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January 29, 2007

Bayes' Rule

John P. Hussman, Ph.D.
All rights reserved and actively enforced.

In the financial markets, investors spend a great deal of effort trying to determine whether stocks are in a bull market or a bear market, whether interest rates are headed higher or lower, and so forth. Unfortunately, these are “hidden variables.” We can't observe them directly, except in hindsight. At best, we can try to figure out the probability of those things that we can't observe, given evidence about things we do observe.

We might be interested, for example, in estimating the probability that stocks are about to enter a bear market, given some amount of observable evidence.

That seems like a difficult question to pin down, but it's easier if we work backwards. Specifically, can look back historically and ask the reverse question: When stocks were about to enter a bear market, what was the probability of observing conditions similar to the present?

Suppose that the unobservable thing, “X,” is whether or not the market will enter a bear market over the next 13 weeks, and the observable evidence is “E.” We can go back historically and count, when X was true (in hindsight), how often we observed E. We can also count, when X was not true (~X), how often we observed E.

We can then apply a calculation called “Bayes' Rule:”

Probability of X, given the Evidence = (Cases where we observe both X and the Evidence) / (All cases where we observe the Evidence)

The geeky math is this. If we write “the probability of X given E” as P(X|E), Bayes' Rule is:
P(X|E) = P(E|X)P(X) / [P(E|X)P(X) + P(E|~X)P(~X)]

A few weeks ago, I noted that historically, we've observed relatively few instances when the S&P 500 traded at a 4-year high, at over 18 times record earnings, with advisory bullishness over 53% and with Treasury bills rising above their level of 6 months earlier. (In addition to the instances noted before, these conditions were also approximately met in December 1961, just prior to a sharp market decline in early 1962).

Of the many abrupt market declines we've observed since 1960, say, where the market dropped by 10% over a period of several weeks, only about half of those instances were preceded by the above conditions. But the outcomes have been unanimously unfavorable when these conditions have been observed.

Mathematically, if X is “bad things happen in the stock market” and “E” is the set of evidence above, P(E|X) has only been about 50%, but P(E|~X) has historically been close to zero, so P(X|E) is about 100%.

Bayes' Rule currently puts the likelihood of a 5% or deeper market correction beginning within the next few weeks at near certainty, the probability of a 10% correction starting during the next 13 weeks at about 65%, and the probability of a bear market beginning within the next 6 months at over 75%. There is, of course, no such thing as certainty in the financial markets. Suffice it to say that the probabilities aren't good.

Accordingly, our investment position is defensive – not based on a specific forecast in this instance, but based on what the evidence suggests as the average outcome from prevailing conditions. Meanwhile, given our defensive investment position and the potential to be frustrated by short-term market fluctuations, it may be helpful to recognize that despite occasional slight new highs, the market has not been running away during the past 8-10 weeks. Most of the excitement during this period has been produced by repeatedly backing away from prior highs and then advancing toward them again.

For example, on November 17 (the first instance we observed the set of evidence above), the S&P 500 traded at 1401.20. It is currently at 1422.18, less than 1.5% above that level. At no time since November 17 has the S&P 500 moved much more than 1.5% over that close on a sustained 5-day basis, nor has it moved more than 2.5% above its October high on a sustained 10-day basis. The trough-to-peak moves create a lot of excitement, but the subsequent peak-to-trough movements combine to produce far smaller net gains.

Importantly, our investment position does not rely on a market decline. A fully hedged position implies only that, on average, the market has historically underperformed Treasury bill yields (currently about 5% annualized) in conditions similar to the present. History indicates that we should not rule out substantial market losses here, but neither should we ever rule out the possibility of further gains (which is why we don't establish hedges in excess of our long stock positions). But as investors who believe that markets tend to experience both advances and declines, we look at market movements in the context of a complete market cycle. The case for the market gaining substantial ground and actually retaining it through the remainder of this market cycle looks very thin, in my view.

In short, our fully hedged position is not based on the expectation of large market losses. Conditions producing market returns below Treasury bill yields, on average, are enough to warrant a hedged position. The tendency for large losses to emerge from the rare set of current conditions only makes us more comfortable with that hedged position. But we would currently have it anyway, based on less extreme and more common criteria.

Fun with Bayes' Rule: Birds and Hamsters

Suppose that you've got a truckload of animals in boxes, but you can't actually see the animals. If the animal is a hamster, there's an 80% chance the box is green. If it's a bird, there's only a 20% chance the box is green. You're given a green box. How likely is it that the animal inside is a hamster?

Though it's tempting to answer that a hamster is most likely, we need more information. We have to know what proportion of the animals are hamsters.

Suppose that only 10% are hamsters. Then given it's a hamster (10%) there's an 80% chance the box is green. So 10% x 80% = 8% of the time we'll have a hamster and a green box. However, we'll also have a green box in 20% of the cases where the animal is a bird (90% of the animals). So 20% x 90% = 18% of the time we'll have a bird and a green box.

Bayes Rule says that the probability it's a hamster when you're handed a green box is:

8% / (8% + 18%) = 31%.

Most likely, there's a bird in the green box. Notice that choosing a green box increases the probability it's a hamster (which you would expect just 10% of the time if you had no information about the box), but it's not enough to make a hamster the most likely expectation.

Proof beyond a reasonable doubt

Suppose that you're a juror deliberating a guilty/not-guilty verdict. You have evidence that's overwhelmingly consistent with the idea that the accused person committed the crime. Suppose you can even say that IF the accused committed the crime, there's 100% chance that you would actually have observed the evidence that was produced. Should you convict?

Not so fast. You've got more questions to ask. First, you need to ask whether the evidence is consistent with a different story: IF the accused didn't commit the crime, how likely is it that you still would have observed the evidence? Let's say that if somebody else committed the crime, you think there's still a 4-in-10 chance you would have observed the evidence that was produced.

Second, if there's a presumption of innocence, you have to begin, before weighing the evidence, with a probability of no more than 50% that the accused is guilty. Let's say 30%.

Finally, if you require proof beyond a reasonable doubt, the final probability of guilt, after weighing the evidence, should be very high, say 80% or more.

In this case, P(Guilty|Evidence) = P(E|G)P(G) / [P(E|G)P(G) + P(E|~G)P(~G)]

= 1.00 x .30 / (1.00 x .30 + .40 x .70) = 51.7%

That's not proof beyond a reasonable doubt. So even though the evidence is consistent with the assumption that the accused is guilty, the consistency of that evidence with alternative explanations, plus the burden of proof beyond a reasonable doubt, should lead you to a not-guilty verdict. This is true even though you believe it's 51.7% likely that the accused is guilty. You don't have a preponderance of evidence. In order to deliver a guilty verdict, the evidence cannot be reasonably consistent with any other assumption except guilt.

Weighing bullish and bearish arguments

Essentially, Bayes' Rule tells us that when we evaluate evidence, we should pay special attention to those pieces of evidence that are largely consistent with only one of the possible conclusions.

An argument that “we expect earnings to grow, so the market should continue rising” is simply not sound Bayesian reasoning, because it turns out that there are many instances when earnings have grown despite a plunging market. Historically, the probability that stocks are in a bull market is unaffected by whether earnings have grown or declined over the prior 6 or 12 months, and is also unaffected by whether earnings have grown or declined over the following 6 or 12 months. There's just no relationship between short-term earnings direction and market direction.

Likewise, to say that “stocks are overvalued, so the market is likely to decline” is also simplistic, because it turns out that there are many historical instances when overvalued markets were accompanied by further market advances, particularly when the internal quality of market action was broadly favorable, and conditions had not yet reached overbought, overbullish extremes.

Of course, some arguments are so vapid that you instantly curl up in a ball from the mental anguish. A few days ago, an analyst on CNBC said that the recent uptick in various inflation measures didn't change his bullish outlook, because “The Fed always cuts rates after a tightening cycle.” Well, you can't really argue with a statement that's true by definition.

Presently, the constellation of evidence is not one that has historically produced strong market returns, but to the contrary, has always (quite literally) produced negative outcomes. Still, nothing in the markets is certain, so as usual, it will suffice to say that current conditions have historically not delivered a favorable return/risk profile, on average.

Market Climate

As of last week, the Market Climate for stocks remained characterized by unfavorable valuations, moderately favorable market action, and overbought, overbullish features that have historically combined with rich valuations to produce unsatisfactory market outcomes, on average.

The Strategic Growth Fund is fully hedged against the impact of market fluctuations, holding long-put / short-call combinations against our diversified portfolio of individual stocks. As always, no more than one of those options is “in-the-money” at the time the position is established. At present, option premiums are very inexpensive due to unusually low implied volatility, so our strike prices are staggered in a way that gives us a reasonable amount of local “gamma.” On balance, this works out to imply a modest reduction in the "implied interest" earned on our hedge, in return for a stronger defense.

Again, in any event, our investment position does not rely on expectations of a significant market decline, nor do we need to rule out a further advance.

In bonds, the Market Climate remains characterized by unfavorable valuations and increasingly unfavorable market action as well. The 10-year Treasury yield broke beyond the highs of its 6-month range last week. While it's always possible that this will mark a short-term high in that yield, there is not much to compel enthusiasm in the present combination of still-insufficient yield levels, continuing inflation pressures, and muted credit spreads. The Strategic Total Return Fund continues to carry a duration of about 2 years, mostly in TIPS.

The Fund also holds about 20% of assets in precious metals shares, based on a combination of factors that has historically been quite favorable for this group. Why precious metals? In the Strategic Total Return Fund, these shares periodically represent the best alternative for inflation hedging as well as implied currency exposure. Though the Fund's allocation is limited to a maximum of 30%, these positions have contributed substantially to the overall return of the Fund since its inception. They do induce some amount of day-to-day volatility in the Fund, but they are often very useful to the Fund's objectives of achieving total return and preserving long-term purchasing power. Presently, the Market Climate for this group continues to be favorable on our measures.

Generally speaking, steep yield curves tend to benefit long bond durations, while inversions tend to favor short durations. That's particularly true when credit spreads are narrow and there's no apparent “flight to quality” toward Treasuries. Meanwhile, true to form, the extremely positive readings from the “Fed Model” in October and November turned out to be bad for bonds and fairly neutral for stocks. As a rule, I've found that “buy signals” from the Fed Model when stock yields are already quite low (meaning that Treasury yields are even lower) tend to be fairly uninformative for stocks, but are generally quite good sell signals for Treasury bonds.

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