THE TIME-SERIES VARIANCE-BOUND FALLACY

In widely cited research into the question of bubbles and irrational behavior in financial markets, Robert Shiller (1981) has suggested that stock market prices clearly exhibit excess volatility compared to their fundamentals. He has recently updated his research up to 2000 in his book Irrational Exuberance. The results are similar to his 1981 findings and are used to discredit the 1990s boom in U.S. and worldwide stock prices.

Whatever the merits of arguments in favor or against the 1990s boom in technology stocks, the variance-bound test as implemented by Shiller and many others is a fallacy.

Shiller's variance-bound model

Shiller’s model is simple. The mathematical equations are as follows

Economic theory suggests that the current rational perfect-foresight price P_t* (i.e. the entire future be known) equals the discounted present value of all future realised dividend payments D_t+k using the discount rate r (equation 1). The current rational market price P_t should equal the market’s expected value of P_t* (equation 2). Using the textbook rational-expectations hypothesis (perfect-foresight-with-error), the actual ex post P_t* can deviate from its rationally expected value only by a random forecast error u_t (equation 3). In this model, the theory of rational stock market prices implies that the variance of actual market prices P is smaller or equal to the variance of the perfect-foresight price P* (equation 5). This key insight is the so-called variance-bound test.

Shiller (1981, 2000) and others obtain the time series data on actual dividends and (using some additional assumptions) calculate values for ex post P*. Comparing P and P* they claim that the data show that in reality var(P) > var(P*). In other words, stock prices exhibit excess volatility; stock market prices defy rational explanations; stock market bubbles exist.

The fallacy: time-series versus cross-sectional variance

Shiller’s conclusion, followed by many economists, is based on a rather silly misunderstanding over what is actually measured by his variances and what is really relevant for the behavior of stock market prices.

Look at the following hypothetical stock (see Figure 2). Assume that a company has a single project that promises a single payoff at some fixed point in the future. The expected value of this payoff or dividend at time t+k is E_tD_t+k. For this expected payoff the rational price equals P_t = E_tD_t+k / (1+r)^k. Because no dividends are paid in the intermediate periods, the total return on this stock will be created by expected capital gains at a rate P_t+i / P_t+i-1 = (1+r). At subsequent moments in time, for example t+1, expected dividends may have changed due to new information about the success or failure of the project. The corresponding rational price equals P_t+1 = E_t+1D_t+k / (1+r)^k-1.

At time t+k the actual dividend is realised. Without implications we may assume it equals its originally expected value D_t+k = E_tD_t+k. Calculating Shiller’s “rational” price P* produces a perfectly smooth line that connects P_t and D_t+k. We know and observe that the actual market price fluctuated due to new information about the future dividend. In the usual application, Shiller’s variance bound is decisively rejected. However, because we have constructed this example on the basis of rational market prices, the variance-bound test as applied by Shiller and others must be wrong!

The trouble and fallacy, as shown by Kleidon (1986), is that Shiller’s time-series variance bound test is simply inappropriate. Economic theory suggests that the correct variance bound test must be applied to dividends and prices in a cross-sectional variance bound test (Figure 3). Var(P_t*) should refer to the variance of P* prices linked to the expected probability distribution of future dividends as expected at every time t. Market price P_t is the expected value of the probability distribution of P* at time t. Because P_t is simply a single observation, var(P_t) at time t is zero. At each time t+j we will find var(P_t*) >= var(P_t). In other words: the time-series variance bound test var(P_t* | P_t-1*, P_t-2*, …) >= var(P_t | P_t-1, P_t-2, …) is simply wrong and does not correspond to the appropriate cross-sectional variance bound var(P_t* | I_t) >= var(P_t | I_t) where I_t represents the information available to investors at time t.

Conclusion

It is time to abandon the (time-series) variance-bound fallacy.

Note: In empirical research many economists have assumed that Kleidon's (1986) critique was concerned with time-series properties of prices and dividends (nonstationarity), and they have attempted to adapt their tests. However, this perception is not correct and their adjusted tests are equally flawed. The cross-sectional variance bound critique is of a much more fundamental nature.

Akdeniz et al (2007) revisit the Kleidon argument and show, using a simulation study, that in a general equilibrium model with stationary variables the unconditional (time-series) variance-bound test still provides misleading results. Their conclusion: "Consequently, we show that one cannot infer any conclusions about market efficiency from the unconditional variance bounds tests."

Literature

• Shiller, R.J., “Do stock prices move too much to be justified by subsequent movements in dividends?” American Economic Review, vol.71 (3) 1981: 421-36.
• Shiller, R.J., Irrational Exuberance. Princeton Univ. Press, 2000.
• Kleidon, A.W., “Variance bounds tests and stock price valuation models,” Journal of Political Economy, vol.94 (5) October 1986: 953-1001.
• Akdeniz, L., A.A. Salih and S.T. Ok, "Are stock prices too volatile to be justified by the dividend discount model?", Physica A, vol.376 March 2007: 433-44.

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