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Consumption Risk?

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 r0) I am indifferent to waiting or consuming. If my return on $X is expected to exceed r0, I’ll wait. If it is expected to be less than r0, 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.

Picture of Einstein writing on blackboard

He would understand I-CAPM.

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.

Ocean lIner sinking at severe angleWe 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 (βD). Different researchers have supplied different technical definitions for βD; here we will use the one provided by Javier Estrada:

βD = downside covariance between asset and market portfolio / downside variance of market portfolio.

There are several ways to calculate βD, 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.

Last time we began our discussion of Behavioral Portfolio Theory with a look at Safety-First Portfolio Theory (SFPT), which basically posits investor motivation to be to avoid ruin.  An extension to SFPT was introduced in 1987 by Lopes, named SP/A Theory. The letters in its title can be summarized:

S = security, a general concern about avoiding low levels of wealth

P = potential, reflects the general desire to maximize wealth

A = aspiration, the desire to reach a specific goal, such as achieving no less than the subsistence level s.

SFPT is a psychological theory of choice under uncertainty. In Lopes’ framework, risk-taking is balanced between fear and hope. Lopes posits that fear is such a strong factor because fearful people overweight the probability of the worst outcomes, underweight those for the best outcomes. This leads individuals to understate the probability of achieving the highest level of expected wealth E(W).  In other words, fearful individuals are pessimistic.

Hope has the inverse effect on individuals – optimism causes hopeful investors to overstate the probability of achieving the highest level of expected wealth.

For those of you who like symbols, wealth W is actually a series of n different levels of wealth, ordered from lowest to highest, designated as W1, W2, W3, … Wn-1, Wn. Lopes denotes a decumulative distribution function[1] D(x) as the likelihood that the wealth W of a portfolio will be greater than the investor’s subsistence level s:

D(x) = Prob {W >= s)

where 0 < x <= n different levels of wealth.

Expected wealth E(W) is the weighted sum of each level of wealth, the weight being the probability pi of achieving a particular level of wealth Wi:

E(W) = ∑piWi

Lopes notes that expected wealth, E(W) = ∑piWi, can be expressed as the sum of the decumulative probabilities ∑Di (Wi - Wi -1), where the summation is from i = 1 to n and W0 is zero. In this expression for E(W),the individual receives W1 with certainty (note that decumulative probability D1 = 1), receives the increment W2 – W1 (that is, an amount over W1) with probability D2, receives the further increment W3 – W2 with probability D3, and so on. Lopes contends that fear operates through an overweighting of the probabilities attached to the worst outcomes (W1). The way that works is that fearful investors set the p1 too high, and pn too low. The reverse is true for hopeful investors.

Lopes concludes that the emotions of fear and hope reside within all individuals, and that each emotion serves to modify the decumulative weighting function D. She suggests that the final shape of the decumulative transformation function h is a convex combination (shaped like a smile) of hs (for fearful investors worried about security) and hp (for hopeful investors looking for maximum potential), reflecting the relative strength of each.

The qualitative lesson from SP/A theory is that you should establish a portfolio in which you have reigned in your emotions of fear and hope, so that neither dominates. Unfortunately, it doesn’t tell you how to accomplish this. Still, not a bad lesson for a hedge fund trader.


[1] The decumulative distribution function is just (1 – cumulative distribution function).  Basically it measures the curve to the right (instead of the left) of a given point.

Man working belt and suspenders

Safty First!

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 μP and return standard deviation σP;
  • the risk-free asset RF is not available;
  • 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 – μP) / σP

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!

New York Stock Exchange

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 infinitely 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.

IAlbert Einstein at blackboardn 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(ri) = Rf +bi1 RP1+. . .+bin RPn where

  • RPn is the premium on factor n
  • bin is asset i’s loading of factor n
  • Rf 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.

Albert Einstein writing on a blackboard.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:

ri= E(ri) +bi1 F1 + bi2 F2 +· · ·+bin Fn + εi for i = 1, 2, . . .n  where:

  • E(ri) is the expected return on asset i
  • F1, . . ., Fn are the latest data on common systematic factors driving all asset returns
  • bin gives how sensitive the return on asset i with respect to news on the n-th factor (factor loading)
  • εi 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 RPi = the risk premium on factor i = the return from factor i above the risk-free rate = (rFn−Rf), then the APT equation is:

E(ri) = Rf +bi1 RP1+. . .+bin RPn where

  • RPn is the premium on factor n
  • bin is asset i’s loading of factor n
  • Rf 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.

Security Market Line

In the first three installments, we looked closely at the assumptions that underlie the Capital Asset Pricing Model (CAPM), as part of our overall project of investigating the role of alpha in hedge fund performance.  Today we will review the basic CAPM equations.  I promise you won’t need a PhD in Mathematics to get the gist.

Recall the purpose of CAPM: to calculate a theoretical price for an asset or portfolio of assets.  In the single asset case, we use the Security Market Line (SML) that projects the risk/reward characteristics of the asset compared to those of the overall market. The x-axis is beta and the y-axis is return. The slope of the SML is the market risk premium (i.e. the return in excess of the risk-free rate, which is the y-intercept). The slope is given by E(Rm) – Rf.  The entire SML equation is:

(E(Ri) – Rf ) / βi = E(Rm) – Rf

Where

E(Ri) is the expected return of asset i

Rf is the risk-free rate

βi , the beta of asset i, the sensitivity of  asset return to market return

E(Rm) is the expected return of the overall market

In other words, the excess return on an asset, deflated by the asset’s beta, equals the expected return of the market in excess of the risk-free rate.

Now, solve for E(Ri) to arrive at the CAPM equation:

E(Ri) = Rf + βi (E(Rm) – Rf)

So the expected return on asset i equals the excess return of the market multiplied by the assets beta, added to the risk-free rate.  This shows that beta is the sole explanation for an asset’s expected return.  As you know from previous blogs, this is a very shaky assumption.

One more rearrangement gives us:

E(Ri) – Rf  = βi (E(Rm) – Rf)

which shows that the asset’s risk premium is equal to the market premium multiplied by beta. In other words, the SML is a single-factor model, and beta, which is the asset’s relative covariance (i.e. covariance / variance) with the market, is that factor.

Now you see why we invested three blogs in gauging the real-world verisimilitude of CAPM.  In the next blog, we’ll explore how CAPM fares when applied to different hedge fund trading strategies.

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” »