Risk For The Mathematically Disinclined

John Blin, chairman and founder of Advanced Portfolio Technologies, explains what's in a covariance number.

Prices move. How much and how often is anyone's guess. That's "risk." Once the market closes, all we are left with is "return"-how much we made or lost. If we buy Microsoft and sell IBM, we expect Microsoft to outperform IBM. As we change our portfolio, we also help move prices. We create risk. Thus risk is born from our quest for better returns.

But how much risk? Statisticians suggest one answer.

Look at IBM and Microsoft over the last couple of years, compute their returns each week, and figure out their average weekly returns. Suppose the answer is 1/8 percent and 1/2 percent. The question is, in any given week, what's the chance IBM is actually up 1/8 percent or Microsoft down 1/2 percent? This is like asking, how "typical" is that number? In other words, how much does the weekly return fluctuate around the average? Statisticians call this the standard deviation. Option traders call it the volatility. Simply put, volatility tells us how far we might end up from the mean return-on average.

But that's only half the story. Volatility measures the average shortfall (or windfall) for that stock, irrespective of any other asset. Yet we don't trade in a vacuum-we trade out of IBM into Microsoft. Or into bonds, cash, yen, Daimler-Benz stock, and so on. Surely their volatilities must be related.

Our actions ensure that they are. No matter how we measure the risk of individual assets, the numbers we get are not completely independent. Simple volatility figures ignore this fact. They are myopic. If Exxon moves 25 percent up or down in any one year, what does it say about Amoco?

Statisticians propose a simple measure of "shared" risk-the covariance between two returns. In other words, how much an average move in Exxon mirrors an average move in Amoco.

But what about Exxon and Chevron? They also share some risk. Hence, so do Chevron and Amoco. Pull this thread and the price web starts unraveling: under the smooth fabric, prices and volatilities are indeed linked. We can't just stop with the first pair, Exxon and Amoco. We must look at all possible pairs at the same time. We must look at risk as a shared attribute.

What Stocks (Bonds, Currencies) Share?

As events unfold we review our portfolio, sell what looks weak and buy what looks promising. In the process, we help move all prices-and create risk and returns. Unwittingly, yet effectively, we enforce "arbitrage pricing." This is the message of the Arbitrage Pricing Theory (APT).

We, the investors, through our every trade, weave the cloth of asset prices. This arbitrage web links all returns. Pull each thread and you get the building blocks of risk. Reassemble them stock by stock into a portfolio, and you get a stunningly accurate measure of your exposure. But in practice, can this be done?

Mathematicians have an answer: pick any two assets and measure how much they move together and how often. This is their covariance. Do this for every pair and record the results in a "covariance matrix." The table quickly grows: for 30,000 stocks worldwide, it contains nearly 450 million different cells. Throw in bonds, currencies and commodities and you top the billion-cell mark. As assets multiply, the table grows factorially. But is there enough history, enough days or weeks, to get a good estimate of each cell?

No, but there's a way around this. Imagine looking up the air mileage between any two cities and storing the numbers in a table. Scanning across the row labeled "New York," under each column heading we find the distance from New York to the city at the top of that column. As we'd expect, flying from New York to London nonstop is shorter than stopping over in Paris on the way. The distances are not totally independent-they are related in some way.

So are the cells of the covariance matrix. Each cell measures how closely two assets shadow each other. As we reshuffle our portfolios, we push prices around and reinforce links across the matrix-in much the same way that simple geometry guarantees consistent distances between cities.

Mathematicians say that an APT-ruled covariance matrix has a "structure." The structure reflects the hopes and fears of investors-their collective wisdom or collective madness. The challenge is finding the key to the structure. That key, the APT signature of every instrument, is nothing short of the DNA risk structure. With it we unlock the elemental makeup, the building blocks of risk. The signature tells us which common genes lie under Exxon and Amoco, Microsoft and Intel, U.S. T-bonds and Commonwealth Edison-what makes up their volatilities.

Mapping the Risk DNA

Mathematicians have studied matrix structures-and covariance matrices in particular. In fact, matrix theory suggests a way to transform this matrix, in order to "boil down" the table, to a small number of basic constituents. There's no black, no data-mining tricks. In an APT-ruled world, covariance matrices effectively involve far fewer cells-mathematicians say they lie in a low-dimensional space. In much the same way we learn to factor a number into the product of smaller numbers, we can factor APT tables into smaller tables that convey their information compactly and accurately.

But let us not delude ourselves. This is not just a convenience, a clever way to store lots of data efficiently-it's first and foremost a necessity. As assets multiply, the table not only balloons into an unwieldy maze, it also becomes grossly inaccurate. We run out of days to get reliable estimates of each cell. But if we know that the cells are "linked" somehow, we no longer drown in noisy data. We can leverage each cell relative to all other cells to map out the whole latent structure.

The Risk Map

To map out risk, we reduce the table to its major dimensions (what mathematicians call its "eigensystem"). Covariances no longer come in disparate pairs: Exxon and Amoco, yen and Deutsche marks, U.S. T-bonds and Commonwealth Edison. Rather, they each boil down to a few key readings of the common risk factors that drive all assets. In the end, each asset reduces to its factor profile-its DNA structure. Less than two dozen numbers capture the volatility makeup of each asset.

Armed with these numbers, we win on three fronts. We get far better measures of each asset's volatility, drawing as we do on the behavior of not just that asset but all others as well. We can speak about disparate assets in a common language (compare how each asset's volatility derives from the same basic building blocks, mixed in different proportions). And we get to play God: we can clone any asset by mixing risk components in the "right" profile.

These components are the risks shared by all assets. A typical shared risk for a stock is its exposure to the stock market as a whole (its "beta") - the mere fact that it is a stock, not a bond or a commodity. Exposure to exchange rates, interest rates, economic activity and inflation are other examples of systematic risks.

But clearly not all risk is shared. No two assets ever move in lockstep. Each must retain some individual traits, some idiosyncrasies unique to that asset.

This raises a key question: In each asset's total volatility, how much is shared? Skipping over algebraic details, we can take each asset risk profile, parse out the shared and the unique components, and express each as a proportion of the total. The answers vary with different instruments. On average, well over half of the risk of large U.S. or foreign stocks is shared risk. In many cases the proportion is more than 70 percent. Put them together in a portfolio, a mutual fund, or an index, and you're left mostly with systematic behavior-stock pickers beware! Small stocks, on the other hand, are far more idiosyncratic.

Earnings surprises, boardroom coups, loss of patent infringement or product liability suits are typical stock-specific risks. Stock analysts are best at ferreting out stock-specific facts and trying to guess what they mean for a company.

Fixed-income instruments share a great deal with each other-and with some stocks (such as utilities). Currencies are even more systematic, sharing more than 80 percent of their total risk.

Whatever the instrument, there's both good news and bad. If growing shared risk weakens the odds for the stock picker, it can also simplify the job of risk managers and VAR analysts. Hardly a small gift from a seemingly arcane bit of mathematics.

John Blin can be reached at: johnb@aptltd.com.