A world with only two moments of the Normal distribution
February 4, 2007
In almost all analytic models of economics or financial economics the analyst is concerned with "what if " type analysis. What if the rate of growth of GDP increased to 3.5% to 3%, what if the inflation in the next quarter decreased to 6%, what if the factory productions doubled, what if, what if,°K..and so on. And this is also the framework that equity analysts in banks and investment houses employ. The whole edifice of DCF technique (discounted cash flow valuation) to value firms is based on what if type financial modeling. Increase the revenue growth rate by 2% and see how the value of the firm changes, or decrease the cost of goods sold by 5% over the next three years and watch its impact on EBITDA, or decrease the capital expenditure for the next couple of quarters and see how the investment cash flow varies, and so on. This kind of what if analysis is concerned with the first moment of a normal distribution (or for that matter any distribution). And the first moment of a normal distribution is trying to figure out what is the expected value of a variable, its mean or the average value. Equity analysts are perennially concerned with the problem of finding out what is the expectation of a certain outcome. When they put a value on the stock index or a stock for a future period - say, an analyst downgrades a stock and puts a value of $35 for the year end value of stock ABC - they essentially calculate an expected value of that stock. What is the expected value of S&P500 stock index in the next quarter? An analyst, like an economist, after producing a 100 page report will say in the end of that report that it will be 1,500. A smart investor may take the average of the last three months value of S&P500 and come to the conclusion that it will be 1,350. Both are toying with the expected value of the S&P500 index.
The expected value is the return calculation. Equity analysts, or financial analysts in general, are mostly concerned with the return (in percentage terms, mostly) of a particular financial asset just as most of their brethren in economics are concerned with the return on the assets of an economy. Their world is complete once the first moment is calculated. For them, once expectation has been priced in there is no reason to worry about the uncertainty of realizing that expectation. Even without any formal or textbook training in finance or economics one can see how tentative such a premise is. Though the Expectation (the mean) does not convey any information about the variability of returns (risk) it is an important measure. But it is important only when we include the outliers in the observations. It is the "rare event" that matters and without the rare event, or the outliers, the mean becomes a rather useless quantity. And most economists, equity analysts and the so called "market strategists" in large investment banks donít take rare events into account when calculating expectation.
We don't know about the economists, but most equity analysts and the market strategists are fools.
Corporate finance is all about the first moment (the mean or the expected value) of a normal distribution; and it would have remained mired in the first moment had Fisher Black, Myron Scholes and Robert Merton not come along in 1973. First moment of a normal distribution, like the people who use it, is a fairly useless concept.
Option traders and derivatives analysts (the quants mostly, but others as well) go beyond the first moment and concern themselves with questions such as: what is the probability that the stock of ABC will lie between $30 and $50 in the next one year? That is, they try to estimate the variability of the price of a financial asset (or its return). The entire theory of financial options and its applications in the real world is mostly concerned about estimating the variability of the underlying financial asset, or more technically the variance of the asset and its square root, the standard deviation of the return of the asset. Standard deviation, or what is termed as "volatility" in financial parlance, is the second moment of the normal distribution. The second moment tries to capture the dispersion around the expected value, the variability of the asset. Thus the second moment captures the risk of a financial asset. It tells you what kind of deviation is expected from the expected value. And it also tells you what is the probability of that deviation happening. The analyst tells you that he expects the stock of GoodCompany to be $12 in the next quarter and it turns out to be $9 or $14. Can this variability in the stock price be explained? Could this variability in the asset price be assumed a priori from its past history?
The second moment - the volatility - of the distribution is one of the most useful concepts in financial theory. Study of financial derivatives is predicated upon this second moment. The second moment of the normal distribution is the "shock" in the system which causes asset prices to deviate from what everyone expects them to be thereby inducing risk in the system. A quintessential observation of Black, Scholes and Merton was that this "shock" is associated with the randomness in asset price and creating any meaningful model of asset price movements (and thereby its pricing) should partake the modeling of this randomness.
Should one stop here, now that the uncertainty in asset prices is fully captured? Most of us will. But at least one question will still linger on: how uncertain is the uncertainty?
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