**"rate of return"**

## 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**.

## Arbitrage Pricing Theory – Part One

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

Arbitrage Pricing Theory (APT) is a multi-factor model conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor.

Last time out, I promised we would look at the risk premia associated with various hedge fund strategies. But first I’d like to finish the discussion of pricing models that we began with our look at the **Capital Asset Pricing Model**.

**Arbitrage Pricing Theory (APT)** is a* multi-factor model *conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own *betas* to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor. Here are the equations:

Suppose that asset returns are driven by a few (*i*) common systematic factors plus non-systematic noise:

r_{i}= E(r_{i}) +b_{i}_{1} F_{1} + b_{i}_{2} F_{2} +· · ·+b_{in} F_{n} + ε_{i}_{ } for i = 1, 2, . . .n where:

- E(r
_{i})*i* - F
_{1}, . . ., F_{n}are the latest data on common systematic factors driving all asset returns - b
_{in}gives how sensitive the return on asset i with respect to news on the n-th factor (*factor loading*) - ε
_{i}is the idiosyncratic noise component in asset i’s return that is unrelated to other asset returns; it has a mean of zero

APT claims that for an arbitrary asset, its expected return depends only on its *factor exposure*:

If we define RP_{i} = the *risk premium *on factor i = the return from factor i above the risk-free rate = (rF_{n}−R_{f}), then the **APT equation** is:

E(r_{i}) = R_{f} +b_{i1} RP_{1}+. . .+b_{in} RP_{n} where

- RP
_{n}is the premium on factor n - b
_{in}is asset i’s loading of factor n - R
_{f}is the risk-free rate

So you can see, the expected return of an asset *i *is a linear function of the assets’ betas to the *n* factors. This equation assumes ideal markets and no surplus of the number of factors above the number of assets.

In Part Two, we’ll explore the assumptions and implications of the APT equation.

## 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” »

## Rate of Return

*by Eric Bank on December 27, 2010 · No Comments · in Documenting the Search for Alpha*

If you have been following our recent blogs, you are by now familiar with the concepts of alpha, beta, and the Efficient Market Hypothesis. Our final goal is to evaluate the role of alpha in hedge fund investing, and to look at trading strategies that do not rely on alpha. Before we can discuss these topics, we need to better understand financial asset pricing models, the role of alpha and beta within these models, and how the models apply specifically to hedge funds. In this installment, we’ll review the concept of rate of return (ROR).

If you have been following our recent blogs, you are by now familiar with the concepts of **alpha**, **beta**, and the **Efficient Market Hypothesis**. Our final goal is to evaluate the role of alpha in hedge fund investing, and to look at trading strategies that do not rely on alpha. Before we can discuss these topics, we need to better understand financial asset pricing models, the role of alpha and beta within these models, and how the models apply specifically to hedge funds. In this installment, we’ll review the concept of **rate of return (ROR)**.

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