Risk Analysis: A Geometric Approach
|
|
- Reynard Bryant
- 6 years ago
- Views:
Transcription
1 Risk Analysis: A Geometric Approach Brian D. Singer, CFA Managing Director Brinson Partners, ncorporated Quantitative methods for risk management should allow investors and portfolio managers to look at, and try to manage, risk in new ways. A geometric approach can help in displaying the risk characteristics of a portfolio and its benchmark and in assessing the impact of portfolio constraints. R isk management and quantitative methods are typically considered to be almost interchangeable, even to the extent that risk management requires or depends on quantitative sophistication. Although a quantitative perspective can certainly be useful in risk management, too often quantitative methods and analytical elegance provide the illusion of controloverrisk. Such approachesmake investors feel as if they have grasped uncertainty and dealt with it simply by the act of quantifying it. But risks, by their very nature, are unexpected. So, quantitative methods should not turn confidence into arrogance; rather, what quantitative methods should do is allow investors (and portfolio managers) to look at risk in new ways-to try to manage risk in ways that they previouslycould notdo, or werenotnecessarily comfortable doing. This presentation discusses a process that uses Euclidean geometry to visualize risk. Such an approach is somewhat avant-garde for risk management and is decidedly quantitative, but the intent is to illustrate a tool that enables quantitative risk managers to communicate with nonquantitative portfolio managers and nonquantitative clients. Although the approach applies to any investment horizon, this discussion takes a relatively long-term perspective on risk-one that is typical of an investment policy perspective-and allows direct analysis of portfolio risk and portfolio constraints. Risk Estimation: Data Risk estimation relies on volatility and correlation data to construct covariance matrixes; one of the key questions, of course, is which data? Historical Data. Risk estimation typically begins with the use of historical data-the computa- tion of historical volatilities and correlations. Historical data are consistent, easy to obtain, and often easy to compute. A manager, for instance, can compute a covariance matrix very easily with Microsoft Excel. The problem with historical data is that the data are almost certain to be inappropriate representations of the future. An investor looking at a broad index of the U.S. bond market, such as the Lehman Brothers Aggregate Bond ndex, would find that the volatility of that index was in excess of 10 percent in the late 1970sand early 1980s. Currently, that same index has a volatility of 4-5 percent. That historical period (late 1970s and early 1980s) was characterized by high and volatile inflation, which is not the case now. Thus, that investor would not want, in any forward-looking sense, to rely on that period as a foundation for his or her risk estimates unless the investor believed, for example, that the U.S. Federal Reserve Board was planning to monetize, or in effect provide an inflation tax for, fiscal policy. This is a very real investment problem: Suppose at Brinson Partners we are trying to set the investment policy-the normal policy mix-for a pension plan, an endowment, or a foundation. n that instance, the client's time horizon is long term, so we do not want to know a daily or weekly value at risk estimate. History might not necessarily represent what we think could happen in the future, but an analysis of monthly or quarterly data going back several decades aids in our understanding of risk in various economic and market environments. Granted, a numberof advances in statisticalmethodology applied to historical data have occurred: the use of volatility clustering, the use of generalized autoregressive conditional heteroscedasticity, and the Association for nvestment Management and Research 73
2 Risk Management: Principles and Practices use of mean reversion for forecasting volatilities. All of those historical approaches have been beneficial for estimating risk, especially over short horizons, but investors are still faced with regime changes-some notable, some not notable; some identifiable, some not identifiable-and every regime change decreases the relevance of historical data. An interesting example comes from New Zealand, which for years suffered from high and volatile interest rates and, therefore, volatile bond returns. n an attempt to change that environment, New Zealand altered the charter for its central bank. The New Zealand central bank now provides an inflation target, and if the head of the central bank does not meet that target, he or she is fired. New Zealand's inflation volatility is now much lower than it was in the past. Forward-Looking Data. Because such regime changes are possible, forward-looking volatilities and correlations can be, although are not always, better representations of the future. That regime change in New Zealand was quick and identifiable, buthistorical data would not have predicted it. Thus, having some type of forward-looking perspective, in terms of the covariance matrix, is a good idea. But forward-looking matrixes also have problems, the biggest one being limitations on human imagination. n a forward-looking sense, people can only incorporate what they imagine,but risks, by their nature, are unexpected. Therefore, it is difficult to incorporate the appropriate forward-looking events or regime changes that might affect the covariance matrix. Geometric Representation. Other important difficulties with using a forward-looking perspective are achieving consistency and intuition, which is where Euclidean geometry comes in. A geometric interpretation of volatilities and correlations has the potential to make risk estimation consistent, practical, and more intuitive to understand and communicate, especiallybetweenquantitative-oriented people and non-quantitative-oriented people. Why take a geometric approach? The mathematician Keith Devlin has commented that mathematicians may be able to express their thoughts using the language of algebra, but generally, they do not think that way. Even a highly trained mathematician may find it hard to follow a long, algebraic argument. But every single one of us is able to manipulate mental pictures and shapes with ease. By translating a complicated problem into geometry, the mathematician is able to take advantage of this fundamental human capability. This ability to manipulate shapes starts in childhood. Children at a very young age learn to put round pegs in round holes, square pegs in square holes, and triangular pegs in triangular holes. They understand and learn to manipulate shapes long before they are able to grasp algebra and other mathematical concepts. Currency Risk Assume that from a U'S. dollar perspective the U.K. pound has a volatility of 12 percent and the German mark also has a volatility of 12 percent. The correlation between the pound and the mark is Although with this information an investor can construct a very simple covariance matrix, another way of looking at the covariance matrix is geometrically, as shown in Figure 1. Volatilities are shown as distances, and correlations are shown as angles. To construct Figure 1 from a U.S. dollar perspective, first made a point for the dollar. Because the volatility of the mark compared with the dollar is 12 percent, drew a line of length 12. That line could be 12 inches, 12 centimeters, or 12 kilometers; it does not matter, just 12 units of length. The volatility of the pound against the dollar is also 12 percent, so needed to draw another line of length 12. The question is, what is the relationship (or the angle) between those two lines? The answer is that the relationship is determined by the correlation, which is 0.71 in this example; specifically, the cosine of the angle is equal to The cosine of 45 is 0.71, so drew the second line at a 45 angle to the first line. Thus, have portrayed the same covariance matrix, but instead of Figure 1. Visual Representation of Volatilities and Correlations from a U.S. Dollar Perspective Dollar 12.0% cos(a) = P( /DM) = % Note: p is the correlation coefficient. Pound Mark 74 Association for nvestment Management and Research
3 Risk Analysis using strictly numbers, have portrayed it as part of a triangle.' One of the tools we use at Brinson Partners is what refer to as the correlation protractor. John Zerolis, one of the more quantitative-oriented people at Brinson Partners, generated the protractorby computing the correlation associated with each angle. We use the protractor for discussions in which immediate visual representations are useful. One of the interesting things people notice when looking at that correlation protractor is that not all correlation changes are created equally in risk space. Suppose a correlation goes from 0.9 to 1.0. This might not seem like a big change, but moving from 0.9 to 1.0. is a 26 angle on the protractor. Similarly, a 26 angle from zero moves the correlation from zero to about n risk space, a movement in correlation from zero to 0.45 is similar to a movement in correlation from 0.9 to 1.0. This relationship is readily apparent from a geometric representation but not at all obvious from a set of formulas. Now suppose want to know the volatility of the pound from a mark perspective. All need to do is draw a line connecting the pound and the mark and measure the length of that line. The dotted line in Figure 1 indicates that the length is 9.1 (hence the volatility is 9.1 percent). n addition, if look at the angle between the dotted line and the solid dollar/ mark line, can tell that from a mark perspective, the dollar and the pound have a correlation of This technique facilitates the ability to understand a single covariance matrix from the perspective of everyinvestorin the world, regardless of the investor's base currency. For U'S. investors, we focus on the dollar vertex of the triangle. For German investors, we focus on the mark vertex, and so on. We can use any number of different base currencies. With just three currencies, we can geometrically represent the correlation matrix on a piece of paper; with four currencies, we would need a three-dimensional tetrahedron. With five currencies, visualization must occur in triangular or tetrahedral subsegments, but the intuition is still the same. One benefit of this approach, in terms of consistency, is being able to see the implications of a covariance matrix from any base currency perspective. Suppose we think that the United Kingdom is going to join theeuropeanmonetaryunion (EMU) andthat thepound'scorrelationwith the euro (represented by the mark) will probably increase to 0.95 as the United For further discussion, see Brian D. Singer, Kevin Terhaar, and John Zerolis, "MaintainingConsistentGlobal Asset Views (with a Little Help from Euclid)," Financial Analysts Journal (January February 1998): Kingdom approaches joining the EMU. Figure 2 shows what happens if the correlation between the pound and the mark is 0.95, which corresponds to an angle of about 18 : A correlation of 0.95 means that the pound must have a volatility of 3.8 from a mark perspective. So, compared with Figure 1 (where the correlation was 0.71), the volatility dropped from 9.1 to 3.8. f we are not comfortable with that change in volatility, then we cannot be comfortable with our correlation estimate of Notice thatifthe correlation between the pound and the mark were 1.0, the line wouldessentiallybecome flat, whichimplies that the pound would have no volatility from a mark perspective. Figure 2. Dollar Effect of Correlation Change Note: p is the correlation coefficient. (T Porrjo/io cos(().) = PPortfolio, ~(T rr <TBmchmark Portfolio Pound.3.8% Mark Portfolio Risk Analysis Risk analysis of a portfolio relative to its benchmark is a simple application of this geometric approach, as shown in Figure 3. The volatility of the benchmark (benchmark risk) is represented by the base of the triangle, the volatility of the portfolio (portfolio risk) is represented by the side of the triangle drawn with the solid line, and the portfolio's tracking error is represented by the side of the triangle drawn with a dotted line. The correlation between the benchmark and the portfolio is represented by the angle o; and Figure 3. Portfolioand Risk Analysis in Geometric Terms (l-l3)u Association for nvestment Management and Research 75
4 Risk Management: Principles and Practices the vertical dashed line indicates the portfolio's residual risk. Portfolio,, and Residual Risks. n the context of a single-index model, the return of the portfolio is equal to a benchmark bet and the residual return, which is uncorrelated with the benchmark. Similarly, portfolio volatility comes from twosources-one that is perfectly correlated with the benchmark bet, or systematic risk, and one that is uncorrelated with the benclunark risk, or residual risk. So, in Figure 3, the line for residual risk is at a right angle to the line for the benchmark risk because a right angle is associated with a correlation of zero. Thus, Figure 3 shows two right triangles, and consequently, the Pythagorean theorem (the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides) can be used to help withthe risk analysis. This approachallows us to look at the risks visually. We do not have to wonder what will happen if the residual risk of our portfolio goes up by 10 percent: The residual risk line will become 10percent longer, the correlation of ourportfoliowith the benchmark will go down because the angle will increase, and the volatility of our portfolio will increase (the portfolio line will lengthen). We do not have to calculate anything to achieve that intuitive understanding. Residual Risk, Relative Bets, and Tracking Error. The right-hand, shaded triangle in Figure 3 shows residual risk, benchmark relative bets, and tracking error. Tracking error can be thought of in this context as value-added risk-the riskof the portfolio from the perspective of the benchmark, or the risk of the difference between the portfolio returns and the benchmark returns. n essence, what this figure indicates is that tracking error is a combination of two things: (1)the benchmarkrelative bet (the base of the shaded triangle), which is the portion of active management that involves an increase or decrease (as in this example) in benchmarkexposure, and (2)the residual risk, whichis that portion of the risk of active management that is not in any way correlated with the benchmark. Ex Post Analysis. n a portfolio performance sense (an ex postsense), this geometric approach is a tool that can help investors understand the overall performance of their portfolios a little more intuitively and clearly. Suppose am a plan sponsor and one of my managers comes to me and says, "The benchmark has a volatility of x, the portfolio has a volatility of y, and the beta is Z." From that information, not only can compute the correlation, determine the cosine, and create the entire triangle, including the tracking error and residual risk (even though the manager did not give me that information), but can also quickly visualize and gauge the tracking error and residual risk without doing any computations. Although could use algebraic or trigonometric formulas to calculate the tracking error, using geometry is often easier because it allows me to visualize how the various risks move relative to each other. As in Figure 3, can see that the residual risk of the portfolio is found by dropping a perpendicular line down from the point for the portfolio; can see that the base of the left-most triangle, found by multiplying the beta times the benchmark risk, is the systematic risk. Ex Ante Analysis. From an ex antestrategyperspective, the geometric view also helps in making decisions about changing the portfolio. For example, it can help investors understand how a certain strategy or change in strategy might affect a portfolio in absolute and/or relative risk terms. Say we have a portfolio that holds some cash, and we think that taking out that cash might reduce the tracking error. fwe take out the cash, we do not change the portfolio's correlation with the benchmark. All we do is increase the risk, so we have to make the line for the portfolio longer. Does increasing the portfolio line decrease the tracking error? Not necessarily. The tracking error decreases to a point but then begins to increase again. Having a risk hedge in the portfolio might reduce risk or it might increase risk relative to the benchmark, whichis easy to see from a geometric, visual perspective. Portfolio Constraints Portfolio managers more often than not operate under a variety of constraints, such as beta, tracking error, or residual risk. But implementing those constraints can cause difficulties. Once again, geometric interpretation can be used to portray feasible sets of alternative portfolios that are consistent with client constraints. Beta Constraint. Suppose a client wants an essentially defensive portfolio. Figure 4 shows four portfolios that have a beta of 0.9. can create many portfolios that have a beta of 0.9, and some of those portfolios might be considered defensive, but some would not. Portfolio A, for example, would likely be considered defensive. t has a low volatility, and it has relatively low tracking error and relatively low residual risk. When the correlation with the benchmark is decreased while maintaining the beta of 0.9, the volatility of the portfolio has to increase. Portfolio D still has a beta of 0.9, but its volatility greatly exceeds that of the benchmark, and it has a relatively 76 Association for nvestment Management and Research
5 Risk Analysis Figure 4. Portfolios with Constant Beta of (J' Ben,hmark J"Bl'lchmark 0.1 (J' Residual Risk DT C 1 B + A+ ~ = il substantial tracking error. All of the portfolios in Figure 4 have a beta of 0.9, but they are very different portfolios; the beta constraint still allows dramatically different levels of portfolio risk, residual risk, and tracking error. Tracking-Error Constraint. Suppose a manager or a plan sponsor wants a portfolio with a tracking error of 5 percent or less. Again, can create a number of portfolios that have a tracking error of 5 percent, as shown in Figure 5. simply draw a circle with a radius length of 5, representing a tracking error of 5 percent, around the benchmark position. Portfolio A, which can be formed by combining the benchmark with cash, has a tracking error of 5 percent; it also has no residual risk and a relatively low volatility. Portfolio D has a beta of 1; its residual risk Figure 5. Portfolioswith ConstantTracking Error Less than 5 Percent Portfolios D.. E B..". :.: 0. '.. ~ " ~ 2\ (1 ~.::.... F 5% would have a volatility of 10 percent and Portfolio F would have a volatility of 20 percent. Figure 5 clearly illustrates that a tracking-error constraint still permits wide variations in volatility, beta, and residual risk. Residual Risk Constraint. can also construct a number of portfolios that have the same residual risk, as shown in Figure 6, but those portfolios are decidedly different from each other. Portfolio A has relatively low volatility but relatively high tracking error. By increasing the risk of the portfolio and its correlation with the benchmark, get to Portfolio C. All three portfolios in Figure 6 have the same residual risk, but Portfolio C has the highest volatility and the lowest tracking error. Thus, a constraint on residual risk places few limits on volatility, beta, or tracking error. Figure 6. Portfolios with Constant Residual Risk (fbellchmqrk Portfolios ~ A. B. ~T 1'1-.,;-.. '. '. ". :' :'1 1..._=-L- -'-::_.~.J 1 (J'ResidualRSk Multiple Constraints. Using this type of analysis allows one to look at the interactions of the investment guidelines imposed on a manager. Figure 7 shows a simple example of this type of interaction, in which the portfolio has two constraints. The first constraint is that the total risk cannot be any greater than that of the benchmark, indicated by the circle with the center at Point A and the radius equal to the Figure 7. Feasible Portfolios with Given Total Risk and Tracking Error 0' lracking Error is equal to its tracking error of 5 percent. ts volatility is slightly greater than that of the benchmark, which is very different from Portfolio A. Portfolio F still has a tracking error of 5 percent and no residual risk, which sounds a lot like Portfolio A, but Portfolio F is much more volatile than Portfolio A. n fact, if the benchmark has a volatility of 15 percent, Portfolio A C B (T Tracking Error Association for nvestment Management and Research 77
6 Risk Management: Principles and Practices benchmark (Point B).The portfolio can be anywhere within that circle. The second constraint is that the tracking error cannot be greater than some specified amount, indicated by a semicircle around the benchmark point (B) with a radius equal to the trackingerror constraint. When combine those two constraints, the only feasible portfolios are within the cross-hatched area. Consequently, the beta cannot be greaterthan 1, and the correlationcannotbeanything less than about 0.9. Portfolio P represents maximum total risk and maximum residual risk, and the minimum risk portfolio is Portfolio C. A simple example provides a good illustration of the interaction of multiple constraints. Consider a portfolio with the same risk as the S&P 500 ndexbut with a beta, with respect to the S&P 500, of 0.8. On the surface, those constraints sound fine, but looking at the implications geometrically may indicate otherwise. First, we draw a line whose length represents the volatility of the S&P 500. Because the portfolio and the benchmark have the same volatility, the beta and the correlation both are 0.8. Second, we draw a line whose angle corresponds to a correlation of 0.8 and that has the same length as the S&P 500. That line represents the portfolio. A straight line to connectthe S&P 500 and the portfolio reveals that the tracking error is 8-9 percent. Thus, the client's constraints seem reasonable-same volatility as the benchmark and a beta of O.8--but those constraints effectively create a disguised risk: tracking error of 8-9 percent. Does the client really want a portfolio with a tracking error of 8-9 percent? Probably not. Conclusion Geometrically displaying the risk characteristics of a portfolio and its associatedbenchmarkis a simplebut powerful tool. The geometric representation of portfolio performance and portfolio strategies helps in simultaneously analyzing multiple risks-absolute risk, relative risk, systematic risk, residual risk, and tracking error. The geometric decomposition also produces an intuitive understanding of the interactions, sometimes subtle and often unintended, that result from imposing portfolio constraints. 78 Association for nvestment Management and Research
7 Question and Answer Session Brian D. Singer, CFA Risk Analysis Question: Can you represent information ratios geometrically? Singer: Yes, information ratios could be represented by drawing what are known as iso-returnlines in Figure 3. These lines might start at zero return (cash return if the analysis is in risk-premium terms) at the dollar vertex of the triangle and go out in parallel fashion at return levels of 5 percent, 10 percent, 15 percent, 20 percent, and so on. Having now superimposed these iso-return lines over the risk triangle, we can begin to do return and risk analysis simultaneously. Question: How do you incorporate fundamental factors with this approach? Singer: We use geometric risk analysis, looking for examples of the risk relationships between a portfolio basket of securities and an industry or other factor basket. n the risk triangle, the benchmark line could be thought of as the industry or factor basket, with the length of the line indicating the volatility of thatindustry or factor. The angle at the dollar vertex represents the portfolio's correlation with respect to the industry or factor and the loading on the industryor factor measuredjustas the benchmark bet (systematic risk) would be measured. This would be a univariate loading on one industry or factor, but we can also do multivariate loadings with multiple industries and factors. n fact, that is how we build our forward-looking covariance matrix-by considering country, currency, equity market, and bond market factors. We do not and could not build thousands of pairwise correlations in any consistent way. Rather, what we do is build aggregate factors, which might be regions, industries, and so on, and think about what the loading of each marketis on thosevariousfactors. Association for nvestment Management and Research 79
Chapter 1 Microeconomics of Consumer Theory
Chapter Microeconomics of Consumer Theory The two broad categories of decision-makers in an economy are consumers and firms. Each individual in each of these groups makes its decisions in order to achieve
More informationAsset Allocation vs. Security Selection: Their Relative Importance
INVESTMENT PERFORMANCE MEASUREMENT BY RENATO STAUB AND BRIAN SINGER, CFA Asset Allocation vs. Security Selection: Their Relative Importance Various researchers have investigated the importance of asset
More informationTHE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management
THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management BA 386T Tom Shively PROBABILITY CONCEPTS AND NORMAL DISTRIBUTIONS The fundamental idea underlying any statistical
More informationCommon Investment Benchmarks
Common Investment Benchmarks Investors can select from a wide variety of ready made financial benchmarks for their investment portfolios. An appropriate benchmark should reflect your actual portfolio as
More informationPortfolio Sharpening
Portfolio Sharpening Patrick Burns 21st September 2003 Abstract We explore the effective gain or loss in alpha from the point of view of the investor due to the volatility of a fund and its correlations
More informationResale Price and Cost-Plus Methods: The Expected Arm s Length Space of Coefficients
International Alessio Rombolotti and Pietro Schipani* Resale Price and Cost-Plus Methods: The Expected Arm s Length Space of Coefficients In this article, the resale price and cost-plus methods are considered
More information1 SE = Student Edition - TG = Teacher s Guide
Mathematics State Goal 6: Number Sense Standard 6A Representations and Ordering Read, Write, and Represent Numbers 6.8.01 Read, write, and recognize equivalent representations of integer powers of 10.
More informationRicardo. The Model. Ricardo s model has several assumptions:
Ricardo Ricardo as you will have read was a very smart man. He developed the first model of trade that affected the discussion of international trade from 1820 to the present day. Crucial predictions of
More informationCommon Core Georgia Performance Standards
A Correlation of Pearson Connected Mathematics 2 2012 to the Common Core Georgia Performance s Grade 6 FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: K-12
More informationModels of Asset Pricing
appendix1 to chapter 5 Models of Asset Pricing In Chapter 4, we saw that the return on an asset (such as a bond) measures how much we gain from holding that asset. When we make a decision to buy an asset,
More informationChapter 6: Supply and Demand with Income in the Form of Endowments
Chapter 6: Supply and Demand with Income in the Form of Endowments 6.1: Introduction This chapter and the next contain almost identical analyses concerning the supply and demand implied by different kinds
More informationPrentice Hall Connected Mathematics 2, 7th Grade Units 2009 Correlated to: Minnesota K-12 Academic Standards in Mathematics, 9/2008 (Grade 7)
7.1.1.1 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational
More informationCUSTOM HYBRID RISK MODELS. Jason MacQueen Newport, June 2016
CUSTOM HYBRID RISK MODELS Jason MacQueen Newport, June 2016 STANDARD RISK MODELS Off-the-shelf or standard equity risk models can be used to forecast portfolio risk and tracking error, to show the split
More informationMath 1201 Unit 3 Factors and Products Final Review. Multiple Choice. 1. Factor the binomial. a. c. b. d. 2. Factor the binomial. a. c. b. d.
Multiple Choice 1. Factor the binomial. 2. Factor the binomial. 3. Factor the trinomial. 4. Factor the trinomial. 5. Factor the trinomial. 6. Factor the trinomial. 7. Factor the binomial. 8. Simplify the
More information1 Interest: Investing Money
1 Interest: Investing Money Relating Units of Time 1. Becky has been working at a flower shop for 2.1 yr. a) How long is this in weeks? Round up. 2.1 yr 3 wk/yr is about wk b) How long is this in days?
More informationUnit 8 - Math Review. Section 8: Real Estate Math Review. Reading Assignments (please note which version of the text you are using)
Unit 8 - Math Review Unit Outline Using a Simple Calculator Math Refresher Fractions, Decimals, and Percentages Percentage Problems Commission Problems Loan Problems Straight-Line Appreciation/Depreciation
More informationStructural positions and risk budgeting
Structural positions and risk budgeting Quantifying the impact of structural positions and deriving implications for active portfolio management Ulf Herold* * Ulf Herold is a quantitative analyst at Metzler
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationGRAPHS IN ECONOMICS. Appendix. Key Concepts. Graphing Data
Appendix GRAPHS IN ECONOMICS Key Concepts Graphing Data Graphs represent quantity as a distance on a line. On a graph, the horizontal scale line is the x-axis, the vertical scale line is the y-axis, and
More informationINTRODUCTION AND OVERVIEW
CHAPTER ONE INTRODUCTION AND OVERVIEW 1.1 THE IMPORTANCE OF MATHEMATICS IN FINANCE Finance is an immensely exciting academic discipline and a most rewarding professional endeavor. However, ever-increasing
More informationBeyond VaR: Triangular Risk Decomposition
Beyond VaR: Triangular Risk Decomposition Helmut Mausser and Dan Rosen This paper describes triangular risk decomposition, which provides a useful, geometric view of the relationship between the risk of
More informationJill Pelabur learns how to develop her own estimate of a company s stock value
Jill Pelabur learns how to develop her own estimate of a company s stock value Abstract Keith Richardson Bellarmine University Daniel Bauer Bellarmine University David Collins Bellarmine University This
More informationWeb Extension: Continuous Distributions and Estimating Beta with a Calculator
19878_02W_p001-008.qxd 3/10/06 9:51 AM Page 1 C H A P T E R 2 Web Extension: Continuous Distributions and Estimating Beta with a Calculator This extension explains continuous probability distributions
More informationFirst Welfare Theorem in Production Economies
First Welfare Theorem in Production Economies Michael Peters December 27, 2013 1 Profit Maximization Firms transform goods from one thing into another. If there are two goods, x and y, then a firm can
More informationDevelopmental Math An Open Program Unit 12 Factoring First Edition
Developmental Math An Open Program Unit 12 Factoring First Edition Lesson 1 Introduction to Factoring TOPICS 12.1.1 Greatest Common Factor 1 Find the greatest common factor (GCF) of monomials. 2 Factor
More informationPearson Connected Mathematics Grade 7
A Correlation of Pearson Connected Mathematics 2 2012 to the Common Core Georgia Performance s Grade 7 FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: K-12
More informationJournal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN ANALYSIS OF VALUE LINE S ABILITY TO FORECAST LONG-RUN RETURNS
Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN ANALYSIS OF VALUE LINE S ABILITY TO FORECAST LONG-RUN RETURNS Gary A. Benesh * and Steven B. Perfect * Abstract Value Line
More informationTheory of Consumer Behavior First, we need to define the agents' goals and limitations (if any) in their ability to achieve those goals.
Theory of Consumer Behavior First, we need to define the agents' goals and limitations (if any) in their ability to achieve those goals. We will deal with a particular set of assumptions, but we can modify
More informationSection 7C Finding the Equation of a Line
Section 7C Finding the Equation of a Line When we discover a linear relationship between two variables, we often try to discover a formula that relates the two variables and allows us to use one variable
More informationChapter 2 Portfolio Management and the Capital Asset Pricing Model
Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that
More informationInternational Financial Markets 1. How Capital Markets Work
International Financial Markets Lecture Notes: E-Mail: Colloquium: www.rainer-maurer.de rainer.maurer@hs-pforzheim.de Friday 15.30-17.00 (room W4.1.03) -1-1.1. Supply and Demand on Capital Markets 1.1.1.
More informationRandomness and Fractals
Randomness and Fractals Why do so many physicists become traders? Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago September 25, 2011 1 / 24 Mathematics and the
More informationRisk-Based Performance Attribution
Risk-Based Performance Attribution Research Paper 004 September 18, 2015 Risk-Based Performance Attribution Traditional performance attribution may work well for long-only strategies, but it can be inaccurate
More informationBest Reply Behavior. Michael Peters. December 27, 2013
Best Reply Behavior Michael Peters December 27, 2013 1 Introduction So far, we have concentrated on individual optimization. This unified way of thinking about individual behavior makes it possible to
More information2 Exploring Univariate Data
2 Exploring Univariate Data A good picture is worth more than a thousand words! Having the data collected we examine them to get a feel for they main messages and any surprising features, before attempting
More informationstarting on 5/1/1953 up until 2/1/2017.
An Actuary s Guide to Financial Applications: Examples with EViews By William Bourgeois An actuary is a business professional who uses statistics to determine and analyze risks for companies. In this guide,
More informationUniversity of Siegen
University of Siegen Faculty of Economic Disciplines, Department of economics Univ. Prof. Dr. Jan Franke-Viebach Seminar Risk and Finance Summer Semester 2008 Topic 4: Hedging with currency futures Name
More informationBack to the Future Why Portfolio Construction with Risk Budgeting is Back in Vogue
Back to the Future Why Portfolio Construction with Risk Budgeting is Back in Vogue SOLUTIONS Innovative and practical approaches to meeting investors needs Much like Avatar director James Cameron s comeback
More information8: Economic Criteria
8.1 Economic Criteria Capital Budgeting 1 8: Economic Criteria The preceding chapters show how to discount and compound a variety of different types of cash flows. This chapter explains the use of those
More informationMathematics of Time Value
CHAPTER 8A Mathematics of Time Value The general expression for computing the present value of future cash flows is as follows: PV t C t (1 rt ) t (8.1A) This expression allows for variations in cash flows
More information2. Criteria for a Good Profitability Target
Setting Profitability Targets by Colin Priest BEc FIAA 1. Introduction This paper discusses the effectiveness of some common profitability target measures. In particular I have attempted to create a model
More informationRisk and Return and Portfolio Theory
Risk and Return and Portfolio Theory Intro: Last week we learned how to calculate cash flows, now we want to learn how to discount these cash flows. This will take the next several weeks. We know discount
More informationProblem set 1 Answers: 0 ( )= [ 0 ( +1 )] = [ ( +1 )]
Problem set 1 Answers: 1. (a) The first order conditions are with 1+ 1so 0 ( ) [ 0 ( +1 )] [( +1 )] ( +1 ) Consumption follows a random walk. This is approximately true in many nonlinear models. Now we
More informationA new tool for selecting your next project
The Quantitative PICK Chart A new tool for selecting your next project Author Sean Scott, PMP, is an accomplished Project Manager at Perficient. He has over 20 years of consulting IT experience providing
More informationPowerPoint. to accompany. Chapter 11. Systematic Risk and the Equity Risk Premium
PowerPoint to accompany Chapter 11 Systematic Risk and the Equity Risk Premium 11.1 The Expected Return of a Portfolio While for large portfolios investors should expect to experience higher returns for
More informationScenic Video Transcript Dividends, Closing Entries, and Record-Keeping and Reporting Map Topics. Entries: o Dividends entries- Declaring and paying
Income Statements» What s Behind?» Statements of Changes in Owners Equity» Scenic Video www.navigatingaccounting.com/video/scenic-dividends-closing-entries-and-record-keeping-and-reporting-map Scenic Video
More informationDoes Asset Allocation Policy Explain 40, 90, or 100 Percent of Performance?
Does Asset Allocation Policy Explain 40, 90, or 100 Percent of Performance? Roger G. Ibbotson and Paul D. Kaplan Disagreement over the importance of asset allocation policy stems from asking different
More informationSymmetric Game. In animal behaviour a typical realization involves two parents balancing their individual investment in the common
Symmetric Game Consider the following -person game. Each player has a strategy which is a number x (0 x 1), thought of as the player s contribution to the common good. The net payoff to a player playing
More informationCABARRUS COUNTY 2008 APPRAISAL MANUAL
STATISTICS AND THE APPRAISAL PROCESS PREFACE Like many of the technical aspects of appraising, such as income valuation, you have to work with and use statistics before you can really begin to understand
More informationA C E. Answers Investigation 4. Applications. x y y
Answers Applications 1. a. No; 2 5 = 0.4, which is less than 0.45. c. Answers will vary. Sample answer: 12. slope = 3; y-intercept can be found by counting back in the table: (0, 5); equation: y = 3x 5
More informationOptimal Portfolio Inputs: Various Methods
Optimal Portfolio Inputs: Various Methods Prepared by Kevin Pei for The Fund @ Sprott Abstract: In this document, I will model and back test our portfolio with various proposed models. It goes without
More informationUniversity 18 Lessons Financial Management. Unit 12: Return, Risk and Shareholder Value
University 18 Lessons Financial Management Unit 12: Return, Risk and Shareholder Value Risk and Return Risk and Return Security analysis is built around the idea that investors are concerned with two principal
More informationThe Fallacy of Large Numbers
The Fallacy of Large umbers Philip H. Dybvig Washington University in Saint Louis First Draft: March 0, 2003 This Draft: ovember 6, 2003 ABSTRACT Traditional mean-variance calculations tell us that the
More informationAxioma s new Multi-Asset Class (MAC) Risk Monitor highlights recent trends in market and portfolio
Introducing the New Axioma Multi-Asset Class Risk Monitor Christoph Schon, CFA, CIPM Axioma s new Multi-Asset Class (MAC) Risk Monitor highlights recent trends in market and portfolio risk. The report
More informationSAMPLE. HSC formula sheet. Sphere V = 4 πr. Volume. A area of base
Area of an annulus A = π(r 2 r 2 ) R radius of the outer circle r radius of the inner circle HSC formula sheet Area of an ellipse A = πab a length of the semi-major axis b length of the semi-minor axis
More informationChapter 19: Compensating and Equivalent Variations
Chapter 19: Compensating and Equivalent Variations 19.1: Introduction This chapter is interesting and important. It also helps to answer a question you may well have been asking ever since we studied quasi-linear
More informationMonetary Economics Measuring Asset Returns. Gerald P. Dwyer Fall 2015
Monetary Economics Measuring Asset Returns Gerald P. Dwyer Fall 2015 WSJ Readings Readings this lecture, Cuthbertson Ch. 9 Readings next lecture, Cuthbertson, Chs. 10 13 Measuring Asset Returns Outline
More informationAnswers to Concepts in Review
Answers to Concepts in Review 1. A portfolio is simply a collection of investment vehicles assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest expected
More informationOPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS. BKM Ch 7
OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS BKM Ch 7 ASSET ALLOCATION Idea from bank account to diversified portfolio Discussion principles are the same for any number of stocks A. bonds and stocks B.
More informationG r a d e 1 1 E s s e n t i a l M a t h e m a t i c s ( 3 0 S ) Final Practice Exam
G r a d e 1 1 E s s e n t i a l M a t h e m a t i c s ( 3 0 S ) Final Practice Exam G r a d e 1 1 E s s e n t i a l M a t h e m a t i c s Final Practice Examination Name: Student Number: For Marker s
More informationIntroduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book.
Simulation Methods Chapter 13 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 26, 2017 Christopher
More informationDescriptive Statistics (Devore Chapter One)
Descriptive Statistics (Devore Chapter One) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 0 Perspective 1 1 Pictorial and Tabular Descriptions of Data 2 1.1 Stem-and-Leaf
More informationHow to Mitigate Risk in a Portfolio of Contracts
How to Mitigate Risk in a Portfolio of Contracts BY dr. mark d antonio Organizational management must use the resources they are entrusted with in the most judicious manner possible. An organization must
More informationCopyright 2009 Pearson Education Canada
Operating Cash Flows: Sales $682,500 $771,750 $868,219 $972,405 $957,211 less expenses $477,750 $540,225 $607,753 $680,684 $670,048 Difference $204,750 $231,525 $260,466 $291,722 $287,163 After-tax (1
More informationBasic Math Principles
Introduction This appendix will explain the basic mathematical procedures you will need to be successful in your new real estate career. Many people are intimidated by the word math, but in this case the
More informationP2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition.
P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju
More informationHow Risky is the Stock Market
How Risky is the Stock Market An Analysis of Short-term versus Long-term investing Elena Agachi and Lammertjan Dam CIBIF-001 18 januari 2018 1871 1877 1883 1889 1895 1901 1907 1913 1919 1925 1937 1943
More informationBusiness Mathematics (BK/IBA) Quantitative Research Methods I (EBE) Computer tutorial 4
Business Mathematics (BK/IBA) Quantitative Research Methods I (EBE) Computer tutorial 4 Introduction In the last tutorial session, we will continue to work on using Microsoft Excel for quantitative modelling.
More informationTurning Points Analyzer
Turning Points Analyzer General Idea Easy Start Going into Depth Astronomical Model Options General Idea The main idea of this module is finding the price levels where the price movement changes its trend.
More informationReal Estate Private Equity Case Study 3 Opportunistic Pre-Sold Apartment Development: Waterfall Returns Schedule, Part 1: Tier 1 IRRs and Cash Flows
Real Estate Private Equity Case Study 3 Opportunistic Pre-Sold Apartment Development: Waterfall Returns Schedule, Part 1: Tier 1 IRRs and Cash Flows Welcome to the next lesson in this Real Estate Private
More informationExercise 14 Interest Rates in Binomial Grids
Exercise 4 Interest Rates in Binomial Grids Financial Models in Excel, F65/F65D Peter Raahauge December 5, 2003 The objective with this exercise is to introduce the methodology needed to price callable
More informationMathematics Success Grade 8
Mathematics Success Grade 8 T379 [OBJECTIVE] The student will derive the equation of a line and use this form to identify the slope and y-intercept of an equation. [PREREQUISITE SKILLS] Slope [MATERIALS]
More informationThe homework assignment reviews the major capital structure issues. The homework assures that you read the textbook chapter; it is not testing you.
Corporate Finance, Module 19: Adjusted Present Value Homework Assignment (The attached PDF file has better formatting.) Financial executives decide how to obtain the money needed to operate the firm:!
More informationActive Portfolio Management. A Quantitative Approach for Providing Superior Returns and Controlling Risk. Richard C. Grinold Ronald N.
Active Portfolio Management A Quantitative Approach for Providing Superior Returns and Controlling Risk Richard C. Grinold Ronald N. Kahn Introduction The art of investing is evolving into the science
More informationFINALTERM EXAMINATION Fall 2009 MGT201- Financial Management (Session - 3)
FINALTERM EXAMINATION Fall 2009 MGT201- Financial Management (Session - 3) Time: 120 min Marks: 87 Question No: 1 ( Marks: 1 ) - Please choose one ABC s and XYZ s debt-to-total assets ratio is 0.4. What
More informationVisualizing 360 Data Points in a Single Display. Stephen Few
Visualizing 360 Data Points in a Single Display Stephen Few This paper explores ways to visualize a dataset that Jorge Camoes posted on the Perceptual Edge Discussion Forum. Jorge s initial visualization
More informationWeek 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals
Week 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg :
More informationApproximating the Confidence Intervals for Sharpe Style Weights
Approximating the Confidence Intervals for Sharpe Style Weights Angelo Lobosco and Dan DiBartolomeo Style analysis is a form of constrained regression that uses a weighted combination of market indexes
More informationArticle from: Risk Management. July 2006 Issue 8
Article from: Risk Management July 2006 Issue 8 Risk Management July 2006 From Pension Risk Management to ERM by André Choquet Pension Risk Management If the field of Enterprise Risk Management (ERM) is
More informationthe display, exploration and transformation of the data are demonstrated and biases typically encountered are highlighted.
1 Insurance data Generalized linear modeling is a methodology for modeling relationships between variables. It generalizes the classical normal linear model, by relaxing some of its restrictive assumptions,
More informationFinancial Mathematics III Theory summary
Financial Mathematics III Theory summary Table of Contents Lecture 1... 7 1. State the objective of modern portfolio theory... 7 2. Define the return of an asset... 7 3. How is expected return defined?...
More informationMGT201 Financial Management All Subjective and Objective Solved Midterm Papers for preparation of Midterm Exam2012 Question No: 1 ( Marks: 1 ) - Please choose one companies invest in projects with negative
More informationIn the previous session we learned about the various categories of Risk in agriculture. Of course the whole point of talking about risk in this
In the previous session we learned about the various categories of Risk in agriculture. Of course the whole point of talking about risk in this educational series is so that we can talk about managing
More informationDoes my beta look big in this?
Does my beta look big in this? Patrick Burns 15th July 2003 Abstract Simulations are performed which show the difficulty of actually achieving realized market neutrality. Results suggest that restrictions
More informationGame Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati
Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati Module No. # 03 Illustrations of Nash Equilibrium Lecture No. # 04
More informationThe mean-variance portfolio choice framework and its generalizations
The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014 Outline and objectives The backward, three-step solution
More informationOptimization 101. Dan dibartolomeo Webinar (from Boston) October 22, 2013
Optimization 101 Dan dibartolomeo Webinar (from Boston) October 22, 2013 Outline of Today s Presentation The Mean-Variance Objective Function Optimization Methods, Strengths and Weaknesses Estimation Error
More informationECO155L19.doc 1 OKAY SO WHAT WE WANT TO DO IS WE WANT TO DISTINGUISH BETWEEN NOMINAL AND REAL GROSS DOMESTIC PRODUCT. WE SORT OF
ECO155L19.doc 1 OKAY SO WHAT WE WANT TO DO IS WE WANT TO DISTINGUISH BETWEEN NOMINAL AND REAL GROSS DOMESTIC PRODUCT. WE SORT OF GOT A LITTLE BIT OF A MATHEMATICAL CALCULATION TO GO THROUGH HERE. THESE
More informationNote on Cost of Capital
DUKE UNIVERSITY, FUQUA SCHOOL OF BUSINESS ACCOUNTG 512F: FUNDAMENTALS OF FINANCIAL ANALYSIS Note on Cost of Capital For the course, you should concentrate on the CAPM and the weighted average cost of capital.
More informationExplanation of Compartamos Interest Rates
Explanation of Compartamos Interest Rates Chuck Waterfield Version 2: 19 May 2008 For a full year, I have seen consistent confusion over what interest rate Compartamos charges its clients. They generally
More informationThe New ROI. Applications and ROIs
Denne_02_p013-026 9/10/03 3:42 PM Page 13 The New ROI If software development is to be treated as a value creation exercise, a solid understanding of the financial metrics used to evaluate and track value
More informationAn Introduction to Breaking Down the Numbers
An Introduction to Breaking Down the Numbers Whether you ve financed equipment before or you re just starting to look into it, you probably noticed some crazy calculations and terms being thrown around
More informationMaximizing Winnings on Final Jeopardy!
Maximizing Winnings on Final Jeopardy! Jessica Abramson, Natalie Collina, and William Gasarch August 2017 1 Abstract Alice and Betty are going into the final round of Jeopardy. Alice knows how much money
More informationDOWNLOAD PDF HOW TO CALCULATE (AND REALLY UNDERSTAND RETURN ON INVESTMENT
Chapter 1 : Return on Investment (ROI) Definition & Example InvestingAnswers The return on investment metric calculates how efficiently a business is using the money invested by shareholders to generate
More informationFE670 Algorithmic Trading Strategies. Stevens Institute of Technology
FE670 Algorithmic Trading Strategies Lecture 4. Cross-Sectional Models and Trading Strategies Steve Yang Stevens Institute of Technology 09/26/2013 Outline 1 Cross-Sectional Methods for Evaluation of Factor
More information(2/3) 3 ((1 7/8) 2 + 1/2) = (2/3) 3 ((8/8 7/8) 2 + 1/2) (Work from inner parentheses outward) = (2/3) 3 ((1/8) 2 + 1/2) = (8/27) (1/64 + 1/2)
Exponents Problem: Show that 5. Solution: Remember, using our rules of exponents, 5 5, 5. Problems to Do: 1. Simplify each to a single fraction or number: (a) ( 1 ) 5 ( ) 5. And, since (b) + 9 + 1 5 /
More informationMarket Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk
Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day
More informationpar ( 12). His closest competitor, Ernie Els, finished 3 strokes over par (+3). What was the margin of victory?
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) Tiger Woods won the 2000 U.S. Open golf tournament with a score of 2 strokes under par
More informationHow Do You Calculate Cash Flow in Real Life for a Real Company?
How Do You Calculate Cash Flow in Real Life for a Real Company? Hello and welcome to our second lesson in our free tutorial series on how to calculate free cash flow and create a DCF analysis for Jazz
More informationTaxation and Efficiency : (a) : The Expenditure Function
Taxation and Efficiency : (a) : The Expenditure Function The expenditure function is a mathematical tool used to analyze the cost of living of a consumer. This function indicates how much it costs in dollars
More information