# Mathematical models: What will investors pay for zero growth?

**Stocks are inherently difficult to value**

A fundamental question that all investors must ask themselves is what to pay for a stock. In theory, a company's value is the present value of all future free cash flows to shareholders. These cash flows are inherently difficult to predict and here lies the cause to why stock prices are very volatile.

Many simple and complex stock valuation models have been developed such as the Gordon growth model, the residual income model, market-based valuations and free cash flow valuation. All these models produce a valuation that can be translated into a market multiple such as Price/Earnings (P/E). The valuation investors are willing to put on stocks must logically be a function of multiple factors (growth, required rate on equity, profit margin etc.) and many of those are estimated so the valuation is inherently uncertain and as result has a wide range. This leads us to the next question.

**What are investors willing to pay for zero growth?**

Instead of going through the various valuation models why not try to answer the question empirically through observations? Our Macro Strategist, Mads Koefoed, accepted the challenge and did a multiple linear regression in order to estimate the valuation for a company with zero growth expectations.

The model aims to explain the earnings yield (inverse of P/E) and is thus our dependent variable. Why use the earnings yield you might ask? For some strange reason stock investors settled early in history on the concept of P/E despite of its mathematical flaw related to earnings approaching zero. When earnings are low the P/E is high and as earnings rise the P/E goes toward zero. Except when earnings are zero (and negative) at which point the P/E does not make sense and cannot be ranked properly. The earnings yield does not have this flaw as it is linear in earnings and is thus better for comparison and ranking.

The earnings yield is regressed against multiple factors*, notably the three-year expected earnings and sales growth rates, but also other control variables such as market capitalisation, profit margin, return on equity, debt-to-assets, real gross domestic product growth, inflation expectations and nine GICS sector dummies with industrials used as base. The growth in earnings and sales were derived from Bloomberg's three-year consensus estimates. Our sample is based on the companies in the S&P 500 Index with quarterly data starting in the fourth quarter of 2005 as estimates are not widely available before then. The results of the regression are shown in the table below.

As the regression output shows, the intercept coefficient (earnings yield) is 9.54 indicating what the estimated valuation on a company is when all factors are zero. The inverse is 10.5 indicating that investors are willing to pay a P/E of 10.5 when all our independent variables are zero.

However, it does not make sense to set either the market capitalisation nor (in general) the debt level of a company to zero. If we set market capitalisation to its mean log value of 2.67 (corresponds to exp(2.67) = $14.5 billion) and debt-to-asset to its mean of 22.4 percent, and hold all the other factors constant at zero (growth, return on equity etc.), then the expected earnings yield is 9.44 with a confidence bound (90 percent) of 9.27 and 9.60 percent. In other words, setting earnings and sales growth, profit margin, return on equity, real GDP growth and inflation expectations to zero the estimated P/E ratio investors are willing to pay lies in the interval of 10.4 to 10.8.

Now this conclusion is of course too simplistic as it only represent the P/E bound for companies with mean market capitalisation and debt-to-assets. The earnings yield has more nuances and has different sensitivities to varies factors. This leads us to our next question.

**How sensitive is the earnings yield to changes in the factors?**

The regression output tells us how the various factors impact the earnings yield. We observe negative coefficients for both EPS and sales growth indicating that higher growth leads to lower earnings yields (higher P/E ratios). Market capitalisation has a positive coefficient indicating that larger companies, on average, have a higher earnings yield (lower P/E), which has also been demonstrated in academic papers for decades. The profit margin also has a negative coefficient indicating that higher profit margins are associated with lower earnings yield (higher P/E). A high profit margin may signal to investors that it is a high quality company that operates in a less competitive industry and therefore deserves a higher premium.

An interesting analysis is to look at how sensitive the earnings yield (P/E ratio) is for various factors when holding the other factors constant at their mean values. The chart below shows the expected P/E ratio with confidence bounds given various expected sales and earnings growth rates. As the chart shows, an average company with zero expected growth in sales and earnings is expected to have a P/E ratio of around 11.3. Based on our sample we can also observe how the sensitivity is exponentially growing as a function of growth. When we move from 20 percent to 30 percent annualised growth rate the P/E ratio explodes, which makes sense when we think of companies such as Salesforce.com and LinkedIn.

**Can profit margin expansion lead to higher stock returns?**

Another interesting factor with a high sensitivity is profit margin. As we have described and can be observed on the chart below, the P/E ratio is positively correlated with profit margins. An interesting concept, given the relationship and sensitivity between profit margin and valuation, is that investing in low profit margin businesses with prospects of improving margins is likely associated with a large expansion in the valuation multiple and thus maybe high ex-post stock returns. The next question is then, of course, to predict future profit margins, but that is not our task here.

**Do sectors matter for earnings yield?**

In order to gauge whether some sectors have** **significantly different earnings yields we include dummy variables for the 10 GICS sectors using industrials as the base. The table of the regression output shows that health care, financials, consumer staples and energy sectors are all significantly different from industrials with energy furthest away from industrials in terms of valuation. Holding all other factors constant, energy stocks have higher earnings yield (lower P/E) compared to industrials. This is clearly the case in today's market. Oil companies such as ConocoPhillips, Marathon Oil and Chevron all have P/E ratios below 9 despite a robust outlook. So for an energy company to have the same valuation as an industrial, the company has to have higher expected growth, a higher profit margin or more leverage.

Financials and health care also have positive coefficients meaning, holding all other factors constant, that they have higher earnings yield (lower P/E). The only sector with a (significant) negative coefficient is the consumer staples sector. This means that consumer staples companies have a lower earnings yield (higher P/E) compared to industrials. The stable businesses operating in this sector often have more predictable cash flows and less sensitivity to the business cycle and thus deserve a higher premium.

** The independent variables are not highly correlated.*