PORTFOLIO THEORY. Master in Finance INVESTMENTS. Szabolcs Sebestyén

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1 PORTFOLIO THEORY Szabolcs Sebestyén Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Portfolio Theory Investments 1 / 60

2 Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 2 / 60

3 Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 3 / 60

4 Introduction Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 4 / 60

5 Introduction Canonical Portfolio Problem for n > 1 Assets (1) Assume n risky assets with returns r 1, r 2,..., r n, and a risk-free asset with return r f The terminal wealth is given by ( ) n ) w 1 = w rf + φ i ( ri r f i=1 The investor s problem is [ ( max E ( ) n ) u w rf + {φ 1,...,φ n } φ i ( ri )] r f i=1 It is convenient to define weights rather ( than ) monetary values, ) so let ω i φ i /w 0 and write w 1 = w rf + n i=1 ω i w 0 ( ri r f The investor s problem becomes { ( [ (1 max E ) n ) u w 0 + rf + {ω 1,...,ω n } ω i ( ri ])} r f i=1 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 5 / 60

6 Introduction Canonical Portfolio Problem for n > 1 Assets (2) Define the portfolio return, r p as r p ω f r f + Since ω f = 1 n i=1 ω i, we have that r p = r f + n i=1 n i=1 ω i r i ω i ( ri r f ) The portfolio problem can be written as [ max E ( ) u (w )] rp {ω 1,...,ω n } This problem is hard to solve without some simplifying assumptions Sebestyén (ISCTE-IUL) Portfolio Theory Investments 6 / 60

7 Introduction Modern Portfolio Theory Modern Portfolio Theory (MPT) explores the details of the above portfolio choice problem, under the mean-variance utility hypothesis for an arbitrary number of risky investments, with or without a risk-free asset Sebestyén (ISCTE-IUL) Portfolio Theory Investments 7 / 60

8 Mean-Variance Preferences Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 8 / 60

9 Mean-Variance Preferences The Importance of Mean and Variance We define the utility function over the portfolio return r p We restrict our attention to utility functions that depend only on the mean and variance of r p This can result from two hypotheses within the expected utility framework: The utility function is quadratic The asset returns are Gaussian Probability distributions are hard to manipulate and estimate empirically, and summarising them by their first two moments is appealing The mean and the variance of the wealth distribution are critical for the determination of expected utility; recall that E [ u (w) ] = u ( E (w) ) u ( E (w) ) Var (w) + R 3 Under quadratic utility R 3 = 0, and under normality R 3 can be expressed in terms of the mean and variance Sebestyén (ISCTE-IUL) Portfolio Theory Investments 9 / 60

10 Mean-Variance Preferences Drawbacks of Mean-Variance Preferences Quadratic utility exhibits IARA, which is unrealistic Normality is attractive because of its additivity property, but it is also an unrealistic assumption for returns Normality is inconsistent with the limited liability of most assets: r i 1, and computing compounded cumulative returns is cumbersome (product of normals is not normal) The continuously compounded return, r i c ln (1 + r i ), is more attractive as it has no lower bound and cumulative returns can jut be added A working assumption in empirical financial economics is that r i c N ( µ i, σi 2 ), implying that ri is lognormal The lognormal assumption is appealing, but is not consistent with all the empirical properties of asset returns: Returns are frequently skewed Returns exhibit excess kurtosis ( fat tails ) Sebestyén (ISCTE-IUL) Portfolio Theory Investments 10 / 60

11 Mean-Variance Preferences Empirical return distributions 0.6 SP 100 return distribution Observed Normal 0.4 Density Standardised return Sebestyén (ISCTE-IUL) Portfolio Theory Investments 11 / 60

12 Mean-Variance Preferences Empirical return distributions 0.6 DEM/USD return distribution Observed Normal 0.4 Density Standardised return Sebestyén (ISCTE-IUL) Portfolio Theory Investments 12 / 60

13 Mean-Variance Preferences Average frequencies for standardised daily returns Observed Range SP 100 DEM/USD Normal [0; µ ± σ] 76.64% 74.36% 68.4% [µ ± σ; µ ± 2σ] 17.98% 20.23% 27.2% [µ ± 2σ; µ ± 3σ] 4.15% 4.13% 4.3% [µ ± 3σ; µ ± 4σ] 0.71% 1% 0.3% > 4σ 0.16% 0.077% 0.003% Sebestyén (ISCTE-IUL) Portfolio Theory Investments 13 / 60

14 Mean-Variance Preferences Mean-Variance Dominance Revisited Proposition Assuming either quadratic utility function or normal returns, the investor maximises a function of the mean and variance of the return distribution, max E [ u ( r) ] = max f ( µ r, σ 2 r ). Moreover, the objective function increases with the expected return, i.e., f µ r > 0, and decreases with the standard deviation, i.e., f σ r < 0. Sebestyén (ISCTE-IUL) Portfolio Theory Investments 14 / 60

15 Mean-Variance Preferences Two Corollaries Corollary Asset a mean-variance dominates asset b if and only if µ a µ b and σ a < σ b, or, equivalently, µ a > µ b and σ a σ b. Corollary A mean-variance investor s portfolio choice problem can be written as max {ω 1,...,ω n } µ p 1 2 g σ2 p, where µ p and σ p are portfolio mean and standard deviation, and g stands for the degree of the agent s risk aversion. Sebestyén (ISCTE-IUL) Portfolio Theory Investments 15 / 60

16 Diversification and the Efficient Frontier Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 16 / 60

17 Diversification and the Efficient Frontier Definitions The expected return to a portfolio is the weighted average of the expected returns of the assets composing the portfolio The variance of a portfolio is generally smaller than the weighted average of the variances of individual asset returns Thus there is gain in diversification The objective of a typical investor is to maximise u ( µ p, σ p ), where u is concave The investor likes expected return, but dislikes standard deviation Define the efficient frontier as the locus of all non-dominated portfolios (in the mean-variance sense) in the σ-µ space By definition, no mean-variance investor would choose to hold a portfolio not located on the efficient frontier Sebestyén (ISCTE-IUL) Portfolio Theory Investments 17 / 60

18 Diversification and the Efficient Frontier Mean-Variance Frontier with 2 Assets Consider two assets with expected returns r 1, r 2, standard deviations σ 1, σ 2, portfolio weights ω 1, ω 2, and return correlation ρ 1,2 The variance of the portfolio is σ 2 p = ω 2 1 σ2 1 + (1 ω 1 ) 2 σ ω 1 (1 ω 1 ) σ 1 σ 2 ρ 1,2 The efficient frontier will depend on the value of return correlations, ρ 1,2 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 18 / 60

19 Diversification and the Efficient Frontier Case 1: Perfect Positive Correlation When the returns of the the two risky assets are perfectly positively correlated (ρ 1,2 = 1), the efficient frontier is linear The two assets are essentially identical, so there is no gain from diversification The portfolio s standard deviation is just the average of the standard deviations of the component assets: σ p = ω 1 σ 1 + (1 ω 1 ) σ 2 The equation of the efficient frontier is µ p = r 1 + r 2 r 1 σ 2 σ 1 ( σp σ 1 ) Sebestyén (ISCTE-IUL) Portfolio Theory Investments 19 / 60

20 Diversification and the Efficient Frontier The Efficient Frontier: Perfect Positive Correlation Source: Danthine and Donaldson (2005), Figure 6.2 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 20 / 60

21 Diversification and the Efficient Frontier Case 2: Imperfect Correlation When the assets are imperfectly correlated (i.e., 1 < ρ 1,2 < +1), we gain from diversification: σ p < ω 1 σ 1 + (1 ω 1 ) σ 2 The smaller the correlation, the more to the left is the efficient frontier from the straight line of the previous case Some portfolios will be dominated by other portfolios; thus not all portfolios are efficient We must distinguish the minimum variance frontier from the efficient frontier Sebestyén (ISCTE-IUL) Portfolio Theory Investments 21 / 60

22 Diversification and the Efficient Frontier The Efficient Frontier: Imperfect Correlation Source: Danthine and Donaldson (2005), Figure 6.3 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 22 / 60

23 Diversification and the Efficient Frontier Case 3: Perfect Negative Correlation With perfect negative correlation, the minimum variance portfolio is risk free and the frontier is linear Source: Danthine and Donaldson (2005), Figure 6.4 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 23 / 60

24 Diversification and the Efficient Frontier Source: Sebestyén Danthine (ISCTE-IUL) and Donaldson (2005), Figure Portfolio 6.5 Theory Investments 24 / 60 Case 4: One Risk-Free and One Risky Asset If one of the two assets is risk free, the efficient frontier is a straight line originating on the vertical axis at the risk-free return Without short sales restrictions, the investor can borrow at the risk-free rate to leverage her holdings of the risky asset Thus, the overall portfolio can be made riskier than the riskiest of the existing assets

25 Diversification and the Efficient Frontier Source: Sebestyén Danthine (ISCTE-IUL) and Donaldson (2005), Figure Portfolio 6.6 Theory Investments 25 / 60 Case 5: n Risky Assets The previous analysis can be generalised as a portfolio is also an asset Adding more assets improves diversification possibilities, and the efficient frontier will have a bullet shape

26 Diversification and the Efficient Frontier Case 6: One Risk-Free and n Risky Assets The investor will choose the tangency portfolio on the mean-variance frontier to combine with the risk-free asset (point T in the previous chart) Thus, the efficient frontier is a straight line again If we allow for short sales, the efficient frontier extends beyond T Formally, with n assets (one possibly risk free), the efficient frontier is the non-dominated part of the minimum variance frontier, which is the solution to the quadratic program min n n {ω 1,...,ω n } i=1 j=1 s.t. n i=1 ω i ω j σ ij ω i r i = µ n ω i = 1 i=1 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 26 / 60

27 Diversification and the Efficient Frontier The Optimal Portfolio: A Separation Theorem The optimal portfolio is the one maximising the investor s (mean-variance) utility With one risk-free and n risky assets, all tangency points must lie on the same efficient frontier, regardless of the risk aversion of the investor If two investors share the same perceptions as to expected returns, variances and return correlations, but they differ in their risk aversion, the efficient frontier will be the same for them, but the optimal portfolios will be different points on the same line Two-fund theorem (Separation theorem): the optimal portfolio of risky assets can be identified separately from an investor s knowledge of the risk preference Sebestyén (ISCTE-IUL) Portfolio Theory Investments 27 / 60

28 Diversification and the Efficient Frontier Illustration: Separation Theorem Source: Danthine and Donaldson (2005), Figure 6.7 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 28 / 60

29 The Mathematics of the Efficient Frontier Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 29 / 60

30 The Mathematics of the Efficient Frontier Notation (1) Assume n 2 risky assets, and that their returns are linearly independent The vector of expected returns is E ( r 1 ) E ( r 2 ) r E ( r) =. E ( r n ) The covariance matrix of returns is σ 2 11 σ 1n V Cov ( r) =..... σ n1 σnn 2 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 30 / 60

31 The Mathematics of the Efficient Frontier Notation (2) The vector of portfolio weights is ω Let 1 denote the vector of ones ω 1. ω n Let µ p be a scalar denoting the required return of the portfolio The expression ω Vω represents the portfolio s return variance Definition (Frontier portfolio) A frontier portfolio is one that displays minimum variance among all feasible portfolios with the same expected return. Sebestyén (ISCTE-IUL) Portfolio Theory Investments 31 / 60

32 The Mathematics of the Efficient Frontier The Investor s Problem A portfolio p, characterised by ω p, is a frontier portfolio if and only if ω p solves min ω 1 2 ω Vω s.t. ω r = µ p ω 1 = 1 Since no non-negativity constraints are present, short sales are permitted Sebestyén (ISCTE-IUL) Portfolio Theory Investments 32 / 60

33 The Mathematics of the Efficient Frontier The Lagrangian and the FOC The Lagrangian of the problem is L = 1 2 ω Vω + λ ( µ p ω r ) + γ ( 1 ω 1 ) The necessary and sufficient FOCs are L = Vω λr γ1 = 0 ω L λ = µ p ω r = 0 L γ = 1 ω 1 = 0 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 33 / 60

34 The Mathematics of the Efficient Frontier Solution The solution to the above problem is where It can be written as ω p = 1 D = g + hµ p ω p = Cµ p A D V 1 r + B Aµ p D A = 1 V 1 r B = r V 1 r > 0 C = 1 V 1 1 > 0 D = BC A 2 > 0 [ B ( V 1 1 ) A ( V 1 r )] + 1 D V 1 1 [ C ( V 1 r ) A ( V 1 1 )] µ p = Sebestyén (ISCTE-IUL) Portfolio Theory Investments 34 / 60

35 The Mathematics of the Efficient Frontier Characterising the Solution Pick the required return µ p, and the solution ω p = g + hµ p yields the weights of the corresponding frontier portfolio Efficient portfolios are those for which µ p exceeds the expected return on the minimum variance risky portfolio If µ p = 0, then g = ω p, so g represents the weights that define the frontier portfolio with zero required return g + h yields the weights of the frontier portfolio with µ p = 1 Proposition The entire set of frontier portfolios can be generated as affine combinations of g and g + h. Proposition The portfolio frontier can be described as affine combinations of any two frontier portfolios, not just the frontier portfolios g and g + h. Sebestyén (ISCTE-IUL) Portfolio Theory Investments 35 / 60

36 The Mathematics of the Efficient Frontier The Variance of the Frontier Portfolio The variance of any portfolio on the frontier is σ 2 p ( µp ) = ( g + hµp ) V ( g + hµp ) = = C D ( µ p A ) C C Properties of the variance: The expected return of the minimum variance portfolio is A/C The variance of the minimum variance portfolio is 1/C The equation defines a parabola with vertex (1/C, A/C) in the σ 2 -µ space, and a hyperbola in the σ-µ space Sebestyén (ISCTE-IUL) Portfolio Theory Investments 36 / 60

37 The Mathematics of the Efficient Frontier The Set of Frontier Portfolios: µ-σ 2 Space Source: Danthine and Donaldson (2005), Figure 7.3 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 37 / 60

38 The Mathematics of the Efficient Frontier The Set of Frontier Portfolios: µ-σ Space Source: Danthine and Donaldson (2005), Figure 7.4 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 38 / 60

39 The Mathematics of the Efficient Frontier Example: Optimal Portfolio (1) Example Assume that there are only two risky assets with E ( r 1 ) = 15%, σ 1 = 25%, E ( r 2 ) = 10%, and σ 2 = 20%. The return correlation is zero, and we require an expected return of µ p = 14% on the portfolio composed by these two assets. Compute the optimal weights, the standard deviation of the portfolio, as well as the expected return and variance of the minimum variance portfolio. Solution The constants are given by A = 1 V 1 r = 4.9 C = 1 V 1 1 = 41 B = r V 1 r = 0.61 D = BC A 2 = 1 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 39 / 60

40 The Mathematics of the Efficient Frontier Example: Optimal Portfolio (2) Solution (cont d) Given the required return of µ p = 14%, the optimal weights are ω p = Cµ p A D V 1 r + B Aµ p D It is easy to verify that ωp r = 14% as required. The variance of the portfolio is σ 2 p = ω p Vω p = , V 1 1 = [ ] so the standard deviation is σ p = The expected return and variance of the minimum variance portfolio are E ( r mvp ) = A/C = and Var ( rmvp ) = 1/C = Sebestyén (ISCTE-IUL) Portfolio Theory Investments 40 / 60

41 The Mathematics of the Efficient Frontier Charaterising Efficient Portfolios Definition (Efficient portfolios) Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio. Proposition Any convex combination of a frontier portfolio is also a frontier portfolio. Corollary The set of efficient portfolios is a convex set. That is, if every investor holds an efficient portfolio, the market portfolio, being a weighted average of all individual portfolios, is also efficient. Sebestyén (ISCTE-IUL) Portfolio Theory Investments 41 / 60

42 Efficient Portfolio and Risk-Free Rate Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 42 / 60

43 Efficient Portfolio and Risk-Free Rate Problem Set-Up and Solution Consider n risky assets with expected return vector r, and one risk-free asset with expected return r f Let ω p be the n 1 vector of portfolio weights on the risky assets of a frontier portfolio p; then ω p is the solution to The solution to the problem is 1 min ω 2 ω Vω s.t. ω r + ( 1 ω 1 ) r f = µ p ω p = µ p r f H V 1( r r f 1 ) where H = ( r r f 1 ) V 1 ( r r f 1 ) = B 2Ar f + Cr 2 f > 0 Sebestyén (ISCTE-IUL) Portfolio Theory Investments 43 / 60

44 Efficient Portfolio and Risk-Free Rate The Frontier Equation The variance of any portfolio on the frontier is σp 2 Var ( r ) p = ω p Vωp = ( ) µp r 2 [ f = V 1( r r f 1 )] [ V V 1( r r f 1 )] = H ( ) µp r 2 f ( = r rf 1 ) V 1 ( r r f 1 ) ( ) 2 µp r f = H H Again we have a linear frontier: µ p = r f + σ p H This line goes through r f and the tangency portfolio (i.e., the set of portfolios with E ( r p ) > rf ) Sebestyén (ISCTE-IUL) Portfolio Theory Investments 44 / 60

45 Efficient Portfolio and Risk-Free Rate Portfolio Frontier When r f < A/C Source: Huang and Litzenberger (1988), Figure Sebestyén (ISCTE-IUL) Portfolio Theory Investments 45 / 60

46 Efficient Portfolio and Risk-Free Rate Portfolio Frontier When r f > A/C Source: Huang and Litzenberger (1988), Figure Sebestyén (ISCTE-IUL) Portfolio Theory Investments 46 / 60

47 Efficient Portfolio and Risk-Free Rate Portfolio Frontier When r f = A/C Source: Huang and Litzenberger (1988), Figure Sebestyén (ISCTE-IUL) Portfolio Theory Investments 47 / 60

48 Efficient Portfolio and Risk-Free Rate Discussion of the Three Cases Case 1: r f < A/C Any portfolio on the line segment r f e is a convex combination of the portfolio e and the riskless asset Any portfolio on the line beyond point e involves short selling the risk-free asset and invest the proceeds in portfolio e Any portfolio on the negatively sloped segment involves short selling portfolio e and investing the proceeds in the risk-free asset Case 2: r f > A/C Any portfolio on the positively sloped segment involves short selling portfolio e and investing the proceeds in the risk-free asset Any portfolio on the half-line r f σ p H involves a long position in portfolio e Case 3: r f = A/C There is no tangency portfolio Invest everything in the risk-free asset and hold an arbitrage portfolio of risky assets (the portfolio weights sum to zero) Sebestyén (ISCTE-IUL) Portfolio Theory Investments 48 / 60

49 Efficient Portfolio and Risk-Free Rate Tangency Portfolio The tangency portfolio is the only frontier portfolio composed only by risky assets, i.e., 1 ω T = 1 This implies that It follows then that µ T r f H 1 V 1( r r f 1 ) = 1 µ T = r f + H A Cr f Substituting this back to the expression for ωt we obtain ω T = 1 V 1( r r A f 1 ) Cr f Sebestyén (ISCTE-IUL) Portfolio Theory Investments 49 / 60

50 Efficient Portfolio and Risk-Free Rate Example: Optimal Portfolio with a Risk-Free Asset (1) Example Consider again the case of two risky assets with E ( r 1 ) = 15%, σ 1 = 25%, E ( r 2 ) = 10%, σ 2 = 20%. The return correlation is zero, and we require an expected return of µ p = 14% on the portfolio composed by these two assets. The risk-free return is 4%. Compute the optimal weights, the standard deviation of the portfolio, the tangency portfolio weights, and the expected return and standard deviation of the tangency portfolio. Solution The constant H is now H = B 2Ar f + Cr 2 f = Sebestyén (ISCTE-IUL) Portfolio Theory Investments 50 / 60

51 Efficient Portfolio and Risk-Free Rate Example: Optimal Portfolio with a Risk-Free Asset (2) Solution (cont d) Given the required return of µ p = 14%, the optimal weights are ω p = µ p r f H V 1( r r f 1 ) = [ ] The variance of the portfolio is σp 2 = ωp Vωp = , so the standard deviation is σ p = The tangency portfolio weights are ω T = 1 A Cr f V 1( r r f 1 ) = [ ] The expected return and variance of the tangency portfolio are E ( r T ) = and Var ( rt ) = Sebestyén (ISCTE-IUL) Portfolio Theory Investments 51 / 60

52 Efficient Portfolio and Risk-Free Rate Example: Plotting the Efficient Frontier Sebestyén (ISCTE-IUL) Portfolio Theory Investments 52 / 60

53 Optimal Portfolio for Mean-Variance Investors Outline 1 Modern Portfolio Theory Introduction Mean-Variance Preferences Diversification and the Efficient Frontier The Mathematics of the Efficient Frontier Efficient Portfolio and Risk-Free Rate Optimal Portfolio for Mean-Variance Investors Sebestyén (ISCTE-IUL) Portfolio Theory Investments 53 / 60

54 Optimal Portfolio for Mean-Variance Investors Problem Set-Up We saw before that a mean-variance investor s portfolio choice problem is max E( r ) 1 p ω 2 g Var( r ) p Assuming n risky assets and one riskless asset, the problem becomes Or, equivalently max ω ω r + ω f r f 1 2 g ω Vω s.t. ω 1 + ω f = 1 max ω ω r + ( 1 ω 1 ) r f 1 2 g ω Vω Sebestyén (ISCTE-IUL) Portfolio Theory Investments 54 / 60

55 Optimal Portfolio for Mean-Variance Investors Solution The FOC of the problem is The solution then becomes r r f 1 gvω = 0 ω p = 1 g V 1( r r f 1 ) To verify that this leads to an efficient portfolio, recall that µ p = ω r + ( 1 ω 1 ) r f = ( r r f 1 ) ω + rf Substituting the optimal weight vector yields µ p = ( r r f 1 ) 1 g V 1( r r f 1 ) + r f = 1 g H + r f Sebestyén (ISCTE-IUL) Portfolio Theory Investments 55 / 60

56 Optimal Portfolio for Mean-Variance Investors Verifying Efficiency There are two alternatives to prove the efficiency of the portfolio: 1 Plug the above expression into the formula for frontier portfolios: ωp = µ p r f H V 1( r r f 1 ) = 1 g = H + r f r f V 1( r r H f 1 ) = 1 g V 1( r r f 1 ) 2 Show that the investor s portfolio verifies the equation for the efficient frontier The portfolio variance is σp 2 = ωp Vωp = 1 g 2 H Plug the variance into the equation for the efficient frontier: 1 µ p = r f + σ p H = rf + g 2 H H = r f + 1 g H Sebestyén (ISCTE-IUL) Portfolio Theory Investments 56 / 60

57 Optimal Portfolio for Mean-Variance Investors Example: Optimal Portfolio for M-V Investor Example Consider two risky assets with E ( r 1 ) = 15%, σ 1 = 25%, E ( r 2 ) = 10%, σ 2 = 20%. Moreover, ρ 1,2 = 0 r f = 4%, and g = 5. Compute the expected return and standard deviation of the optimal portfolio. Solution The optimal portfolio weights are ω p = 1 g V 1( r r f 1 ) = [ ] and ω f = The expected return and standard deviation of the optimal portfolio are µ p = ω r + ( 1 ω 1 ) r f = 9.67% σ 2 p = ω p Vω p = or σ p = 10.65% Sebestyén (ISCTE-IUL) Portfolio Theory Investments 57 / 60

58 APPENDIX Sebestyén (ISCTE-IUL) Portfolio Theory Investments 58 / 60

59 Moment-Generating Function Definition (Moment-Generating Function) The moment-generating function of a random variable x is M (t) = E ( e t x) t R, whenever the expectation exists. The name comes from the property that, if it exists on an open interval around t = 0, it generates the moments of the probability distribution: where n is a non-negative integer. E ( x n) = M (n) (0) = dn M dt n (0), Sebestyén (ISCTE-IUL) Portfolio Theory Investments 59 / 60

60 MGF of a Gaussian Random Variable Example (Moment-generating function of a Gaussian random variable) The moment-generating function of a Gaussian random variable x N ( µ, σ 2) is M (t) = e tµ+ 1 2 t2 σ 2. Return Sebestyén (ISCTE-IUL) Portfolio Theory Investments 60 / 60