# Rate of convergence between price and value

In my experience, there are two primary methods of alpha generation. In both cases, assume we know what price is.

Method 1: Inference on what the price/payoff will be.

Method 2: Inference on what the underlying (“intrinsic”) value is (I.e., what the price/payoff should be).

Generally, method 1 regresses variables (e.g., factors and/or anomalies) to infer what the price/payoff will be (and/or what the returns will be) within a given time frame. Under method 1, the speed and likelihood of convergence is implied.

Method 2 may be broadly referred to as "valuation". Canonically, it is underpinned by a conviction that price and value will at some point converge (read margin of safety). As such, value investors are primarly concerned with real world probabilities (since, in the risk-neutral world, $P_t \equiv \mathbb{E}\left[V_T\right]$ for an underlying asset). For example, in the equities world, analysts utilize various measures for net present value to infer what the price should be. Such methods include discounted cash flow analyses, precedent transactions, comps (i.e., peer group benchmarking), etcetera. However, valuation only tells us what the price should be, but nothing about likelihood or rate of the price-value convergence. Even if we knew with absolute certainty that price would converge to value, this says nothing about when it will converge or how.

For example, let's say we have an instrument which continuously pays $X_\tau$ over the interval $(t,\infty] \, \forall \, \tau \in T$. The NPV can then be expressed as such:

$$\mathbb{E}\left[V_t \right] =\int_{t}^\infty m(\tau)X_\tau \,d\tau$$

where: $m(\tau)$ is the discount factor (i.e., "deflator").

This will give us an expected net present value, which tells us whether the instrument is under or overvalued versus its price, $P_t$. Canonically, we would interpret a large enough discrepancy between $V_t$ and $P_t$ as an opportunity to go long or short to capture the difference. But it appears that we do not have enough information to assess the rate of return (let alone the likelihood).

Are there any theories of value or valuation methods which indicate both the likelihood and rate of value-price convergence?

References are always appreciated.

I performed spectral analysis on the stock market for disaggregated returns. If $\mu$ is the center of location and anything away from $\mu$ is an "error", then the stock market is in equilibrium once every 20-21 years as an aggregate whole. But, like a musical instrument, the periods of the individual firms could be relatively small. Still, that would imply that if you bought the "market" then present value and price would meet only every two decades. Every 41-42 years you get one sweep of the spectrum of returns. I performed the spectral analysis both on the log values and on ranks. You can find literature on spectral analysis by ranks in the geology literature. It has some rather interesting additional properties.

EDIT

Begin with

Slutzky, Eugen, The Summation of Random Causes as the Source of Cyclic Processes, Econometrica, 5(2), Apr.1937, pp. 105-146

If you use a process such as $x_{t+1}=\beta{x}_t+\epsilon_{t+1}$ then you end up with a cyclical process. The time to span one entire spectrum of possible values of $\beta$ is between 40 and 41 years for the broad market. It would be in equilibrium approximately twice in that time period though it could be close much of the entire time. If you assume that $k_t=p_t$ in equilibrium or equivalently that the present value of cash flows equals price, where $k_t$ is the replacement cost of capital, then price equals value in the economy has a whole about once a generation.

EDIT I am not actually assuming that value is fixed, merely that it cannot be contemporaneously observed. The strong assumption I am making is that $\mu$ as the center of location has an error of zero. If its error is zero, it is a correct price. What is being ignored is the cost of funds, not the underlying valuation. As to citation, I have never published it. I ran the spectral analysis in two ways.

First, I ran it using logs on the S&P 500, the Dow and GDP. They have the same periodicity. Then I converted returns into ranks and ran spectral analysis on the ranks.

As to $\beta$, it does not exist, or at least it cannot exist as understood in mean-variance finance. It is improper to make an assumption as to the distribution of returns. The distribution of returns are not data, they are a statistic. You can make assumptions about the nature of the data, but not of transforms of the data. To do so would be no different than assuming the test distribution that we would normally use Student's t distribution on is Weibull distributed because someone did not bother to derive the correct distribution.

In the specific and very narrow case mentioned above of $x_{t+1}^i=\beta^i{x_t^i}+\epsilon_{t+1}^i,\beta^i>1$, then no covariance matrix exists among assets $i,j$. By theorem, there is no Frequentist solution to this equation that can converge to a population parameter. $\beta^i>1$ as the alternative is to plan to lose money in every period. The test statistic that also conforms to the assumptions of mean-variance finance is ordinary least squares. The sampling distribution of $\hat{\beta}-\beta$ is the Cauchy distribution.

The Cauchy distribution has no mean and the least squares estimator is a variant on the mean. Therefore the sampling distribution has zero power. This is well known. One could abandon the maximum likelihood estimator, which should be the minimum variance unbiased estimator, in favor of something such as Theil's regression. To do so would be to reject mean-variance finance. You would end up with median-interquartile range finance and the summary equations would vanish as they depend upon a covariance. Nonetheless, Theil's regression and quantile regression result in inadmissible statistics.

This result follows from two directions. The first is that the distributions involved lack a sufficient statistic and so any point estimator loses information. The Bayesian estimator is always minimally sufficient, therefore it loses no information. Consequently, for that reason alone, there is no non-Bayesian estimator for use with most stocks and with the per capita GDP of a growing nation. The second reason is that in the above AR(1) equation if follows from rationality that $\beta>1$. If that is the case then the prior density over the space $(-\infty,1]$ is zero.

That is prior information and in the presence of real prior information, non-Bayesian estimators are inadmissible.

• Is there any relationship between spectral analysis and spectral risk measures? Mar 1, 2018 at 17:45
• It seems you assume that price is the only stochastic variable and that value is stationary -- I can't imagine a scenario where I would permit the assumption. Moreover, only price can be directly observed, whereas intrinsic value must be inferred. But that's an interesting thought because if you assume the distance between market price and intrinsic value is an AR process cointegrated of order 1 or greater, you will get these prolonged periods where price d.n.e. value. If, however, you have beta of less than 1, then price and value will cross frequently. Now, what is beta? Mar 10, 2018 at 23:18
• Also, how does it follow that the replacement cost of capital equals the price of capital only once in a generation? Do you have a reference? Mar 10, 2018 at 23:31
• I understand your rejection of mean-variance finance as "unfounded". I have sometimes wondered why the statistical world went the way of squared error terms... Alas, there are many investors who are (at least partly) motivated by mean and variance. The reason for this is not due to infallibility of this assumption, but rather due to the analytical tractability. Moreover, the normal distribution is the maximum entropy distribution given only mean and variance. If, however, you are motivated by something else (e.g., raw data) you may end up with something less convenient. Mar 13, 2018 at 22:03
• While we are adrift here and going into chat, the focus on squared error is that for a broad class of problems squared error is the risk-minimizing solution. It is not a universal solution and a statistician would tell you that. Non-statisticians don't usually realize there is a reason and just assume it is a universal rule. Least squares isn't a tractable solution unless it is thought of as a tractable solution to a problem other than those faced in finance. To be a tractable solution, it must first be a valid solution or at least approximation. Mar 14, 2018 at 3:59