Abstract
This chapter presents the mathematical prerequisites for measuring risk and return in asset management. It provides basic information together with many exercises and case studies and describes traditional and non-traditional measures: volatility and Sharpe ratio are traditional measures for absolute portfolio management, whereas tracking error, covariance, correlation, beta, bull and bear market beta, information ratio and Treynor ratio apply for relative portfolio management. The non-traditional measures include maximum absolute drawdown for absolute portfolio management and maximum relative drawdown, semi-deviation and -variance, shortfall risk and Sortino ratio for relative portfolio management. The chapter concludes with the calculation of the return and the volatility of a portfolio.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
An integer is a number that can be written without a fractional or decimal component. For example, 1, 55, −11, 0, and 1,270 are integers, but 0.25 or 9.76 are not. The name derives from the Latin word integer, meaning literally untouched, hence whole. Source: http://www.wikipedia.org.
- 2.
- 3.
Esch, Kieffer, and Lopez (2005, p. 36).
- 4.
Esch et al. (2005, p. 36).
- 5.
Chincarini and Kim (2006).
- 6.
See the non-formal definition of random variable in DeFusco, McLeavey, Pinto, and Runkle (2004, p. 232). The book mentioned here is the standard reference for the quantitative part of the CFA program. It offers a good summary of the basic mathematical/statistical methods in finance. This book is helpful as a starting point for those with a non-mathematical background who would like to get some basic understanding in this topic without too much mathematical formality. The material in this book is also useful for the preparation of the CFA exam if you need to review quantitative methods.
- 7.
DeFusco et al. (2004, p. 194).
- 8.
The use of bull and bear to describe markets comes from the way the animals attack their opponents. A bull thrusts its horns up into the air while a bear swipes its paws down. These actions are metaphors for the movement of a market. In a bull market, the trend is up, in a bear market, the trend is down. Source: http://www.investopedia.com.
- 9.
- 10.
Lowenstein (2002).
- 11.
- 12.
DeFusco et al. (2004, p. 195).
- 13.
DeFusco et al. (2004, p. 195).
- 14.
Also simply called mean or average.
- 15.
Esch et al. (2005, p. 41).
- 16.
Compare Esch et al. (2005, p. 41).
- 17.
Please note that no further mathematical analysis on this will be provided here. For further details, please see the basic introductions in Schulmerich (2010a, Chap. 2), Schulmerich (2005, Chap. 2), Schulmerich (2008a, Chaps. 1 and 2) or Schulmerich (1997, Chaps. 1 and 2). For a rigorous mathematical introduction to this topic see, for example, Neftci (2000) or Øksendal (1995).
- 18.
Lhabitant (2004, p. 59).
- 19.
Lhabitant (2004, p. 59).
- 20.
Natenberg (1994, pp. 60–61).
- 21.
A fundamentally managed portfolio is a portfolio that follows the fundamental approach to investing. This means that fundamental portfolio managers when constructing a portfolio use the evaluations of fundamental analysts who gather and analyze information on potential investments, for example, by examining balance sheets and income statements.
- 22.
A quantitatively managed portfolio is a portfolio that follows the quantitative approach to investing. Quantitative analysts develop computer algorithms that evaluate the potential return of an investment. Quantitative portfolio managers use this return estimation in order to achieve an optimal portfolio using portfolio construction software that includes trading costs and risk limits.
- 23.
A detailed comparison between the fundamental and quantitative approach to asset management can be found in Glavin and Reinganum (2013), Schulmerich, Hooker, McGoldrick, and Mallik (2008), Hooker and Schulmerich (2008) and Schulmerich and Hooker (2008). Interesting articles about quantitative equity investing in crisis times or shortly thereafter can be found in Schulmerich (2008b), Schulmerich et al. (2009), Schulmerich (2009) and Schulmerich (2010b).
- 24.
Grinold (1989).
- 25.
Richard C. Grinold, Ph.D., was until 2009 managing director of the Advanced Strategies and Research group at Barclays Global Investors . Dr. Grinold was for 20 years on the faculty at the School of Business Administration at the University of California, Berkeley. He has published extensively and is widely known in the industry for his pioneering work on risk models, portfolio optimization, and trading analysis; equity, fixed income, and international investing; and quantitative approaches to active management.
- 26.
Roger G. Clarke, Ph.D., is the chairman of Analytic Investors. Recognized as an authority with more than 20 years experience in quantitative investment research, Roger Clarke has authored numerous articles and papers including two tutorials for the CFA Institute. He also served on the faculty of Brigham Young University for 8 years where he specialized in investment and options theory and continues to lecture as a guest professor.
- 27.
Harindra de Silva, Ph.D., CFA, is the president of Analytic Investors and a portfolio manager. De Silva has authored several articles and studies on finance-related topics including stock market anomalies, market volatility and asset valuation.
- 28.
Steven Thorley, Ph.D., CFA, is the H. Taylor Peery Professor of Finance at the Marriott School of Management at Brigham Young University in Provo, UT.
- 29.
This is most common in retail funds, i.e., funds available for public distribution.
- 30.
Analytic Investors, LLC was founded in 1970. The original firm was known for its expertise in derivatives strategies. Nowadays, Analytic is part of Old Mutual Asset Management (U.S.), a group of affiliate firms selected by Old Mutual that have complementary investment styles (non-overlapping) and are considered top-quality investment management firms.
- 31.
This is, for example, done by using risk models like BARRA. It is a highly mathematical task, so we do not go into further detail here.
- 32.
Sometimes also called bet.
- 33.
If these costs are already subtracted, alpha is called net-of-fee alpha .
- 34.
Assuming the portfolio manager is not restricted to invest only in benchmark securities, i.e., off-benchmark positions are allowed.
- 35.
Also known as IMA, Investment Management Agreement .
- 36.
DeFusco et al. (2004, p. 204).
- 37.
DeFusco et al. (2004, p. 207).
- 38.
Based on Lhabitant (2004, p. 127).
- 39.
Esch et al. (2005, p. 42).
- 40.
Hull (2009, p. 284). We assume 252 trading days per year.
- 41.
DeFusco et al. (2004, pp. 207–208).
- 42.
Also called scatter chart, scattergram, scatter diagram or scatter graph.
- 43.
Esch et al. (2005, p. 91).
- 44.
For more information see, for example, Woodward and Anderson (2009).
- 45.
Lhabitant (2004, p. 65).
- 46.
If we include a benchmark, the risk-adjusted return measure is a relative measure versus a benchmark called information ratio . This concept is explained in Sect. 1.3.8.
- 47.
William F. Sharpe was born on June 16, 1934, in Boston, MA/USA. He is the STANCO 25 Professor of Finance, Emeritus at Stanford University’s Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences. Sharpe was one of the originators of the capital asset pricing model (CAPM) and created the Sharpe ratio for risk-adjusted investment performance analysis. He contributed to the development of the binomial method for the valuation of options, the gradient method for asset allocation optimization, and returns-based style analysis for evaluating the style and performance of investment funds.
- 48.
Sharpe (1966, p. 123).
- 49.
Lhabitant (2004, p. 67).
- 50.
Jack Treynor was born 1930 and is a U.S. financial engineer and portfolio manager. He studied mathematics at Haverford College and completed the MBA program at Harvard Business School 1955. In 2007, the International Association of Financial Engineers (IAFE) named Treynor as the 2007 IAFE/SunGard Financial Engineer of the Year (FEOY), recognizing him for his preeminent contributions to financial theory and practice, particularly the essence of the CAPM.
- 51.
Jorion (2001, p. 395).
- 52.
Lhabitant (2004, p. 75).
- 53.
Spremann (2008, p. 381).
- 54.
A drawdown diagram shows the development of an index whereby positive subinterval returns are only taken into account if the index value is below 1. If a subinterval return is positive while the index value is already 1, the index value will remain at 1. A drawdown diagram shows the losses an index suffers and how long it takes to regain these losses and reach the initial investment of 1.
- 55.
Portugal, Italy, Ireland, Greece and Spain were often referred to as the PIIGS countries. Common to all of them were the increasing debt ratios that in many cases already had been high even before the crisis.
- 56.
See page 77 for more details.
- 57.
Nassim N. Taleb (born January 1, 1960, in Amioun, Lebanon) is an essayist, scholar and former practitioner of mathematical finance. He is best known as the author of the book The Black Swan. Taleb has pursued three distinct careers. Firstly, he is a bestselling author with about 3 millon copies sold in over 30 languages. Secondly, he is a university professor in risk engineering, a scholar, an epistemologist and a philosopher of science. Finally, he is a former senior Wall Street trader, risk expert, and practitioner of mathematical finance. The Black Swan has been described by The Times as one of the 12 most influential books since World War II.
- 58.
The Black Swan theory or theory of Black Swan events, was developed by Nassim N. Taleb. It explains the disproportionate role of high-impact, hard-to-predict, and rare events that are beyond the realm of normal expectations in history, science, finance and technology. Unlike the earlier philosophical Black Swan problem, Taleb’s Black Swan theory refers only to unexpected events of large magnitude and consequence and their dominant role in history. Such events, considered extreme outliers, collectively play vastly larger roles than regular occurrences.
- 59.
CRSP is the Center for Research in Security Prices at the University of Chicago.
- 60.
Lhabitant (2004, pp. 55–56).
- 61.
Lhabitant (2004, p. 51).
- 62.
In this example a static target rate of return is used. However, the mechanics remain the same for a non-static return target. Often, the benchmark return is used as the target.
- 63.
Bacon (2008, p. 94).
- 64.
The Sortino ratio was devised by Brian M. Rom, founder and president of the software development company Investment Technologies, in 1983. The ratio is named after Dr. Frank A. Sortino, an early popularizer of downside risk optimization.
- 65.
Sortino and Price (1994).
- 66.
For the formula, see Reilly and Brown (1997, p. 254).
- 67.
Reilly and Brown (1997, p. 261).
References
Bacon, C. R. (2008). Practical portfolio performance measurement and attribution (2nd ed.). Chichester: Wiley Finance.
Chincarini, L. B., & Kim, D. (2006). Quantitative equity portfolio management. New York: McGraw-Hill.
Clarke, R. G., de Silva, H., & Thorley, S. (2002, September/October). Portfolio constraints and the fundamental law of active management. Financial Analysts Journal, 58, 48–66.
DeFusco, R. A., McLeavey, D. W., Pinto, J. E., & Runkle, D. E. (2004). Quantitative methods for investment analysis (2nd ed.). Charlottesville: CFA Institute.
Esch, L., Kieffer, R., & Lopez, T. (2005). Asset and risk management. Hoboken: Wiley Finance.
Glavin, B., & Reinganum, M. R. (2013). Get active: Two approaches to actively managing equities. State Street Global Advisors.
Grinold, R. C. (1989, Spring). The fundamental law of active management. Journal of Portfolio Management, 15(3), 30–37.
Hooker, M., & Schulmerich, M. (2008, March). Aktives quantitatives Aktienmanagement in 2008. Germany: DPN.
Hull, J. C. (2009). Options, futures, and other derivatives (7th ed.). Upper Saddle River: Pearson Prentice Hall.
Jorion, P. (2001). Financial risk manager handbook (1st ed.). New York: John Wiley & Sons.
Lhabitant, F. S. (2004). Hedge funds quantitative insights. Chichester: John Wiley & Sons.
Lowenstein, R. (2002). When genius failed: The rise and fall of Long-Term Capital Management. London: Fourth Estate.
Markowitz, H. M. (1959). Portfolio selection. New Haven: Yale University Press.
Markowitz, H. M. (2008). Portfolio selection. München: FinanzBuch Verlag.
Natenberg, S. (1994). Option volatility and pricing. New York: McGraw-Hill.
Neftci, S. N. (2000). An introduction to the mathematics of financial derivatives (2nd ed.). San Diego: Academic Press.
Øksendal, B. (1995). Stochastic differential equations (4th ed.). New York: Springer.
Reilly, F. K., & Brown, K.C. (1997). Investment analysis and portfolio management. Chicago: The Dryden Press.
Reinganum, M. R. (2010). Overcrowding or dynamic missteps? Why active quant equity performance was challenged. State Street Global Advisors.
Schulmerich, M. (1997, September 9). Statistische Verfahren für Diffusionsprozesse mit Anwendung auf stochastische Zinsmodelle der Finanzmathematik. Master’s thesis in Mathematics.
Schulmerich, M. (2005). Real options valuation (1st ed.). New York: Springer.
Schulmerich, M. (2008a). Statistische Verfahren für Diffusionsprozesse mit Anwendung auf stochastische Zinsmodelle der Finanzmathematik. GRIN Verlag. Master’s thesis in Mathematics from 1997 published 2008, School of Mathematics, Johannes Gutenberg-University Mainz, Germany.
Schulmerich, M. (2008b, January 3). Quantitative Anlagemodelle in einem unsicheren Umfeld. Neue Zürcher Zeitung.
Schulmerich, M. (2009, November). Quantitative investing: Seizing opportunities in the European equities market. IPE Magazine.
Schulmerich, M. (2010a). Real options valuation (2nd ed.). Berlin: Springer.
Schulmerich, M. (2010b, June/July). Warum quantitative Anlagen? Wir sind nicht voreingenommen!. Germany: DPN.
Schulmerich, M. (2013). Can the Black-Litterman framework improve asset management outcomes? State Street Global Advisors. SSgA Capital Insights.
Schulmerich, M., & Hooker, M. (2008). Quantitative and fundamental active equities in institutional portfolios. In SSgA Investment Quarterly, State Street Global Advisors (pp. 1–3).
Schulmerich, M., Andres, P., Baur, B., Bornemann, H.-O., Kempf, A., Kleeberg, J., et al. (2009, April/May). Roundtable discussion: Quant management. Germany: DPN.
Schulmerich, M., Hooker, M., McGoldrick, H., & Mallik, G. (2008). Comparing quantitative and fundamental approaches to active equity investing. State Street Global Advisors.
Sharpe, W. F. (1966, January). Mutual fund performance. Journal of Business, 39(1), 119–138 [Part 2: Supplement on Security Prices].
Sortino, F. A., & Price, L. N. (1994, Fall). Performance measurement in a downside risk framework. Journal of Investing, 3(3), 59–64.
Spremann, K. (2008). Portfoliomanagement (4th ed.). München: Oldenbourg Verlag.
Taleb, N. N. (2010). The black swan (2nd ed.). New York: Random House Trade Paperback.
Woodward, G., & Anderson, H. (2009). Does beta react to market conditions? Estimates of bull and bear betas using a nonlinear market model with endogenous threshold parameter. Quantitative Finance, 9(8), 913–924.
Zagst, R., Goldbrunner, J., & Schlosser, A. (2010). Zu nah an der Sonne. München: FinanzBuch Verlag.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Schulmerich, M., Leporcher, YM., Eu, CH. (2015). Risk Measures in Asset Management. In: Applied Asset and Risk Management. Management for Professionals. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55444-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-55444-5_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-55443-8
Online ISBN: 978-3-642-55444-5
eBook Packages: Business and EconomicsEconomics and Finance (R0)