Return and Risk in a Portfolio Setup

Παρουσίαση με θέμα: "Return and Risk in a Portfolio Setup"— Μεταγράφημα παρουσίασης:

1 Return and Risk in a Portfolio Setup
Κωνσταντίνος Δράκος Αναπληρωτής Καθηγητής Τμήμα Λογιστικής & Χρηματοοικονομικής Οικονομικό Πανεπιστήμιο Αθηνών Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

2 Indifference Curve Increasing Utility Expected Return E(r)
Represents individual’s willingness to trade-off return and risk Assumptions: 5 Axioms Prefer more to less (Greedy) Risk aversion Assets jointly normally distributed Expected Return E(r) Increasing Utility Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος Standard Deviation σ(r)

3 Dominance • 2 dominates 1; has a higher return
Expected Return 4 2 3 1 Standard Deviation • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

4 Statistics: Some Definitions
port1 16/04/2018 Statistics: Some Definitions Expected Return of Portfolio E(RP) = w1 ER1 + w2 ER2 Variance of Portfolio s2P = w21 s21+ w22 s w1 w2 s12 s2P = w21 s21+ w22 s w1 w2(  s1 s2) Also, ‘proportions’ are: w1 + w2 = 1. Note s12 =  s1 s from statistics Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος Copyright K. Cuthbertson

5 The Expected Return of a Portfolio
So to calculate a portfolio’s expected rate of return, we weight each individual investment’s expected rate of return using the fraction of money invested in each investment. Example: If you invest 25%of your money in the stock of Citi bank (C) with an expected rate of return of -32% and 75% of your money in the stock of Apple (AAPL) with an expected rate of return of 120%, what will be the expected rate of return on this portfolio? Expected rate of return = .25(-32%) (120%) = 82% Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

6 Evaluating Portfolio Risk
Unlike expected return, standard deviation is not generally equal to the weighted average of the standard deviations of the returns of investments held in the portfolio. This is because of diversification effects. The diversification gains achieved by adding more investments will depend on the degree of correlation among the investments. The degree of correlation is measured by using the correlation coefficient Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

7 Measures of Association
* 07/16/96 Measures of Association Correlation shows the association across random variables Variables with Positive correlation: tend to move in the same direction Negative correlation: tend to move in opposite directions Zero correlation: no particular tendencies to move in particular directions relative to each other Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

8 Covariance and Correlation
* 07/16/96 Covariance and Correlation Covariance in returns, sAB, is defined as: The correlation, rAB, is defined as: rAB is in the range [-1,1] Often estimated using historical averages (excel function: “correl”) Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

9 Correlation and diversification
The correlation coefficient can range from -1.0 (perfect negative correlation), meaning two variables move in perfectly opposite directions to (perfect positive correlation), which means the two assets move exactly together. A correlation coefficient of 0 means that there is no relationship between the returns earned by the two assets. As long as the investment returns are not perfectly positively correlated, there will be diversification benefits. However, the diversification benefits will be greater when the correlations are low or negative. The returns on most stocks tend to be positively correlated. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

10 Standard Deviation of a Portfolio
For simplicity, let’s focus on a portfolio of 2 stocks: Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

11 Diversification effect
Investigate the equation: When the correlation coefficient =1, the portfolio standard deviation becomes a simple weighted average: If the stocks are perfectly moving together, they are essentially the same stock. There is no diversification. For most two different stocks, correlation is less than perfect (<1). Hence, the portfolio standard deviation is less than the weighted average. – This is the effect of diversification. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

12 Example Determine the expected return and standard deviation of the following portfolio consisting of two stocks that have a correlation coefficient of .75. Portfolio Weight Expected Return Standard Deviation Apple .50 .14 .20 Coca-Cola Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

13 Answer Expected Return = .5 (.14) + .5 (.14)= .14 or 14%
Standard deviation = √ { (.52x.22)+(.52x.22)+(2x.5x.5x.75x.2x.2)} = √ .035= .187 or 18.7% Lower than the weighted average of 20%. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

14 Gains from Diversification
Diversification gain is related to correlation coefficient value. The degree of risk reduction increases as the correlation between the rates of return on two securities decreases. r = +1, Risk reduction does not occur by combining securities whose returns are perfectly positively correlated. Risk reduction occurs by combining securities whose returns are less than perfectly positively correlated. 0 < r < 1, If the correlation coefficient is less than 1, the third term in the portfolio variance equation is reduced, reducing portfolio risk. r = –1, If the correlation coefficient is negative, risk is reduced even more, but this is not a necessary prerequisite for diversification gains. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 23 23 22 23 22 23 23 23 23 23

15 Diversification with Multiple Assets
The more assets we incorporate into the portfolio, the greater the diversification benefits are. The key is the correlation between each pair of assets in the portfolio. With n assets, there will be an n × n covariance matrix. The properties of the variance-covariance matrix are: It will contain n2 terms. The two covariance terms for each pair of assets are identical. It is symmetrical about the main diagonal that contains n variance terms. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 25 25 24 25 24 25 25 25 25 25

16 The Concept of Risk With N Risky Assets
* 07/16/96 The Concept of Risk With N Risky Assets As you increase the number of assets in a portfolio: the variance rapidly approaches a limit, the variance of the individual assets contributes less and less to the portfolio variance, and the interaction terms contribute more and more. Eventually, an asset contributes to the risk of a portfolio not through its standard deviation but through its correlation with other assets in the portfolio. This will form the basis for CAPM. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

17 Diversification with Multiple Assets (cont.)
For a diversified portfolio, the variance of the individual assets contributes little to the risk of the portfolio. For example, in a 50-asset portfolio there are 50 (n) variance terms and 2450 (n2 − n) covariance terms. The risk depends largely on the covariances between the returns on the assets. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 26 26 25 26 25 26 26 26 26 26

18 Variance of a naively diversified portfolio of N assets
Portfolio variance consists of two parts: 1. Non-systematic (or idiosyncratic) risk and 2. Systematic (or covariance) risk The market rewards only systematic risk because diversification can get rid of non-systematic risk Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

19 Systematic and Unsystematic Risk
Intuitively, we should think of risk as comprising: Systematic risk: Component of total risk that is due to economy-wide factors. (non-diversifiable risk) Unsystematic risk: Component of total risk that is unique to firm and is removed by holding a well-diversified portfolio. The returns on a well-diversified portfolio will vary due to the effects of market-wide or economy-wide factors. Systematic risk of a security or portfolio will depend on its sensitivity to the effects of these market-wide factors. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 27 27 26 27 26 27 27 27 27 27

20 Feasible portfolios with N risky assets
* 07/16/96 Feasible portfolios with N risky assets We will refer to the convex surface as the feasible set. Notice the boundary of the feasible set starting from the minimum variance portfolio and moving upward and toward the right. This boundary represents the most efficient risky portfolios that an individual can hold. Why? Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

21 Dominated and Efficient Portfolios
* 07/16/96 Dominated and Efficient Portfolios Give me a portfolio (say, portfolio A) strictly within the feasible set and I can find you another portfolio B which has the same variance but higher expected return (by moving along a vertical line passing through portfolio find you another portfolio C which has a lower variance but same expected return as portfolio A (by moving along a horizontal line passing through portfolio A). In our jargon, we say that portfolio B and C dominate portfolio A. Notice then that the portfolios along the boundary of the feasible set, upward and to the right of the minimum variance portfolio, are the most efficient in the sense they dominate all other portfolios with the same variance and also all other portfolios with the same expected return. That is why we call the boundary as the efficient frontier. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

22 How would you find the efficient frontier?
* 07/16/96 How would you find the efficient frontier? 1. Find all asset expected returns and standard deviations. 2. Pick one expected return and minimize portfolio risk. 3. Pick another expected return and minimize portfolio risk. 4. Use these two portfolios to map out the efficient frontier. To maximize utility, an investor would prefer to hold a risky portfolio on the highest indifference curve. This means that the investor will prefer to hold the portfolio that is represented by the point of tangency of an indifference curve with the feasible set. This, of course, is a portfolio on the efficient frontier. There is little reason to believe, however, that different investors will hold the same portfolio on the efficient frontier. Our typical investor chooses a portfolio that we will call D. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

23 * 07/16/96 Three Important Funds The riskless asset has a standard deviation of zero The minimum variance portfolio lies on the boundary of the feasible set at a point where variance is minimum The market portfolio lies on the feasible set and on a tangent from the riskfree asset Suppose we allow investors to hold combinations of a riskfree asset and a risky portfolio. To understand the decision problem in this case we need to define three funds (one asset and two portfolios) that are of critical importance. The first fund is the riskless asset and it lies on the y-axis because it has zero standard deviation. The second fund is the minimum variance portfolio and it lies on the boundary of the feasible set at a point where variance is minimum. You can locate this fund by finding the point of tangency between a vertical straight line and the feasible set. Finally, if you draw a line from the riskless asset toward the feasible set and rotate it until it is tangent to the feasible set, the point of tangency is a fund known as the market portfolio. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

24 Tobin’s Two-Fund Separation
* 07/16/96 Tobin’s Two-Fund Separation When the riskfree asset is introduced, All investors prefer a combination of 1) The riskfree asset and 2) The market portfolio Such combinations dominate all other assets and portfolios This gives a very strong result often referred to as Tobin's Two-Fund Separation Result. We are finding that, when the riskfree asset is introduced, all investors prefer to move from their originally optimal portfolios to a combination of the riskfree asset and the market portfolio. Alternatively stated, combinations of the riskfree asset and the market portfolio dominate all other assets and portfolios. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος *

25 Systematic Risk and Market Portfolio
It would be an onerous task to calculate the correlations when we have thousands of possible investments. Capital Asset Pricing Model or the CAPM provides a relatively simple measure of risk. CAPM assumes that investors choose to hold the optimally diversified portfolio that includes all risky investments. This optimally diversified portfolio that includes all of the economy’s assets is referred to as the market portfolio. According to the CAPM, the relevant risk of an investment relates to how the investment contributes to the risk of this market portfolio. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

26 Systematic versus Idiosyncratic Risk
An investment’s systematic risk is far more important than its unsystematic risk. If the risk of an investment comes mainly from unsystematic risk, the investment will tend to have a low correlation with the returns of most of the other stocks in the portfolio, and will make a minor contribution to the portfolio’s overall risk. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

27 Systematic Risk and Beta
Systematic risk is measured by beta coefficient, which estimates the extent to which a particular investment’s returns vary with the returns on the market portfolio. In practice, it is estimated as the slope of a straight line: Beta could be estimated using excel or financial calculator, or readily obtained from various sources on the internet (such as Yahoo Finance and Money Central.com) Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

28 Beta Beta is a measure of a security’s systematic risk, describing the amount of risk contributed by the security to the market portfolio. Cov(Ri , RM) can be scaled by dividing it by the variance of the return on the market. This is the asset’s beta (i): Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 30 30 29 30 29 30 30 30 30 30

29 Portfolio Beta The systematic risk (beta) of a portfolio is calculated as the weighted average of the betas of the individual assets in the portfolio: Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος 41 41 40 41 40 40 40 40 40 40

30 Portfolio Beta The beta of a portfolio measures the systematic risk of the portfolio and is calculated by taking a simple weighted average of the betas for the individual investments contained in the portfolio. Example: Consider a portfolio that is comprised of four investments with betas equal to 1.5, .75, 1.8 and .60. If you invest equal amount in each investment, what will be the beta for the portfolio? Portfolio beta= 1.5*(1/4)+.75*(1/4)+1.8*(1/4)+.6*(1/4) =1.16 Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

31 The CAPM CAPM also describes how the betas relate to the expected rates of return that investors require on their investments. The key insight of CAPM is that investors will require a higher rate of return on investments with higher betas. The relation is given by the following linear equation: Rmarket is the expected return on the market portfolio Rf is the risk-free rate (return for zero-beta assets). Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

32 Example Example: What will be the expected rate of return on AAPL stock with a beta of 1.49 if the risk-free rate of interest is 2% and if the market risk premium, which is the difference between expected return on the market portfolio and the risk-free rate of return is estimated to be 8%? AAPL expected return = 2% *8% = 13.92%. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

33 CAPM’s Answers E(return) = Risk-free rate of return + Risk premium specific to asset i = Rf + (Market price of risk)x(quantity of risk of asset i) The equilibrium price of risk is the same across all marketable assets In the equation, the quantity of risk of any asset, however, is only PART of the total risk (s.d) of the asset. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

34 Total risk = systematic risk + unsystematic risk
CAPM’s Answers Specifically: Total risk = systematic risk + unsystematic risk CAPM says: (1)Unsystematic risk can be diversified away. If there is something you bear but can be avoided by diversifying at NO cost, the market will not reward the holder of unsystematic risk at all. (2)Systematic risk cannot be diversified away without cost. In other words, investors need to be compensated by a certain risk premium for bearing systematic risk. Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

35 CAPM results E(return) = Risk-free rate of return + Risk premium specific to asset i = Rf + (Market price of risk)x(quantity of risk of asset i) Precisely: [1] Expected Return on asset i = E(Ri) [2] Equilibrium Risk-free rate of return = Rf [3] Quantity of risk of asset i = COV(Ri, RM)/Var(RM) [4] Market Price of risk = [E(RM)-Rf] Thus, the equation known as the Capital Asset Pricing Model: E(Ri) = Rf + [E(RM)-Rf] x [COV(Ri, RM)/Var(RM)] Where [COV(Ri, RM)/Var(RM)] is also known as BETA of asset I Or E(Ri) = Rf + [E(RM)-Rf] x βi Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

36 Pictorial Result of CAPM
E(Ri) Security Market Line E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

37 CAPM in Details: What is an equilibrium?
CONDITION 1: Individual investor’s equilibrium: Max U Assume: [1] Market is frictionless => borrowing rate = lending rate => linear efficient set in the return-risk space [2] Anyone can borrow or lend unlimited amount at risk-free rate [3] All investors have homogenous beliefs => they perceive identical distribution of expected returns on ALL assets => thus, they all perceive the SAME linear efficient set (we called the line: CAPITAL MARKET LINE => the tangency point is the MARKET PORTFOLIO Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

38 CAPM in Details: What is an equilibrium?
Two-Fund Separation: Given the assumptions of frictionless market, unlimited lending and borrowing, homogenous beliefs, and if the above 4 equilibrium conditions are satisfied, we then have the 2-fund separation. TWO-FUND SEPARATION: Each investor will have a utility-maximizing portfolio that is a combination of the risk-free asset and a portfolio (or fund) of risky assets that is determined by the Capital market line tangent to the investor’s efficient set of risky assets Analogy of Two-fund separation Fisher Separation Theorem in a world of certainty Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

39 CAPM in Details: What is an equilibrium?
CONDITION 1: Individual investor’s equilibrium: Max U E(Rp) Capital Market Line B Q E(RM) Market Portfolio A Rf Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος σM σp

40 Stock Performance Measurement
Sharpe Index Assumes total variability is the appropriate measure of risk A measure of reward relative to risk Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

41 Stock Performance Measurement
Treynor Index Assumes that beta is the appropriate type of risk Measure of risk-adjusted return Higher the value; the higher the return relative to the risk-free rate Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

42 Portfolio VaR: more concepts
VaR on portfolio of assets Similar to standard VaR with new complications (Covariance, Dependence, Portfolio weights) Computational burden increases rapidly if each component treated alone: n assets imply n volatilities and n*(n-1) / 2 correlation coefficients to be estimated, with n=10 we would need 10 volatilities and 45 correlations. While in larger (more diversified) portfolio with 100 or 500 stocks, the number of correlations would become, respectively, 4950 and 124,750 Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

43 Portfolio VaR Individual VaR:
As we know as correlations become lower the diversification benefits increase. Portfolio VaR (“diversified VaR”, taking into account all diversification benefits between components: Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

44 When the correlation is zero, the portfolio VaR reduces to:
When the correlation is unity then the portfolio VaR is: Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

45 Mapping exposures Portfolio mapping is the process of transforming and decomposing a vector of exposures to a certain limited number of random variables (stock indices, exchange rates, interest rates etc) This allows calculating the VaR on the mapped portfolio This reduces the computational burden but can lead to oversimplifications Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

46 Mapping equity positions through Beta
Consider a portfolio of 1.5 million euros invested in 4 stocks: Alpha Bank (300 thousand), Ethniki (500 thousand), Eurobank (400 thousand), and ATE (300 thousand) Using the market model (CAPM), we may use the General Index as the relevant risk factor. Then we run a time series regression of each stock’s return on the market portfolio return in order to obtain its beta (systematic risk measure): Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

47 BETA (point estimate from regression) 1.15 1.41 1.05 0.69
ALPHA BANK ETHNIKI EUROBANK ATE Total amount invested BETA (point estimate from regression) 1.15 1.41 1.05 0.69 Position size by stock (in euros) 300000 500000 400000 Equivalent Index Position (in euros) 345000 705000 420000 207000 Total (index equivalent) position Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

48 So the 99% daily VaR for the portfolio of the four stocks is:
The equivalent market portfolio position is determined by simply multiplying the exposure in each stock by the stock’s beta, which we then sum up. So the 1.5 million euros invested in the four banks is therefore considered equivalent to million euros invested in the index directly (this reflects the fact that the average position in our portfolio is aggressive). The daily standard deviation of the market portfolio return for this period was 1.47 % So the 99% daily VaR for the portfolio of the four stocks is: Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος

49 This example shows that mapping has made the calculation of VaR easier
At the same time, mapping comes at the cost of accepting some simplification We have transformed the portfolio to a position with the same systematic risk, but… Our actual position is only on four stocks, and therefore is far from being diversified Thus, the true risk is underestimated to the extent that idiosyncratic risk is not taken into account Ειδικά Θέματα Χρηματοοικονομικής, Κωνσταντίνος Δράκος