Mean-Variance Portfolio Theory Lakehead University Winter 2005
Outline Measures of Location Risk of a Single Asset Risk and Return of Financial Securities Risk of a Portfolio The Capital Asset Pricing Model 2
Measures of Location How to calculate the return on a single asset? Let W End-of-period wealth I Initial investment The holding period return is then R = W I I 3
Measures of Location Future holding period returns are rarely known with certainty. The best we can do is to assign probabilities to various possible outcomes. Suppose, for example, that the stock price of Bayside Smoke is currently P 0 = $25. At the end of the period, the stock can either be $20.00, $22.50, $25.00, $30.00 or $40.00 according to the probabilities in the following table. 4
Measures of Location Hypothetical Prices for Bayside Smoke Co. Probability End-of-Period Return (%) p i Price per Share R i.1 20.00 20.2 22.50 10.4 25.00 0.2 30.00 20.1 40.00 60 5
Measures of Location In a world with N states denoted i = 1,2,...,N, where p i denotes the probability of state i, the expected value of the random variable X is given by E[ X] = p 1 X 1 + p 2 X 2 +... + p N X N = N i=1 p i X i, where X i denotes the value of X in state i. 6
Measures of Location In the case of Baysmoke, the expected end-of-period price is E[ P] =.1 20.0 +.2 22.5 +.4 25.0 +.2 30.0 +.1 40.0 = $26.50 and thus the expected return is E[ R] = E[ P] P 0 P 0 = 26.50 25.00 25.00 = 6%. 7
Measures of Location Note that E[ P] =.1 20% +.2 10% +.4 0% +.2 20% +.1 60% = 6%. 8
Measures of Location Properties of the Operator E[ ] Let X and Ỹ be random variables and let a be a constant. Then E[ X + a] = E[ X] + a E[ X +Ỹ ] = E[ X] + E[Ỹ ] E[a X] = ae[ X] 9
Measures of Location The Greek letter µ is often used to represent expected value. The expected value of a random variable X, for instance, can be expressed as µ X. That is, E[ X] µ X. 10
Measures of Location The expected value is often referred to as the mean, or the average, of a distribution. The median and the mode are other measures of location. The median is outcome in the middle. The mode is the most frequent outcome. 11
Measures of Location Consider the following set of numbers: 17 0 7 10 13 3 15 4 6 1 17 13 13 25 13 150 1 6 8 2 54 32 202 16 13 21 120 24 29 37 12
Measures of Location Assuming the numbers are all equally likely, what is the mean, median and mode of this distribution? The mean is the sum of all numbers divided by 30 (the number of observations), which gives 28.13. The median is 13. The mode is also 13. Note that this distribution is skewed to the right. 13
Measures of Dispersion The range is the difference between the highest and lowest outcomes. The semi-interquartile range is the difference between the observation of the 75th percentile (X.75 ) and the 25th percentile (X.25 ) divided by 2: Semi-interquartile range = X.75 X.25 2 14
Measures of Dispersion Using the above 30 numbers, we find, since 30/4 = 7.5, and thus X.25 = 3 + 6 2 = 4.5 and X.75 = Semi-interquartile range = 27 4.5 2 25 + 29 2 = 27 = 11.25. 15
Measures of Dispersion The most frequently used statistic to measure dispersion is the variance. The variance of a random variable X is given by [ = VAR( X) = E ( X µ X ) 2]. σ 2 X 16
Measures of Dispersion If there are N states of the world, p i being the probability of state i and X i being the outcome in state i, the variance of X is given by σ 2 X = VAR( X) = N i=1 p i (X i µ X ) 2, where µ X = E[ X] = N i=1 p i X i. 17
Measures of Dispersion In the Bayside Smoke example, the variance of P is VAR( P) =.1 (20 26.5) 2 +.2 (22.5 26.5) 2 +.4 (25 26.5) 2 +.2 (30 26.5) 2 +.1 (40 26.5) 2 = 29.00. 18
Measures of Dispersion To calculate the variance of a sample of N observations drawn from a population, the monthly stock price of a company over the last five years, for example, we assume that all observed prices are equally likely. A variance in this case is calculated as VAR( X) = N i=1 1 N (X i µ X ) 2, where µ X = E[ X] = 1 N N X i. i=1 19
Measures of Dispersion Note that observations are squared to calculate the variance. This implies that the variance does not have the same units as the random variable. It is then usually more convenient to refer to dispersion using the standard deviation, which is the square root of the variance. In the Bayside Smoke example, the standard deviation of P is σ P = 29.00 = $5.39. 20
Measures of Dispersion Let X and Ỹ be random variables. The covariance between X and Ỹ is given by [ ] COV( X,Ỹ ) = E ( X µ X )(Ỹ µ Y ). 21
Measures of Dispersion Properties of the Operator VAR( ) Let X and Ỹ be random variables and let a and b be constants. Then VAR( X + a) = VAR( X) VAR(a X) = a 2 VAR( X) VAR(a X + bỹ ) = a 2 VAR( X) + b 2 VAR(Ỹ ) + 2abCOV( X,Ỹ ) 22
Measures of Dispersion Using these properties, we can calculate the variance of the return to investing in Bayside Smoke from the variance of the stock price. That is, since R = P P 0 = P 1, P 0 P 0 ( ) ( ) P P VAR( R) = VAR 1 = VAR P 0 P 0 = VAR( P) P0 2. 23
Other Measures of Dispersion The variance gives equal weight to possibilities above and below the mean. Risk-averse investors may be more concerned with downside risk and thus could be more interested in knowing the semi-variance of a stock return. Let X i = { X i µ X if X i < µ X, 0 if X i µ X for all i and let X represent the random variable obtained through this transformation. Then SEMIVAR( X) = E[ X 2 ]. 24
Other Measures of Dispersion Another measure of dispersion is the average absolute deviation (AAD), which is calculated as AAD( X) = E [ X µ X ] 25
Measuring Portfolio Risk and Return If the only statistics considered to characterize an asset are its mean and variance, we must assume that no other statistic is necessary. If we assume that asset returns have a normal distribution, then the mean and variance are the only statistics we need to compute in order describe an asset. Another property of the normal distribution is that it is symmetric around its mean. 26
Measuring Portfolio Risk and Return If an asset return R is normally distributed with mean µ R and variance σ 2 R, then the random variable z = R µ R σ R is normally distributed with mean 0 and variance 1. This distribution is called the standard normal. 27
Risk of a Single Asset If asset X s return is assumed to be normally distributed, then all we need to know about this asset is the pair (σ X,µ X ), where σ X represents the risk of the asset and µ X its expected return. 28
Risk Preference It is generally assumed that individuals are risk averse. Hence return is good but risk is bad. Note that risk may be enjoyable for small gambles. 29
Risk and Return of a Portfolio The goal of a financial manager serving risk-averse investors in world where asset returns are normally distributed is to create an efficient portfolio. An efficient portfolio is such that is provides the desired expected return at the lowest possible risk. Diversification, due to the correlation between assets, helps reduce the risk of a portfolio. 30
Return of a Portfolio Let p be a portfolio composed of n assets, the fraction of wealth invested in assets i being represented by w i, with the condition that w 1 + w 2 +... + w n = n w i = 1. i=1 Let R p denote the return to portfolio p and let R i denote the return to Asset i. 31
Return of a Portfolio If the expected return on Asset i is µ i, the expected return of portfolio p is µ p = E[ R p ] = E[w 1 R 1 + w 2 R 2 +... + w n R n ] = E[w 1 R 1 ] + E[w 2 R 2 ] +... + E[w n R n ] = w 1 E[ R 1 ] + w 2 E[ R 2 ] +... + w n E[ R n ] = w 1 µ 1 + w 2 µ 2 +... + w n µ n n = w i µ i i=1 32
Risk of a Portfolio Correlation To determine the risk of a portfolio of assets, we need to use the concept of correlation. When two assets tend to move in the same direction, they are positively correlated. When two assets tend to move in different directions, they are negatively correlated. Two assets are not correlated when there is no relationship between their movements. 33
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Risk of a Portfolio Correlation Consider two assets, x and y, let R i, µ i and σ i represent the return, expected return and standard deviation of asset i = x,y, respectively. The correlation between the two assets is given by ρ xy = σ xy σ x σ y, where σ xy is the covariance of x and y s returns. 36
Risk of a Portfolio Correlation By definition, 1 ρ 1. ρ = 1 if x and y s returns are perfectly positively correlated, 1 if x and y s returns are perfectly negatively correlated. 37
Risk of a Portfolio Let p = (w x,w y ) denote the portfolio that allocates a fraction w x of every dollar invested in asset x and a fraction w y of every dollar invested in asset y, with w x + w y = 1. The expected return of portfolio p is µ p = w x µ x + w y µ y 38
Risk of a Portfolio The variance of p is [ ( ) ] 2 σ 2 p = E R p µ p [ ( ) ] 2 = E w x R x + w y R y w x µ x w y µ y [ ( ) ] 2 = E w x ( R x µ x ) + w y ( R y µ y ) [ ( ) 2 ( ) 2 ( )( ) ] = E w 2 x R x µ x + w 2 y R y µ y + 2wx w y R x µ x R y µ y [ ( ) ] [ 2 ( ) ] 2 [( )( )] = w 2 xe R x µ x + w 2 ye R y µ y + 2w x w y E R x µ x R y µ y = w 2 xσ 2 x + w 2 yσ 2 y + 2w x w y σ xy. 39
Risk of a Portfolio Since σ xy = ρ xy σ x σ y, the variance of p can be rewritten as σ 2 p = w 2 xσ 2 x + w 2 yσ 2 y + 2w x w y ρ xy σ x σ y. 40
Risk of a Portfolio A Portfolio of 3 Assets Suppose a portfolio contains 3 assets, x, y and z. The weight on each assets is given by w i, i = x,y,z and the correlation between the return of two assets is ρ i j. 41
Risk of a Portfolio A Portfolio of 3 Assets The variance of this portfolio is σ p = w 2 xσ 2 x + w 2 yσ 2 y + w 2 z σ 2 z + 2w x w y ρ xy σ x σ y + 2w x w z ρ xz σ x σ z + 2w y w z ρ yz σ y σ z. 42
Risk of a Portfolio Back to the two-asset portfolio. If w x is the weight on x, then the weight on y is 1 w x. The standard deviation of portfolio p is then σ p = w 2 xσ 2 x + (1 w x ) 2 σ 2 y + 2w x (1 w x )ρ xy σ x σ y. 43
Risk of a Portfolio Note that σ p = w x σ x + (1 w x )σ y If ρ xy = 1, wx σ x (1 w x )σ y If ρ xy = 1. where w x σ x (1 w x )σ y = w x σ x (1 w x )σ y when w x σ y σ x +σ y, (1 w x )σ y w x σ x when w x < σ y σ x +σ y. 44
Risk of a Portfolio Suppose µ x = 8%, σ x = 20%, µ y = 18% and σ y = 34%. Let p be a portfolio investing in x and y only. What is the relationship between the standard deviation of p (σ p ) and the weight on x (w x )? 45
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Risk of a Portfolio The following graph looks at µ p in terms of σ x using the same parameters as above for µ x, µ y, σ x and σ y. 47
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Risk of a Portfolio Note that we can form a risk-free portfolio only if ρ xy = 1. The risk-free portfolio is such that w x = σ y σ x + σ y. If ρ xy > 1, then it is not possible to form a risk-free portfolio with x and y. 49
Optimal Portfolio Choice: Many Assets If, using all available risky assets, we construct all possible portfolios, then we end up with a portfolio possibilities set. Note that not all risk-return combinations are feasible. This also gives us a portfolio frontier, i.e. a curve giving the maximum expected return for each level of realisable risk. 50
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Optimal Portfolio Choice: Many Assets The upper part of the curve on the preceding graph is called the efficient portfolio frontier. A portfolio on the efficient portfolio frontier provides the highest possible return for a given level of risk. Only portfolios of risky assets on the efficient frontier should be chosen by investors, regardless of their risk tolerance. 52
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Optimal Portfolio Choice: Many Assets If only risky portfolios are available, then individuals will choose different portfolios on the efficient frontier depending on their risk tolerance. An individual will choose a portfolio that reaches his highest possible indifference curve. 54
E[ R] A B Efficient Frontier σ 55
Optimal Portfolio Choice: Many Assets If Joe is less risk averse than Bob, then Joe will select a portfolio with more risk than Bob s. Given the shape of the efficient portfolio frontier, risk neutral and risk loving individuals will select portfolios with the highest feasible level of risk. 56
E[ R] Bob Joe A B Efficient Frontier σ 57
Optimal Portfolio Choice: Many Assets Suppose there exists a risk-free asset with return denoted r f. The standard deviation of the risk-free return is zero. Let p be a portfolio containing asset x = (σ x,e[ R x ]) and the risk-free asset f = (0,r f ). The expected return of this portfolio is E[ R p ] = w x E[ R x ] + (1 w x )r f. 58
Optimal Portfolio Choice: Many Assets Since the risk-free return is not random, its covariance with asset a s return is zero. That is, ρ x f = 0 The standard deviation of p is then σ p = w 2 xσ 2 x + (1 w x ) 2 σ 2 f + 2w x(1 w x )ρ x f σ x σ f = w 2 xσ 2 x = w x σ x w x = σ p σ x. 59
Optimal Portfolio Choice: Many Assets Replacing w x with σ p σ x in E[ R p ] = w x E[ R x ] + (1 w x )r f gives E[ R p ] = σ pe[ R p ] σ x Note that ρ xp = 1 and thus + (σ x σ p )r f σ x = r f + σ p σ x ( E[ R x ] r f ). σ p σ x = ρ xpσ p σ x σ 2 x = cov( R x, R p ) σ 2. x 60
Optimal Portfolio Choice: Many Assets This gives us E[ R p ] = r f + cov( R x, R p ) σ 2 x ) (E[ R x ] r f. Similarly, the expected return of a portfolio q containing asset y, say, and the risk-free asset is given by E[ R q ] = r f + cov( R y, R q ) σ 2 y ) (E[ R y ] r f. 61
Optimal Portfolio Choice: Many Assets If one can borrow and lend without restrictions at the risk-free rate, the weight on the risky asset can vary from zero to infinity. More risk averse investors with put more weight on the risk-free asset. Less risk averse investors will put less weight on the risk-free asset. 62
E[ R] p (w x > 1) r f x y q (w y < 1) σ 63
Optimal Portfolio Choice: Many Assets Of all the risky assets available, an individual forming a portfolio including the risk-free asset will use the risky asset allowing him to reach the highest possible indifference curve. We can see on the preceding graph that combining x and f offers better possibilities than combining y and f : for every portfolio on the line r f y, there exists a portfolio on the line r f x that offer a higher return at a lower risk. 64
Optimal Portfolio Choice: Many Assets The slope of the line r f x is E[ R x ] r f σ x, which is the Sharpe ratio of asset x. The slope of the line r f y, on the other hand, is E[ R y ] r f σ y, which is the Sharpe ratio of asset y. 65
Optimal Portfolio Choice: Many Assets Assuming that all investors have the same beliefs about asset returns, they will all hold the same portfolio of risky assets. Let m denote this portfolio and let s call it the market portfolio. Combining m with the risk-free rate allows any investor to reach his/her highest indifference curve regardless of his/her risk tolerance. 66
E[ R] r f m Capital Market Line (CML) Efficient Frontier σ 67
Optimal Portfolio Choice: Many Assets The market portfolio is the portfolio of risky assets with the highest Sharpe ratio. Since there is no other portfolio offering a higher Sharpe ratio, all investors hold the same portfolio of risky assets. The capital market line is the efficient set for all investors. 68
Optimal Portfolio Choice: Many Assets Two-Fund Separation: Each investor has a utility-maximizing portfolio consisting of the risk-free asset and the market portfolio. 69