# Under the CAPM, how do I deal with market returns being below the risk-free rate?

Let's say I'm using CAPM to estimate the cost of equity, so I need expected market returns for the calculations.

The standard approach is simply to compute arithmetic mean of an index (or rather its returns) that represents the market well. There is no problem in doing so when the returns trend is upward (like with S&P 500), but for some indexes (like MICEX) it's downward or roughly horizontal, and then the calculations make no sense since we get the market returns lower than the riskfree rate (returns on government bonds). How should I go about that?

I'm thinking about weighting techniques but not sure which one to implement, the literature review got me nowhere in particular.

If I understand you properly you’re wondering if it is possible to have negative beta’s or a negative market factor (Rm-Rf<0) in the context of CAPM.

One could consider buying such assets as buying insurance against macroeconomic risk that adversely relates to the rest of your portfolio. A common example is gold, since the could be seen as an insurance against inflation. The cost of that insurance in therefore (Rf-Rm if Rm-Rf<0). So indeed, you might find Rm

Regarding weighing techniques, value weighting is most common since it limits the effect of small stock on your index.

I have to agree with Tim as Most books also suggest the same. But, realistically, considering the cost of Insurance and there being no significant returns from the underlying asset, the investor should question himself whether "Is the asset Worth investing?"

Considering the huge risk of invested amount & current macro economic events, the underlying isn't worth the price. Insurance is depreciating. If you opt for futures, you end up loosing your returns on underlying in case of a upward movement.

For such cases, there exists debt markets, which often guarantee risk free rate and a slight premium over risk free rates.

So, let me begin by stating that the distribution for returns has been derived and solved. The good news is that it solves your problem, the bad news is that you can also prove that the CAPM, even if strictly true, cannot be solved. There is in fact a 1958 non-existence proof, once you link them together.

The good news is the missing return is dividends. The distribution of returns, or price returns plus dividend returns as in IRR, in equilibrium, ignoring bankruptcy, merger and liquidity risks, is $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]\frac{\sigma}{\sigma^2+(r-\mu)^2}.$$

This only holds for stocks sold in a double auction, some assets, such as antiques, have a very different distribution. You will have to use a Bayesian method because there is no sufficient statistic and the maximum likelihood estimator has not been solved. There is a wonderful paper in it for you if you can solve the intense polynomial that would be created. There is no admissible Frequentist estimate.

The distribution differs if you use logs, and you can make an argument for using OLS, but because the underlying likelihood function lacks a covariance matrix, you cannot create a $\beta$ in the sense of the CAPM. Assets can comove, but cannot covary. The log-distribution, the hyperbolic secant distribution, violates the definition of covariance.

The positive news is that it only requires the subjective intent to make a profit, it does not prohibit long term down trends.

Consider these papers:

https://ssrn.com/abstract=2828744

https://ssrn.com/abstract=2656681

As to the cost of capital, it is the marginal cost for the firm to acquire its next dollar of capital. If it has a line of credit then it is pretty simple.

Russia is a difficult case to analyze. According to Dimson, Staunton and Marsh the real annualized return on Russian equities from 1995 to the end of 2014 is 3.5% a year. Source: 2015 Global Investment Return Yearbook page 52. This is the longest and best continuous estimate that I am aware of. After subtracting a real return on Tbills in this period of approx -2.2%, they arrive at an Equity Premium (i.e. $R_m-R_f$) of 5.8% per year. In a case like this, with relatively short stock market history, there is a lot of uncertainty associated with this estimate.

• Great source, thanks. I wonder if they have any documentation on how they derived those estimates, I don't think it's just a simple arithmetic mean over the period. – Always Right Never Left Nov 9 '15 at 9:12
• @AlwaysRightNeverLeft Concerning your Q. how they estimated the return. As far as Russia is concerned, there are 2 broad market indices: ruble denominated MICEX and dollar denominated RTS. 3.5% real dollar denominated return, for RTS index I suspect, seems quite plausible to me. However, if you're concerned with equity risk premium (ERP), you need to take into account other risks investors need to be compensated for like currency and Russia sovereign default risk. Damodaran has a really nice section for calculating ERP for foreign (to US) countries in one of his books. – Sergey Bushmanov Jan 11 '16 at 23:32

In my opinion previous answers are a bit off goal. The CAPM is, at least in your primarily role, an equilibrium model. Is shared opinion that the investors are "risk adverse"and, as a consequence, the risk premium $R_m - R_f$ cannot be negative, but strictly positive. If your target is estimate the risk premium you are not constrained to use the data in backward looking manner. Even better you are not constrained to use the historical data at all. Historical mean is only the easiest way that many books show. You must have forward looking perspective. If in equilibrium (long term) the common opinion became for indefinitely down trend ... stock market is end. Nevertheless if you don't care for equilibrium ... $R_m - R_f<0$ is not a problem. In Mean Variance and related CAPM framework the "best results" will be long position in risk free asset and, eventually, short position in risky.