**"portfolios"**

## 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**.

## 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.

## 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!

## Arbitrage Pricing Theory – Part Two

*by Eric Bank on February 10, 2011 · No Comments · in Documenting the Search for Alpha*

Arbitrage Pricing Theory (APT) is a multi-factor model in which a series of input variables, such as macroeconomic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset.

In our previous blog on this subject, we identified **Arbitrage Pricing Theory (APT)** as a* multi-factor model* in which a series of input variables, such as macroeconomic indicators and market indices, are each assigned their own *betas* to determine the expected return of a target asset. Recall from its equation that the expected return of an asset *i *is a linear function of the assets’ betas to the *n* factors:

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

Let’s explore the theory in a little more depth. It is based on the concept of arbitrage, which is when a trader makes a risk-free profit when a single asset is priced differently in different markets. By shorting the higher price while simultaneously buying at the lower price, the price differential is locked in. Unfortunately, in today’s lightning-fast global markets, this is a very rare opportunity.

The theoretical value of the APT is its ability to detect a *mispriced asset*: if the weighted sum of each factor’s risk premium is different from the asset’s current risk premium (as indicated by the asset price), then an arbitrage opportunity exists. If you think as the expected return on an asset as a *discount rate* on its future cash flows, a price can be calculated and compared to the actual current price.

We use the term asset here to encompass both single assets and portfolios. If a single asset is being modeled, then the asset is exposed to each macroeconomic factor to the extent of its beta with that factor. Let’s assume we identify a mispriced asset based on a 5-factor ABT model. An *equivalent portfolio* would contain a set of 6 correctly-priced assets, one asset per factor plus one extra. Each asset would have the same exposure to a single factor as the mispriced asset. By varying the amount of each asset in the portfolio, you can arrive at a portfolio beta/factor that is the same as that for the mispriced asset. You can now treat the portfolio as a *synthetic* version of the asset, and take a short or long position in the portfolio to offset the opposite position in the mispriced asset, theoretically capturing a risk-free profit.

Of course, identifying the important factors associated with a particular asset is no small undertaking, and APT does not offer a roadmap. However in general, one would have to take into account, for each of the factors affecting a particular asset, the expected return of each factor and the asset’s sensitivity to the factor.

We’ll complete our look at APT next time by comparing it to the Capital Asset Pricing Model.

## 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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