The S&P 500 has been down 13 out of the last 16 days, an impressive run to be sure. The question on most investors’ (and traders’) minds is whether we’re close to the bottom, an interim bottom, or whether there’s substantial downside left. While some pundits argue that the index is fairly-valued or undervalued, others (mainly technicians) point to a lack of technical support and predict further pain (for those who are long). Who’s right?

In light of this intense interest in prognostication, we attempt to estimate the cumulative distributions (rather than point estimates) of the S&P 500’s intrinsic value and its 10-year forward annualized return (including dividends), as a means of evaluating the relative likelihood of future returns and future levels of the index (under both a somewhat optimistic scenario and a possibly more realistic one).

The analysis is relatively rudimentary. First, we take the current level of S&P 500 earnings and grow those earnings over the next 10 years at a presumed compounded annualized growth rate to arrive at 10-year forward earnings. We then apply a P/E multiple to those terminal earnings to give us the “terminal” value of the index 10 years hence. Finally, we discount that terminal value to the present (adjusting for accrued dividends) using the long-term expected return of the S&P 500 (cum dividends) to arrive at an estimate of fair value.

The twist is that we allow the earnings growth rate and the P/E multiple to be random variables with prescribed means and standard deviations. We then generate multiple realizations of the earnings growth rate and the P/E multiple to arrive at multiple possible fair value estimates of the S&P 500, from which we can analyze the distributional properties of its intrinsic value.

We assume that the earnings growth rate and terminal P/E multiple are normally distributed. However, because normal random variables can become infinitely small or infinitely large, we truncate these distributions at what we believe are reasonable lower and upper bounds. When a realization of one of these variables falls below or exceeds a threshold, we “reflect” the value back into the heart of the distribution (for example, if the lower bound of the earnings growth rate is 3% and a realization of 1% is generated, we set that realization to 5%--the mirror image about the 3% threshold). This reflection shifts the means and standard deviations of the random variables, but (in our case) not materially.

**Earnings Growth Rate**

According to John Hussman, the long-term peak-to-peak annualized earnings growth rate for the S&P 500 is approximately 6%, with a maximum annualized growth rate over a 30-year horizon of 7.8%. Thus, our initial analysis assumes a mean earnings growth rate of 6%, with a standard deviation of 2%, a maximum value of 8%, and a minimum value of 0% (some may take exception to this minimum value in light of the potential for massive deflation that we currently face—notwithstanding the Fed’s attempts to stimulate the economy and the potential monetization of debt down the road—thus we will relax this assumption a bit later).

**P/E Multiple**

Referring to Professor Robert Shiller’s analysis, we see that the average P/E multiple on trailing 10-year *average* earnings is roughly 15, which we take as the terminal multiple on our 10-year forward projected earnings (which may be construed as a discrepancy, but the forward projection is meant to smooth the earnings, so we don’t want to double smooth them). We set the standard deviation of the P/E multiple to 5 (out of thin air), with a lower bound of 5 and an upper bound of 25.

**Current S&P 500 Earnings**

We assume that the earnings from the last ten years or so have been inflated by leverage and are not valid representations of the earnings power of the index. Thus, we go back twenty years to 1989, take the Bloomberg EPS estimate of $15, and project that value forward to the present at a 6% annualized growth rate—which gives us a starting point of $48 for S&P 500 earnings.

**Dividend Yield and Long-Term Expected Return**

We set the dividend yield on the index at 3% and the long-term annualized return at 7% (including dividends). While each of these quantities could have been modeled as random variables, we take them as deterministic for simplicity.

**Results**

With the foregoing assumptions, we arrive at the following cumulative distribution for the fair value of the S&P 500:

The table suggests that there is a 50% chance that the index is worth less than 830 and a 30% chance the index is worth less than its current value of 660 (just below the S&P’s Friday low). With a mean at about 840 and a standard deviation of approximately 300, there’s roughly a 70% chance (if you buy the input assumptions) that the “fair value” of the S&P 500 lies within the range 500 – 1100. From the current level of 676, the analysis suggests the 10-year expected annualized return (including dividends) is about 9%. (Note: in the link to the Hussman article above, written in 2006, he said there was a high probability the index was worth between 700-800 dollars—talk about being spot-on).

If these assumptions are valid (or in the ballpark of validity), it looks like the S&P 500 might be a decent buy at current levels—as the analysis suggests a 70% chance the index is “worth” more than 660.

**Changing the Assumptions**

Now one may argue that the aforementioned assumptions are a bit optimistic; that the economy is undergoing a profound structural change and thus there is no reason for us to expect that earnings will grow at an average of 6% over the next 10 years. Moreover, one may argue that there is no justification for a lower bound of 0% on the earnings growth rate. One might also take issue with the starting point of $48 for the earnings on the index. Given that forecasted earnings for 2009 are coming in at $30, a starting point of $48 may be too aggressive (the market is certainly telling us this). (Bloomberg lists the trailing 12-month EPS as $60, whereas Political Calculations lists the value at $45).

- Average earnings growth rate set to 4%
- No lower bound on the earnings growth rate
- Initial earnings on the S&P 500 set to $30 (we don’t make an allowance for the fact that the forecasted $30 will be realized one-year forward)

These adjustments, while potentially more descriptive of reality, significantly alter the analysis and provide justification for the current level of the index and fodder for the bears.

With these assumptions we see that there is a 50% chance the index is worth less than 440 and a 90% chance the index is worth less than 690 (roughly its current level). Commensurately, the expected 10-year annualized return (with dividends) is approximately 2.5%, with a 90% chance that return will be less than 7%. Interestingly these numbers gibe with The Big Picture’s Barry Ritholtz’s suggestion that the “fair value” on the S&P 500 may be 440.

If pressed, we would say the latter set of assumptions might be more descriptive of the future probabilities than the initial assumptions. Thus, while the index has been down 13 out of the last 16 days, and could very well be due for a bounce (particularly given that the “smart money” has turned bullish while the “dumb money” has finally become bearish), the collapse of earnings and potential structural change in the economy may be inexorable forces that drive the index lower in the months ahead.

That being said, if the economy does in fact recover, we may see a burst of earnings growth coming out of the ruins, in which case the lower growth assumptions may be overkill on the pessimism—so be sure to take the analysis with the appropriate grain of salt.

*Disclosure: The author is long QLD, DIG, UYG, and EEV and has offsetting options short put positions in SDS and SSO.*