**"systematic risk"**

## Consumption CAPM

*by Eric Bank on March 13, 2011 · No Comments · in Documenting the Search for Alpha*

When we recently examined Intertemporal Capital Asset Pricing Model (I-CAPM), we understood it to be an extension of CAPM in which investors establish additional portfolios to hedge specific risks representing short-term idiosyncratic tastes and preferences. Typical examples of these specific risks include inflation, job loss, economic downturn, etc. The model is somewhat incomplete, in that it doesn’t specify how to arrive at the proper factors to quantify these additional hedges. Another approach, called the Consumption CAPM (C-CAPM) posits a single additional hedge portfolio based on consumption risk, which is a hedge against future consumption rates.

When we recently examined **Intertemporal Capital Asset Pricing Model (I-CAPM),** we understood it to be an extension of CAPM in which investors establish additional portfolios to hedge specific risks representing short-term idiosyncratic tastes and preferences. Typical examples of these specific risks include inflation, job loss, economic downturn, etc. The model is somewhat incomplete, in that it doesn’t specify how to arrive at the proper factors to quantify these additional hedges. Another approach, called the **Consumption CAPM (C-CAPM)** posits a single additional hedge portfolio based on** consumption risk**, which is a hedge against future consumption rates. Let’s see how that works.

C-CAPM assumes that investors are not interested in maximizing their portfolio, but rather in *maximizing over their lifetimes the consumption they get from their portfolio*. Therefore, there is a tradeoff between current and future consumption. The model assumes that investors will sell assets during bad economic times and buy them when times are good. That means that, say, a yacht, will have to have a very high future expected return (a high risk premium) to induce the owner to hold onto it when times are bad. Therefore the systematic risk of the asset (the yacht) is tied to the state of the economy.

An illustration will clarify.

Each person has their own set of **utility functions**. For instance, if I buy an asset (such as a yacht) today, it will cost me $X. If I am to hold off on buying the asset, it had better be worth it to me to wait. In other words, I should receive a return on the amount, $X, that I didn’t consume today. If that return, which we’ll call the consumption premium, is high enough, I’ll wait; if not, I’ll buy the asset today. At some rate of return (let’s call it r_{0}) I am indifferent to waiting or consuming. If my return on $X is expected to exceed r_{0}, I’ll wait. If it is expected to be less than r_{0}, I’ll consume today.

Now, if I expect a rosy economic future, I will assign a *higher* risk premium to the yacht; conversely, if I see the future economy tanking, I *lower* risk premium on the yacht, making it more likely to consume the asset now rather than waiting. Of course, if I’m anticipating a severe depression, I just might keep the $X in Treasury bonds and adjust my utility function accordingly. Since the price of my current Treasury bonds will rise during a depression (because of their relatively high interest rates), they are a good hedge against bad times – something that cannot be said about a yacht. In terms of C-CAPM theory, the returns on the Treasury bonds have a *negative correlation* with consumption, and are thus are worth more during times of depressed consumption. The yacht’s returns have a *positive correlation* with consumption (yachts sell for more in times of consumption growth), and are therefore worth less during low-consumption periods.

Bottom line: hedge your portfolio with assets that will do well when economic consumption rates drop.

In C-CAPM, one has to measure the consumption premium accurately in order to have a valuable model. One problem that arises is that of **satisfice**: are consumers really looking to optimize consumption, or are they merely satisfied with achieving some minimum constraint? This will have considerable impact on the rate at which future consumption is discounted to arrive at its value today.

Now that we’ve looked at CAPM and several alternatives, we want to next explore the topic of specific **risk premia**, which we’ll loosely define as the return (above the risk-free rate) that can be associated with taking on certain systematic risks. We want to see whether, for any given hedge fund strategy, returns can be ascribed solely to the systematic risks (i.e. **beta**) of the strategy, or whether some portion of the return is due to a superior fund manager (**alpha**). This will lead to our ultimate topic: **hedge fund replication**.

## Capital Asset Pricing Model, Part Three – Other Assumptions

*by Eric Bank on January 22, 2011 · No Comments · in Documenting the Search for Alpha*

In Parts One and Two of our examination of the Capital Asset Pricing Model (CAPM), we evaluated two major assumptions:

1) Market returns are properly modeled by a normal distribution

2) Beta (systematic risk) is the sole source of priced risk for an asset or portfolio of asset

In Parts One and Two of our examination of the **Capital Asset Pricing Model (CAPM)**, we evaluated two major assumptions:

1) Market returns are properly modeled by a **normal distribution**

2) **Beta (systematic risk)** is the sole source of priced risk for an asset or portfolio of asset

As you recall, we found several weaknesses in both assumptions as they may apply to hedge funds. This time, we’ll examine the remaining assumptions underlying CAPM, and see if each is reasonable when applied to hedge fund trading.

CAPM assumes: Continue reading “Capital Asset Pricing Model, Part Three – Other Assumptions” »

## 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.

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