## Ahistorical Covariance

As anyone who’s studied finance even casually knows, the most celebrated principle of investing is diversification. In particular, purchasing multiple securities with low or negative *covariance* diminishes the risk an investor faces in owning a portfolio. Risk, in financial parlance, is operationalized as volatility, the variability of a security’s value. Covariance means the tendency for two securities to move simultaneously. Holding multiple securities of low covariance reduces risk, because if one security drops in value, low or negative covariance implies that it’s unlikely that all will drop in value simultaneously. The the risk — the variability in value — associated with a particular security is diluted in the portfolio by the likelihood that when one security falls, others are will rise or at least hold their value, damping any change in value of the portfolio as a whole. This is the principal of diversification.

In practice, investment managers construct portfolios by estimating the return prospects, the risk (volatility), and the covariances of various securities, and computing the portfolio that, based on these estimates, maximizes the ratio of risk to return. If managers were able to forecast expected return, variance, and covariances with 100% accuracy, rational investment would be a science, and there would be a clear “best” portfolio. To the degree that within a universe of securites, there are ample choices with low or negative covariances, investors would see great benefits to diversification. But unfortunately, future return, risk, and covariance can be forecast imperfectly at best. Portfolio managers understand this, especially with respect to return. Few professionals would rely on the naive forecasting strategy of guessing that this year’s return on some stock will simply be the same as last year’s. But variance and covariance are different. Historical values for the volatility and comovement of securities are in fact very frequently used as “best guesses” of future covariance. Investors are urged to hold portfolios that include stocks, bonds, commmodities, precious metals, currencies, and real estate, because historically these asset classes don’t typically move together.

But guessing future covariance from past covariance can be as hazardous as expecting a stock to do great this year because it did last year. Over the past while, there has been an odd spate of covariances. US stocks, real estate, precious metals, and oil have all done well. The US dollar has also done well, despite the historical negative covariance between USD and precious metal. Bonds have, until very recently, been flat. US wage perpetuities, the stream of future wages that represents most Americans’ core asset, are not directly priced. But wage growth, the value driver of this asset, has been flat, unusual in the face of rising stocks, commodities, and real estate.

In other words, covariances over the last year, viewed in terms of annual return, have been weird. There’s been a lot of simultaneous zigging where ordinarily a bit of zig and a bit of zag would have been expected. These are ahistorical covariances. Most investors haven’t minded, because nearly everything in their diverse portfolios stayed put or went up, rather than the usual won-some-lose-some scenario. It’s almost as if there’s some magnet or wind pulling normally independent spirits in the same general direction. Covariance is nice when it’s a general updraft.

But beware ahistorical covariance if the wind changes direction. The most professionally crafted portfolio won’t protect an investor from wild swings in value, if its holdings were based on historical covariance assumptions that suddenly fail to hold.

**Update History:**

- 23-Apr-2006, 12:45 a.m. EET: Changed “vaunted” to “celebrated”.

Ha, could one get alpha without paying with beta? :-) I’d assume all risk free instruments should return about the same.

On a different tangent: we discussed cost of shorting before. You indicated that the cost makes derivatives inefficient. I am curious on why that is the case. Selling a put is more or less buying (or promise to buy) the security. If hedging is needed one can buy call instead (collar trade).

April 13th, 2006 at 6:42 pm PDT

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HZ — Futures pricing is easier than options pricing. Consider the future price of a liquid stock, whose spot price today is $100. Let’s say the current 1 year risk free interest rate is 5%. What is the price of a future, that is, of a commitment to buy the stock 1 year from now?

Intuitively, one might think that it’s unknowable, that reasonable people can disagree. Who knows how much the stock will be worth in a year? But the intuition is wrong, at least in a world where shorting is cheap and some market participants can both lend and borrow at about the risk free rate. Given the current spot price of 100 and risk free interest rate of 5%, the price of a future is absolutely determined to be very near $105. Why should this be?

If nature hates a vacuum, then markets hate an arbitrage opportunity. Suppose the future price were $106. Then a market participant could do the following: 1) Borrow $100 at 5% interest; 2) Buy the stock today, and hold it; 3) Sell the future for $106 (to be paid in a year). There is no risk in this strategy. The arbitrageur purchases the stock today, so whatever happens to its price, he’ll have a share to sell in a year. What are the arbitrageur’s cash flows? Today he receives $100 loan and pays $100 for the share. Today’s cashflows net to 0. What about one year from now? The arbitrageur will sell his share for $106. He’ll pay off his $100 loan, plus $5 interest. So, he knows with 100% certainty that in one year, he’ll earn a profit of $1. Any profit without risk is very desirable. Given a futures price of 106, spot of 100, and 5% interest rate, arbitrageurs will rush to implement this strategy until they have bid the spot price of the stock up and/or the futures price down to its no-arb bound, spot + 5%.

There was no shorting in this scenario. But now consider the same stock, spot price $100, risk free rate 5%, future price $104. Can the futures price be

belowspot + risk-free-rate? Not if shorting is cheap. If shorting is cheap, arbitrageur shorts the stock today, lends the proceeds at risk-free-rate, and purchases the future. He thus commits to paying $104 in a year to cover his short position. But in a year, he has $105 cash, $100 from the short sale, $5 interest. So again, with no risk whatsoever, he realizes a $1 profit. Thus, for a liquid security, with no cash flows or cost of carry, in a world with zero transaction costs, free shorting, and the ability to borrow and lend at the risk free rate, we find the future price must be exactly $105.Cash flows (if the stock pays dividends, for example) alter the value in a deterministic way. The fact that in the real world every player faces some transaction costs, no one but the gov’t can borrow or lend at precisely the risk free rate, and shorting has costs means that the future price fluctuates within some band around the theoretical price of $105 where the cost of implementing this strategy exceeds its profit. But in practice this band is very small. If there were no players who could short at a low cost, the lower bound on security futures price would not be tight. Since it is in practice, someone somewhere must be able to short cheap.

Note that futures pricing is only deterministic when the thing being bought/sold in the future can be held or stored, and when the cost or benefit of holding the item is predictable. Electrical power future prices, for example, can’t be set so neatly, because you can’t store the juice in order to sell it a year from now. Oil futures are also not so deterministic, because oil storage has costs, and holding oil has unpredictable but exploitable benefits when there are spikes in demand. The simple story above works best when pricing futures of financial instruments – stocks, bonds, indexes, etc.

Options pricing is also based on no arbitrage bounds that assume cheap shorting for the bounds to hold. But the scenarios there are a bit more complex. If you really wanna know, I’ll hit the books to remind myself, then whip one up for you.

April 13th, 2006 at 7:25 pm PDT

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Steve,

As you said there are many pratical issues to gum up the ideal arbitrage. Options markets are often very illiquid. Tax treatments on interest earned and capital gain/loss are different. Big players use LEAPs or swaps that are not accessible to retail customers. Shorting can be done notionally.

April 14th, 2006 at 2:04 am PDT

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No disagreement. Given the imperfections of the real world, it’s remarkable that market prices are as close to theoretical no arbitrage prices as they are. As you suggest, the enforcment of no-arb pricing is basically free money for big players, who, for example, can borrow their customers shares without their customers having any idea or demanding any sort of interest.

No arb bounds should be self-enforcing by security design. But they’re not. I’m sure at all the major financial conglomerates, there’re computers busily comparing the spot and future prices, bond yields, and option pricing, and plucking off occasional mismatches. I bet it’s a pretty good living too, if your transaction costs are low and you can borrow securities for free.

April 14th, 2006 at 7:43 am PDT

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