 by Merton. It is an extension of CAPM which recognizes not only the familiar time-independent CAPM beta relationship, but also additional factors that *change over time* (hence “intertemporal”). Specifically, Merton posited that investors are looking to hedge risks based on current and projected **factors**, such as changes in inflation, employment opportunities, future stock market returns, etc. Thus, an I-CAPM portfolio contains a core investment tied to the market portfolio, plus one or more hedge portfolios that mitigate an investor’s currently-perceived risks. Since each investor has his/her own perceptions of risk, I-CAPM is hard to generalize to a population of investors. Also, the correct factor to use for any given hedge is *ambiguous*: if you think you might be unemployed 3 years from now, which factor do you choose to hedge?

So, I-CAPM is a multi-factor model that attempts to capture more determinants of risk than just beta, but does not provide concrete guidance for identifying which additional factors to use, or even how many of them to include.

A popular “fix” to this problem has been to employ the **Fama-****French**** Three-Factor Model (FF)**, which states that the two additional factors beyond beta should be used to improve CAPM. These two factors relate to specific **investment styles**:

**Liquidity**: small capitalization stocks outperform large-cap stocks**Value**: value stocks outperform growth stocks

FF speculates that these two stock groupings, small cap and value stocks, are inherently more risky to macroeconomic downturns and thus are required by investors to provide higher returns (hence portfolios built on these factors should have betas higher than 1.0, the market beta). However, FF posits that the risk premia provided by these two factors *exceeds* that provided by a higher beta alone – that there is an **alpha** component (representing *non-systematic risk*) to the returns generated by portfolios built on these factors. But I-CAPM requires **efficient markets,** which would mean that each factor’s risk premium is due *only* to its beta. Hence, I-CAPM argues *against* alpha-based returns, which indeed is the overall conclusion we have been seeking to evaluate over the past months. We will keep returning to this question, as it has serious implications for our ultimate topic: alternative beta strategies and hedge fund replication.

## 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.

## Arbitrage Pricing Theory – Part Three

*by Eric Bank on February 14, 2011 · No Comments · in Documenting the Search for Alpha*

Having previously discussed them, let’s compare them. How does the Arbitrage Pricing Theory (APT) stack up against the Capital Asset Pricing Model (CAPM)?

Having previously discussed them, let’s compare them. How does the **Arbitrage Pricing Theory (APT) **stack up against the **Capital Asset Pricing Model (CAPM)**?

Both theories are important mechanisms for pricing assets. The APT assumes less than CAPM. Whereas CAPM relies on probabilities and statistics, APT provides for cause and effect to explain its predictions of asset returns. CAPM assumes investors hold a mix of cash and the market portfolio, whereas APT allows each investor to hold his own portfolio with its own unique beta. So if you applied APT to the market portfolio, the example would degenerate into CAPM, because the securities market line would be a single factor price model in which beta is exposure to price changes in the market portfolio.

Since the beta in CAPM is correlated to asset demand based on each investors incremental utility for each asset, it is considered a demand-side model. In contrast, APT is a supply-side model, because its betas correlate an asset’s return to specific economic factors. Any sudden change to one of these factors would ripple through the market containing the asset, and thus are inputs to asset returns.

Both models assume **perfect competition** within markets:

- Buyers are
*price-takers*, and are too small to affect prices. - Assets are i
*nfinitely liquid*, and supply and demand will reach equilibrium at a certain price. - There are no
*barriers*to entering or leaving a market. - There are no
*trading costs*such as commissions, taxes or fees. - All investors share a sole motivation: to
*maximize returns*. - A given share of a security is the
*same*as any other share, regardless of where (i.e. on which exchange) the security is exchanged.

The only factor in CAPM is the market portfolio. ABT is a multi-factor model and has one additional requirement: the number of factors must be no greater than the number of assets. Certain assumptions are made about APT factors:

- The factors are
*macroeconomic*in nature, and thus create risks that cannot be easily avoided through diversification. - Factors affect asset prices through
*shocks*(unexpected changes). - All information about each factor is
*well-known and accurate.* - There is some real-world
*relationship*between a factor and an asset.

So, which model better predicts asset returns? The answer is not clear. Quoting from a study published in the Asia-Pacific Business Review, March, 2008 by Rohini Singh: “*The macroeconomic factors used in this study were able to explain returns marginally better [using APT] than [CAPM] beta alone. While this confirms that risk is multidimensional and that we should not depend on beta alone, further research is required to identify other variables that can help explain the cross section of returns.*”

## Modern Portfolio Theory – Part Three

*by Eric Bank on January 5, 2011 · No Comments · in Documenting the Search for Alpha*

The Capital Market Line represents a portfolio containing some mixture of the Market Portfolio (MP) and the risk-free rate.

Our quest continues to find out whether hedge fund alpha really exists or is just hype. Recall from last time our documentation of the **Capital Market Line (CML)**. The CML represents a portfolio containing some mixture of the **Market Portfolio (MP)** and the **risk-free rate**. It is a* *special version of the **Capital Asset Line**, ranging from the risk-free rate tangentially to the **Efficient Frontier** at the Market Portfolio, and then extending upwards beyond the tangent point. **Modern Portfolio Theory (MPT)** posits that any point on the CML has superior risk/return attributes over any point on the Efficient Frontier. Let’s ponder that for a second – just adding some T-Bills to, say, S&P 500 baskets (our proxy for the Market Portfolio) will improve the risk/return characteristics of your portfolio.

If your entire portfolio consisted only of the cash-purchased Market Portfolio (i.e. the tangent point on the Efficient Frontier), your **leverage ratio** would be 1 – you are **unleveraged**. The points on the CML below the Market Portfolio represent **deleveraging**: adding cash to your portfolio. You are lowering risk and expected return when you deleverage. If you borrowed and sold TBills, and used the proceeds to buy additional Market Portfolio, your new portfolio would be **leveraged**, and would be a point on the CML above the tangent. Leveraging increases your risk and expected return. If you disregard the effects of borrowing (or margin) costs, then all points on the CML share the maximum **Sharpe Ratio**, a popular formula for expressing risk/return. Continue reading “Modern Portfolio Theory – Part Three” »

## Modern Portfolio Theory – Part Two

*by Eric Bank on January 3, 2011 · No Comments · in Documenting the Search for Alpha*

We continue our journey into the wonderland of alpha, taking up with leverage and the Efficient Frontier.

We continue our journey into the wonderland of alpha, taking up with leverage and the Efficient Frontier. We documented last time that a mix of a risky portfolio and the **risk-free rate (Rf)** yields a linear** Capital Asset Line (CAL) **within **risk-return space**. When the mix is varied to decrease the amount of Rf, the riskiness and expected return of the portfolio increase. The mirror image occurs as we increase the relative percentage of Rf in the portfolio mix. We can even borrow at the risk-free rate to purchase additional risky assets for our portfolio – one form of a practice known as **leverage**. Continue reading “Modern Portfolio Theory – Part Two” »

## Beta

*by Eric Bank on December 20, 2010 · No Comments · in Documenting the Search for Alpha*

Hedge funds use an array of strategies to guide trading. Most of these strategies seek to decouple returns from those of the overall market, as measured by a statistic called “beta” (β).

Hedge funds use an array of strategies to guide trading. Most of these strategies seek to decouple returns from those of the overall market, as measured by a statistic called “**beta**” (β). Beta is calculated by dividing the **covariance** of an investment’s return by the **variance** of a portfolio or market return:

β_{i} = Cov (r_{i}, r_{m}) / Var(r_{m}) where i = an investment, m = market portfolio, and r = return

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