Первый слайд презентации
1 Class Risk and Return Historical returns in the USA. Sigma. CV. Security Characteristic Line (SCL). Beta. Security Market Line (SML). CAPM formula. Study materials: KR: Ch. 11, 12. RWJ: Ch. 9, 10 BM: Ch. 7, 8
Слайд 3: Risk and Return
3 Risk and Return Risk - The chance that an investment's actual return will be different than expected. This includes the possibility of losing some or all of the original investment. Total risk measured by the standard deviation of the historical returns. A fundamental idea in finance is the relationship between risk and return. The greater the amount of risk that an investor is willing to take on, the greater the potential return. The reason for this is that investors need to be compensated for taking on additional risk. E.g., a U.S. Treasury bonds (bills) is considered to be the safest investment and, when compared to a corporate bond, provides a lower return. The reason is that a corporation is much more likely to go bankrupt than the U.S. government.
5 Risk and Return P 1 – P 0 ( + ∑ D) HPR, % = P 0 Return, % = P 1 / P 0 - 1 CV = sigma / mean return
6 Measuring Returns Historical returns: X 1 =10%, X 2 = (-5%), X 3 = 20% Arithmetic Average: [0.10 + (-0.05) + 0.20] / 3 = 8.33% - usual way Geometric Average: [(1.10)*(0.95)*(1.20)] 1/3 - 1 = 7.84% - more correct
7 7 Общий риск: стандартное отклонение
8 8 2. Risk and return relationship Общий риск: стандартное отклонение
9 Return (actual), mean return, % CAPM-return (model return), % Risk-free return, % Excess return, % Abnormal return, % Total risk (sigma), % Systematic risk (beta), t Unsystematic risk, % Coefficient of correlation, t Coefficient of determination, t Coefficient of variation (CV), t Sharpe’s measure (S), t Treynor’s measure (T), t Appraisal ratio (AR), t Z - beta, t Z - alpha, % Degree of volatility, t Risk and Return measures
10 Measuring Risk: ERR Expected return: The return for an asset is the probability weighted average return in all scenarios.
11 Variance: A measure of the dispersion of a set of data points around their mean value. Variance measures the variability (volatility) from an average. Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security. The variance of an asset’s expected return is the expected value of the squared deviations from the expected returns. Measuring Risk: ERR
12 Standard deviation: Square root of Variance. A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. In finance, it is a measure of total risk of a financial asset. Measuring Risk: ERR
13 Measuring Risk: ERR Example: Calculating ERR, Variance, and Standard deviation
14 Measuring Risk: Historical RR Variance: The variance of an asset’s historical returns is the sum of the squared deviations divided by (n-1). Standard deviation: Square root of variance.
15 Measuring Risk: Historical RR Example: Calculating Mean, Variance, and Standard deviation
16 Risk and Return Relationship Coefficient of Variation (CV) = Standard deviation / Expected (historical) rate of return. Measures risk-return relationship, i.e., sigma per 1 % of return. better choice
18 1 2 3 4 Return, % Sigma, % • 2 vs. 1; return is higher • 2 vs. 3; risk is lower • 4 vs. 3; return is higher Types of investors: Risk-averse, ( Risk-neutral ), Risk-seeker. A rational investor would choose the securities that locate in the left upper corner. CV MIN. R&RR and types of investors
22 Beta Beta: A measure of systematic risk, of a security (or a portfolio) in comparison to the market (market index, proxy). Measure of elasticity. Equals the variable coefficient “b” in the linear regression equation: Y = a + b*X + ε Graphically, Beta = Y / X A beta of greater than 1 indicates that the security's price is more volatile than the market, more risky. “Aggressive” security. The Security characteristic line is more steep. A beta of less than 1 indicates that the security's price is less volatile than the market, less risky. “Defensive” security.
23 Total Risk: Components Systematic Risk - The risk inherent to the entire market, or market segment. Also known as “non-diversifiable risk" or "market risk." E.g., interest rates, recession and wars represent sources of systematic risk because they affect the entire market and cannot be avoided through diversification. Affects a broad range of securities. Even a portfolio of well-diversified assets cannot escape all risk. Measured with BETA. Unsystematic Risk – Company-specific risk that is inherent in each investment. Can be reduced through appropriate diversification (=strategy designed to reduce risk by spreading the portfolio across many investments. Also known as "specific risk", "diversifiable risk“, or "residual risk". E.g., a sudden strike by the employees of a company is considered to be unsystematic risk.
24 Сигма портфеля Количество акций в портфеле, шт. Риск рынка ( Market Risk, Риск компании Общий риск (Diversifiable risk, Unsystematic risk) Nondiversifiable, Systematic risk) Total Risk: Components
26 Безрисковая доходность
27 Correlations of indices with S&P500 2010 2009 North America: 0,80 – 0,95 0,88 – 0,99 South America: 0,70 – 0,76 0,83 – 0,88 Western Europe: 0,80 – 0,87 0,79 – 0,92 Australia and NZ: 0,76 – 0,80 0,47 – 0,65 Russia: 0,71 – 0,76 0,53 – 0,59 Asia: 0,45 – 0,77 0,27 – 0,76 Ukraine: 0,46 – 0,51 0,44 – 0,48
29 Security Market Line (SML) Return,% . r f Risk-free rate Market Portfolio Market rate r m Beta 1.0 SML SML equation = r f + ( r m - r f ) = CAPM
Слайд 30: Sample Calculations for SML
30 r m = 13%, r f = 3% x = 1.20, then: E(r x ) = 3% + 1.2*(13%-3%) = 15% y = 0.80, then: E(r y ) = 3% + 0.8*(13%-3%) = 11% Sample Calculations for SML
31 Alpha Alpha (abnormal return; excess return) = r act - r f r p = r f + p ( r m - r f ) (+ /- alpha )
34 Гарри Марковиц, « Выбор портфеля», 1952. Нобелевский лауреат ( 1990 г. ) Markowitz’ Efficient Frontier
Слайд 35: Coefficient of Determination
35 Coefficient of Determination
36 Stock BEHAVIOUR – from the SCL equation Find and describe: Beta Alpha Correlation coef. Sigma Mean return CV
38 Capital Allocation Line (CAL) A line created in a graph of all possible combinations of risky and risk-free assets. The graph displays to investors the return they can make by taking on a certain level of risk. Also known as the "reward-to-variability ratio". E(r) E(r p ) = 15% r f = 7% p = 22% 0 P F E(r p ) - r f = 8% CAL
39 Portfolio risk and return: Return The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions ($) as weights. r p = w 1 r 1 + w 2 r 2 w 1 = Proportion of funds in Security 1, % w 2 = Proportion of funds in Security 2, % r 1 = Expected return on Security 1, % r 2 = Expected return on Security 2, %
40 Portfolio risk and return: Covariance Covariance is a measure of the linear association between the 2 variables. A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means that returns move inversely. Covariance between Stock 1 and Stock 2 = OR:
41 Portfolio risk and return: Covariance Covariance between Stock 1 and Stock 2 = = Cov 1,2 = ρ 1,2 σ 1 σ 2
42 Portfolio risk and return: Correlation Correlation is a measure of the linear association between the 2 variables. In the world of finance, a statistical measure of how two securities move in relation to each other. ρ 1, 2 = Cov 1,2 / σ 1 σ 2 Correlation coefficient is ALWAYS [-1; 1]. Perfect positive correlation (a correlation coefficient of +1) implies that as one security moves, either up or down, the other security will move in lockstep, in the same direction. Alternatively, perfect negative correlation means that if one security moves in either direction the security that is perfectly negatively correlated will move by an equal amount in the opposite direction. If the correlation is 0, the movements of the securities are said to have no correlation; they are completely random.
43 Portfolio risk and return: Sigma When two risky assets with variances 1 2 and 2 2, respectively, are combined into a portfolio with portfolio weights w 1 and w 2, respectively, the portfolio variance is given by: p 2 = w 1 2 1 2 + w 2 2 2 2 + 2w 1 w 2 Cov(r 1 r 2 ) = = w 1 2 1 2 + w 2 2 2 2 + 2w 1 w 2 ρ 1 2 Hence, the portfolio’s standard deviation - p - is the square root of the above formula.
44 Portfolio risk and return: Sigma Rule: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.
45 DIVERSIFICATION Coca Cola Reebok Standard Deviation 35% in Reebok, 65% in Coca-Cola Expected Returns and Standard Deviations vary given different weighted combinations of the stocks Expected Return, % See: file “port.xls”
Слайд 46: Efficient Frontier
Example Correlation Coefficient = 0.4 Stocks s % of Portfolio Avg Return ABC Corp 28 60% 15% Big Corp 42 40% 21% Standard Deviation = weighted avg = 33.6 Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4%
Слайд 47: Efficient Frontier
Example Correlation Coefficient = 0.4 Stocks s % of Portfolio Avg Return ABC Corp 28 60% 15% Big Corp 42 40% 21% Standard Deviation = weighted avg = 33.6 Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4% Let’s Add stock New Corp to the portfolio
Слайд 48: Efficient Frontier
Example Correlation Coefficient = 0.3 Stocks s % of Portfolio Avg Return Portfolio 28.1 50% 17.4% New Corp 30 50% 19% NEW Standard Deviation = weighted avg = 31.80 NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%
Слайд 49: Efficient Frontier
Example Correlation Coefficient = 0.3 Stocks s % of Portfolio Avg Return Portfolio 28.1 50% 17.4% New Corp 30 50% 19% NEW Standard Deviation = weighted avg = 31.80 NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20% NOTE: Higher return & Lower risk How did we do that? DIVERSIFICATION
51 Сигма портфеля Количество акций в портфеле, шт. Риск рынка ( Market Risk, Риск компании Общий риск (Diversifiable risk, Unsystematic risk) Nondiversifiable, Systematic risk) Diversification
53 ОНД, % Стандартное отклонение, % Stock B Stock A Corr = +1 Corr = -1 Corr = 0 Portfolio diversification : 2 stocks
54 Efficient Frontier Standard Deviation Each half egg shell represents the possible weighted combinations for two stocks. The composite of all stock sets constitutes the efficient frontier Expected Return, %
55 Efficient Frontier A B N AB ABN Goal is to move up and left Sigma Return, %
56 Efficient Frontier Return, % Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Sigma Goal is to move up and left
57 Efficient (Markowitz) frontier A line created from the risk-reward graph, comprised of optimal portfolios. The optimal portfolios plotted along the curve have the highest expected return possible for the given amount of risk. Harry Markowitz, "Portfolio Selection" (1952). Nobel prize 1990.
58 Political Risks : Mechel Story “В четверг ( 2 5 июля) премьер-министр Владимир Путин встретился в Нижнем Новгороде с представителями крупнейших российских металлургических и угольных компаний и высказался о ситуации на российском рынке стали. П ремьер-министр неожиданно сделал ряд жестких заявлений в отношении Мечела (ОАО «ЧМЗ»). По сути, он обвинил компанию в нерыночном поведении и предположил, что прокуратуре следуетобратить внимание на торговую деятельность компании. Этих заявлений оказалось достаточно, чтобы обрушить котировки Мечела – с четверга они упали на 37,6%, заставив инвесторов гадать о будущем компании – ведущего производителя коксующегося угля в России.” «Тройка-Диалог», 2 5 июля 2008 г. ( www.troika.ru )
60 Political Risks: MICEX Index, 2008 Inauguration of president Medvedev Putin accuses “Mechel” of fishy business Russian intrusion into Georgia Putin’s speech in Sochi, VII Economics Forum Medvedev: to establish a missile complex in Kaliningrad
61 Portfolio risk and return: Beta Systematic Risk - The risk inherent to the entire market, or market segment. Also known as "un-diversifiable risk" or "market risk." E.g., interest rates, recession and wars represent sources of systematic risk because they affect the entire market and cannot be avoided through diversification. Affects a broad range of securities. Even a portfolio of well- diversified assets cannot escape all risk. Measured with BETA. Still, one can diversify the systematic risk – in the sense that one can choose securities with different betas (defensive vs. aggressive stocks) and combine them.
62 Portfolio risk and return: Beta The Beta of a portfolio is the weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions ($) as weights. b p = w 1 b 1 + w 2 b 2 w 1 = Proportion of funds in Security 1, % w 2 = Proportion of funds in Security 2, % b 1 = Beta of Security 1 b 2 = Beta of Security 2