Standard Deviation Of Returns Research Articles

Geometric and Arithmetic Volatility

Volatility as a risk measure has been criticized and dismissed on various grounds. Yet, volatility is the dominating risk measure in the investment management industry. Volatility is typically calculated as the arithmetic standard deviation of returns. At the same time, it is industry practice to use discrete returns, chain-linked annualized discrete returns and geometric average discrete returns in investment reporting. Unfortunately, mixing arithmetic and geometric calculations potentially distort risk and return comparisons. In this research note, we derive geometric and arithmetic versions of volatility. We discuss how they relate and their proper context.

An Overview of the Determinants of Financial Volatility: An Explanation of Measuring Techniques

The majority of asset pricing theories relate expected returns on assets to their conditional variances and covariance’s. Since conditional variances and covariance’s are not observable, researchers have to estimate conditional second moments relying on models. An important concern is the accuracy of these models and how researchers may estimate them more accurately. In this paper, various measures of volatility have been examined ranging from time invariant to time variant measures. In the former case one of the simplest measures examined was the standard deviation. A weakness of this measure is the assumption that volatility is constant, this being due to the standard deviation of returns increasing with the square root of the length of the period. Empirical evidence, however, shows us that the behavior of asset returns in the real world changes randomly over time. This led us to an examination of time variant models for measuring volatility.

Risk, Book to Market and Size Effects in the South African Stock Market

Anecdotal evidence suggests that investment in high risk shares will yield superior returns. In this study companies listed on the Johannesburg Stock Exchange were reviewed over a ten year period. The return of each company, over the first five year period, was used to calculate a standard deviation of return and a beta value for each company. These values represent the risk associated with the shares. The correlation coefficient between the risk and the return over the following five years was calculated. (r = -0.1296, not significant at a 5% level of significance). The study also looks at firm size and the book to market ratio, to determine if those factors are predictors of returns. A significant negative relationship was found between Firm Size and Return (r = -0.1995, which is significant at a 5% level). The Return of small firms significantly outperform large firms. It is recommended that investors should use firm size when making investment decisions.

Annualising Volatility with Serial Correlation: Correcting Sharpe and Information Ratios

Investors who use a risk-adjusted return approach to decision-making, could be making significant errors should they fail to adjust for serial correlation in fund, index and relative return data. The standard deviation of daily, weekly and monthly returns are often annualised using what is known as the “square root of t rule” where the standard deviation of returns is multiplied by the square root of the frequency of the data in order to arrive at an annualised volatility or σ_a=σ_m×√t for monthly data. It is the logical result of additive variances and is only correct under specific circumstances, which are surprisingly rare. As a result, serial correlation in returns data requires an adjustment to the annualised volatility calculation. This paper describes the rationale for this methodology and simple but necessary adjustments for serial correlation which results in a more accurate calculation of annualised volatility, where: σ_a=σ_m×√(t 2(t-1) ρ_(t,t-1) ) or in the extreme σ_a=σ_m×12 in the case of money market returns.

The Statistics of Cross-Sectional Information Coefficient

Information coefficient (IC) is one of the most commonly used statistics in quantitative financial analysis that is usually defined as the correlation coefficient between a variable's predicted and actual values. In this paper, I derive the asymptotic distribution of the sample average cross-sectional information coefficient (IC) when the true underlying IC is time varying. I show that sample average IC divided by sample IC standard deviation approaches the ex ante expected portfolio IR as derived in Ding (2011). I extend the result to cross-sectional factor return and factor return standard deviation. Simulation result is strikingly close to what theory suggests. I also conduct empirical simulation using actual quantitative factors and the relationships between cross-sectional IC, IC standard deviation and sample size are all as predicted by a linear one factor model with time varying IC.

Calibrating Risk Preferences with Generalized CAPM Based on Mixed CVaR Deviation

The generalized Capital Asset Pricing Model based on mixed CVaR deviation is used for calibrating risk preferences of investors protecting investments in S&P500 by means of options. The corresponding new generalized beta is designed to capture tail performance of S&P500 returns. Calibration is done by extracting information about risk preferences from option prices on S&P500. Actual market option prices are matched with the estimated prices from the pricing equation based on the generalized beta. In addition to the risk preferences, an optimal allocation to a portfolio of options for the considered group of investors is calculated. I Introduction The Capital Asset Pricing Model (CAPM, see Sharpe (1964), Linther (1965), Mossin (1966), Treynor (1961, 1962)) after its foundation in the 1960’s became one of the most popular methodologies for estimation of returns of securities and explanation of their combined behavior. This model assumes that all investors want to minimize risk of their investments, and all investors measure risk by the standard deviation of return. The model implies that all optimal portfolios are mixtures of the Market Fund and risk free instrument. The Market Fund is commonly approximated by some stock market index, such as S&P500. An important practical application of the CAPM model is the possibility to calculate hedged portfolios uncorrelated with the market. To reduce the risk of a portfolio, an investor can include additional securities and hedge market risk. The risk of the portfolio in terms of CAPM model is measured by “beta”. The value of beta for every security or portfolio is proportional to the correlation between its return and market return. This follows from the assumption that investors have risk attitudes expressed with the standard deviation (volatility). The hedging is designed to reduce portfolio beta with the idea to protect the portfolio in case of a market downturn. However, beta is just a scaled correlation with the market and there is no guarantee that hedges will cover

Predicting Extreme Returns and Portfolio Management Implications

We consider which readily observable characteristics of individual stocks (e.g., option implied volatility, accounting data, analyst data) may be used to forecast subsequent extreme price movements. We are the first to explicitly consider the predictive influence of option implied volatility in such a framework, which we unsurprisingly find to be an important indicator of future extreme price movements. However, after controlling for implied volatility levels, other factors, particularly firm age and size, still have additional predictive power of extreme future returns. Furthermore, excluding predicted extreme return stocks leads to a portfolio that has lower risk (standard deviation of returns) without sacrificing performance.

Jump Robust Two Time Scale Covariance Estimation and Realized Volatility Budgets

We estimate the daily integrated variance and covariance of stock returns using high-frequency data in the presence of jumps, market microstructure noise and non-synchronous trading. For this we propose jump robust two time scale (co)variance estimators and verify their reduced bias and mean square error in simulation studies. We use these estimators to construct the ex-post portfolio realized volatility (RV) budget, determining each portfolio component's contribution to the RV of the portfolio return. These RV budgets provide insight into the risk concentration of a portfolio. Furthermore, the RV budgets can be directly used in a portfolio strategy, called the equal-risk-contribution allocation strategy. This yields both a higher average return and lower standard deviation out-of-sample than the equal-weight portfolio for the stocks in the Dow Jones Industrial Average over the period October 2007-May 2009.

Is Information Risk Really a Determinant of Security Returns? Evidence from TORQ

Using a sample of NYSE securities, this article investigates whether there is an information risk premium in securities return. The author uses two proxies for information— probability of information trading (PIN) and institutional trading—and determines how those impact securities return after controlling for other known risk factors. Intraday transactions data from TORQ are used to compute PIN, denoting information content in trades from the maximum-likelihood estimates of the parameters of a mixture model. The main finding is that only beta and volatility, denoted by price inverse and standard deviation of returns, respectively, are the primary determinants of security returns, but PIN and institutional trading, proxies for information risk, are not priced in security returns. Beta and volatility are derived from market prices and hence fall under price risk. Turnover, a likely proxy for both liquidity and price risk, is also a significant determinant of return. Contrary to the findings of Easley et al. [2002], this article shows that information risk is not a significant determinant of security return. <b>TOPICS:</b>VAR and use of alternative risk measures of trading risk, statistical methods, exchanges/markets/clearinghouses

Does Prior Performance Affect a Mutual Fund’s Choice of Risk? Theory and Further Empirical Evidence

AbstractRecent empirical studies of mutual fund competition examine the relation between a fund’s performance, the fund manager’s compensation, and the fund manager’s choice of portfolio risk. This paper models a manager’s portfolio choice for compensation rules that can be either a concave, linear, or convex function of the fund’s performance relative to that of a benchmark. For particular compensation structures, a manager increases the fund’s “tracking error” volatility as its relative performance declines. However, declining performance does not necessarily lead the manager to raise the volatility of the fund’s return. The paper presents nonparametric and parametric tests of the relation between mutual fund performance and risk taking for more than 6,000 equity mutual funds over the 1962 to 2006 period. There is a tendency for mutual funds to increase the standard deviation of tracking errors, but not the standard deviation of returns, as their performance declines. This risk-shifting behavior appears more common for funds whose managers have longer tenures.

Indications of Risk: Standard Deviation of Returns or VaR of Terminal Wealth?

Investors with long time horizons are more concerned with the dispersion of their terminal wealth than with within-period volatility of asset returns. The risk of terminal wealth will indicate whether the liquidated value of the portfolio will be sufficient to finance liabilities. In this paper we apply three different strategies to construct portfolios using the historical returns of five asset classes and report their performance over several time horizons. These three investment strategies are: mean-variance optimized portfolios, Roy’s safety first rule, and Value at Risk, VaR to minimize the impact of downsize risk when selecting portfolios. The distributions of terminal wealth generated from these portfolios are used to compare these investment strategies.

Correlation, Return Gaps, and the Benefits of Diversification

Correlation is the common indicator for the benefits of diversification, but it is not a good indicator for two reasons. First, the benefits of diversification depend not only on the correlation between returns, but also on the standard deviation of returns. Second, correlation does not provide an intuitive measure of the benefits of diversification. Return gaps are better indicators. A return gap is the difference in return between two assets or two portfolios. For example, as the authors show, the estimated 12-month return gap between the S&amp;P 500 Index and the Russell 2000 Index from February 2002 to January 2007 was 8.90%. This return gap implied that investors who concentrated their portfolios in one index or the other should have expected to lead or lag investors who diversified between the two in equal proportions by 4.45%. The realized 12-month return gaps ranged from 0.1% to 28.7%. It is difficult to intuitively deduce these figures from the relatively high 0.82 correlation between the two indices. Likewise, it is difficult to intuitively deduce from the relatively high 0.86 correlation between the S&amp;P 500 and EAFE indices that, over this same period, their estimated 12-month return gap was 6.86% and realized 12-month return gaps ranged from 1.8% to 23.0%. Moreover, the figures belie any claim that these assets9 risk-reduction benefits have largely vanished. <b>TOPICS:</b>Accounting and ratio analysis, in markets, global

Option Pricing in P-Space: An Actuarial Approach

Option pricing in the broadest sense of pricing contingent cash flows is clearly a major focus of Actuarial Science. Most of this actuarial pricing work is done using realistic probability distributions and realistic return expectations, yet one of the cornerstones of modern finance, the Black-Scholes equation, uses risk-neutral probabilities and the risk-free rate. This paper provides a methodology for pricing options using realistic probabilities (P-Space) and state-price deflators. It also shows that these P-Space option prices match Black-Scholes option prices when stock returns are assumed to be normally distributed, and proves that in this case the P-Space call equation is equivalent to the Black-Scholes call equation. The paper also provides the connection among the key option pricing variables: expected return, standard deviation of return, and the risk-free rate.

Measuring the economic significance of mean-variance spanning

This paper investigates the economic significance of mean-variance spanning tests using three classical statistical tests in a unified framework. I show how to compute confidence intervals about the Sharpe ratios of tangent portfolios, the variance of return of minimum variance portfolios, as well as the certainty equivalent utility gains. I apply this statistical framework to the question of whether US investors should diversify internationally. The analysis suggests that a strong statistical rejection of the hypothesis that there is no improvement in the minimum variance portfolio’s standard deviation of return does not imply that there are no significant economic benefits to be made in terms of a substantial risk reduction. These results have important implications for empirical tests of mean-variance spanning as well as empirical assets pricing tests and minimum variance bounds on stochastic discount factors.

Day-of-the-Week Effect and Volatility in Stock Returns: Evidence From East Asian Financial Markets

The presence of the day-of-the-week effect has been documented in finance literature. This paper investigates the presence of the day-of-the-week effect and return volatility in ten East-Asian financial markets in the post Asian financial crisis period, after 1998. A set of parametric and non-parametric tests is used to test the equality of mean returns and standard deviations of returns. The results indicate the presence of the day-of-the-week effect and insignificant daily returns volatility in most markets. Some of these results reinforce some previously documented evidence and others are at variance with published results for the same markets. This effect, unlike in devloped markets, is still persistent.

Correlation, Return Gaps and the Benefits of Diversification

Correlation is the common indicator for the benefits of diversification, but it is not a good indicator. This is for two reasons. First, the benefits of diversification depend not only on the correlations between returns but also on the standard deviations of returns. Second, correlation does not provide an intuitive measure of the benefits of diversification. Return gaps are better indicators. Return gaps are the difference between the returns of two assets or between two portfolios. For example, the estimated 12-month return gap between the S&P 500 Index and the Russell 2000 Index and during February 2002 - January 2007 was 8.90%, implying that investors who concentrated their portfolios in one index or the other should have expected to lead or lag investors who diversified between the two in equal proportions by 4.45%. The realized 12-month return gaps ranged from 0.1% to 28.7%. It is hard to deduce these figure intuitively from the relatively high 0.82 correlation between the two. Similarly, it is hard to deduce intuitively from the relatively high 0.86 correlation between the S&P 500 and EAFE Indexes that their estimated 12-month return gap was 6.86% and their realized 12-month return gaps ranged from 1.8% to 23.0%. Moreover, the figures belie any claim that these assets' risk-reduction benefits have largely vanished.

A Simple Model for the Expected Premium for Hedge Fund Lockups

What excess return should a fund of funds expect to earn for investing in a hedge fund with an extended lockup? In this paper, we present a simple model for estimating the premium for long-term lockups. Because there is a demonstrated statistical persistence to the quality of hedge fund returns within a particular hedge fund strategy — above average funds tend to continue to do well, and below average funds continue to falter — a lockup deprives an investor of the opportunity to redeem an investment in poorly performing funds and reinvest the proceeds in successful ones. The value of that lost future opportunity is the expected premium for committing to the lockup. We estimate the value of the premium for multi-year lockups in a variety of strategies using a discrete-time Markov chain model for the evolution of hedge funds, in which a hedge fund at any time can be in one of three states: Good, Sick or Dead. For convertible bond funds, for example, the return premium for a two-year lockup over a one-year lockup is approximately 48 basis points (b.p.) per year. The premium rises to 76 b.p. per year for a three-year lockup, and approaches a limit of 156 b.p. as the duration of the lockup becomes infinite. The premium is proportional to the standard deviation of the returns themselves, so that strategies with greater standard deviations of returns require greater compensating lockup premiums.

Time Series Volatility Forecasts for Option Valuation and Risk Management

Option pricing models and longer-term value-at-risk models typically require volatility forecasts over horizons considerably longer than the data frequency. These are generally generated from short-horizon forecasts by successive forward substitution. We document deficiencies with the resulting long-horizon volatility predictions generated by GARCH type models, such as GARCH(1,1), EGARCH, and GJR. One, since volatility forecasts for forward periods are functions of forecast volatility for the next period, this recursive procedure keeps the relative weights of recent and older observations the same whether forecasting volatility in the near or distant future. In contrast, we find that older observations are relatively more important in forecasting at long horizons, e.g., more important in forecasting volatility next month than in forecasting volatility tomorrow. Two, forecasts of the return standard deviation - the most appropriate volatility measure for option valuation and value-at-risk models - are strongly positively biased. Three, GARCH(1,1) and GJR forecasts are especially biased following high volatility days. We find that the ARLS model of Ederington and Guan corrects these three deficiencies and generally forecasts better out-of-sample than GARCH, EGARCH, AGARCH and the GJR models across a wide variety of markets and forecast horizons.

The risk and return characteristics of developed and emerging stock markets: the recent evidence

Finance theory suggests that the higher volatility typically associated with emerging stock market returns translates into higher expected returns in those markets. This study compares the risk and return profile of emerging and developed stock markets over the period from 1988 through April 2003. Specifically, this study investigates whether a difference in risk characteristics exists between the two markets and whether the realized rates of return in these two types of markets reflect these risk characteristics. The results show that the risk associated with emerging markets, as measured by the standard deviation of returns, is higher than the risk in developed markets in most periods. Also, the returns in emerging markets have been higher than those in developed markets for most of the time frames examined. The findings suggest that risk-averse investors seeking higher returns in emerging markets have been compensated for assuming the higher risk associated with these markets.

Equity Returns at the Turn of the Month: Trading Strategies and Implications for Investors and Managers

We propose to extend and complete the attached study of the effect in equity returns in two ways. First, we propose to study whether the effect is more pronounced among certain industries where industries are determined by SIC codes and/or the Fama-French industry classifications and/or whether the effect is more pronounced in certain time periods especially periods of 'high' investor sentiment. Second, we propose to investigate whether the effect can be used to generate a profitable trading strategy after trading costs. This latter investigation will examine three low cost strategies: 1. a strategy of switching between an equity fund and a bond fund of the same mutual fund family; 2. a strategy of buying and selling exchange traded funds (ETFs), either specific styles of ETFs or market-based ETFs; and 3. a strategy of buying and selling equity index futures. We propose to consider the implications of these for fund managers especially funds that might be exposed to being picked off by a turn-of-the-month trading strategy. The turn-of-the-month effect in U.S. equity returns was initially identified by Ariel (1987), who studies CRSP market returns over the period 1963-1982, and Lakonishok and Smidt (LS whereas over the other 16 trading days, it is -0.001%. When our results are combined with those of LS it is not concentrated at calendar year-ends or quarter-ends; and it is not confined to the US: of the 34 countries we study, the effect occurs in 30 of them. Further the effect is not due to risk as measured by standard deviation of returns. Finally, we find that the effect is not due to increased buying pressure measured by trading volume and net flows to mutual funds. We propose to extend and complete this study as described above.