# Modeling The Fair Value Of The S&P 500

## Summary

- A methodology to model the value of the S&P 500 is presented.
- The model relies on macroeconomic factors: gross domestic product, corporate bond yield relative to 10 year Treasury bond, money stock index and consumer price index.
- The model accuracy (R-squared) exceeds 95%.
- The model was used to provide the 2021 S&P 500 price outlook under three scenarios.

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Large and small investors are quite often faced with the question: Where is the equity market heading? Generally speaking nobody knows what will happen tomorrow or in a year from now. I am confident that any of the reader did not come across the idea of world lockdown prior this COVID pandemic.

What we know is that the market is influenced by factors like the gross domestic product, treasury rates, inflation and so on. For instance, it is given that when the gross domestic product grows, all other factors being equal, then the equity market goes up. The question then becomes, how much?

In this article, I am showing that the S&P 500 can be modeled as function of the main macroeconomic parameters of the USA economy. These factors are: gross domestic product (GDP), money stock index (M1), BAA corporate bond yield relative to 10 year Treasury bond (BAA-10Y) and consumer price index (CPI).

In Figure 1, the actual and modeled S&P 500 index is presented. The model uses monthly data tracing back to the 1960's. The model has an accuracy of 95.72%; R^{2}. The 2 standard deviation of the error has been calculated to be 10.48%. The error is defined as the relative difference between actual and modeled value.

**Figure 1: Actual and modeled S&P 500 index. Source: Created by the author.**

In Figure 2, it is shown how these four macroeconomic variables are affecting the value of the S&P 500. The variables were varied one at the time ±20% from their reference value. The reference value was assumed to be the one available on fred.stlouisfed.org at the end of November 2020.

If an analyst and/or an investor are interested in understand where the market is heading in e.g. one year from now, this model can be used to create outlooks based on estimations of future values of: GDP, M1, BAA-10Y and CPI.

The next question that might come to mind is whether the predictions are robust. To answer this question we need to keep two things in mind. First, the prediction is heavily dependent on the guess of the future value of the variable. If the guess is off then the prediction will be off. Second, this is a neutral model. Geopolitical events and buyers/sellers momentum are not accounted. If for any reason the GDP goes down and investors keep on heavily buying equities then the market will trend higher instead of going down as the model would have estimated.

**Figure 2: Calculated S&P 500 as function of macroeconomic variables variations from their end of November 2020 values. Source: Created by the author.**

How to include geopolitical and market sentiment? The approach that was used in this work has been to calculate the coefficients of the model recursively. Figure 3 shows the accuracy of the model as a function of the rolling sample size. Each sample is equivalent to one month. As it can be observed, by having a smaller rolling training period the accuracy of the model improves. 733 samples (the entire time series) has an accuracy of 10.48%, with 120 samples (10 years), the accuracy is as low as 1%. This latest case is shown in Figure 4. With an improved in accuracy, such approach can be used to assess whether the market is fairly valued both based on fundamentals as well as other factors such as investor sentiment. Based on such information investors might decide whether to take a directional bet.

**Figure 3: Model er****ror as function of the number of samples used to train the model. Source: Created by the author.**

**Figure 4: Actual and modeled S&P 500 index. The modeled S&P 500 was created using a rolling regressive approach with 120 samples (10 years). Source: Created by the author.**

One question that might come to mind is: how far off is the S&P 500 from its fair value? At the time of the writing (end of November 2020), the S&P 500 was valued 3621.63 points. The fair value was estimated to be 3030.20 points ±11.28% and 2591.32 points ±4.99% for the non-rolling and rolling models respectively. In this estimate, the following numbers were used: GDP = 2.116T$, M1 = 5,733B$, BAA-10Y = 2.32% and CPI = 260.325. The numbers were taken from fred.stlouisfed.org.

What about 2021? Outlooks are based on assumptions and 2020 thought us that the un-imaginable can happen e.g. the world went on a standstill for a few months and the Federal Reserves printed more money in a few weeks than what it did during the 2007/09 financial crises. For the end of 2021, the author is proposing the following outlook scenarios:

**Scenario 1: The world goes back to normal**. In this scenario the unemployment level goes back to its pre-crisis level, GDP & CPI growth follow their historical trend, the rate of increase in M1 is the same as in the last 10 years while BAA-10Y will be the same as its last 10 years average.

**Scenario 2: Large Fiscal Stimulus.** In this scenario, the un-employment fails to get back to the pre-2020 level. As result a fiscal stimulus is proposed and M1 increases by 50%. GDP, CPI and BAA-10Y will be the same as in Scenario 1.

**Scenario 3: Black Swan.** In this last scenario, the world goes back in a lockdown for a few months, the unemployment rises and the US government fails to put in place an adequate fiscal stimulus. The following assumptions are made: GDP = -30% vs. 20Q3, BAA-10Y rises to 3.5%, M1 grows at the rate of the last 10 years and CPI will be the same as in 20Q4.

In Table 1, the forecast value of the S&P 500 at the end of 2021 is presented for the above mentioned three scenarios. Because this was a longer term forecast, the non-rolling model was used. The non-rolling model would have given a negative value for the Scenario 3 thus highlighting the unsuitability of this approach when it comes to performing longer term outlooks.

**Table 1: Forecast S&P 500 value at the end of 2021 for three different scenarios**

Concluding, the two models presented in this article are able to fairly well reproduce the trend of the S&P 500. The second one does a better job since it also takes into account the market dynamic on top of macroeconomic factors. The models can not predict where the market is heading, the future is not written. Despite they might be able to provide insights on how far off is the fair value of the market in relation to its actual value. The models have estimated that at the time the article was written, the S&P 500 was overvalued 19.52 and 39.76% for the non-rolling and rolling model, respectively. An outlook for the end of 2021 was also provided. In the business as usual case, the S&P 500 might close the year at 3513 points, with a large fiscal stimulus at 4083 points and if a second black swan event will take place at 2242 points.

This article was written by

**Analyst’s Disclosure:** I am/we are long TQQQ, NDAQ. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

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