Monday, April 16, 2012

Price and volatility in the great crash

I made the following cool graph last week. (Well, I think it's cool):

The black line is the VIX volatility index. You can think of it as a market forecast of volatility over the next month. It starts at about 25, reflecting a 25% per year standard deviation of stock returns. In the financial crisis, it shoots up to 80%. Yes, 80% annualized standard deviation. Then it tails back again with another jump in early 2010. (The VIX tracks realized volatility almost perfectly in this episode, so it's not about volatility risk premiums.)

The blue line is the cumulated return on the Fama-French total stock market index. (Data from Ken French's website.)  If you had a dollar in the market in January 2008, it shows you the percent gain or loss through time. (The level of the S&P500 shows almost exactly the same pattern.) That's also pretty dramatic: you lose half your money by March 2009, before the market recovers.

The negative correlation between these two lines is striking. Yes, we've known for a long time that lower prices are associated with higher volatility, but it's not often that you see such a striking correlation.

What do we make of it? It seems to revive the idea that mean returns and volatility are related. If volatility goes up, then mean returns must go up too, so that the average investor keeps holding the market portfolio. The only way for mean returns to go up is for the price to decline. You can almost blame the fall in prices on the rise in volatility.

The challenge is to get the numbers to add up.  The standard portfolio allocation rule says

share in risky assets = 1/(risk aversion) x mean return / variance of return

Variance is volatility squared, so if volatility goes from 20 to 80, the denominator rises by a factor of 80^2/20^2= 16!  If you have all your money in stocks, share = 1, we need the mean also to rise by a factor of 16; say from 6% to nearly 100%. Did  participants expect the entire 50% stock market decline to be reversed in 6 months? I've written about time-varying expected returns, but even for me a market risk premium of +100% seems like a lot.

I think the answer is, this is the wrong equation. It's time to get serious about Merton portfolio theory for long-lived investors. The real (Merton) portfolio theory adds to the last equation

... + (aversion to volatility risk) x (covariance of return with changes in volatility)

So, something about this term must be screaming "get in" to counteract the last equation's advice to "get out."  Why do people care about volatility risk? ("aversion") Why does this term vary strongly over time?

Something in this event seems to be crying to explain "state variable risk" in an intuitive way, but doing so is just out of my reach. (I've been puzzling about this for a while, see p. 1082 of "Discount Rates". )

Of course, volatility is not the ultimate "state variable," and understanding the movement of stock prices will eventually means we need to dig deeper to underlying events. 

Context: I was discussing two nice papers at the NBER asset pricing meetings:  "Volatility, the Macroeconomy and Asset Prices,  by  Ravi Bansal, Dana Kiku, Ivan Shaliastovich, and Amir Yaron, and  "An Intertemporal CAPM with Stochastic Volatility" by John Y. Campbell, Stefano Giglio, Christopher Polk, and Robert Turley.  Both papers explore  time-varying volatility and attempt to answer this puzzle. (Google for latest versions of the papers.)

Cambpbell et. al. also argue that the value effect (higher returns for value stocks than growth stocks) is explained by value stock's tendency to move with changes in volatility.  That's why I included the value stock cumulative return in the graph.


Pedro Santa-Clara sent this graph:

It shows a nice correlation between the VIX and the earnings/price ratio. In turn, the earnings/price ratio is one of the best return forecsaters. So, conditional mean and conditional variance of returns do move together.

That's nice, as it sometimes seems that volatility and mean return wander off in different directions, in response to different state variables, and at different frequencies. A united view of the two moments is essential.

But don't get too exited. The graph certainly does not document a constant Sharpe ratio, or even a constant mean to variance ratio. The earnings yield corresponds roughly 1 to 1 ("roughly" means between 1 to 1 and 1 to 3) with one-year expected returns, so you're seeing expected returns vary roughly from 4 to 7 percent. The VIX is moving orders of maginitude more. So one-year Sharpe ratios and mean/variance ratios are still moving a lot over time! But perhaps we can coalesce the state variables somewhat, and find common factors in conditional mean and variance. 


  1. I agree with the general thrust of this post, but have quibbles with some other parts.
    1) When you show that graph in the future, you might want to put the VIX on the right hand axis. And for whatever reason, it looks like Rm and HML factors don't both start at 0.
    2) Garch-in-mean models don't particularly do a good better job in explaining asset returns. Given these charts, one would think they would work very well.
    3) Another way to think about it is like a term structure. Implied volatility has a term structure such that volatility tends to revert to the mean when it is elevated. Applying the portfolio formula to this would suggest that the returns have a term structure as well. Those who had longer horizons may not have changed their expected return forecasts, but those with short horizons may have been expecting a strong rebound.
    4) A more general approach (though not as well suited to a blog post) is a Black-Litterman reverse optimization. I would guess that market share of equity significantly declined over this period, which would have some offsetting effect to your calculations.

  2. Regarding the static portfolio allocation rule, doesn't the LHS move too? That is, the market value of risky assets will fall, so the share of risky assets will fall as well. Wouldn't that help counteract the RHS volatility effect?

    Thanks for the blog!!

  3. How does this model look in an ECM ?

    Volatility appears to have a linear relationship with asset prices. Something like ln(MV) = ln(X) - (mean_duration)*(.06/.2)ln(VIX). Shocks in volatility should be reflected in changes in asset prices.

    If we decompose the expected return of an asset to: vol*risk_premium_per_unit_of_volatility*beta, we should see a linear relationship between levels of VIX and expected market returns. Something like dln(MV) - dln(X) = (.06/.2)ln(VIX)

  4. First, VIX is a relatively short-term (30 days) proxy for the market risk. Yes, it peaked at 80, but it quickly went down. If the modern portfolio theory is correct, portfolio managers base their holdings on E( future variance of returns) for the duration of the holding period, which is different for different investors - from seconds to months to a year or so. Most likely the majority of big market players expected volatility to subside within a few months, and they were right.

    This can be easily modeled: E(future variance of returns) = a VIX^2 + (1-a) G^2, where a is the ratio of VIX horizon (30 days) to the participant's horison, and G^2 is long-range expectation of variance.

    Next, the "risk aversion" parameter is not constant. If "mean return" and "variance of return" are expressed in dollars, then risk aversion is in dollars, too, so it should depend on the dollar value of one's assets.

    Then markets participants are heterogeneous. Risk aversion is different for different investors. What if pension funds move out (which they did) and short-term institutions that prefer (not necessarily can afford) more risk – move in?

  5. Crisis equals opportunity, especially for those with cast-iron stomachs and no problems with insomnia.

  6. "Yes, we've known for a long time that lower prices are associated with higher volatility, but it's not often that you see such a striking correlation. "

    True, but this is not that universal as many think. Volatility clustering eg. in shipping markets tell the opposite. In tight markets, markets become an auction and freight rates move a lot (high volatility) while in depressed markets with overcapacity, freight rates move almost nothing (= homogeneous cost structure fleet)
    Am sure you all know this (fleet need time to be expanded by newbuildings), but somehow it is an interesting thought to compare...


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