Free Essay

Submitted By raza19086

Words 1872

Pages 8

Words 1872

Pages 8

by Steen Koekebakker and Valeri Zakamouline

Introduction

The risky assets available to investors are numerous: mutual funds, hedge funds, structured products, equity-linked notes to name a few. The characteristics of each asset class can be summarized in the different return distributions. Even within a single asset class the return distributions of assets are not alike. We assume that the return distributions of all risky assets are known and would like to choose the best asset to invest to, meaning that the risky assets are mutually exclusive investment alternatives. How to do this? The standard approach in financial theory and practice is to employ some portfolio performance measure to rank the various risky investments. Each portfolio performance measure calculates a score for each asset using its probability distribution of returns. The best asset to invest to is the asset with the highest score.

The Sharpe ratio is a commonly used measure of portfolio performance. But because it is based on mean-variance theory, this measure can only be used in some restrictive cases, for example, when return distributions are normal. When return distributions are non-normal, the Sharpe ration can lead to misleading conclusions and unsatisfactory paradoxes, see Bernardo and Ledoit (2000) and Hodges (1998). There have been proposed numerous universal performance measures that, in one way or the other, are alternatives to the Sharpe ratio and try to take into account non-normality of return distributions. For some examples, see Sortino and Price (1994), Dowd (2000), Stutzer (2000), Keating and Shadwick (2002), Gregoriou and Gueyie (2003), Kaplan and Knowles (2004), and Ziemba (2005). The main drawback of many of these alternative performance measures is that they lack a solid theoretical underpinning. In this paper we review the latest results on portfolio performance measures based on either expected utility theory or non-expected utility theory (the latter is opposed to the von Neumann and Morgenstern expected utility theory). The main purpose of this paper is to show that unless we know exactly the investor’s preferences, and unless all investors share the same preferences, a single performance measure that is suitable for all investors cannot exist.

Performance Measures Based on Expected Utility Theory

Expected utility theory of von Neumann and Morgenstern has long been the main workhorse of modern financial theory. A von Neumann-Morgenstern utility function is defined over the investor's wealth as U(W), where U(.) is some function and W is the investor’s wealth. The celebrated modern portfolio theory of Markowitz and the use of mean-variance utility function can be justified by approximating a von Neumann-Morgenstern utility function by a function of mean and variance, see, for example Samuelson (1970) and Levy and Markowitz (1979). Besides, the use of mean-variance utility can also be justified when either the return distribution is normal or the investor's utility function is quadratic. But when it comes to quadratic utility, it is also well known that it has anomalous properties such as increased risk aversion as wealth increases, to say nothing of ultimate satiation.

Mean-variance utility function is given by

[pic]

Where E[W] and Var[W] are the expected wealth and the variance of the wealth, and γ is the investor’s risk aversion coefficient. If the investor’s utility is given by some function, then the risk aversion coefficient is computed as

[pic].

This is also known as the Arrow-Pratt measure of absolute risk aversion. Mean-variance utility function leads to a performance measure known as the Sharpe ratio

[pic],

where μ and σ are the expected returns and the standard deviation of returns of some risky asset, and r is the risk-free rate of return. The Sharpe ratio is appealing, and rather popular in practice, because the investor’s preferences magically disappear from the performance measure. That is, irrespective of the level of risk aversion, all investors with either quadratic or mean-variance utility will rank indentically different investment alternatives. But what will be the ranking of alternative investments in the general case, that is, for a general utility function? For a general utility function, the explicit (closed-form) solution for the performance measure does not exist. Observe that in principle, in the framework of expected utility theory, the portfolio evaluation problem is rather trivial, though quite demanding in implementation. We need first to specify the investor’s utility function and then we choose the risky asset/portfolio that maximizes the investor’s expected utility. Also, in principle, we do not need an explicit formula for the performance measure. All we need is to specify the investor’s utility which can rank the risky alternatives in terms of expected utilities they give. However, the number of possible utility functions is, in principle, unlimited. Will the ranking remain basically the same or will it be substantially different for different utility functions? The analysis of Koekebakker and Zakamouline (2007a) suggests that the ranking of alternative investment alternatives might be very different for different types of utility functions.

In particular, Koekebakker and Zakamouline (2007a) suggested approximating a general utility function using the first three moments of the return distribution, namely the mean, variance, and skewness. They arrived to the measure that they denote as the Adjusted for Skewness Sharpe Ratio (ASSR) given by

[pic],

where SR is the standard Sharpe ratio, S is the skewness of return distribution, and b is a parameter that reflects the investor’s preference to the skewness of return distribution. This parameter is defined in terms of first three derivatives of the investor’s utility function

[pic].

Observe that, in contrast to the Sharpe ratio, the value of the ASSR is not unique and depends on the investor’s preference for skewness as specified by a utility function. In using the ASSR for practical purposes one cannot avoid the ambiguity in ranking different risky assets. Koekebakker and Zakamouline (2007b) illustrated that the ranking of alternative investments depends quite substantially on the value of the skewness preference parameter b, especially when either the return distributions are highly skewed or the value of parameter b is rather large with respect to 1.

Performance Measures Based on Non-Expected Utility Theory

Not very long ago after expected utility theory was formulated by von Neumann and Morgenstern (1944), questions were raised about its value as a descriptive model of the choice under uncertainty. Allais (1953) and Ellsberg (1961) were among the first to challenge expected utility theory by showing that some of the assumptions behind this theory cannot be justified by empirical studies. From the other side, the mean-variance analysis of Markowitz was criticized because many investors do not associate the risk with the standard deviation of returns, rather with the possibility of loss. Actually, even Markowitz recognized that the standard deviation is not a suitable measure of risk. Markowitz (1959) also proposed to use the semi-variance as an alternative measure of risk. Semi-variance is like variance, except that it considers only returns below some target level. Technically, aggregating semi-variances from assets to portfolios is extremely difficult. That is probably why this idea was not pursued further. Later on the notion of a downside semi-variance was generalized by Fishburn (1977) and Bawa (1978) who introduce the notion of a lower partial moment as a risk measure. The definition of a lower partial moment of order n at some level tg is

[pic] where x is some random variable and F(x) is the cumulative probability distribution of x. In a similar manner one can define an upper partial moment UPMn(x,tg). Fishburn (1977) and Bawa (1978) proposed the mean - lower partial moment model for portfolio selection. These authors show that the usage of the mean - lower partial moment objective corresponds to a specific utility function of the investor. As defined over the asset’s random return x, the utility function is given by

[pic]

where γ is the measure of the investor’s risk aversion, and n is the order of the lower partial moment. If the risk-free rate of return is used as the target level, that is, tg=r, and n=2, then the investor’s performance measure becomes

[pic].

This performance measure was introduced by Sortino and Price (1994) and Ziemba (2005). The utility function of Fishburn (1977) and Bawa (1978) is rather specific and restrictive, though it leads, as mean-variance utility, to a performance measure with no investor’s preferences. Below we show that this utility function is a special case of a more general behavioral utility function.

Influential experimental studies have shown the inability of expected utility theory to explain many phenomena and reinforced the need to rethink much of the theory. Kahneman and Tversky (1979) propose an alternative descriptive model of the choice under uncertainty that they call prospect theory. Prospect theory can predict correctly individual choices even in the cases in which expected utility theory is violated (for a brief description see, for example, Camerer (2000)). In prospect theory, the utility function is defined over gains and losses relative to some reference point, as opposed to wealth in expected utility theory. More formally, the utility function is defined as

[pic]

where U+(.) is the utility function for gains, U-(.) is the utility function for losses, and W0 is the reference point. The current level of the investor’s wealth (so-called the “status quo”) serves usually as the reference point. However, as Kahneman and Tversky point out “gains and losses can be coded relative to an expectation or aspiration level that differs from the status quo” (see Kahneman and Tversky (1979) page 286). The behavioral utility function has a kink at the origin, with the slope of the loss function steeper than the gain function. This is what is called loss aversion which is an important element of prospect theory. In addition, in prospect theory the investor transforms the objective probability distribution into a subjective probability distribution.

Recently, Koekebakker and Zakamouline (2007c) performed the approximation analysis of the investor’s optimal capital allocation problem and showed that the utility function in prospect theory is equivalent to the following utility

[pic],

where [pic]and [pic] are the positive and negative parts, respectively, of the difference W-W0, λ is the measure of the investor’s aversion to losses defined by

[pic],

γ+ and γ- are the investor’s measures of aversion to uncertainties in gains and losses respectively

[pic], [pic],

and where [pic] are the left-sided derivatives of U(.), and [pic] are the right-sided derivatives of U(.) at the reference point. If the reference point is assumed to be the investor's initial wealth scaled up by the risk-free rate, that is, W0=W(1+r), then it is possible to arrive at the explicit solution of the optimal capital allocation problem, and to the closed-form solution for the investor’s performance measure. If γ+>0 (which means that the gain function is concave), then the performance measure of a risky asset becomes

[pic],

where the parameter θ is the relation between the investor’s aversion to uncertainty in losses and the investor’s aversion to uncertainty in gains

[pic].

If γ+=0 (which means that the gain function is a straight line), then the performance measure of a risky asset becomes

[pic].

If PM

Premium Essay

...How to Measure Diversification • All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M • A completely diversified portfolio would have a correlation with the market portfolio of +1.00 Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return Standard Deviation of Return Unsystematic (diversifiable) Risk Total Risk Standard Deviation of the Market Portfolio (systematic risk) Systematic Risk Number of Stocks in the Portfolio The CML and the Separation Theorem • The CML leads all investors to invest in the M portfolio • Individual investors should differ in position on the CML depending on risk preferences • How an investor gets to a point on the CML is based on financing decisions • Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio The CML and the Separation Theorem Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio The CML and the Separation Theorem The decision of both investors is to invest in portfolio M along the CML (the investment decision) E ( Rport ) B CML M A PFR port The CML and the Separation Theorem The decision to borrow or lend to obtain a point on the CML is a separate decision based on risk preferences (financing decision) The CML and the Separation Theorem Tobin refers to this......

Words: 1858 - Pages: 8

Premium Essay

...TOPIC THREE PORTFOLIO THEORY AND CAPITAL ASSET PRICING MODEL (CAPM) Reading : BKM: Chapters 7&9 Pilbeam: Chapters 7&8 OUTLINE Section I: The concept of portfolio and diversification Calculate portfolio expected return Measuring portfolio total risk: variance and standard deviation Market portfolio Measuring systematic risk: Beta Section II: Markowitz Portfolio Theory Efficient portfolio and Efficient Frontier Capital Asset Pricing Model - CAPM CAPM lines: CML and SML PORTFOLIO A portfolio is a collection of assets Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments. Diversification reduces risk because prices of different securities do not move exactly together. - The amount of possible risk reduction through diversification depends on the correlation (see later) An asset’s risk and return are important in how they affect the risk and return of the portfolio The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets PORTFOLIO EXPECTED RETURN The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio Portfolio rate of return = fraction of portfolio rate of return + x in second asset on second asset ( ( fraction of portfolio in first asset )( )( x s i 1 rate of return on first asset )......

Words: 5021 - Pages: 21

Premium Essay

...Foundations of Finance: The Capital Asset Pricing Model (CAPM) Prof. Alex Shapiro Lecture Notes 9 The Capital Asset Pricing Model (CAPM) I. II. III. IV. V. VI. Readings and Suggested Practice Problems Introduction: from Assumptions to Implications The Market Portfolio Assumptions Underlying the CAPM Portfolio Choice in the CAPM World The Risk-Return Tradeoff for Individual Stocks VII. The CML and SML VIII. “Overpricing”/“Underpricing” and the SML IX. X. Uses of CAPM in Corporate Finance Additional Readings Equilibrium Process, Supply Equals Demand, Market Price of Risk, Cross-Section of Expected Returns, Risk Adjusted Expected Returns, Net Present Value and Cost of Equity Capital. Buzz Words: 1 Foundations of Finance: The Capital Asset Pricing Model (CAPM) I. Readings and Suggested Practice Problems BKM, Chapter 9, Sections 2-4. Suggested Problems, Chapter 9: 2, 4, 5, 13, 14, 15 Web: Visit www.morningstar.com, select a fund (e.g., Vanguard 500 Index VFINX), click on Risk Measures, and in the Modern Portfolio Theory Statistics section, view the beta. II. Introduction: from Assumptions to Implications A. Economic Equilibrium 1. Equilibrium analysis (unlike index models) Assume economic behavior of individuals. Then, draw conclusions about overall market prices, quantities, returns. 2. The CAPM is based on equilibrium analysis Problems: – – There are many “dubious” assumptions. The main implication of the CAPM concerns expected......

Words: 1605 - Pages: 7

Premium Essay

...re-written as follows: In this subsection, we apply the CAPM to the pricing of securities. The security market line (SML) is a graphical representation of the capital asset pricing model with beta, reflecting systematic risk, on the x-axis and expected return on the y-axis. Using the same concept as the capital market line, the SML intersects the y-axis at the risk-free rate of return, and the slope of this line is the market risk premium, Rm – Rf. Recall that the capital allocation line (CAL) and the capital market line (CML) does not apply to all securities or assets but only to efficient portfolios on the efficient frontier. The efficient frontier gives optimal combinations of expected return and total risk. In contrast, the security market line applies to any security, efficient or not. The difference occurs because the CAL and the CML use the total risk of the asset rather than its systematic risk. Because only systematic risk is priced and the CAL and the CML are based on total risk, the CAL and the CML can only be applied to those assets whose total risk is equal to systematic risk. Total risk and systematic risk are equal only for efficient portfolios because those portfolios have no diversifiable risk remaining. We are able to relax the requirement of efficient portfolios for the SML because the CAPM, which forms the basis for the SML, prices a security based only on its systematic risk, not its total risk. o Practice problem 1 (p. 384) and its......

Words: 1446 - Pages: 6

Premium Essay

...security market line (SML) equation is the Capital Asset Pricing Model. It is used to price risk, i.e., it is used to specify the risk/return relationship of a particular asset or portfolio, regardless of the level of diversification. The SML equation (provided with the CFP Board Exam) is: ri = rf + (rm - rf) βi The SML equation states that the return of a specific investment is equal to the risk-free rate plus a market risk premium multiplied by the investment’s beta (βi). By definition, the beta of the market is 1. Unlike the CML which uses standard deviation (σ) to measure risk, the SML uses beta (βi), i.e., systematic risk, to measure risk. Given a stock’s beta, the risk-free rate, and the market’s expected return, the SML equation will solve for the stock’s required rate of return. • Undervalued stocks will have an expected return greater than the SML’s required return; if a security plots over the SML it is undervalued and should be purchased. • Overvalued stocks will have an expected return less than the SML’s required return; if a security plots under the SML it is overvalued and should be sold or shorted. • A stock that plots on the SML has an expected return than is equal to the SML’s required return and can be bought or sold – the investor is indifferent. [pic] Which stock should be purchased, which should be sold? Stock plots over the SML therefore it is undervalued and should be purchased. Stock B plots under the SML therefore it is......

Words: 331 - Pages: 2

Free Essay

...tài sản rủi ro * Khi bạn kết hợp tài sản phi rủi ro với một danh mục đầu tư các tài sản rủi ro trên đường Markowitz efficient frontier thì danh mục đầu tư sẽ như thế nào? * Thiết lập ban đầu một danh mục đầu tư có thể với một tài sản phi rủi ro và điều gì xảy ra khi bạn dung đòn cân nợ * Danh mục đầu tư thị trường là gì? Tài sản nào sẽ nằm trong danh mục này, trọng số tương đối của các loại tài sản có liên quan * Đường thị trường vốn (CML) là gì? * Thế nào là đa dạng hóa hoàn toàn? * Làm thế nào để có thể đo lường đa dạng hóa đối với một danh mục đầu tư riêng lẻ (cá nhân) * Rủi ro hệ thống và rủi ro phi hệ thống * Nếu có đường CML thì đo lường rủi ro tương ứng của một tài sản rủi ro riêng lẻ thế nào? * Đường thị trường chứng khoán (SML) và sự khác biệt với đường thị trường vốn (CML)? * Beta là gì và vì sao beta được xem như công cụ đo lường đươc chuẩn hóa của rủi ro hệ thống * Bạn có thể sử dụng đường SML để xác định suất sinh lợi kỳ vọng đối với 1 tài sản rủi ro * Sử dụng đường SML để xác định giá cổ phiếu trên hay dưới giá trị cũng như tài sản undervalued hay overvalued * Thế nào là đường đặc điểm của tài sản và cách tính toán căn cứ trên đường đặc điểm của tài sản * Các ảnh hưởng của đường đặc điểm tài sản khi bạn tính toán nếu như sử dụng các return intervals khác nhau (ví dụ: tuần với tháng) hoặc khi bạn sử dụng các tiêu chuẩn khác nhau cho danh mục đầu tư thị trường (ví dụ: S&P 500 versus a global stock index) ...

Words: 7112 - Pages: 29

Premium Essay

...investment giving the highest expected return for a given level of risk or one that has the lowest risk for a given level of expected return. For instance, if you were told that the return on a share is expected to be 20% and you have been given three risk ratings, 30%, 15% and 10%; assuming that the 10% risk rating is associated with the lowest risk, then you will select this level of risk for the given level of return. CAPITAL ASSET PRICING MODEL(CAPM): A model that describes the relationship between risk and expected return and that is used in the pricing of risky securities.CAPM uses beta to link formally the notions of risk & return. CAPM can be viewed both as a mathematical equation & graphically, as the security market line,(SML). Assumptions of CAPM: • Investors are risk averse. They take decision based upon risk and return assessment. • The purchase or sale of a security can be undertaken in infinitely divisible units. • Purchase and sale by a single investor can not affect prices. • There are no transaction costs. • There are no taxes. • Investor can borrow and lend freely at a risk less rate of interest. • Investors have homogeneous expectations - they have identical, subjective estimate of the means, variances among returns. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates......

Words: 4408 - Pages: 18

Free Essay

...đồ thị tỷ suất sinh lợi và rủi ro có thể có của danh mục sẽ có dạng đường thẳng. Σp = (1- wf ) * σi GVHD: LÊ ĐẠT TRÍ Trang 2 MÔ HÌNH ĐỊNH GIÁ TÀI SẢN VỐN CAPM Khi không có tài sản phi rủi ro thì danh mục nằm trên đường markowitz là danh mục tốt nhất. Bây giờ chúng ta giả sử nhà đầu tư có thể cho vay và đi vay tiền với lãi suất phi rủi ro. d) Sử dụng đòn bẩy tài chính sẽ có ảnh hưởng gì lên rủi ro và tỉ suất sinh lợi của danh mục Một nhà đầu tư có thể muốn đạt được một tỷ suất sinh lợi cao hơn tại điểm m nhưng phải chấp nhận mức độ rủi ro cao hơn. Nhà đầu tư sẽ sử dụng đòn bẩy tài chính bằng các đi vay ở lãi suất phi rủi ro và đầu tư số tiền này vào danh mục tài sản rủi ro M Tỉ suất sinh lợi kì vọng(r),% Đi vay Cho vay CML M rf Độ lệch chuẩn( ),% + Nếu nhà đầu tư, đầu tư 50% số tiền vào danh mục m và cho vay phần còn lại. Giả sử danh mục m có tỷ suất sinh lợi mong đợi là 15% và độ lệch chuẩn là 16%, trái phiếu kho bạc có lãi suất phi rủi ro là 5%. Lúc này rp = wf * rf + (1- wf ) * rM = 0.5*5 + 0.5*15 = 10% Độ lệch chuẩn danh mục có đòn bẩy: Σp = (1- wf ) * σM = 0.5*16 = 8% + Nhà đầu tư đi vay một số tiền bằng 50% số tiền bạn có với lãi suất bằng với lãi suất trái phiếu và đầu tư tất cả vào danh mục M. Lúc này, nhà GVHD: LÊ ĐẠT TRÍ Trang 3 MÔ HÌNH ĐỊNH GIÁ TÀI SẢN VỐN CAPM đầu tư sẽ có gấp đôi số tiền của mình để đầu tư vào M, nhưng lại phải chi trả lãi vay. Do đó Tỷ suất sinh lợi: rp = 2 * r m - rf = 2*15 - 5 =......

Words: 6023 - Pages: 25

Premium Essay

...doesn't act much like the index. Our R2s are low because the company does not act like the market in most of the past years. Sometimes the company’s return is positive when the market’s is negative and vice versa. 4. Managerial finance is just suggestions. We can’t predict exactly how the market is going to act, but we can try to predict it as best we can. Expected returns and actual returns are most likely going to be different, but that does not mean that the data is wrong. There are several ways of calculating statistics of assets and portfolios and they are not all going to say the same thing. 5. It appears from our calculations that both stocks fall on the SML because for the Beta’s of stocks of 1.5 1.75 they have returns of 12% and 13% respectively which are stated as fairly valued on the SML. 6. The two instances both cause the required risk premium to go higher, but not in the same magnitude. 7. The covariance was -.01446 which means that if the market in general is going down than outplace will tend to go up as a hedge against the market. The correlation is -.63489 which states that they move in opposite directions of each other and that are good hedging candidates for each other. The correlation coefficient and the covariance are factors of each other, and if you have one, you can find the other relatively easily. 8. It would appear that as the portfolio becomes more weighted towards outplace the gap between the SD line and the E(r) line......

Words: 2068 - Pages: 9

Premium Essay

... 6. Since a stock’s beta coefficient determines how the stock affects the risk of a diversified portfolio, beta is the most relevant measure of any stock’s risk. The Relationship Between Risk and Return Market risk premium is the premium that investors require for bearing the risk of an average stock. RPm=rM-rRF where rM is required rate of return of the market and rRF is the risk free rate of return Risk premium for a stock = RPi=(rpM)bi Required return = Risk-Free return + Premium for risk The relationship between risk and required returns can be found as specified in the Security Market Line (SML): SML=Required Return on Stock i= RFR+(Market Risk Premium)(Beta of i) Impact of Inflation and Interest Rate Changes RFR=r*+IP where r* is the real inflation-free rate of return and IP is the inflation premium Changes in Risk Aversion The slope of the SML reflects how risk averse investors are. Changes in Beta Beta changes impact the amount of return required. Concerns These are not perfect measures, unfortunately and come debate persists over whether beta or CAPM are accurate theories. Implications 1. Trade off between risk and return 2. Diversification is crucial 3. Real returns are what matters 4. The risk of the investment often depends on investment horizon 5. PAST PERFORMANCE IS NOT A GUARANTEE OF FUTURE RETURNS!!!!!!! Chapter 7 Though shareholders have voting rights, most go via proxy. Proxy fights, however, can occur......

Words: 1934 - Pages: 8

Premium Essay

...market portfolio. Therefore, the full CML becomes a line from the RFR to the M portfolio and continuing upward. 6. You can measure how well diversified a portfolio is by computing the extent of correlation between the portfolio in question and a completely diversified portfolio - i.e., the market portfolio. The idea is that, if a portfolio is completely diversified and, therefore, has only systematic risk, it should be perfectly correlated with another portfolio that only has systematic risk. 7. Standard deviation would be expected to decrease with an increase in stocks in the portfolio because an increase in number will increase the probability of having more inversely correlated stocks. There will be a major decline from 4 to 10 stocks, a continued decline from 10 to 20 but at a slower rate. Finally, from 50 to 100 stocks, there is a further decline but at a very slow rate because almost all unsystematic risk is eliminated by about 18 stocks. 8. Given the existence of the CML, everyone should invest in the same risky asset portfolio, the market portfolio. The only difference among individual investors should be in the financing decision they make, which depends upon their risk preference. Specifically, investors initially make investment decisions to invest in the market portfolio, M. Subsequently, based upon their risk preferences, they make financing decisions as to whether to borrow or lend to attain the preferred point on the CML. 9. Recall that the......

Words: 4239 - Pages: 17

Premium Essay

...return and standard deviation of his portfolio? Expected return Standard deviation a) 6.0% 6.8% b) 8.0% 4.8% c) 10.0% 6.6% d) 8.0% 6.6% 6) The risk-free rate is 6%, and the expected market return is 15%. A stock with a beta of 2 is selling for $25 and will pay a $ 1 dividend at the end of the year. If the stock is priced at $30 at year-end, it is: a) Overpriced, so short it. b) Underpriced, so buy it. c) Underpriced, so short it. d) Priced fairly, so don’t bother buying or shorting it. 7) Which of the following statements about the Securities Market Line (SML) and the Capital Market Line (CML) is the least accurate? a) Securities that plot above the SML are undervalued. b) Investors expect to be compensated for systematic risk. c) Securities that plot on the SML have no value to investors. d) CML represents risk-return characteristics of optimal portfolios. 8) An investor buys 1,000 shares of a stock on margin at a price of $50 per share. The initial margin requirement is 40% and the margin-lending rate is 3%. The investor's broker charges a commission of $0.01 per share on purchases and sales. The stock pays an annual dividend of $0.30 per share. One year later, the investor sells the 1,000 shares at a price of $56 per share. The investor's rate of return is closest to: a) 6% b) 12% c) 27% d) 36% Use the following information to answer Questions 9-11: | As of January 1,......

Words: 2309 - Pages: 10

Premium Essay

...securities in the portfolio • SML (Security Market Line) – shows the relationship between an expected return on an asset to its systematic risk. CAPM – Capital Asset Pricing Model • Security Market Line ▫ Formula of SML : ri = rRF + (rM – rRF) bi CAPM – Capital Asset Pricing Model • Security Market Line Required Rate of Return ▫ Formula of SML : ri = rRF + (rM – rRF) bi ▫ Remember that rRF or nominal RFR = r* or real risk free rate + IP or inflation premium; risk free rate (based on financial instruments with no default risk, typically represented by a 3 month US T-bill) ▫ rM – rRF = Market risk premium = the premium that investors require for bearing the risk of an “Average Stock” ▫ (rM – rRF) bi = Risk premium on the stock Movement along SML Expected Return SML More Risk Less Risk Beta (Systematic Risk or Non-diversifiable Risk) Shift of SML Expected Return SML Beta (Systematic Risk) This indicates increase in nominal risk free rate of return. It is either due to increase in Real risk free rate or an increase in inflation rate. Shift of SML Expected Return SML Beta (Systematic Risk) Changing of slope of SML indicates change in risk taking capacity of investors. Steeper slope indicates that investors are more risk averse now hence they require more premium for bearing same risk. Shift of SML Expected Return SML Beta (Systematic Risk) How would you interpret the shift of this......

Words: 4013 - Pages: 17

Premium Essay

...publicly traded financial assets such as stocks and bonds o Limits the scope of CAPM to financial assets traded in an organised exchange o Market participants can ensure that assets are priced at equilibrium levels o When an asset is overpriced, homogeneous expectations assumption implies all investors will sell, until the price reaches equilibrium The CAPM focuses on quantifying the amount of risk an asset contributes to the portfolio of risky assets that every investor should hold and the pricing of this risk. Thus, derivative securities, contingent on a number of factors in addition to the risk of the underlying assets are not considered. 8 Cheryl Mew FINS2624 – Portfolio Management Semester 1, 2011 D ERIVING THE CML AND THE MEANING OF SHARPE RATIO T HE MARKET PORTFOLIO Assumptions 1, 2, 3 imply: • • All investors see the same investment opportunity set – all attainable portfolios of risky assets with risk return combinations with the minimum variance set All investors see the same efficient frontier – line that shows risk-return combinations of portfolios of risky assets that offer the largest level of expected return for a given level of risk Assumptions 4,5, further imply that investors should all choose the same portfolio of risky assets, as this portfolio offers investors the largest excess return per unit of risk among all the risky portfolios lying on the EF. Assumption 6 restricts the set of risky assets to bonds and stocks......

Words: 14579 - Pages: 59

Free Essay

...Maraschiello (maraschiello@vodafone.it) Si è visto nel capitolo relativo al Capm come le scelte di investimento di un investitore razionale possano essere analizzate attraverso l’approccio media/varianza. In particolare si è visto come tra i diversi mix disponibili di portafogli rischiosi, alcuni sono preferibili in quanto associano ad un rendimento maggiore un rischio minore. Tra questi il portafoglio preferito sulla frontiera è quello di tangenza con la Capital Market Line. Nel foglio excel CAPM[1] vedremo come, a partire dai rendimenti di alcune attività rischiose, sia possibile con excel stimare la matrice di varianza covarianza e costruire la frontiera dei mix di investimenti possibili. Analizzeremo quindi la creazione della CML e della SML. Dati [pic] Nel foglio dati, partendo dalla matrice di rendimenti compresi tra la cella C14 e la cella H73, si stimano varianza, covarianza e rendimento medio relativi. Riguardo ai rendimenti questi sono ottenuti attraverso le differenze prime dei logaritmi dei prezzi (lnP(t)–lnP(t–1)) . Considerando ad esempio i primi rendimenti del titolo a: [pic] In excel la stima della varianza è ottenuta tramite la funzione =VAR.POP(), per la covarianza si utilizza l’omonima funzione =COVARIANZA(). La media è calcolata con la funzione media (=MEDIA() ). Chiaramente calcolando la covarianza tra i rendimenti dello stesso titolo si ottiene lo stesso risultato del comando VAR.POP applicato sugli stessi dati. Gli input vanno inseriti come......

Words: 1805 - Pages: 8