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I compare the five-factor model of Fama and French (2015) and the q5-factor model of Hou et al. (2018a) and find that the five-factor model outperforms the q5-factor model in terms of having closer-to-zero estimated error terms, time-series and cross-sectional covariance-to-variance ratios, and time-series variance of coefficient in most of the cases. The results thus suggest that the explanatory variables of the five-factor model are more accurate proxies of the factors stock returns depend on.

This paper reports implications from an assumption that total value of equity regresses toward total value of debt and loan liabilities over time. I show that, for example, change in value of equity markets equals change of total value of debt and regression factor determined ten years earlier plus change of value of residual factor and that residual factor explains short and mid-term variance of value of equity markets. Furthermore, I show that expected change in value of equity markets equals constant changes of total value of debt and regression factor and expected change of residual factor and that the constant changes drive the outperformance of equity markets over government bond yields.

I study if changes in total value of debt ten years earlier explain expected value or mean of changes of value of US equity markets in a long run. I find that, when total value of debt is controlled over time, mean of quarter, annual, five-year and ten-year changes of Russell 3000 and S&P 500 price indices decreases near to zero. Results are statistically significant at 5 to 0.1 % level. I thus conclude that changes in total value of debt ten years earlier do explain expected changes in value of US equity markets in a long run.

Fundamental theory of asset pricing states that current price of an asset equals expected value of multiplication between stochastic discount factor and future cash flows of the asset in a discrete period. If we assume that such theory holds, 1) discount factor and cash flow of an asset are independent in which case expected value of returns of the asset equals risk-free rate of return if discount factor and cash flow realize at the same point in time, 2) discount factor and cash flows are interdependent via dependence of risk-free rate of return on cash flows in which case expected asset return is explained by risk-free rate of return and its dependence on asset return if realization of cash flows precedes realization of discount factor, or 3) discount factor and cash flows are interdependent via dependence of expected value of cash flows on stochastic discount factor in which case expected asset return over a period is, via the dependence, explained by expected asset returns over sub-periods if realization of discount factor precedes realization of cash flows. In this paper, I provide the rationale behind these conclusions.

In this paper, I derive properties of expected value of asset returns and present their implications. I show that expected value of asset returns has no probability dimension and thus equals asset-specific risk-free rate of return and that any dependence between asset returns is captured by the expected value of asset returns. Moreover, I conduct that market returns indeed follow a normal distribution with time-varying expected value of returns, that risk factors determine the shape of probability distribution of assets and that expected value of returns measures the relative skill of investors.

This paper discusses the theory that risk factors divide to the company specific and asset specific risk factors. The first group affects to the expected value of an equity of a company whereas the second only to the positive cash outflows for a specific asset. I find that equity market, value, and quality factors are indeed possible company specific risk factors with influence on an expected equity of a company and dividend and volatility factors are possible stock specific risk factors affecting positively to dividends and other cash payments from a company to shareholders. These results are statistically significant and important for our understanding of risk factors and their characteristics.

Define total risk as all possible outcomes and their probabilities. In this case, expected value – probability weighted average – of outcomes represents average figure around which the risk is distributed. Assume also that total return from an asset represents compensation for the total risk of holding the asset and that true expected values of future prices and cash flows of the asset can be derived from yield curve of risk-free returns. With these assumptions, I derive a theory of fair asset pricing in which any total return is fair compensation for taken risk if the expected value of total returns is equal for all assets in the secondary markets.

The purpose of this paper is to aggregate some of my thoughts and perspectives, which are continuation to my previous papers, from the last months. The paper consists of different independent parts that are separated by headings. These parts are however somewhat linked to each other, and together they constitute a whole. Hence, I have excluded forewords, closing words, and other words not included in these parts from the paper. This paper starts with a relationship between stock price and true value.

I share some thoughts regarding realized, required, and expected returns and risk-free rate. I provide simple mathematics to divide return to change in real equity value and change in demand-supply curve of an asset. I also consider risk-free rate in terms of inflation and additional interest rate that compensates for liability risk. Finally, I show mathematically how this risk-free rate affects implications about realized and expected returns of assets with certain payoff. This is a perspective paper.

Higher returns are not compensation for a higher uncertainty but rather for a higher probability of losing a part or the whole investment. With uncertainty I refer to a situation where an investor does not know what will happen in the future and, at the same time, he cannot expect a specific outcome to happen more probably than another. If risk is nothing but uncertainty, higher risk is not rewarded with higher returns. Otherwise there would be an anomaly in the required return curve. I, however, leave space for a debate how to define risk.I also conclude that expected return is actually a risk-free rate of return. Because expected return covers all the possible returns of a probability space, by gaining all these returns an investor gets for certain the expected return. This is a case when probabilities are dependent on each other. If probabilities are independent, expected return is gained only when repeats approach to infinite. This is a perspective paper.