**"covariance"**

## Capital Asset Pricing Model, Part Two – Systematic Risk

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

We are reviewing the underlying assumptions made by the Capital Asset Pricing Model (CAPM). Recall from last time the assumption that returns are distributed normally (i.e. a bell-shaped curve) and how this fails to account for skew and fat tails. Today we’ll look at CAPM’s assumption that there is but a single source of priced systematic risk: market beta.

We are reviewing the underlying assumptions made by the **Capital Asset Pricing Model (CAPM)**. Recall from last time the assumption that returns are distributed normally (i.e. a bell-shaped curve) and how this fails to account for **skew** and **fat tails**. Today we’ll look at CAPM’s assumption that there is but a single source of priced **systematic risk**: market **beta**.

Beta, of course, refers to the correlation between the returns on an asset and the returns on the broad market. Specifically, it is defined as:

βi = Cov(Rm, Ri) / Var(Rm) for asset i

that is, the **covariance** of the asset’s returns with the those of the market divided by the **variance** of the market returns. Another way to think about beta is as a discount factor: when divided into an asset’s expected return, it adjusts the expected return so that it equals the market’s expected return.

By making this assumption, CAPM is able to simplify the universe of different risks undertaken by a portfolio, such as a hedge fund, to one global number. That’s good for ease of calculation but bad for reflecting reality. It is also assumed that all non-systematic risk (**alpha**) has been diversified away by having a sufficiently large portfolio, so that the expected return on the portfolio is a function of its beta alone.

Many academics challenge the use of variance in the formula for beta. They point out that variance is a symmetrical measure of risk, whereas investors are only concerned about loss (just the left side of the curve). Instead of variance, they point to other measures of risk, such as **coherent risks**, that may better reflect investor preferences. Coherent risks have four attributes:

1) When comparing two portfolios, the one with better values should have lower risk

2) The joint risk of two portfolios cannot exceed the sum of their individual risks

3) Increasing the size of positions within a portfolio increases its risk

4) Picture two identical portfolios except one contains additional cash. The one with the extra cash is less risky, to the exact extent of the extra cash.

Hedge funds clearly take on risks that are not reflected in beta. These risks are the basis for the returns of many strategies, such as long/short (liquidity risk), fixed income (credit risk), and sector trading (sector risk) to name a few. To the extent that a hedge fund takes on risks that are not reflected by beta, it is modifying its long-term expected return for better or worse. For instance, a portfolio may add an asset with a negative expected return in order to hedge another asset that has an expected positive return. This has the effect of lowering the expected return of the portfolio as it lowers the portfolio’s risk.

Recall that the expected market return above the risk-free rate is called the market’s **risk premium**. Many hedge funds seek returns independent of the market risk premium. Rather, hedge funds supposedly choose their risk exposures, and hence their risk premia. Ideally, their market beta would be zero, so that hedge fund returns would reflect only risks other than market risk. Note that these non-market risks are often hard to diversify away. For instance, it is difficult to diversify away the risks involved in “risk arbitrage” – a misnomer for the strategy of taking positions on mergers and acquisitions. By assuming this “event risk” (i.e. by betting on whether a proposed merger will succeed), fund managers hope to capture the associated risk premium, independent of market beta.

There are a large number of hedge fund strategies, and most are exposed to their own systematic risks. Later on, we will review different strategies in detail, and look at the associated risk premia at that time. *The important point here is that systematic risk, whether captured in market beta or due to strategy-specific risk(s), should not be conflated with *alpha* – the returns due to superior skill and/or timing.* Ultimately we will be exploring the case for and against alpha when we examine **hedge fund replication strategies**.

Next time, we’ll finish discussing the assumptions underlying CAPM.

## 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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