The APT Approach

The APT System is a full, genuine and uncompromising implementation of the Arbitrage Pricing Theory. While the APT theorem (Ross, 1976) focuses on expected asset prices in an arbitraged market, one of its most powerful implications concerns price fluctuations across these assets - risk.

A Word about Portfolio Risk

Before we explore the theory, recall a few key facts about portfolio risk:

A portfolio return is simply the weighted average of the individual asset returns, using the portfolio holdings as weights.

Portfolio risk is a bit more involved: it's the weighted sum of the individual asset variances and covariances with all other assets, using as weights the squared portfolio weights. For easier interpretation, it's best to use the original return units, rather than the squared returns used to compute variance. Thus risk is usually reported as the square root of the variance, the "volatility" (i.e., the standard deviation) of portfolio returns. The risk relative to a benchmark is then the volatility of the "hedged portfolio", the combination of the original portfolio and a benchmark.

In short, to compute portfolio risk, you (i) collect return variances and covariances in a table - the "covariance matrix"; (ii) read in the portfolio holdings weights; and (iii) apply the portfolio risk formula - a function dubbed a quadratic form by mathematicians.

Why restate a forty-year-old bit of algebra?

Simply for this reason: whatever method we use to analyze portfolio risk, in the end it must be consistent with that matrix. Until its re-discovery by the Value-at-Risk architects (1994), this basic fact seemed all but forgotten by most popular risk analysis systems.

Estimating the Covariance Matrix

Looking forward, to estimate prospective portfolio risk, we need estimates of the variances and covariances. We need an estimate of the covariance matrix.

How can we get it?

One method is to treat each asset independently and use some simple volatility estimator - combinations thereof (Garman-Klass, 1980), or even GARCH estimators (Engle, 1982; Bollerslev, 1986). For assets with traded volatility markets, i.e., option markets, we could even take the market forecasts implied by option prices "implied volatility". Whatever we choose though, each asset is treated in isolation. The only data used in the estimate is that asset history, and none other.

Likewise for the covariance: for any stock pair, we can use the classical covariance between the two assets, for instance between Microsoft and IBM, ignoring all other stocks. In this case, the only data used is for those two stocks, and none other. Each pair is treated in isolation.

In short, these are local methods.

They are myopic, wasteful and inaccurate. Worst of all, there is no underlying theory. They are purely data-driven. And indeed their shortcomings do show: covariance matrices built in this way are notoriously wobbly, and often outright misleading.

Another method is to assume we know the "drivers" of volatility. If we pick some variables and specify the relation between each driver and each stock, we can estimate the contribution of that driver from historical data. For instance, if we say that Microsoft is a "growth" stock, and we come up with some definition of a "growth" stock, we can try to measure the "growth" component in Microsoft's performance. However, basic econometrics raises a formidable caveat: to trust the estimate, we have to assume the model is correctly specified. In plain English, if we use driver X when we should really be using Y, the estimates are no better than a wild guess. If we take other sample points, the figures change, often drastically. And so does the risk estimate. This type of error, the bane of econometrics, has a name: specification error. No amount of data mining will correct it. The data is not to blame - it's the lack of a theory. The only way out is to have a theory - here, an asset pricing model. Statistical analysis in a theory vacuum is not just worthless, it is dangerous: it suggests knowledge where there is only data fits. That lesson was learned twenty years ago with the spectacular failure of macro-econometric models.

Note also that even if we knew the specification to be correct, from some theory, we would still need to be able to measure the drivers correctly. Otherwise the estimates would also be flawed.

The final blow with this method is equally deadly: it actually ignores the covariance matrix. Whatever drivers we pick, we measure their contribution to each stock "in a vacuum" so to speak totally independently of the actual covariance with all other stocks. In fact, the covariance matrix never enters the picture. It is literally gone.

But there is no avoiding the math: portfolio risk is an explicit function of the covariance matrix. To estimate portfolio risk irrespective of the covariance matrix is folly. To ignore what defines portfolio risk is to guarantee disappointing estimates.

In all fairness, popular risk models based on this approach were built before Ross's discovery of the APT Theorem (1976). But the fatal flaws remain in all their subsequent variations. But today this need not be the case.

The APT Theorem and Risk

We are no longer stuck in a theory void. Since Ross's discovery there is in fact a remarkably strong theory: a theory based on the most fundamental model of asset pricing - arbitrage pricing. It is the same arbitrage pricing that has spawned the financial modeling breakthroughs of the derivatives era in the last 25 years.

The APT theorem establishes an equilibrium pricing relation between each asset's expected return and all others. The relation is embedded in the covariance matrix. Put another way, it is the very structure of the covariance matrix which enforces the arbitrage pricing. Specifically, the theorem shows that an asset's expected return beyond the risk-free rate will simply be the sum of its exposure to some shared sources of risk, weighted by the prices the market assigns to these risks - the risk premia.

What defines a risk in this equation isn't some specific variable(s) in the real world but rather its being shared. At different times, investors will focus on different asset features. They will often disagree on what does, or doesn't, matter. They will change their focus and their mind time and again. Capital markets are born of their discord. Different themes, old and new, will ebb and flow at different times. This means that we can't ever hope to observe these common features directly Nor should we try. Not at this stage, when we want to estimate the risk accurately - not attribute it yet. The theorem has nothing to say about it. It is a straight mathematical result relating each stock performance to uncorrelated portfolios of all stocks, each portfolio "mimicking" their independent contribution due to shared features. All the theorem says is that on average, the returns will obey the pricing relation. Without delving into complex mathematical arguments, this, in fact, is a statement about the structure of the covariance matrix. Formally it says that rather than being just any square symmetric matrix, it will have an effective rank far smaller than the number of assets being priced. It will lie in a lower-dimensional subspace than the asset space.

Representing matrices in different subspaces and different coordinate systems is an area with a long illustrious history in mathematics (e.g., eigenspace analysis, spectral decomposition, Eckart and Young, 1936). So, in principle, extracting that structure seems clear enough. But in practice, when the data are sample observations generated by some stochastic process, it is a major statistical challenge (Blin, 1997, Blin and Bender 1995).

One thing is certain, though: if we succeed we will have no specification error - by construction, since we simply apply the theorem. And we will be totally consistent with the covariance matrix - again by construction, since we will start with the covariance matrix.

About Estimation

The standard approach is to map the matrix in an eigenspace, for instance through principal component analysis (PCA). In fact, ten years before the discovery of the APT theorem, King (1966) had applied PCA to the Dow Jones Industrial Average Stock Universe. While the goal of the exercise was to try to uncover some broad common statistical factors rather than measure the risk of stock portfolios drawn from the DJIA universe, the exercise did illustrate the strong shared behavior driving their performance.

Little noticed, however, was the fact that there were far fewer stocks in the sample (30) than return observations. In general, the reverse situation is the rule: since we must use all the traded assets (stocks, bonds, commodities, currencies, etc.) to create the matrix, and since the returns frequency is limited by the non synchronicity of the prices, there are always far more assets than there are time periods to observe returns. The "concentration ratio" - the number of assets over the number of time observations - is extremely lopsided. Without going into the mathematical developments at this time, suffice it to say that the resulting matrix estimate is then hopelessly biased. So factorizing that matrix (for instance through PCA) will produce equally flawed results. Although little known outside the mathematical literature, this problem plagued the original attempt by Ross and Roll (1980) to apply APT to the U.S. stock universe.

We use a robust efficient and accurate algorithm to factorize very large asset matrices while avoiding the concentration ratio trap. Starting from three and a half years of weekly returns on all U.S. securities (over 10,000 items) for the U.S. model, or all World securities (over 40,000 items) for the World model, we produce unbiased estimates of the covariance matrix.

We factorize the matrix into twenty to twenty five components (depending upon the market). These components form an orthonormal basis - a right-angled, coordinate system measured in units of standard deviation. Mapping each security return in that space through robust regression produces: (i) the "systematic" portion of the stock return variance shared with the other stocks; and (ii) the "security-specific" return variance, which is wholly driven by the characteristics of that security.

The APT risk profile of each security (the "map") is simply a vector of twenty-odd numbers representing the coordinates of that security on each axis. The last figure represents the "residual" (security-specific) component that lies outside that space.

To measure total risk on any security we take the square root of the sum of the squares of these coefficients. Systematic risk is that portion which lies in that space, and specific risk is that portion which lies in the complement space.

What We Have Achieved?

At this point, we have indeed achieved what we set out to do:

Taking any portfolio of securities, we can measure its overall risk and its components:

1. Systematic Risk shared by all securities - to different degrees. Since it is shared and thus not diversifiable, this risk is shunned by portfolio managers as they strive to keep the systematic risk profile of their portfolio in line with that of their target index.

2. Security-/Company-Specific Risk. This is the risk that comes with the specific strategies, earnings surprises, legal entanglements and other idiosyncratic events that differentiate a firm from its competitors.

Portfolio risk follows naturally. First we take the weighted sum of the individual APT profiles (using the portfolio holdings as weights) to obtain the portfolio APT profile. Then treating this profile as a single (synthetic) security, we apply the same square-root-of the-sum-of-squares formula as for any other security.

With this information, the APT model can calculate a portfolio's tracking error to any target index. Extensive backtests and out-of sample simulations have demonstrated that the APT tracking error is the most accurate estimate available.

Risk Measurement vs. Risk Attribution

While we have achieved our initial goal of measuring portfolio risk and breaking it down into shared and specific components, these components are wholly statistical at this point. They are often called "statistical factors" as they arise from a factorization of the covariance matrix. Their meaning is that they span the covariance matrix best; simply put, they reflect the actual covariance between all stocks, irrespective of what real world variable may have led to such covariance at any one point. From the standpoint of the APT theorem, that is all that's needed. And that's what guarantees consistent accurate risk estimates.

But from the standpoint of testing whether the portfolio is exposed to long rate spikes, yield curve shifts, credit squeezes, drops in the dollar, resurgence of "value" stocks etc. we have one more step to complete. We need to map any of these variables, or even user-proprietary variables, in the same space as the APT factor space, to analyze their own variance and relate it to the portfolio variance. In short, we need to link our estimates to the variable of interest to the manager.

To do so, APT has developed a unique risk attribution technique that decomposes a portfolio's risk or tracking error into factors based on actual observable data.

Note a key point in this exercise: we are not looking to alter our basic risk estimates. These are robust, covariance consistent, theory driven and above all accurate. We are simply looking to apportion whatever fraction of these estimates can be explained by the variables (the "factors") we choose. This distinction is essential. Popular risk models confuse risk measurement and risk attribution. As we have demonstrated, and simple algebra illustrates, risk measurement comes first and is completely independent from risk attribution. Mixing the two, or reversing the order, will produce bad attribution and bad risk measures.

By designing an open system, allowing users great latitude to pick the set of risk factors they deem most suitable for a given purpose, we allow them to explore alternative "factor decompositions" of their portfolio, while not subjecting them to the vagaries of different risk figures as the specification changes. In fact, we explicitly show the gap between a given specification and the true portfolio risk - effectively measuring the specification error.

Contact APT for a presentation, demonstration and example risk report for your fund or book.

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