Part II Financial Markets and Portfolio Theory
7.6 Comparing CAPM and APT
Using CAPM and APT makes them appear like close relatives. Analyzing their foundations reveals fundamental differences. But let’s first look at the similarities in their application. We have seen that the β-coefficient of a risky security in the framework of the CAPM can be estimated by linear regression. The λs in the APT can be estimated by multiple linear regression
where
If we identify the realized values of the response variable and the regressors, as we did before, with
then we obtain the estimate of the coefficient vector |λ by least-squares,
As in case of the CAPM, we certainly expect the estimate to be approximately zero, if the APT is correct. Everything we said about the statistical properties still remains valid. From (7.86) we can only tell the difference, because we called the coefficients of the APT λ and not β. This is a very close resemblance.
What about the differences? The CAPM was derived, departing from modern portfolio theory. The framework included multivariate normal and independently distributed returns, the same risk-free interest rate for borrowing and lending, and equilibrium prices of assets. All of these features have been the subject of harsh criticism in the past, although empirical results do not support rejection of the CAPM. The APT on the other hand is based only on the assumption that in very large markets there should not be any arbitrage opportunities. The absence of arbitrage is an extremely general requirement. Realize that in the presence of arbitrage opportunities, there cannot be an equilibrium, because agents would trade in strategies to exploit them, until price adjustments remove these opportunities. But the absence of arbitrage does not require an equilibrium. The bottom line is that APT operates under extremely mild and barely criticizable conditions. On the other hand, its implications only apply to large portfolios and markets. In small markets, APT can be violated quite seriously. The theoretical implications are also far more general than those of the CAPM. The APT merely provides the shape of the relation between latent factors and the security returns. It says neither what they are, nor how many of them are needed. The fact that the CAPM is reproduced as one particular manifestation of the APT, clearly strengthens the theoretical rationale of the CAPM.
7.1
7.2
7.4
7.7
7.8
7.3
Further Reading
The classical references for the capital asset pricing model (CAPM) are Sharpe (1964), Lintner (1965), and Mossin (1966). For an excellent review including all technical assumptions, see Gatfaoui (2010). For a non-technical discussion of the CAPM see Estrada (2005, chap. 6 & 7). The former source also discusses the three-factor extension of Fama and French (1993, 1996). The original sources for the arbitrage pricing theory (APT) are Ross (1976), and Roll and Ross (1980). A compressed discussion of the concepts involved can be found in Shiryaev (1999, sect. 2d). A very careful and accessible treatment of this subject is Huberman (1982), and also Ingersoll (1987, chap.
7). Statistical and econometrical issues of regression analysis and parameter estimation are treated thoroughly in Greene (2003). For the factor analytic model see Mardia et al. (2003, chap. 8 & 9).
Problems
Practitioners often assume that the true yearly β of a security decays towards βMP 1 with time. A simple model for such a mean reversion process is
where . Show that is given by
Prove that the half life of the difference in Problem 7.1 is
years.
Look at the general mean reversion structure
where 0 ≤ λ ≤ 1 is an arbitrary coefficient. How is λ to be chosen if the intrinsic period is one month, to maintain the term structure of the yearly period model?
For an arbitrary random variable Y and a σ-algebra , generated by observing some event A, the variance decomposition
7.5
1 2
7.6
holds. Show that the unconditional variance of the least-squares estimator is
w her e X is the usual data matrix, containing a column of ones and the regressors Xt, for t = 1, . . . , T.
The original three-factor portfolio model of Fama and French (1993, 1996) is formulated in the form
where SMB indicates the market capitalization spread (“small minus big”), and HML is the spread in the book-to-market ratio (“high minus low”). Show that this model is empirically indistinguishable from a three-factor APT- model, if the restriction
holds, and βq = λq for q = 1, 2, 3.
Show that the modified factor model
w i th E[Fq] = μFq, Cov[Fp, Fq] = σpq, , , and , generates the correct APT-equation (7.57).
Even if is not normally distributed, is still asymptotically normal under fairly mild conditions, see Greene (2003, chap. 5).
The exact distribution is Student’s t-distribution, with T − 2 degrees of freedom. For T > 30, this one virtually coincides with the normal distribution.
(8.1)
(8.2)
8
8.1
8.1.1
Portfolio Performance and Management
Portfolio performance is usually measured in terms of certain ratios like the Sharpe- or the Treynor-ratio. But there is much more to this subject than just comparing statistics. There are questions like: What proportion of the available capital should be invested in the risky part of the portfolio? This is a matter of optimal money management. Another important question is how future expectations affect today’s optimal portfolio decisions. Those are important issues of portfolio selection and control.
Portfolio Performance Statistics
Assessing the performance of a portfolio is more than comparing past returns over a given time horizon. Imagine two managers of different mutual funds. Both have realized precisely the same annual returns over the last 10 years, but the first one generated only half the volatility in terms of yearly standard deviation, compared to the second one. Which fund would you buy? Which portfolio manager did the better job? Indeed, due to their risk-averse attitude, most people would prefer the first mutual fund, because they conclude from observation that the same expected return is realized with half the risk exposure. This is of course an educated guess, because we cannot guarantee returns and volatilities of both funds not to evolve in a completely different way in the future. The lesson here is that portfolio statistics are always based on past performance and cannot predict the future development of a portfolio. Nevertheless, they can be helpful in gaining or losing trust in the abilities of the portfolio manager. Subsequently, three of the most commonly used performance statistics are introduced.
Jensen’s Alpha
Jensen’s alpha (Jensen, 1968) is named after the usual Greek letter for the intercept in the linear regression model. Let’s focus on the CAPM for the moment. If we estimate the regression coefficients for the return series of an arbitrary portfolio P, we get an equation of the form
or after rearranging
(8.3)
(8.4)
(8.5)
8.1.2
In the framework of linear regression, Jensen’s alpha is most easily computed from the least-squares estimator, , with e1| = (1 0). In the APT-case we would obtain , with e1| extended to the dimension of .
Obviously, represents a kind of extra return above the β-weighted excess return, predicted by the CAPM. We can make this point even more transparent by introducing an estimated extra return , with , and substituting in (8.1),
It is disputable, if such an extra return can be achieved permanently. Of course the market portfolio proxy is never an exact and exhaustive representation of the true market portfolio. But usually it is close enough to prevent from being significantly different from zero. In his original article, Jensen (1968) analyzed 115 mutual funds in the period from 1945 to 1964. He found that the vast majority generated a negative estimate for αP, with an average of −1.1% after fees. He only found three funds, with alpha statistically significant greater than zero, on a significance level of 5%. But as Jensen pointed out himself, when analyzing 115 funds with true αP = 0, one would expect five or six of them to generate a significant result on a 5%-level purely by chance.
There is another possible explanation for a positive alpha. Portfolio managers have to specify an index, against which their fund is to be benchmarked. They are not free to choose such an index, because a high percentage of the securities have to be listed in this index. On the other hand, they are free to include a small proportion of foreign securities, possibly from riskier markets. Those are more volatile and if they realize higher returns, the return series generated by the portfolio supports a positive estimate for αP. Thus, Jensen’s alpha is also a measure for the selection success.
Treynor-Ratio
Unlike Jensen’s alpha, the Treynor-ratio (Treynor, 1966) is inextricably linked to the CAPM. To understand Treynor’s measure, first observe that the slope of the security market line (SML) with respect to β is
because βMP = 1 (see Figure 8.1). We can of course indicate a point (βP, μP) beyond the SML, representing the beta-factor and the expected return of a hypothetical portfolio P. Assume for the moment that such a violation of the CAPM is possible and call the imaginary line from (0, r) through (βP, μP) the portfolio market line (PML). The Treynor-ratio is defined as the estimated slope of the PML with respect to β,
(8.6)
8.1.3
8.2
where is an unconditional estimate of the expected return of the portfolio P. For all practical purposes, is the least-squares estimate of the portfolio beta, and is the mean of the observed portfolio returns.
Fig. 8.1 Treynor-ratio as slope of the “portfolio market line” in the μ-β-diagram
Sharpe-Ratio
The Sharpe-ratio (Sharpe, 1966) can be motivated in exactly the same way as the Treynor-ratio, but in the μ-σ-diagram, not in the μ-β-diagram. Thus, the Sharpe-ratio for an arbitrary portfolio P is
where and are the usual sample estimators for the mean and the standard deviation of the portfolio return Rp. The Sharpe-ratio is therefore a completely model independent statistic. It also plays an important role in the economic theory of asset pricing, as we shall see.
As pointed out in the last chapter, the confidence bands for predicted returns can be huge compared to the magnitude of the returns themselves. This degree of uncertainty also affects the reliability of coefficients like the Treynor- or Sharpe-ratio. One should always keep that in mind, before relying on such statistics too heavily.
Money Management and Kelly-Criterion
Money management is a concept often more familiar to professional gamblers than to portfolio managers. For example Long Term Capital Management (LTCM), a hedge fund managed, among others, by two Nobel Prize laureates, crashed in 1998 due to massive liquidity problems. Paul
Wilmott (2006b, p. 742) even coined the term Short Term Capital MisManagement (STCMM). This is not the only historical example of a disastrous outcome due to poor money management.
You can see the paramount importance of good money management in a very simple example.
Assume, you have to decide which fraction π of your wealth w0 to invest in a successive coin flip game, where the outcome is in your favor with probability p. You either double the amount of your bet, or you lose your wager, independently in each round all over again. Now assume the odds are extremely in your favor, say p = 90%. Would you put your whole wealth at stake in a repeated game like this? I suppose not. If you repeatedly put 100% of your wealth in jeopardy, the one unfavorable outcome will eventually occur and you lose everything. That much is easy to see. But which fraction π of your wealth should you choose? That is the difficult question. The answer depends on what objective you pursue. If you are interested in maximizing the expected long-term capital growth rate, then the Kelly-criterion (Kelly, 1956) gives the right answer. Let’s see how this works in our simple coin flip setup.
Example 8.1
Look at the repeated coin flip game with probability p, not necessarily 50%, for the favorable outcome. Which fraction π of the initial wealth w0 should be invested in each round, to maximize the long-term growth rate?
Solution
First examine what happens in the first round. If you win, your wealth is
If you lose, your remaining wealth is
It is easy to see that if you have played N successive rounds of the game, and say you won n of those N rounds, your wealth is
The order in which you win or lose obviously does not make any difference. The average growth g of your capital after N rounds of the game is not given by the arithmetic mean, but by the geometric mean
or after taking the logarithm on both sides, making it a growth rate
Now remember that we are talking about long-term effects and thus, , and
(8.7)
(8.8)
(8.9)
(8.10)
To maximize this quantity, we have to solve the following optimization problem
This is easily solved after applying a few algebraic manipulations, and one obtains
which is the Kelly-criterion for the simple coin flip game. It is easily shown that the first order condition is sufficient for 0.5 ≤ p ≤ 1. For p < 0.5, you simply should not participate in the game.
Quick calculation 8.1 Convince yourself that the Kelly-fraction of Example 8.1 is perfectly sensible for the fair coin, p = 0.5, and also for the always winning coin, p = 1.
Let’s now turn to a more realistic scenario. Think of a portfolio P of risky securities. Every day, this portfolio is subject to a randomly driven return process. Let’s assume for the moment that daily returns are identically and independently distributed and call the realized return rP,t on day t. Again, starting with an initial wealth w0 and a fixed fraction π, the wealth at time T is
Quick calculation 8.2 Convince yourself that (8.7) is correct.
By completely analogous arguments to those presented in Example 8.1, we can calculate the average growth of capital by the geometric mean
Let’s proceed further with our analogy and calculate the average growth rate of the portfolio by taking logarithms
Now realize that the right hand side of (8.9) is the usual arithmetic mean, and in the limit T → ∞ we obtain
The next step is to Taylor-expand the logarithm in (8.10). As usual, we assume that on average daily
(8.11)
(8.12)
returns are positive but very small, E[RP] = μP ≪ 1, such that holds. Expanding to second order, one obtains
All that is left to do is to set the derivative of (8.11) with respect to π equal to zero, and to calculate the Kelly-fraction for the portfolio problem
Quick calculation 8.3 Verify the Kelly-criterion (8.12) for the portfolio problem.
Example 8.2
Assume you have the opportunity to invest in a mutual fund that offers an expected return of 5% per year and a standard deviation of 31.63%. Trading is only on a daily basis (250 trading days per year). What is the Kelly-fraction and what yearly capital growth rate can you expect?
Solution
The first thing to do is to convert the mean and standard deviation into daily quantities
From this we get immediately the Kelly-fraction
Plugging this ratio into the long-term average growth rate (8.11) yields
Finally, reconverting this growth rate into yearly terms yields an average capital growth rate of 1.25%.
Example 8.2 indicates that the average long-term growth rate of capital is significantly smaller than the expected return of the portfolio. This is partly due to the fraction invested in the fund, which is one half in this case. But there is another effect, best understood in a heuristic way. Assume you invested
(8.13)
8.3
an arbitrary amount of money w0 at time t = 0. The return at t = 1 is 10% and the return at t = 2 is
−10%. What does your account look like after the second day? You might be tempted to say it is still w0, because you realized a loss and a gain in capital of the same magnitude, but this is not correct.
The right answer is
Fig. 8.2 Kelly-criterion for the portfolio problem
You have indeed realized a loss of 1% after the second day. This may seem utterly unfair, but such is life.
It gets even worse. If you plot the average long-term growth rate against the Kelly-fraction, as in Figure 8.2, it becomes evident that if you willingly or by accident go beyond double-Kelly, you only can expect capital losses. In reality most of the time you do not know the parameters μP and σP, but have to estimate them from a finite sample of past realizations. Conservative investors therefore often try to invest a half-Kelly fraction, because the expected growth rate is still , but there is a comfortable margin for error before you enter the danger zone. For investments in ordinary securities or stock indices, the Kelly-fraction is most of the time something between one half and one, ruling out all kinds of leveraged strategies.
Finally, returning to our initial discussion of Long Term Capital Management, one may conclude from the analysis of their positions, that they were highly over-leveraged. As pointed out by Wilmott (2006b, p. 741), just their notional position in swaps was $1.25 trillion, which was 5% of the entire market volume at that time, providing a leverage ratio of more than 20:1. It has been estimated that they were committed to about double-Kelly.
Adjusting for Individual Market Views
(8.14)
(8.15)
(8.16)
By now it should be clear that managing a portfolio requires strategic decisions. Maybe the simplest decision of this kind is to choose a buy and hold strategy. This is by no means a dull strategy, because we saw earlier that there is no convincing evidence that managed funds are able to beat their reference index over an extended period of time. Last but not least, this was one of the favorite strategies of the Hungarian stock market wizard André Kostolany. The next more challenging strategy could be to modify the portfolio composition such that in times of bullish market perspectives, the portfolio beta is considerably larger than one, and in bearish periods the beta is smaller than one.
With this strategy you should on average participate stronger in market upturns than the index, but weaker in downturns. The problem in both strategies is to decide what the perspective of a single security, or the whole market, respectively, is. This is the point, where individual expectations and beliefs come in. As we shall see, there is a way to combine such individual views in an optimal way with the information the market has provided so far.
The first to consider individual views in the portfolio selection decision were Black and Litterman (1992). Their goal was to improve the estimator for the expected return of an arbitrary asset, provided by an equilibrium model like the CAPM, by incorporating information from individual investors’ views. As pointed out by Meucci (2010a), processing the return estimators results in two puzzles with respect to scenario analysis and completely uninformative views. To avoid these kinds of problems, we consider here what is called the market formulation of the Black–Litterman- approach by Meucci. This modification is more general and completely consistent.
Departing from a given distribution of an N-dimensional vector of market returns |R ∼ N(|μ , Σ), assume that an investor has individual views on the future outcome of some of these random returns where P is a K × N “pick”-matrix. The random error associates uncertainty with the views |V , and is assumed to be independently
distributed, where Ω is a K × K covariance matrix. A convenient choice for Ω is
with c representing an overall confidence parameter. For c → 0, the variance of grows to infinity, which renders the views completely uninformative, whereas c → ∞ expresses complete confidence, in which case we are in fact doing scenario analysis.
Quick calculation 8.4 Convince yourself that with (8.16), holds.
By now you should have two questions about what we have said so far. The first is, what is the pick-matrix exactly and how is it applied to express a particular view? The second is, how does this framework help us at all? So let’s answer the first question by an example.
Example 8.3