Mean Variance Portfolio Theory PDF

4 Mean Variance Portfolio Theory Michalis Chronopoulos [email protected] AS2109 2 Learning Outcomes After this session you will understand the notion of diversifica...

4 Mean Variance Portfolio Theory Michalis Chronopoulos [email protected] AS2109 2 Learning Outcomes After this session you will understand the notion of diversification know the form of the efficient frontier with/without short selling and with/without risk-free asset be able to calculate – the minimum variance portfolio for the case of no risk free asset & short-selling allowed Munir Hiabu, Michail Chronopoulos AS2109 3 – the minimum variance portfolio which gives a certain expected return for the case of no risk-free asset & short-selling allowed – calculate the efficient frontier and tangency portfolio for the case with risk-free asset Munir Hiabu, Michail Chronopoulos AS2109 4 Introduction In the previous section, we constructed portfolios consisting of 2 securities. In this section, we consider portfolio diversification: constructing portfolios with a large number of securities to spread the risk and we construct portfolios consisting of N securities (N = 2, 3,... ) portfolios with and without short-selling portfolios with and without riskless lending portfolios with and without borrowing Munir Hiabu, Michail Chronopoulos AS2109 5 4.1 N −asset portfolio Munir Hiabu, Michail Chronopoulos AS2109 6 Consider a portfolio with N securities, N ≥ 2 N X Rp = xiRi i=1 X µp = x i µi i X σp2 = xixj σij i,j N X N X −1 N X = x2i σi2 + 2 xixj σij i=1 i=1 j=i+1 Munir Hiabu, Michail Chronopoulos AS2109 7 Munir Hiabu, Michail Chronopoulos AS2109 8 4.2 Diversification Demonstration of diversification (approximate) Consider a portfolio with N securities: N X N X −1 N X σp2 = x2i σi2 + 2 xixj σij i=1 i=1 j=i+1 1 Invest equally in the N securities: xi = N for all i. We get: N N −1 XN 2 X 1 2 X 1 σp = 2 σi + 2 2 σij i=1 N i=1 j=i+1 N Munir Hiabu, Michail Chronopoulos AS2109 9 Return on security i has variance σi2. Let the average of these variances be: N 1 X 2 σ = σi2 N i=1 There are N (N − 1)/2 pairs of (non-coincident) securities, each pair has a covariance σij. Let the average of these covariances be: N −1 N 1 X X σ = σij N (N − 1)/2 i=1 j=i+1 Statistical data on stock returns shows that: Munir Hiabu, Michail Chronopoulos AS2109 10 they are usually positively correlated σ > 0 σ 2 > σ N N −1 XN X 1 2 X 1 σp2 = 2 σi + 2 2 σij i=1 N i=1 j=i+1 N N N −1 XN 1 X 1 2 N −1 X 1 = σi + σij N i=1 N N i=1 j=i+1 N (N − 1)/2 Munir Hiabu, Michail Chronopoulos AS2109 11 Thus, we finally have: 1 N −1 σp2 = 2 × σ + × σ N N 1  2  = σ − σ + σ N Munir Hiabu, Michail Chronopoulos AS2109 12 Question What happens to σp2 as N → ∞? Answer It approximates σ , i.e: σp2 → σ Interpretation: Individual risk does not matter in a ‘well diversified’ portfolio, only covariances matter. Munir Hiabu, Michail Chronopoulos AS2109 13 1  2  σp2 = σ − σ + σ N Portfolio risk does not fall to zero (in general) as N → ∞ Nondiversifiable risk arises from covariances; it remains, even in a well-diversified portfolio Diversifiable risk arises from variances; it falls to zero as N →∞ Munir Hiabu, Michail Chronopoulos AS2109 14 Munir Hiabu, Michail Chronopoulos AS2109 15 Definition (Portfolio Diversification) Portfolio diversification is achieved by investing in a portfolio with a large number of securities (whose returns are not heavily positively correlated) thereby reducing portfolio risk Colloquially: “Spread the risk” or “Don’t put all your eggs in the same basket” Munir Hiabu, Michail Chronopoulos AS2109 16 Question: Describe some practical ways of diversifying an investor’s stock portfolio Answer: Buy stocks from different countries Buy stocks from different industries Mix the stock portfolio with other types of assets Rebalance your portfolio over time Keep track of and reassess your portfolio Munir Hiabu, Michail Chronopoulos AS2109 17 4.3 Efficient Frontier (I) Recall: Efficient portfolio: A portfolio is efficient if an investor cannot find another portfolio with both less variance higher expected return and strict inequality for at least one of the two. Efficient frontier: The set of efficient portfolios. Munir Hiabu, Michail Chronopoulos AS2109 18 In previous section, we considered a 2-asset portfolio and found that Rp vs. σp plot is a hyperbola (in general). In this section, we consider a portfolio of N securities and investigate the shape of the efficient frontier evaluate the efficient frontier, i.e., specify the set of efficient portfolios Munir Hiabu, Michail Chronopoulos AS2109 19 Start with the simplest case (case I): no short-selling no riskless lending and borrowing Later we extend to other cases by removing above restrictions. Munir Hiabu, Michail Chronopoulos AS2109 20 Key observation 1: With N securities, there is a global minimum risk portfolio G a global maximum expected return asset M1 a global minimum expected return asset M2 Munir Hiabu, Michail Chronopoulos AS2109 21 Key observation 2: A combination of G and any other asset A such that RG < RA results in an increasing concave hyperbolic Rp vs. σp plot. Munir Hiabu, Michail Chronopoulos AS2109 22 Key observation 3: A combination of G and any other asset B such that RG > RB results in an increasing concave hyperbolic Rp vs. σp plot. Munir Hiabu, Michail Chronopoulos AS2109 23 With N securities in portfolio P , the Rp vs. σp space looks like this: Munir Hiabu, Michail Chronopoulos AS2109 24 Minimum variance frontier: set of portfolios with the lowest risk for a given expected return; here it is M1GM2 Efficient Frontier: part of MVF above G; here it is M1G; it is increasing, concave and hyperbolic. Munir Hiabu, Michail Chronopoulos AS2109 25 Recall MPT assumptions: Time horizon: single-period investment Asset risk characteristics: mean and variance only needed Investor risk preference: Non-satiation (“go north”) Investor risk preference: Risk aversion (“go west”) Result: A MPT investor only invests in the portfolios on M1G (the efficient frontier). Munir Hiabu, Michail Chronopoulos AS2109 26 Optimal portfolio: Single optimal portfolio for a particular investor lies on the efficient frontier depends on investor’s degree of risk aversion is portfolio where efficient frontier is tangential to investor’s indifference curve of highest expected utility Munir Hiabu, Michail Chronopoulos AS2109 27 Munir Hiabu, Michail Chronopoulos AS2109 28 Short-selling Earlier we considered efficient frontier for simplest case: no short-selling no riskless lending and borrowing Portfolio A collection or combination of N securities with proportion xi invested in security i (where i = 1,... , N ). Proportion xi is portfolio weight P Question: i xi. Is xi < 0? Is xi > 1? Short-selling: Selling an asset i that an investor does not own, meaning that xi < 0 for at least one i. Munir Hiabu, Michail Chronopoulos AS2109 29 4.4 Efficient Frontier (II) Case I (earlier) no short-selling no riskless lending and borrowing Case II short-selling is allowed no riskless lending and borrowing Allowing short-selling simplifies minimisation Munir Hiabu, Michail Chronopoulos AS2109 30 Precluding short-selling imposes constraints X xi = 1 and 0 ≤ xi ≤ 1 i Allowing short-selling leaves only one constraint X xi = 1 i Munir Hiabu, Michail Chronopoulos AS2109 31 2-asset portfolio (x2 = 1 − x1) µp = x1µ1 + x2µ2, σp2 = x21σ12 + x22σ22 + 2x1x2ρ12σ1σ2 Munir Hiabu, Michail Chronopoulos AS2109 32 N -asset portfolio Munir Hiabu, Michail Chronopoulos AS2109 33 G is global minimum risk portfolio, MVF is minimum variance frontier, EF is efficient frontier When short-selling is allowed, the optimal portfolio for a very risk-tolerant investor could involve investing more in a more risky asset by short-selling a less risky asset. Munir Hiabu, Michail Chronopoulos AS2109 34 Riskless Lending and Borrowing Riskless asset Asset with fixed return over given period. Has zero risk: σf = 0. Riskless lending Buying the riskless asset, i.e. lending to Treasury at fixed interest rate Rf over the single investment period Riskless borrowing Taking out a loan at the fixed rate Rf over the single investment period Munir Hiabu, Michail Chronopoulos AS2109 35 Question: Can investors engage in riskless borrowing in practice? Answer: It is possible for professional investors, but often expensive and difficult for private investors. Munir Hiabu, Michail Chronopoulos AS2109 36 4.5 Efficient Frontier (III) Case I (earlier) no short-selling no riskless lending and borrowing EF is a constrained hyperbola Case II (earlier) short-selling is allowed Munir Hiabu, Michail Chronopoulos AS2109 37 no riskless lending and borrowing EF is a unconstrained hyperbola Case III short-selling is allowed riskless lending and borrowing are allowed EF is a straight line Munir Hiabu, Michail Chronopoulos AS2109 38 2-asset portfolio Asset 1 and asset 2 µp = x1µ1 + x2µ2, σp2 = x21σ12 + x22σ22 + 2x1x2ρ12σ1σ2 Suppose: asset 1 is risky asset, but asset 2 is risk-free (σ2 = 0) µ p = x 1 µ 1 + x2 µ 2 , σ p = x 1 σ 1 Rename asset 2 as asset f (where f refers to risk-free asset): µp = x1µ1 + (1 − x1)Rf , σp = x1σ1 Munir Hiabu, Michail Chronopoulos AS2109 39 µp = x1µ1 + (1 − x1)Rf and σp = x1σ1 As x1 varies: straight line in µu-σp space. Munir Hiabu, Michail Chronopoulos AS2109 40 Lending and borrowing at the risk-free rate Rf. Munir Hiabu, Michail Chronopoulos AS2109 41 N -asset portfolio A, B and T are three risky assets Question: Does an investor prefer portfolios along Rf A, or those along Rf B, or those along Rf T ? Answer: Rf T is best. Munir Hiabu, Michail Chronopoulos AS2109 42 Tangency portfolio Portfolio T of risky securities such that line Rf T is tangent to efficient frontier. Aka: Optimal portfolio of risky securities. Efficient frontier: Straight line from Rf through T Munir Hiabu, Michail Chronopoulos AS2109 43 Interpretation An efficient investment consists of: either lending at riskfree rate Rf (i.e. buying riskless asset) or borrowing at riskfree rate Rf (i.e. shorting riskless asset) investing remaining wealth in tangency portfolio T of risky securities Munir Hiabu, Michail Chronopoulos AS2109 44 One-Fund Theorem (aka “Separation theorem”) A single fund of assets (composed of the tangency portfolio T of risky securities) is required for the construction of all efficient portfolios for all investors All efficient portfolios are combinations of T and Rf. Munir Hiabu, Michail Chronopoulos AS2109 45 Evaluating the Efficient Frontier Example: Given riskless lending and borrowing at 2% and three securities with R1 = 0.04, R2 = 0.05, R3 = 0.06     σ12 σ12 σ13 0.12 0.01 0.01 σ21 σ22 σ23 = 0.01 0.22 0.01     σ31 σ32 σ32 0.01 0.01 0.32 calculate composition of tangency portfolio Munir Hiabu, Michail Chronopoulos AS2109 46 Solution: To get x1, x2 and x3 in T , solve these 4 equations simultaneously: x1 + x2 + x3 = 1        0.12 0.01 0.01 x1 0.04 0.02 k 0.01 0.22 0.01 x2 = 0.05 − 0.02        0.01 0.01 0.32 x3 0.06 0.02 Munir Hiabu, Michail Chronopoulos 5 Modern Portfolio Theory AS2109 48 Learning Outcomes After this session you will know the assumptions underlying MPT know the definitions of: a portfolio, an opportunity set, an efficient portfolio, an efficient frontier know how to calculate the expected return and variance of a portfolio understand how correlation affects the risk in a two asset portfolio choice know why utility theory is more individual specific compared to mean variance analysis Munir Hiabu, Michail Chronopoulos AS2109 49 5.1 Investment Portfolio Definition (Portfolio): A collection or combination of N securities with proportion xi invested in security i (where i = 1,... , N ). Proportions, xi, are sometimes called portfolio weights. P We have i xi = 1. Proposition (Portfolio return): The return on a portfolio is given by X Rp = x i Ri. i Munir Hiabu, Michail Chronopoulos AS2109 50 Question: A owns £200 of BT shares and £800 of M&S. B owns £2000 of BT shares and £8000 of M&S. Are their portfolios different? Answer: They are the same because each portfolio consists of the same securities with the same proportions: 0.2 BT and 0.8 M&S. Munir Hiabu, Michail Chronopoulos AS2109 51 Question: A owns £200 of BT shares and £800 of M&S. C owns £2000 of BT shares and £8000 of BP. Are their portfolios different? Answer: Portfolios are different, because securities are different. Munir Hiabu, Michail Chronopoulos AS2109 52 5.2 Modern Portfolio Theory Application of Modern Portfolio Theory (MPT) Specification of opportunity set: 1. specify investment universe, i.e., specify securities i = 1,... , N that are available 2. estimate securities’ risk characteristics, i.e. expected return and variance. Determination of efficient portfolios: choice of {xi} Munir Hiabu, Michail Chronopoulos AS2109 53 Assumptions of MPT Summary: 1. Time horizon: single-period investment 2. Asset risk characteristics: mean and variance of return only are needed 3. Investor risk preference: Non-satiation 4. Investor risk preference: Risk aversion Munir Hiabu, Michail Chronopoulos AS2109 54 Assumption 1: Investors select their portfolios based on a single period. We ignore dynamic portfolio selection. Assumption 2: Investors select their portfolios based on the expected return (“reward”) and the variance of return (“risk”) only. Question: When is this assumption optimal, if believing in utility theory? Answer: It is optimal for investors with quadratic utility function or normally distributed returns. Munir Hiabu, Michail Chronopoulos AS2109 55 Investor Preferences Definition (Efficient Portfolio): A portfolio is efficient if there is no other portfolio with both less variance higher expected return and strict inequality for at least one of the two. Munir Hiabu, Michail Chronopoulos AS2109 56 Definition (Efficient frontier): The efficient frontier is the set of all efficient portfolios Non-satiation ⇒ prefer portfolios to the “north” Risk aversion ⇒ prefer portfolios to the “west” Efficient portfolios lie to the “northwest” Munir Hiabu, Michail Chronopoulos AS2109 57 Question: Explain why an investor prefers asset A to B Answer: A has strictly less variance while having the same expected return. Munir Hiabu, Michail Chronopoulos AS2109 58 Question: Explain why an investor prefers asset A to C Answer: A has strictly higher expected return while having the same variance. Munir Hiabu, Michail Chronopoulos AS2109 59 Question: Do investors prefer B to C? Answer: Depends on the investor. Munir Hiabu, Michail Chronopoulos AS2109 60 5.3 Mean, Variance, Covariance, Correlation of Returns Suppose that there are N securities in the investment universe or opportunity set. For each security, indexed by i, we define Ri = random variable denoting return µi = E(Ri) = Ri = expected return σi2 = Var(Ri) = variance of return For each pair of securities i and j, let σij = Cov(Ri, Rj ) = covariance of returns ρij = correlation coefficient of returns Munir Hiabu, Michail Chronopoulos AS2109 61 Covariance measures the extent to which two variables “move together” in the same direction: σij = Cov(Ri, Rj ) = E[(Ri − µi)(Rj − µj )] The correlation coefficient standardises the covariance: σij ρij = , −1 < ρij < 1 σi σj Munir Hiabu, Michail Chronopoulos AS2109 62 5.4 Basic Portfolio Maths P Question: What is i xi ? Answer: 1 Example: For portfolio {20%BT, 80%BP}, x1 = 0.2, x2 = 0.8, x1 + x2 = 1 Question: Suppose BT shares return 10% and BP shares return 5%. Calculate the return on portfolio {20%BT, 80%BP} P Answer: In general Rp = i xiRi, hence, Rp = 0.2 × 0.1 + 0.8 × 0.05 = 0.06 Munir Hiabu, Michail Chronopoulos AS2109 63 5.5 2-asset portfolio: Mean and Variance Rp = x1R1 + x2R2 E[Rp] = µp = x1µ1 + x2µ2 Var(Rp) = σp2 = x21σ12 + x22σ22 + 2x1x2σ12 Remark: Expected return on portfolio does not depend on correlation between security returns; the variance does depend on the correlation. Munir Hiabu, Michail Chronopoulos AS2109 64 Perfectly positively correlated securities (ρ = 1) 2-asset portfolio with perfectly positively correlated assets: σp2 = x21σ12 + x22σ22 + 2x1x2σ12 σp2 = x21σ12 + x22σ22 + 2x1x2σ1σ2 σp = |x1σ1 + x2σ2| σp = |x1σ1 + (1 − x1)σ2| Munir Hiabu, Michail Chronopoulos AS2109 65 Plot as x1 varies of: µp = x1µ1 + (1 − x1)µ2 σp = |x1σ1 + (1 − x1)σ2| Munir Hiabu, Michail Chronopoulos AS2109 66 Perfectly uncorrelated securities (ρ = 0) 2-asset portfolio with perfectly uncorrelated assets: σp2 = x21σ12 + x22σ22 + 2x1x2σ12 σp2 = x21σ12 + x22σ22 σp2 = x21σ12 + (1 − x1)2σ22 Munir Hiabu, Michail Chronopoulos AS2109 67 Plot as x1 varies µp vs. σp2: a parabola µp vs. σp: a hyperbola Munir Hiabu, Michail Chronopoulos AS2109 68 Perfectly negatively correlated securities (ρ = −1) 2-asset portfolio with perfectly negatively correlated assets: σp2 = x21σ12 + x22σ22 + 2x1x2σ12 σp2 = x21σ12 + x22σ22 − 2x1x2σ1σ2 σp = |x1σ1 − x2σ2| σp = |x1σ1 − (1 − x1)σ2| Munir Hiabu, Michail Chronopoulos AS2109 69 Plot as x1 varies of: µp = x1µ1 + (1 − x1)µ2 σp = |x1σ1 − (1 − x1)σ2| Munir Hiabu, Michail Chronopoulos AS2109 70 Remarks: Correlation has no effect on expected return, but does affect risk. Portfolio risk can be reduced by choosing securities that are not very positively correlated. Munir Hiabu, Michail Chronopoulos AS2109 71 5.6 MPT and Utility Theory Consider an investor with quadratic utility, e.g. U (W ) = W − 4W 2, f or 0 < W < 1/8 His expected utility is a function of µ and σ 2 (expectation and variance of return on an investment) Munir Hiabu, Michail Chronopoulos AS2109 72 Indifference: Investor is indifferent among assets that have the same expected utility. Indifference curve: plot of constant expected utility in R − σ space. Lines of higher expected utility are above and to the left Munir Hiabu, Michail Chronopoulos AS2109 73 Indifference curves for investor X are steeper than indifference curves for investor Y X is more risk averse than Y X demands a higher additional reward than Y for taking the same additional risk Munir Hiabu, Michail Chronopoulos AS2109 74 Investor X prefers asset C to B C has a higher expected utility for him. Investor Y prefers asset B to C B has a higher expected utility for him. Munir Hiabu, Michail Chronopoulos AS2109 75 MPT determines a set of efficient portfolios or efficient frontier for all investors but utility theory is required to find the single optimal portfolio for a particular investor Munir Hiabu, Michail Chronopoulos