Excess volatility project
Definition of volatility: http://en.wikipedia.org/wiki/Coefficient_of_variation
Original motivation: confirm folk wisdom that prediction markets have less excess volatility (less prone to speculative bubbles) than the stock market
Conjectures: 1. Sports markets won’t exhibit excess volatility 2. Options prices won’t exhibit excess volatility —too obvious to publish? (Preston says no, he’s not even certain it’s true.) 3. The lack of short/known time horizons for payoffs is a necessary and sufficient condition for excess volatility
Shiller (JEP’03) seems to assume that all cash is eventually paid out in dividends: is this reasonable? If dividend payouts do eventually happen, but take many years, then we should expect the measured excess volatility to be less in earlier time periods. In other words, the measured excess vol may be less in earlier periods, because by then we really have observed the true dividend stream well into the future. In later periods, the true complete future dividend stream upon which the present value calculation is based still remains partially hidden. Eyeballing Fig 1 in Shiller JEP’03, this does seem to be true. Another factor exacerbating this problem: the authors assume dividends after 2002 are contsant off to infinity, of course reducing the volatility of present values near the end of the time period.
Theoretical question: Suppose you know a forecaster is more volatile than the variable he is forecasting, but you are otherwise completely uninformed. What is the optimal way to arbitrage the forecaster?
An alternate test for (or definition of) excess volatility: a contrarian strategy must make money: buy when price is below moving avg, sell when price is above moving avg. The question is whether this strategy’s risk/return dominates “buy & hold S&P”. Almost by definition if contrarian doesn’t dominate buy and hold, then there is no “excess” volatility. Sharad and Preston argue that it still might be possible for a moving avg to be a better predictor of price but the risk of the strategy is higher. Dave doesn’t undertand this argument.
The apparent “excess” volatility may be due to uncertainty in dividend streams: when a company is young, it could go in many directions and its future dividend stream could be very uncertain, however once it does start producing dividends they are often very uniform. So observed volatility of dividends is much less than actual uncertainty about dividends at the time of prediction. Ie, ex ante volatility is high, ex post is low. Preston thinks this explanation may be hard to reconcile with the Shiller data, which is for the whole market.
Q: Could we come up with a better predictive model of dividends using Yahoo! data or other means? One company’s dividend announcement affects others.
One idea for paper: posit a standard noise trader model and examine whether short term markets (like prediction markets) exhibit less excess volatility than long term markets (like stock markets). Alternative: run laboratory experiments comparing short and long term markets. Charlie Plott was able to systematically produce speculative bubbles in the lab.
Conjecture: 1. Contrarian strategy makes money (consistent with excess volatility). 2. Contrarian strategy does not dominate equivalent risk investments.
In case it’s useful, the USD-EURO forex market is probably one of the thickest in the world and has wildly high volatility.
There are some interesting tracking indices whose behavior Dave doesn’t quite understand. SDS tracks twice the inverse of the S&P500 (if S&P goes up 1%, SDS goes down 2%) and SSO tracks twice the S&P500. Note their ADDITIVE nature! s(1+e) -> t(1-2e). You might expect multiplicative: s*d -> t/2d. The fact that they are additive implies some strange things. For example, if the S&P is up more than 50%, SDS should be down more than 100%, which is undefined/impossible. Also it seems that SDS will inexorably go to zero. It seems that every time the two indices SDS and SSO cross, the crossing point is lower than before (conjecture: this is provable). So if there is any excess volatility (reversion to the mean), both indices will trend down. Dave has a Mathematica file exploring these.
TODO List: 1. Create a live version of Figure 1 in Shiller JEP’03. Updated data is here: http://www.econ.yale.edu/~shiller/data/ie_data.xls
Scratch:
http://www.bloomberg.com/apps/news?pid=20601213&sid=aEZlqEfkKhbk&refer=home Feb 23 2009: “Ultimately, what you get out of investing in stocks is the cash flow from dividends,” said Laurence Booth, finance professor at University of Toronto… The discounted dividends model “which values a stock as the sum of all its future dividends, shows equities are still overpriced. With S&P 500 companies projected to pay a combined $25.27 in dividends this year, the index would need to fall to 526.46 before investors are compensated for owning shares.” I think that any cash reserves of a company should be treated as if they will be eventually paid as dividends.
Preston wrote in email:
Cash and profits are supposed to be invested in things at least as valuable (to the shareholders) as cash paid to shareholders. Failure to follow that prescription is a violation of management’s fiduciary responsibility to the shareholders, who own the company. So there is a legal requirement that all the profits eventually go to the shareholders. Moreover, if you have $1B, and management can invest it or pay it to shareholders, the expected value of the investments had better exceed $1B in present value or mgmt has violated their legal responsibility.
So all cash is eventually paid to shareholders. There is uncertainty in how much cash there is, of course.
I don’t see why excess volatility should be less in earlier periods. In the life-cycle of the firm, the uncertainty about future profitability is greater for young firms; the effect of news should be larger.
A noisy forecaster doesn’t induce excess volatility in a rationally formed estimate. If everything is normally distributed, the variance enters the denominator of the estimate. That is, if our prior has variance S1 and the forecaster has variance S2, and e1 and e2 are the prior and forecaster’s estimates of value, the Bayes update is
(e1/S1 + e2/S2)/ (1/S1 + 1/S2)
And this has a lower variance than either estimate. This is a general property of Bayes updates.
A strategy to “proving” theoretically that prediction markets have less excess volatility than equity markets is to first hypothesize the reason why there is excess volatility. We should just take an explanation from the literature (e.g. noise traders), but then ask whether there is less excess volatility in prediction markets using the same set of agents.