Is it possible to reliably estimate the cost of capital with CAPM (or any other factor model)?

In a previous post I discussed why the cost of debt has little influence on investments. What about the cost of equity? Firms typically use (much) more equity than debt to finance their investments. So the cost of equity should matter more. In a recent study, Murray Frank and Tao Shen investigate how the cost of equity and the weighted average cost of capital (WACC) influence investments of US firms.  Remarkably, they find that the cost of equity and the WACC are positively related to corporate investments. Firms with a higher estimated cost of equity and WACC tend to invest significantly more. That is a very strange result. We would expect firms with a high cost of capital to invest less, not more.

Perhaps something is wrong with the model that Frank and Shen use to estimate the cost of equity? They focus on the Capital Asset Pricing Model (CAPM) which is still the most commonly used model for determining the cost of capital. In a recent survey of finance professionals, Lilia Mukhlynina and Kjell Nyborg find that 76% of these professionals use CAPM. Nevertheless, CAPM has been criticized a lot. The model assumes that the expected return on investments is determined by their systematic risk, i.e. the extent to which the return to the investment is correlated with overall market return. Researchers including Nobel price winner Eugene Fama argue that other factors such as market capitalization and the market-to-book ratio also affect the expected return of investments, up and above systematic risk. These factors should therefore be included in any empirical model that estimates the cost of equity. However, the findings of Frank and Shen not only hold for CAPM but also for the two most commonly used alternative factor models: the Fama and French three factor model and the Carhart version with four factors. Frank and Shen additionally check whether their results hold for different time periods, they take into account differences in firm size and leverage, add additional price factors, consider the possibility of autocorrelation which may affect their findings and use different estimation periods to determine the cost of equity. Whatever they do, the positive relation between the equity cost of capital and corporate investments remains significant. 

However, when they try a different approach to estimate the equity cost of capital, the implied cost of equity capital, the relation between cost of capital and corporate investments becomes negative. This is what we would expect: a lower cost of equity capital makes it cheaper to finance investments and hence induces firms to invest more. Calculation of the implied cost of equity capital is based on valuation models such as the Gordon growth model and the residual income model, which value equity by discounting expecting future cash flows or earnings. If you have estimates of equity value (based on the stock market price) and expected future cash flows or earnings (based on analyst forecasts), you can 'back out' the implied cost of equity capital. This is the cost of equity that stock market investors use to value the stocks (assuming that their expectations about future cash flows / earnings tend to be in line with those of analysts).

Why then is there a positive relation between factor model estimates of the cost of equity and corporate investments? From an economic point of view this does not make sense, and Frank and Shen are not able to explain the positive relation. Anyway, their findings raise serious doubts about the validity of factor models for determining the cost of capital. This is troubling, since these models still are the most common approach to estimate the cost of equity and WACC. The findings of Frank and Shen suggest that the implied cost of equity is a better approach. Unfortunately there are limits to this approach. Estimation requires not only the availability of stock market prices but also reliable estimates of expected future cash flows or earnings, which are often not available. A major advantage of CAPM is that it is a fairly easy model to estimate, which explains why it is still so popular with finance professionals.

Interest rates are very low: why don't firms invest more?

When interest rates are low, firms can borrow cheap, which lowers their cost of capital. It can therefore be expected that firms invest more when interest rates are down, since they will have more projects for which the expected return is higher than the cost of the capital. However, while interest rates have been at historically low levels in recent years, the effect on investments seems to be very limited. Why is that?

First, firms not only fund their investments with interest paying debt but also with equity. The overall cost of capital is the weighted average of the cost of debt (i.e. the interest rate, taking into account the tax deductibility of interest payments) and the cost of equity. Firms typically use (much) more equity than debt to fund their investments. The cost of equity is substantially higher than the cost of debt, because the cost of equity includes an risk premium that compensates investors for the volatility of equity returns. This equity risk premium is not fixed but varies over time. As a result, the cost of equity, and therefore also the weighted average cost of capital (WACC), is not closely tied to current market interest rates. This weakens the relation between interest rates and investments.

Second, when firms decide about an investment project, they often use a hurdle rate that is much higher than the estimated cost of capital. In a survey of US firms, Jagannathan and colleagues find an average hurdle rate of 15%, while the average WACC of these firms is only 8%. The hurdle rate also tends to be stable over time and firms care little about changes in interest rates. One reason why firms use such a high hurdle rate could be that they are financially constrained. That is, they cannot find enough funding to finance all valuable projects. As a result, they have to restrict themselves to the projects with the highest returns. However, Jagannathan et al. find that financially constrained firms actually use hurdle rates which are closer to their cost of financial capital, while firms with ample financial flexibility in the form of low debt ratios and high cash balances use higher hurdle rates. Their results suggest that it is nonfinancial, operational constraints, such as managerial or organizational requirements, that make firms more selective in making investments.

Interestingly, they also find that hurdle rates reflect idiosyncratic risk, i.e risk that is specific to individual firms. For example, the CEO of the firm might unexpectedly die, which could have large consequences for the firm, but matters little or nothing for the rest of the economy. According to standard portfolio theory, idiosyncratic risk does not affect investors who hold a diversified portfolio. If your portfolio consists of many different securities, the idiosyncratic risk of the different securities will cancel out if your portfolio is large enough. As a result, investors do not have to be compensated for idiosyncratic risk, and this risk should not affect the cost of equity. However, if the firm’s shareholders are not well-diversified, it makes sense to add a premium for idiosyncratic risk in the hurdle rate for investments: the shareholders need to be compensated for idiosyncratic risk that they cannot diversify away. Actually, many firms are controlled by shareholders which are not well diversified. Most firms are not listed on a stock exchange, and non-listed firms typically have a limited number of shareholders who have a substantial part of their wealth invested in an equity stake that they cannot easily sell. Even many (actually most) listed firms around the world are controlled by shareholders who have a substantial stake invested in the company. Idiosyncratic risk matters for those shareholders, and it makes sense to take it into account when setting the hurdle rate for new investment projects.

Finally, the value of investments not only depends on the required rate of return, but also on the expected return of investment projects. If there are fewer good investment opportunities in times when interest rates are low, firms in the aggregate will invest less even if it is apparently cheap to invest. So, the current lack of investments may also reflect a (perceived) lack of good investments opportunities.

The art (or is it science?) of company valuation

When it comes to valuing companies, some people seem to suffer from what could be called 'reverse number blindness': if there's a lot of numbers and formulas involved, it must be good. However, as Aswath Damodaran points out, 'the test of a valuation is not in the inputs or in the modeling, but in the story underlying the numbers and how well that story holds up to scrutiny'. If the numbers you put in your valuation model do not reflect reasonable expectations about future cash flows and risk, your valuation is worthless however sophisticated your model is. Which raises the question: what are reasonable expectations? As a wise man once said: 'it's tough to make predictions, especially about the future'. When valuing a company, it is important to understand that valuations are only informed guesses, typically based on limited and (very) noisy information.

The Financial Times recently discussed an interesting illustration of this problem. In 2011, a company called Rural-Metro was acquired by private equity firm Warburg Pincus. Afterwards, shareholders sued Rural-Metro's directors for breaching their fiduciary duties, and Rural Metro's bankers - including Royal Bank of Canada - for aiding the directors in selling the company too cheaply, supposedly to get new lucrative financing assignments. While the other parties settled before a trial, Royal Bank of Canada contested the charges in a court in Delaware.

This led to a discussion about the valuation of Rural-Metro. In any discounted cash flow valuation (DCF), expected future cash flows are discounted at the cost of capital. According to the Capital Asset Pricing Model (CAPM) (basic version), the cost of equity = the risk free rate + beta x equity risk premium. In this model, beta measures the 'systematic risk', i.e. the extent to which the equity returns co-vary with the overall market return. There are two big problems with this approach. To begin with, it isn't even clear whether CAPM is a good approach to measure the cost of equity. Basically, people use it because (a) it is very simple to apply, and (b) there do not seem to be any significantly better approaches when it comes to valuation in the real world.

A second problem is that beta can be measured in different ways, and different approaches can lead to very different valuations. In the case of Rural-Metro, both parties made different assumptions with respect to beta. While they both used regression analysis and historical returns to estimate beta, the plaintiffs based their analysis on the weekly returns over a two year period, and the Royal Bank of Scotland advocated the use of monthly returns over a five year period. Both approaches are considered valid ways to estimate beta, but in this case the results were markedly different. The weekly data yielded a beta of 1.2, while the monthly data generated a higher beta of 1.454. It will probably not surprise you that the approach chosen be the parties confirmed their respective view points. The plaintiffs argued that the price at which Rural-Metro was sold was too low, which was disputed by the bank. A higher beta implies a higher cost of capital, and hence a lower DCF-valuation. The higher the DCF-value of Rural-Metro, the greater the damages to which the plaintiffs were entitled. The beta of 1.2, advocated by the plaintiffs, yielded a value of $21.42 per share, while the beta of 1.454, defended by the Royal Bank of Canada, yielded a 21% lower value of only $16.91 per share, decreasing total DCF value by no less than $100 million.

Which of these two approaches is the best? Personally, I would use neither: beta coefficients of individual stocks are often very unstable over time. You may try it yourself: calculate the beta of any stock over different periods of time, and you often end up with very different beta estimates. An extreme example is the beta of bank stocks, which for some banks exploded during the recent financial crisis. (*) A more reliable way to estimate beta is to use the mean or median 'unlevered beta' (i.e. adjusted for differences in leverage) of the industry in which the firm is operating. This is a much more stable measure, and it has been found that the betas of individual stocks tends to evolve towards industry averages.

However, the broader problem here is the reliability of DCF valuations. Any valuation will depend on the underlying assumptions, and a minor change in assumptions can lead to very different valuations. Does this mean that DCF valuations are worthless? According to Marc Andreessen, DCF valuations only serve to justify existing market valuations. I think mr. Andreessen is a bit too cynical here. (**) The big advantage of DCF valuation is that it allows to assess the assumptions on which a company valueis based. Are the underlying assumptions about the cost of capital and growth prospects in any way realistic? Let's say you would consider to buy Apple stocks: what does the current market valuation of Apple imply with respect to growth rates? DCF will help you understand this and will you help to decide whether to buy or sell Apple.

(*) Yes, I am aware that the 2007-2008 crisis was an unexpected, exceptional event, but unexpected exceptional events are basically happening all the time: any approach which does not take into account the possibility of such events is problematic in my view.

(**) Although I do like Andreessens' statement that the 'Efficient Market Hypothesis is correct if for "all information" you substitute "all information, theories, noise, and bullsh*t"'. It's a strange beast, this so-called Efficient Market.