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It is often said that the most accurate prediction of the equity market is that it will fluctuate. Nevertheless, attempts to predict the direction and level of equity prices is a time honored tradition, occupying the time and resources of macro and micro analysts. It preoccupies the efforts of professional asset managers, private investors, the general public, and of course Wall Street strategists.

Not to be left out, this report describes a quantitative method for estimating and forecasting the S&P 500 index. It has two basic pillars, the first of which we outlined in two earlier reports. This was a method for estimating the market multiple or the amount the equity market is willing to pay for a given level of earnings. To forecast equity prices this multiple or price-earnings ratio is multiplied by the amount of S&P 500 earnings to derive an estimate of the overall index.

Needless to say, then, the second pillar of our approach and the subject of this report is an estimating procedure for S&P 500 earnings. Of course, the caveat is that any estimate of the future is only as good as the estimating procedure itself and the accuracy of the input variables on which it is based.

In our previous reports Update On S&P 500 P/E Ratios Are Inversely Related To Future Federal Spending and The Market Multiple On The S&P 500 Can Be Explained: P/E Ratios Are Inversely Related To Future Federal Spending we estimated the market multiple by using the yield on the Treasury's ten year note and a forward looking moving average of the ratio of federal government outlays to GDP. The interest rate variable is inverse to the multiple as is the government spending variable meaning that a rising rate environment and an expanding role of government in the economy would be consistent with a declining multiple and vice versa.

The second pillar is to forecast the level of S&P 500 profits. We experimented with various permutations and combinations of likely explanatory variables and we found the best historical fit to be that using nominal or current dollar GDP, the percent change in real GDP, and a four year moving average of manufacturing unit labor cost which is the difference between output and productivity changes in this sector.

This model thus incorporates the effect of price and volume changes on profits and the principal cost factor for corporations which is labor. The relationships are direct respecting output and inverse respecting labor costs. Labor cost in manufacturing is the best measure of labor cost pressure in our view because manufacturing is the only sector of the economy in which productivity may be accurately measured.

This model explained about 98% of the variation in profits when estimated over the 1990-2012 period. We think this is a particularly relevant time frame because it coincided with the collapse of the Soviet Union and the increasing globalization of economic activity. These events are embedded directly in the GDP variables, particularly real GDP.

Our regression analysis for profits followed the standard procedure of specifying the natural log of S&P earnings and the natural log of nominal GDP. This mitigates the issue of scale when using both the level and percent change as explanatory variables. Over the 1990-2012 period earnings increased by about 400% while the natural log of earnings increased by only 55%. Once this was done the results were converted back from logs to actual data.

The accompanying chart shows the actual historical fit. As is evident the model tracked very closely in the 1990s and in the 2002-2007 business cycle expansions. It underestimated the timing of the downturn in profits during the 2000-2001 recession which incidentally coincided with the disruption caused by the terrorist attack on the World Trade Center and also the bursting of the internet bubble. Quantitative models are by their nature incapable of capturing the effects of such extraordinary events.

S&P 500 Earnings as a Function of GDP, % change real GDP, 4-year moving average Unit Labor Cost % mfg

(click to enlarge)

In any event Table I sketches all the relevant explanatory variables that are needed for our models of the market multiple and profits. These data forecasts reflect those from the most recent update of the Congressional Budget Office (CBO). The forecast variables are presented through 2018 because our forward looking government/GDP variable requires estimates that extend beyond the CBO current forecast horizon.

A word about the models' input variables is in order. The CBO forecasts a drop in GDP growth this year as a result of recent tax increases and the federal government sequestration order. It then forecasts a steady acceleration in GDP growth in 2014-2016 before growth trails off. Throughout the forecast period the U.S. economy avoids recession.

Government's share in the economy holds roughly steady throughout the forecast period according to the CBO. But on a moving average basis it actually rises modestly. Interest rates are presumed to normalize over the forecast period while inflation and unit labor cost pressure is hardly visible. Low inflation in turn holds down nominal GDP growth.

Utilizing these input variables the model predicts an earnings recovery in 2013 to about $104 per share. This is lower than the rate at which earnings are currently tracking, and it is about 5% below current full year consensus estimates. This could mean that our input variable estimates may prove too pessimistic; it could mean that consensus estimates are too optimistic; or some combination of the two.

An alternative explanation may also lie in the nuances of the corporate sector. Nonfinancial corporate earnings are relatively weak to date while there is a sharp recovery in financial sector earnings which is almost entirely related to a reduction of loan-loss reserves. Additionally share buybacks have proliferated this year, bolstering reported earnings. In other words, technical factors may be unduly skewing the profits picture.

Using the CBO baseline forecast, Table I shows that our earnings model estimates $104 for this year, $106 in 2014, and $146 by 2018. The market multiple is computed as 18.6 this year, falling to 18.4 next year and 15.2 in 2018 as interest rates fully normalize. Thus, as shown on the table the S&P index is projected at 1927 this year, 1951 in 2014, and 2207 by 2018.

As documented in our previous report on the market multiple a caveat is in order. The market multiple is rising sharply this year but not by as much as predicted by our model. This we attribute to the notion that market participants en masse do not believe current interest rates are sustainable or warranted under normal business conditions.

In view of this we ran two simulations. One was to determine what interest rate would comport to the current multiple of about 15 and we found that a required yield on the ten year Treasury would approximate 4%. That rate will not occur in 2013 but a nearly 3% rate is not out of the ballpark. So in a normalized rate environment shown at the bottom of Table I the rate is assumed at 3% this year, 3.5% next year, rising to 5% by the end of the forecast period. With this assumption the normalized market multiple would be 18 this year, 17.7 next year and 15.3 by 2018. Applying this to the earnings estimate generated in the CBO baseline shows an S&P index value of 1893 this year, 1878 next year, and 2231 in 2018.

Tables II and III show some alternative economic environments and the effect on the S&P. Table II assumes a more vibrant business climate which is characterized by higher GDP growth than forecast by the CBO and consequently higher interest rates, a stronger labor market, and thus higher unit labor cost. The result is as one might expect which is a lower market multiple and higher earnings such that the S&P ends the forecast period 30% higher than depicted in the CBO baseline. Interestingly the models' earnings forecast for 2013-2015 is fairly close to current consensus expectations. Perhaps stronger economic growth acts as a proxy for the sharp increase in share buybacks.

Table III depicts a more pessimistic business environment. Labeled as a stagnation setting, GDP growth stalls, barely rising in real terms. The interest rate environment matches that shown for nominal GDP, meaning there is no emergency rate. Labor costs are sticky because of a poor climate for productivity growth. The effect of stubborn interest rates is to lower the multiple while stagnant activity amidst higher labor costs depress earnings. The result is that S&P earnings peak this year and steadily slide over the forecast period until recovering slightly in 2017.

These directional results are quite consistent with a rational expectation given the three alternative business environments that are depicted. Needless to say government policy could influence outcomes beyond what any model might infer. But the point is that as long as inflation is low and interest rates are well behaved and the economy is unencumbered by restrictive policy, equity markets will respond positively.

CBO Baseline Table I

2010

2011

2012

2013

2014

2015

2016

2017

2018

GDP % change

4.2

3.9

4.1

2.9

4.4

6.2

6.6

6.0

4.7

Real GDP % change

3.1

1.6

2.3

1.4

2.6

4.1

4.4

3.8

2.6

10-year treasury

3.2

2.8

1.8

2.5

2.7

3.5

4.3

5.0

5.2

Unit Labor Costs mfg

-5

1.4

1.3

1.1

1.4

1.6

1.6

1.7

1.8

Consumer prices

1.6

3.1

2.1

1.6

1.9

2.1

2.1

2.2

2.3

S&P 500 p/e ratio

15.0

13.0

13.9

18.6

18.4

17.6

16.6

15.6

15.2

S&P 500 earnings

84

97

102

104

106

123

136

143

146

S&P 500 index

1258

1258

1426

1927

1951

2153

2251

2238

2207

Normalized 10-year treasury

3.2

2.8

1.8

2.9

3.5

4.5

4.7

5.0

5.0

Normalized S&P 500 p/e ratio

15.0

13.0

13.9

18.3

17.7

16.7

16.3

15.6

15.3

Normalized S&P 500 index

1258

1258

1426

1893

1878

2050

2210

2238

2231

Higher Growth Table II

GDP % change

4.2

3.9

4.1

3.5

6.5

6.5

6.5

6.5

6.5

Real GDP % change

3.1

1.6

2.3

2

3

4

4

4

4

10-year treasury

3.2

2.8

1.8

3

4

5

5.5

5.5

5.5

Unit Labor Costs mfg

-5

1.4

1.3

1.1

2.5

3

3

3

3

Consumer prices

1.6

3.1

2.1

1.6

3

3

3

3

3

S&P 500 p/e ratio

15.0

13.0

13.9

19.2

18.6

18.0

17.9

18.1

18.4

S&P 500 earnings

84

97

102

107

110

124

133

142

156

S&P 500 index

1258

1258

1426

2047

2049

2230

2382

2581

2867

Stagnation Table III

GDP % change

4.2

3.9

4.1

2.9

3

3

3

3

3

Real GDP % change

3.1

1.6

2.3

1.4

2

2

1.5

1.5

1.5

10-year treasury

3.2

2.8

1.8

2.1

2.5

2.7

2.9

3

3

Unit Labor Costs mfg

-5

1.4

1.3

1.1

2.5

3

3

3

3

Consumer prices

1.6

3.1

2.1

1.6

2

1.5

1.5

1.5

1.5

S&P 500 p/e ratio

15.0

13.0

13.9

16.2

15.0

14.1

13.1

12.1

11.3

S&P 500 earnings

84

97

102

103

100

102

102

104

108

S&P 500 index

1258

1258

1426

1676

1501

1440

1345

1265

1226

Please note that this article was written by Dr. Vincent J. Malanga and Dr. Lance Brofman with sponsorship by BEACH INVESTMENT COUNSEL, INC. and is used with the permission of both.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

Additional disclosure: Please note that this article was written by Dr. Vincent J. Malanga and Dr. Lance Brofman with sponsorship by BEACH INVESTMENT COUNSEL, INC. and is used with the permission of both.