In March 2014, when this book was completed and handed over to publishers, more than five years had passed since the collapse of the US. Yet the consequences of this bankruptcy, which was the height of the subprime crisis, are still being felt. Today.
Introduction
As such, this book is ideally suited for bachelor's and master's students in the field of modern portfolio management as well as for young professionals in the asset and risk management industry. Therefore, in the following sections we present the most important risk and risk-adjusted return measures used in asset management practice.
Measuring Investment Returns .1 Notation.1Notation
- Basic Performance Measures
- Random Variable and Expected Value
Citk D cash amount that the asset pays out at the end of the periodk (eg coupon, dividend). For the month of February 2009, the absolute return of the Lufthansa share can be calculated as follows:
Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
- Sample vs. Population
- Conclusion
- Tracking Error
- Note
- Interpretation
- Conclusion
- Relationship of Tracking Error and Alpha
- Covariance and Correlation
- Notes
- Properties of Covariance and Correlation A covariance has the following properties
- Beta
- Interpretation
- Note
- Conclusion
- Bull and Bear Market Beta
- Interpretation
- Conclusion
- Sharpe Ratio
- Notes
- Interpretation
- Information Ratio
- Note
- Interpretation
- Treynor Ratio
- Interpretation
- Note
- Conclusion
T to N equal subintervals that are usually days or months. benchmark subperiod percentage returns. In this situation, we need to compare the Sharpe ratios of the portfolio and the benchmark.
Advanced Risk and Risk-Adjusted Return Measures
- Maximum Absolute Drawdown
- Maximum Relative Drawdown
- Interpretation
- Conclusion
- Semi-deviation and Semi-variance
- Interpretation
- Conclusion
- Shortfall Probability
- Interpretation
- Sortino Ratio
- Interpretation
- Conclusion
Choice of N, i.e. the choice of sub-periods (daily, weekly, monthly, etc.) plays a key role in the calculation of the maximum absolute drawdown. As with absolute drawdown, the length of the historical time period and the choice of sub-periods are key. Using the same notation as before, the shortfall Pf of the portfolio due to shortfall risk can be calculated as 63.
In practice, it is the relative frequency of a portfolio return that is below the target rate within N subperiods of the historical time periodŒ0; T. For example, a probability of failure of 50% means that in the past the portfolio missed the target return 50% of the time.
Portfolio Return and Volatility
Therefore, the Sortino ratio is a key ratio when evaluating the risk-return ratio of a portfolio in times of crisis, that is, when downside risk is the risk measure to look at and not simply volatility. Then the portfolio volatility can be calculated as. 1.82), is the expected return of the portfolio. The lower the correlation, the lower the volatility of the portfolio, the better the diversification benefits.
To conclude this chapter, we return to our business case and calculate portfolio volatility. If we invest 70% in oil and 30% in Delta, what is the volatility of the portfolio. 1.85) to calculate the volatility of a portfolio where oil is the first asset and Delta shares are the second asset.
Summary
Let's continue to look at our business case using the Delta (DAL) stock and oil. This portfolio has lower volatility than its individual components oil and Delta because the correlation between oil and Delta is negative.
Introduction
Today, the results of the CAPM (and its extended versions) are widely used to describe the risks and returns of portfolios and to measure performance. The concept of regression will be demonstrated by extending our business case from the first chapter. Stock market anomalies will be described in detail in Chapter 3, and historical stock market crashes will be summarized in Chapter 4.
Empirical evidence shows that the returns of a risky asset/portfolio are driven not only by market movements, but also by other risk factors that are not included in the CAPM. As in the CAPM discussion, empirical tests of the FF3M will be presented along with some critical views of this model.
A Quick Review of Regression Analysis
- Simple Linear Regression
- Multi-Linear and Non-Linear Regression
The line (2.2) is drawn in such a way that the error terms"k, i.e. the distances of the points .rOilk; rLCCk /from the line, are kept small on average. 34;k D Yk.aCbXk/: (2.6) The number andbfor the regression lineY DaCbX is chosen as follows, that the sum of the squared error terms Our interpretation of the result is: For every 1% increase (decrease) in the price of oil in a given month, US Airways shares tend to decrease (increase) by the variation in US Airways can be explained by movements in oil prices.
Our interpretation of the result is: For every 1% increase (decrease) in Delta Airlines stock in a given month, US Airways stock tends to increase (decrease) by the variation in US Airways stock can be explained by the movements of Delta Shares of airlines. Therefore, we will present an extension of the CAPM: the Fama–French three-factor model.
The Capital Asset Pricing Model (CAPM) .1 Introduction.1Introduction
- Assumptions
- The Model
- Empirical Tests
- The Testable Hypotheses
- Regressions
- Beta Stability
- Roll’s Critique: The Market Portfolio Problem
- Evaluation of the CAPM
- A Critical View of the CAPM Assumptions
- Zero-Beta CAPM
- Transaction Costs
- Heterogeneous Expectations, Investment Horizons and Taxes The CAPM assumes homogeneous expectations (A1), while in reality, investors
The result was that “the proportional composition of the non-cash assets is independent of their total share in the investment balance. Given the risk-free returnrrf and the market return RMkt, the portfolio's expected return EŒRPf depends only on the systematic risk, the portfolio betaˇPf (i.e., the covariance of the portfolioPf with the market portfolioMkt divided by the variance of the market portfolio), which satisfies the equation . The beta premium,40 ie the difference between the expected return of the market portfolio and the expected return of assets that are not correlated with the market, is greater than zero.
CAPM implies that the beta premium is the excess market return, i.e., the difference between the expected return on the market portfolio and the risk-free rate. The dotted line is the theoretical CAPM line (ie, the SML) which crosses 0.0; 0/ and the market portfolio is also drawn to compare it with the regression line. This process is repeated for the other eight test periods (see table 2.4), the regression coefficients are averaged over all the months of the period and the result for the whole period is obtained.
Roll argues in Roll (1977, p. 130): “The theory is not testable unless the exact composition of the true market portfolio is known and used in the tests.
The Fama–French Three-Factor Model .1 Introduction.1Introduction
- The Model
- Theoretical Explanations of the Fama–French Three-Factor ModelModel
- Empirical Tests
- Other Empirical Tests
Note that this generally differs from the CAPM ˇwhich only takes into account the market sensitivity of the stock. Last but not least, there are also explanations for the size effect of behavioral finance. One behavioral view of the value premium is that investors like growth stocks and value stocks.
Small businesses are most affected by the delay, which covers part of the size premium.122. Compared to the CAPM regression in Table 2.13, the R2 values for the size-only regression and B/M are low.
Summary
Beta and revised returns: Evidence from the German stock market. Journal of International Financial Markets, Institutions and Money. Systematic risk, total risk and size as determinants of stock market returns.Journal of Banking and Finance November–December) On the short-term stationarity of beta coefficients. The Effect of Personal Taxes and Dividends on Capital Asset Prices: Theory and Empirical Evidence. Journal of Financial Economics.
Further evidence on the stationarity of beta coefficients. Journal of Financial and Quantitative Analysis November).Comparative study of CAPM, Fama and French model and reward beta approach in the Brazilian market. On the exclusion of assets from tests of the two-parameter model. Journal of Financial Economics.
Introduction
This chapter provides a summary of the main stock market anomalies that have occurred to date. An example of such anomalies would be the P/E (price-to-earnings) ratio.2 The second type of stock market anomaly refers to the calendar. Stocks seem to perform differently depending on the time of year, holidays, end of month, etc.
For example, unfair competition, regulations or market transparency can be the origin of exploitable deviations. Each section describes the anomaly and provides evidence and explanations on the issue when available.
Weekend Effect
- Description
- Evidence
- Explanations
- Persistence
- Summary
The weekend effect can be defined as a Friday's return minus the following Monday's return for a single security or a portfolio of securities.3 Under normal conditions, there should be no significant difference between each day of the week over a long period of time. The second category includes the shares traded on the value-weighted (VW) index, where the weight of each share is proportional to the company's capitalization in relation to the capitalization of the entire traded index. The discovery of the weekend effect led Kamara to study the S&P 500 from 1962 to 1993.6 He found no significant Monday effect after April 1982 except for a portfolio of smaller U.S.
They are a milestone to buy back what was sold short earlier in the week. However, news releases are not limited to a given type of firm and do not explain more than 3.4% of the weekend effect.15 Therefore, the timing of bad news may not be sufficient to explain this anomaly.
January Effect
- Description and Evidence
- Explanations
- Persistence
Bhardwaj and Brooks as well as Eleswarapu and Reinganum have proven it to occur in the periods 1977–1986 and respectively.24 These findings allowed Robert Haugen and Philippe Jorion to note that “the January effect is perhaps the best-known example of abnormal behavior in the security sector. Other properties of the January effect were revealed in 1986 by Chang and Pinegar. Yet Chen and Singal28, among others, have also identified a December effect, which seems to stem from the requirement that many funds report their investments. at this time of year, but also from investors anticipating possible January gains.29 Fund managers should follow their investment objectives. He is well known and widely quoted for discovering the January effect.
That is why experts believe that these effects should disappear over time.32 Meanwhile, buying stocks in early December or 6 days before the end of the year and selling them on the last trading day of the year can generate consistent abnormal returns. In addition to the January effect, a comparable anomaly occurs at each end of the month: the month-end effect.
Turn-of-the-Month and Holiday Effect
- Description
- Evidence
It is also important to note that the stocks affected by the January effect are quite small. It only reaches the bottom 20% of all stocks traded on organized exchanges and NASDAQ: out of all 6,500 stocks, the anomaly affects only 1,300 stocks with a median market capitalization of about USD 25 million.31 Due to high trading costs, investors will not be able to take advantage of this anomaly. Even index futures, options and mutual funds are useless for reducing trading costs because their capitalization is too small to take advantage of.
