Wednesday, August 26, 2015

Negative skewness, or: bulls walk up stairs, bears jump out of windows

Recent market action is a good reminder of the asymmetry in markets. In general, stock market rises don't look like stock market declines. Stock indexes slowly eke out gains over a period of months, but lose all of those gains just a few days. There are plenty of famous meltdowns in stocks, including 1914, 1929, 1987, and 2008, but almost no famous "melt ups."

Just like the Inuit have multiple words for snow because they are surrounded by the stuff, equity commentators have many words for crashes (panics, selloff, etc). These events are not uncommon. In the same way that many indigenous African languages have no word for snow, we lack a good word to describe one or two day melt-ups in equity markets since these aren't part of our landscape.

There are a number of trader's adages that describe this pattern, including bulls walk up the stairs, bears jump out the window and variations on that theme. In the economic literature, this phenomenon is referred to as negative skewness. If you look at the distribution of daily percent returns for the S&P 500 Index over a long period of time, you'll notice that there are more extreme negative results than extreme positive results, with the majority of results being slightly positive. Whereas a normal distribution, or the bell shaped curve we've all seen in statistics class, is symmetrical with 95% of values dwelling within two standard deviations of the mean, a negatively skewed distribution has a fat left tail where declines extend far beyond what you would expect for a normally distributed data set.

The chart below illustrates this. Out of 22,013 trading days going back to 1928, just 47.8% of days resulted in negative outcomes while 52.2% resulted in positive outcomes. This makes sense given the generally upward trajectory of equity markets over that period. If we sort each day's return into buckets, we start to see asymmetries develop. For instance, there were 10,973 days on which markets moved higher or lower by by 0.5%, just 48.4% of which were lower. The majority of 0.5 to 1% and 1 to 2% changes were to the positive side as well. The distribution changes once we look at the 2% and over bucket. Out of 1485 days with "extreme" returns, the majority (51.9%) of changes were declines of 2% or greater rather than rises of +2% or greater.

Figure: Distribution of daily changes in the S&P 500 index going back to 1928

Financial economists have a number of hypothesis for negative skewness. One theory blames leverage, whereby a drop in a firm's equity price raises its leverage, or the amount of debt it uses to finance itself. This makes an investment in the company more risky and leads to higher volatility of its shares. Conversely, when a stock rises, its leverage decreases, making the shares less risky. For that reason, rises in equities are tame while falls are wild. While an attractive theory, data shows that as stock prices decline, all-equity financed companies experience jumps in volatility of the same magnitude as leveraged companies, indicating that leverage is not a good explanation for a pattern of negative skewness.

Another explanation is the existence of "volatility feedback." When important news arrives, this signals that market volatility has increased. If the news is good, investor jubilation will be partially offset by an increase in wariness over volatility, the final change in share price being smaller than it would otherwise have been. When the news is bad, disappointment will be reinforced by this wariness, amplifying the decline.

Other theories blame short sale constraints for the asymmetry. If bearish investors are restricted from expressing their pessimism, they will be forced to the sidelines and their information will not be fully incorporated into prices. When the bulls start to bail out of equities, the bearish group becomes the marginal buyer, at which point bearish information is finally "discovered" by the market, the result being large price declines.

Putting the reasons aside, behavioral finance types have some interesting things to say about how investors perceive skewness. According to prospect theory, investors are not perfectly rational decision makers. To begin with, returns are not appraised in a symmetrical manner; a 5% loss hurts investors more than a 5% gain feels good. Next, investors overweight unlikely events and underweight average ones. Given these two quirks, investors may prefer positively skewed assets (like government bonds), which have far fewer large declines than normally skewed assets, as this distribution reduces the potential for psychological damage. The possibility of large lottery-like returns, the odds of which investors overweight relative to the true odds of a positive payout, also drive preferences for positive skew assets. Negatively skewed assets like equity ETFs, which expose investors to tortuous drops while not offering much potential for large melt-ups, are to be avoided.

Put differently, positive skew is a feature that investors will pay to own. Negative skew is a "bad" and people need to be compensated for enduring it.

If you buy this theory, then in order to coax investors into holding negatively skewed assets like stocks, sellers need to offer buyers a higher expected return. The presence of this carrot could be one of the reasons why equities tend to outperform bonds over time. For equity owners who are suffering through the current downturn, here's the upshot: negative skew events like the current one, while stressful, may be the price you have to pay in order to harvest the superior returns provided by stocks over the long term.


  1. Hi JP,

    "...a drop in a firm's equity price raises its leverage.."

    How does a drop in firm's equity price raise its leverage.

    1. It comes from Fischer Black. Here I am quoting the "Legacy of Fischer Black":

      "When stock prices fall, the firm's debt-to-equity ration in market value terms tends to rise since its denominator falls faster than its numerator. If returns from assets remain the same, the increasing debt-to-equity ratio magnifies the influence of return from assets on stock returns, thereby increasing volatility. Thus, indirectly through automatic changes in the debt-to-equity ratio, a fall in stock prices causes an increase in stock volatility."

  2. Good post. Because I learned from it.

    The volatility theory strikes me as more plausible. I wonder if there's any way to test it?

    1. "Because I learned from it."

      Glad you liked it. I learnt plenty writing this post. I always knew about the distribution of equity returns from observation and personal experience, but was not aware that such a huge academic literature on skewness in financial markets existed. Thousands of man hours have probably been spent on this esoteric subject (and this blog post only touches on a tiny fraction of the material).

  3. "5% loss hurts investors more than a 5% gain feels good"

    I'd expect this to be amplified as the magnitude of the percentage increases. For example, a 100% loss (Oops! time to sell the wife and kids into slavery) probably hurts a LOT more than a 100% gain feels good (I will buy that slave family after all!).

    Perhaps a better way to look at it is as multipliers (and their reciprocals):

    Does a multiplier of 0.5 hurt about as much as a multiplier of 1/0.5 = 2 feels good?

  4. Good post on a very interesting question. I have no idea what the best answer is.

    I’m contemplating your point on leverage.

    I think it’s fair to make a distinction between the leverage of the firm and the leverage inherent in its stock price.

    A firm’s leverage is fundamentally a book value concept – along the lines of recognizing the debt/equity mix and the amount of debt interest that must be paid before book profit is netted out.

    The leverage inherent in the firm’s stock price essentially recognizes the stock as an option on the value of the firm’s assets at a strike price equal to the value of its debt. As the stock price moves toward zero, the optionality inherent in the stock gets much more volatile as the same underlying asset volatility becomes more dominant when measured against the lower intrinsic value of the equity as an option. This changing volatility effect is captured in by option gamma measures, I believe. I think a good example of this today is the case of the oil industry, where an argument can be made that most of the industry is bankrupt at a $ 40 dollar oil price and that equity values today measure the option value associated with the “risk” of higher oil prices. So those stocks are really volatile right now.

    A couple of points:

    I’m not sure how higher volatility at lower stock prices translates directly to a negative skew in that volatility. In fact, at sufficiently low stock price levels, one might expect a positive skew due to the existence of the zero lower bound on the stock price. I’m not dismissing the nature of the skew, but I’m not sure how this point alone would explain it.

    Regarding the effect of capital structure, suppose that a firm with a 50:50 debt equity mix is twice the size of an all equity firm, but both have the same asset mix. Would we expect a different stock price volatility or skewness result for these two firms?

    1. This comment has been removed by the author.

    2. - JKH

      In you example the volatility is a measure of a change in the asset value in proportion to the equity, and so smaller equity equals higher volatility. But in JP's example the share price moves without reference to the assets of the firm, and so then the volatilaty would not be effected by small equity , except from the effect that a very small share price is volatile due to any movement being significant..

      But as the equity is also affected by market sentiment, and the funds in the market, such as QE, rather than a change of the market value of the assets, the assets are the same value, the money originally raised from selling the shares is unchanged and the debt is unchanged. I don't see how a higher share price makes it easier for a firm to pay its debts, ie it doesn't change its indebtedness. So what is the precise definition of leverage.

    3. Good comment. I think you know more about this stuff than me.

      "I’m not sure how higher volatility at lower stock prices translates directly to a negative skew in that volatility."

      You may find it interesting that some studies find that while stock market indexes display negative skew, individual stocks tend to display either no or slightly positive skew.

      I recall reading a paper that mentioned that companies near bankruptcy display more positive skew than companies who are not.

      There is a huge amount of research being done on this subject, much of it is contradictory.

    4. Dinero,

      I’m essentially agreeing with you in my comment above and here.

      But my option based interpretation is also consistent with the Fischer Black quote provided by JP above. It is a market value of equity approach, framed in option terms (Black was an options guy among other things).

      My first point in my first comment above is that the normal context for the concept of firm leverage is measurement based on the book value of debt and equity. The higher the ratio of book debt to book equity, the greater the leverage, and the greater the relative burden of interest payments to be deducted in calculating earnings. That the essence of firm leverage, where one speaks of leverage ratios based on book values.

      And you are quite correct to say “I don't see how a higher share price makes it easier for a firm to pay its debts, i.e. it doesn't change its indebtedness.”

      That is correct insofar as the market value of equity has no direct effect on the cash flow available to the firm in order to pay interest and earn a return on equity. And it has no direct effect on the book value capital structure that is the basis for the calculation of the firms earnings after payment of interest.

      But the market value of equity may well reflect the risk in that cash flow expectation, so it is an indicator of risk emerging as a result of the book value capital structure.

      One can then use the market value of equity in the Black sense instead of book value, but that is not the normal expression of leverage. I’ve indicated how that market value approach can be translated to an option framework, where there is leverage inherent in the market value of the stock itself due to the embedded optionality implicit in the capital structure. That is consistent with the Black quote.

      But again, that market value of equity approach is not what firm capital structure leverage normally refers to.

      Think of the Black approach as a market value refinement of the standard book value concept of leverage.

      I think both interpretations are useful.

    5. Thanks for the explanation. Which analysis is the definition used for Bank Regulatory Capital.

    6. Regulatory capital is based on the book value of equity.

    7. For an interesting example of embedded optionality and volatility in oil stocks with oil at $ 40, check out the trading action in Crescent Point Energy over the last 10 days or so.

    8. How does that tally with the definition in the Basel 3 document ,

      "Definition of capital"

      page 12 half way down the index . - "common shares issued by the bank..."

      Here -

    9. the issuance of new common shares increases the book value of capital

      (as does retained earnings)

      new shares are issued at the market price, and the result is an increase in the aggregate book value of common equity capital. The new book value per share will depend on the difference between the previous book value per share and the market/book value per share of the shares just issued

    10. OK

      So is it right that the Basel definition of shares value is the book value of the equity , which is the assets, not including retained earnings, less the debt.

      And so the Basel definition of capital is the assets minus the debt , plus the retained earnings.

    11. There is no short cut to the precise definition of capital. Have a close look at your Basel document for the precise definition and qualifying components for capital.

      I've simplified in the extreme here for purposes of discussing the subject of the post (which is not the Basel rules for capital adequacy).

      Actual capital may include elements of common equity, preferred equity, and possibly certain other qualifying components depending on their design. I haven't kept up with the precise rules.

      Common equity includes paid in capital and retained earnings.

      All this stuff is valued at book value for purposes of capital calculations. The firm's capital is not the market value of its stock.

    12. I mean above "common equity includes paid in common equity (from shares originally issued) and retained earnings".

  5. Construction is a slow process, while destruction is fast. A city that took 1000 years to build can be destroyed in a moment. If you could buy shares of stock in that city, you'd see the stock slowly rising over 1000 years, and then crashing when the bombs start falling.

  6. Making up a 5% loss takes a little more than 5% in profit. The difference get's huge when you loose 50%, then you have to make 100% to get back to 0%. But I assume the measured assymetry is adjusted for something as basic as that?

    1. That fact could be enough in itself to give an equity chart a shape that appears as a sawtooth. If you sketch a chart that decreases by 10 percent repeatedly and then increases by 10 percent repeatedly you get a sawtooth shape.

  7. JP: I suddenly remembered that I had my own theory on why the news is skewed. I had never thought of applying it to stock markets.

  8. "Just like the Inuit have multiple words for snow because they are surrounded by the stuff,"

    But they DON'T have more words for snow!

  9. Does the negative skew compensation offset the liquidity premium of the price of stocks?

  10. you could say its just spiky selling