RISK, RETURN AND EQUILIBRIUM: SOME CLARIFYING COMMENTS

PurposeMany studies have shown that from a theoretical and empirical point of view, downside risk-based measures of risk are better than the traditional ones. Despite academic appeal and practical implications, downside risk has not been thoroughly examined in markets outside developed country markets. Using downside beta as a measure of downside risk, this study examines the relationship between downside beta and stock returns in Indian equity market, an emerging market with unique investor, asset and market characteristics.Design/methodology/approachThis is an empirical study done by using ranked portfolio return analysis and regression analysis methodologies.FindingsThe study results show that downside risk, as measured by downside beta, is distinctly priced in the Indian equity market. There is a direct positive relationship between downside beta and contemporaneous realized returns, indicating a premium for downside risk. Downside risk carries a higher weightage than upside potential in the aggregate return of the stock portfolios. Downside beta is a better measure of systematic risk than conventional market beta and downside coskewness.Practical implicationsThe empirical results support the adoption of downside beta in practice and provide a case for replacing traditional beta with downside beta in asset pricing applications, trading and investment strategies, and capital allocation decision-making.Originality/valueThis is one of the first in-depth studies examining downside beta in Indian equity markets using a broad sample of individual stock returns covering a wide time range of 22 years. To the best of our knowledge, this study is the first one to compare downside beta and downside coskewness using individual stock data from the Indian equity market.

(ProQuest: ... denotes formulae omitted.)1.INTRODUCTIONThe Capital Asset Pricing Model (CAPM) sets the expected return on any asset as a positive linear function of its systematic risk measured by means of the beta coefficient (β). This concept emphasizes the importance of systematic risk as a measure of non-diversifiable risk, the only risk that is remunerated in financial markets.The basic underlying notion of this model is that every asset is affected by the market's general movements, assuming that the market factor is a systematic force. Other effects are assumed to be specific or unique to an individual asset and they diversify in a portfolio.The beta coefficient is the ordinary least squares (OLS) estimator of the return on asset j on the portfolio return over a period of time. This estimation, besides using historical returns, requires other practical assumptions. Each assumption can significantly affect results.An essential requirement for using beta to obtain the future risk of a financial asset is that it has predictive power. Since future values are calculated from past data, they must be stable over time so that the estimation is correct and precise. Therefore, the more stable a value is over time, the more useful it will be. Although beta is an indicator of risk, its value is not unique and its result will depend on the hypothesis and data that are used. Many authors have studied beta's historical evolution, and analyzed its capacity to make predictions from empirical and theoretical points of view.The first decision that must be considered when calculating betas is the length of the sample period. A longer period provides more data, but the company itself could have changed its risk characteristics. Various studies, Levy (1971), Blume (1971), Altman et al. (1974), Eubank and Zumwalt (1981), Armitage and Brzeszczynski (2009), KiHoon and Satchell (2014), analyze the relation between the length of the estimation period and beta stationarity. They find that the prediction ability of betas (and consequently their stationarity) increases with the length of the period. However, this increase decreases in more diversified portfolios.A conceptual problem arises when we try to determine the return on an asset. Financial theory does not specify if returns should be considered on a daily, weekly or even monthly basis. The calculation of betas will depend on which price is considered: closing price, average daily price, and so on.Beta assets vary from one period to another because, in the first place, the risk measured by the beta coefficient of a value can vary over time. In the second place, each period's beta is calculated with a random error which increases as the coefficient goodness and the prediction power decrease. If we consider a portfolio, random errors committed in the calculation of individual betas will tend to cancel each other out, so a portfolio beta is more stable than a single beta value.Two studies, Levy (1971) and Blume (1971), analyze the seasonality of betas of individual securities and portfolios. They observe that, whereas betas of portfolios with a high number of securities provide a considerable amount of information about future betas, the betas of individual securities provide much less. This result suggests that a portfolio's beta is more stable than a single security's beta. The same direct relationship between the portfolio size and the beta stationarity has been observed in various studies: Altman et al. (1974), Eubank and Zumwalt (1981), Tole (1981), Iglesias Antelo (1999).In Jagannathan and Wang (1996) a conditional CAPM is specified, on the basis that the beta and expected returns vary over time. The results are better than those of the static model. Similarly, Mergner and Bullab (2008) uses 6 different techniques to make a study of 18 sectors in Europe, and shows that variable betas estimate the profitability of the sector, explained in terms of market movements, more efficiently than OLS. …

The ability to price risks and devise optimal investment strategies in the presence of an uncertain random market is the cornerstone of modern finance theory. We first consider the simplest such problem of a so-called European call initially solved by Black and Scholes using Ito stochastic calculus for markets modelled by a log-Brownien stochastic process. A simple and powerful formalism is presented which allows us to generalize the analysis to a large class of stochastic processes, such as ARCH, jump or Levy processes. We also address the case of correlated Gaussian processes, which is shown to be a good description of three different market indices (MATIF, CAC40, FTSE100). Our main result is the introduction of the concept of an optimal strategy in the sense of (functional) minimization of the risk with respect to the portfolio. If the risk may be made to vanish for particular continuous uncorrelated 'quasiGaussian' stochastic processes (including Black and Scholes model), this is no longer the case for more general stochastic processes. The value of the residual risk is obtained and suggests the concept of risk-corrected option prices. In the presence of very large deviations such as in Levy processes, new criteria for rational fixing of the option prices are discussed. We also apply our method to other types of options, `Asian', `American', and discuss new possibilities (`doubledecker'...). The inclusion of transaction costs leads to the appearance of a natural characteristic trading time scale.

An Asymmetric Approach to the Cost of Equity Estimation: Empirical Evidence from Russia

(1988). Traditional versus Theoretical Risk Measures. Financial Analysts Journal: Vol. 44, No. 2, pp. 52-57, 67.

Purpose – The paper examines the conditional capital asset pricing model (CCAPM) of Jagannathan and Wang using the UK data and develops a data-driven measure of beta instability risk that is pertinent to the UK stock market. In contrast to the view that the main part of the Jagannathan and Wang's model is the inclusion of human capital, however, the paper finds that human capital remains insignificant in most tests. Design/methodology/approach – Data were taken from the London Share Price Database and Datastream. This paper therefore examines the premium labour (PL) model of Jagannathan and Wang using the UK data, while the paper attaches particular importance to the measure of beta instability as a source of time variation in betas. In analysing the measure of beta instability risk, this study considers a testable measure of instability risk that varies across markets and across time as the interaction between the stock market and the economy varies across different time periods. Hence, this paper develops a data-driven measure of beta instability risk that is pertinent to the UK stock market. Findings – The results confirm the premium version of the model, that is, the CCAPM without a proxy for human capital. In particular, the paper finds that over the entire time period of this study, the measure for beta instability risk and market portfolio has significant explanatory power for the variations of returns. More specifically, when using the average earnings index as a proxy for human capital in the PL model, the premium model performs better than the PL model. When total income from employment is used as a proxy for human capital, the performance of the PL model improves for the full period. However, the results for the two sub-periods are less favourable for the PL model as, again, labour income is not priced for these periods. These results indicate that the PL model is sensitive to proxies used for human capital. Originality/value – The results revive the importance of beta instability risk in CCAPM of Jagannathan and Wang's model and suggest that the beta instability drives this model.

Computer Science has historically been strong on data structures and weak on inference from data, whereas Statistics has historically been weak on data structures and strong on inference from data. One way to draw on the strengths of both disciplines is to develop “inferential methods for data structures”; i.e., methods that update probability distributions on recursively-defined objects such as trees, graphs, grammars and function calls. This is the world of “nonparametric Bayes,” where prior and posterior distributions are allowed to be general stochastic processes. Both statistical and computational considerations lead one to certain classes of stochastic processes, and these tend to have interesting connections to combinatorics. I will give some examples of how this blend of ideas leads to useful models in some applied problem domains, including natural language parsing, computational vision, statistical genetics and protein structural modeling.

This work initiates a general study of learning and generalization without the i.i.d. assumption, starting from first principles. While the standard approach to statistical learning theory is based on assumptions chosen largely for their convenience (e.g., i.i.d. or stationary ergodic), in this work we are interested in developing a theory of learning based only on the most fundamental and natural assumptions implicit in the requirements of the learning problem itself. We specifically study universally consistent function learning, where the objective is to obtain low long-run average loss for any target function, when the data follow a given stochastic process. We are then interested in the question of whether there exist learning rules guaranteed to be universally consistent given only the assumption that universally consistent learning is possible for the given data process. The reasoning that motivates this criterion emanates from a kind of optimist's decision theory, and so we refer to such learning rules as being optimistically universal. We study this question in three natural learning settings: inductive, self-adaptive, and online. Remarkably, as our strongest positive result, we find that optimistically universal learning rules do indeed exist in the self-adaptive learning setting. Establishing this fact requires us to develop new approaches to the design of learning algorithms. Along the way, we also identify concise characterizations of the family of processes under which universally consistent learning is possible in the inductive and self-adaptive settings. We additionally pose a number of enticing open problems, particularly for the online learning setting.

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

This chapter is a continuation and extension of modern portfolio theory presented in Chapter 3, with an emphasis on risk measures and risk management of a portfolio. It introduces the capital asset pricing model (CAPM), linear factor models, and several approaches to portfolio risk measures such as value-at-risk, conditional value-at-risk and the concept of coherent risk measures, as well as a variety of portfolio evaluation techniques such as the alpha and beta, the Sharpe ratio, the Sortino ratio and maximum drawdown. The introduction to factor models is brief and from intuitive perspective.

This part is concerned with the applications of the theorems on error estimates for asymptotic expansions in the central limit theorem for general linear stochastic processes of Part I to two important typical models. These are the general random noise process and a class of pulse train processes that occur in stochastic signal theory and in a variety of digital communication schemes. The assumptions on the inherent structure of the general theorems turn out to be simple here.

Value at Risk (VaR) is a measure of risk that is able to calculate the worst possible loss that can occurs to stock prices with a certain level of confidence and within a certain period of time. The purpose of this study was to determine the VaR estimate from PT. Indonesian Telecommunications by using Displaced Diffusion volatility. The Displaced Diffusion Model is a stochastic volatility model that describes changes in a financial asset assuming volatility is not constant, but follows a stochastic process. Displaced Diffusion model are capable of modelling skewed implied volatility structures and frequently applied by interest rate quants. Based on the estimation of Displaced Diffusion volatility, it is found that volatility for PT. Indonesian Telecommunications is 0.010168 and VaR estimation using Displaced Diffusion volatility with a confidence level of 95 percent of 1.63%.

This article is the first step toward integrating in a single framework two previously separate lines of research on major structural decisions of life insurers. The literature has previously studied the relation between capital structure and asset risk on the one hand, and the relation between organizational form and distribution system on the other hand, without integrating them. Using life insurer data for 1993–1999, we model the four key insurer decisions of capital structure, asset risk, organizational form, and distribution system as endogenous choices in a single interrelated set of simultaneous equations. The model assesses the nature of the interactions among these decisions. The model also assesses the impact of insurers' fundamental business strategy (treated as predetermined) on these choices. The business‐strategy hypothesis views other key decisions as jointly determined and driven by the fundamental business strategy, once the latter is set in motion. Confirming previous studies, we find a positive relation between capital ratios and asset risk. We also find an association in the simultaneous context between stock ownership and brokerage distribution, which was not found in prior studies. Stock ownership is related to greater financial and asset risk taking, whereas brokerage distribution is associated with lower risk taking. These and other results are interpreted in light of several theories, including transaction‐cost economics (TCE), agency theory, and regulatory and bankruptcy cost avoidance. Deriving from these theories, the finite risk paradigm emerges as the most comprehensive interpretation of the results, as opposed to the risk‐subsidy hypothesis of the impact of guarantee funds. We also find support for the notion that the business strategy drives the capital and distribution decisions, as predicted by TCE.

One of the basic types of portfolio valuation as well as valuation of individual company shares is the CAPM (Capital Asset Pricing) model, which uses a well-known measure of systemic risk in its analysis, which is beta. The CAPM model in its analysis uses the link between the systemic risk measure (beta) and the expected market return. Guided by this model, the analysis of monthly returns of selected shares on the Belgrade Stock Exchange in the period from 2011 to 2021 was performed in this research. In the research, the beta coefficient of selected shares was calculated with the help of the covariance of market return and stock return. The results and their statistical value were confirmed by the linear regression test. The rest of the research tests the applicability of the CAPM model to selected actions and in the same way, the SML (security market line) is devised, which is a graphical representation of the model. The research indicated that the basic assumptions of the CAPM model are not applicable as a predictor of future expected returns of selected shares on the Belgrade Stock Exchange due to various other elements that affect price movements and returns of selected shares not covered by the model.

I. INTRODUCTION The consumption-based capital asset pricing model (CCAPM) has become the dominant paradigm in asset pricing theory because of its intuitive characterization of asset risk. That is, unlike expectations-based asset pricing theories (e.g. the present value model of stock prices) which inherently assume risk neutrality, the CCAPM is predicated upon risk-averse agents attempting to achieve a smooth consumption profile through the trading of assets. From this perspective, the risk of an asset is captured by the covariance of the asset's return with agents' marginal utility of consumption, or, more broadly, agents' intertemporal marginal rate of substitution of wealth. With this characterization, the CCAPM extends the results of the standard CAPM, i.e. the risk of an asset is reflected in the covariance of the asset's return with the market portfolio, to a general equilibrium setting. However, this intuitively appealing theory has not found much empirical support. Perhaps the most famous demonstration of inconsistency between the theory and data was Rajnish Mehra and Edward Prescott's [1985] analysis of the equity premium, i.e. the difference between the average return on equity and that on government bonds. Using calibration methods rather than a formal statistical approach, they showed that a standard CCAPM severely underpredicts the size of the risk premium on equity. The important implication of this finding is that the basic CCAPM incorrectly models the covariance structure between the return on equity and asset holders' intertemporal marginal rate of substitution. Economists' response to this empirical rejection has been to either identify new sources of risk (e.g., that implied by a particular monetary environment as in Labadie [1989]) or define alternative preferences toward risk (e.g., the use of non-expected utility theory as presented in Weil [1989] and Epstein and Zin [1989]). The potential for this research was illustrated recently by George Constantinides's [1990] resolution of the equity premium puzzle in a model that included habit formation in consumption. By relaxing the standard CCAPM's assumption of time-separable preferences, Constantinides demonstrated that a relatively smooth consumption profile and large risk premium on equity are consistent with equilibrium in a complete markets, rational-expectations framework.(1) I use this insight to study another risk premium puzzle; specifically the behavior of the nominal term premium. The existence of a nominal term premium puzzle was demonstrated in papers by Backus, Gregory, and Zin [1989] and Salyer [1990] in which the term structure implications within a Lucas [1982] type cash-in-advance (CIA) economy were studied. While some equilibrium characteristics of this model were consistent with the data, both papers demonstrated that the model predicts the wrong sign of the term premium: the theory implies the term premium should be negative while, based on quarterly data for yields of three- and six-month U.S. Treasury Bills over the period 1959.1-1989.4, the observed term premium is positive. Similar to the equity premium puzzle as posed by Mehra and Prescott [1985], the prediction of a negative average term premium signals a critical discrepancy between the risk associated with nominal bonds as perceived by market participants and as captured by the models. Moreover, this term premium puzzle is in some sense a stronger rejection of the underlying asset pricing model than that embodied by the equity premium puzzle. To see this, recall that within the consumption-based asset pricing model, the sign of the risk premium on any asset depends on the sign of the covariance between the marginal utility of consumption and that asset's return. Applying this reasoning to the case of the equity premium suggests that, while the discrepancy in size between the observed equity premium and that generated within the model is troubling, the model's prediction of a positive equity premium implies that the basic payout structure of equity within the model is consistent with that in the data (e. …