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Appendix: the Mechanics of Mean-Variance Optimization1 Appendix: The Mechanics of Mean-Variance Optimization1 In the following, we will assume that investment returns are completely described by the two moments of mean and variance. To think in terms of asset allocation, we will refer to the elements in the portfolio as “assets” rather than “securities”. We will begin this review with the Markowitz model2 and round off with the capital asset pricing model (CAPM). Throughout, risk will be understood to mean the standard deviation of the return, or volatility. We will begin with a portfolio consisting of two risky assets and later generalize to n assets. Two Risky Assets Notation Ra: Return on asset a Rb: Return on asset b Ra : Expected (mean) return on asset a Rb : Expected (mean) return on asset b Rp : Expected (mean) return on portfolio wa: Weight in the portfolio of asset a wb: Weight in the portfolio of asset b (=1 − wa) 2 σ a : The variance of the return on asset 1 1 This Appendix is meant as a quick refresher. For a more complete coverage, see, for example, Elton et al. (2014) or Bodie et al. (2014). 2 Markowitz (1952, 1959). © The Author(s) 2018 211 H. Lumholdt, Strategic and Tactical Asset Allocation, https://doi.org/10.1007/978-3-319-89554-3 212 Appendix: The Mechanics of Mean-Variance Optimization 2 σ b : The variance of the return on asset 2 σab: The covariance between the returns on assets a and b 2 σ p : The variance of the return on the portfolio All wealth is invested in the two assets, implying that wa + wb = 1. The expected return of the portfolio is a linear combination of the expected return on each asset: Rw=+Rw1− R (A.1) paaa()b The variance of the returns on the portfolio is given by: 2 σ 2 =−ER R pp p Expanding this expression, we get: 2 σ 2 =+Ew Rw11− Rw−+Rw− R pa aa()ba()aa()b 2 =−Ew RR+−1 wR− R aa()aa()()bbb 2 2 2 2 =−Ewaa()RRaa+−()12wR()bb− Rw+−a ()1 wRaaa()− R ()RRRbb− 2 2 Note that ER()aa− R and ER()bb− R are the variances of the returns ER − RR( − R on assets a and b, respectively, while ()aa()ab is their covari- ance. We can therefore write: 2 σσ22=+ww2 12− σσ2 +−ww1 (A.2) paaa()ba()aab and 12/ 22 2 2 σσpa=+wwaa()12− σσba+−ww()1 aab (A.3) The covariance can be expressed as the product of the standard deviation of each asset and the correlation coefficient between the assets (ρab) i.e. σab = σaσbρab. We can therefore write (A.2) as Appendix: The Mechanics of Mean-Variance Optimization 213 2 σσ22=+ww2 12− σσ2 +−ww1 σρ (A.4) paaa()ba()aabab and 12/ 22 2 2 σσpa=+wwaa()12− σσba+−ww()1 aaσρbab (A.5) We see that portfolio risk depends both on the risk of the individual assets and on the correlation of their returns. This is what gives rise to a diversifica- tion effect. To examine this more closely, we begin with the case of perfect posi- tive correlation. Substitute ρab = 1 into (A.4) to get: 2 σσ22=+ww2 12− σσ2 +−ww1 σ (A.6) paaa()ba()aab which corresponds to 2 σσ2 =+ww()1− σ pa aab hence σσ=+ww1− σ (A.7) paaa()b The standard deviation of the portfolio is now a simple weighted average of the standard deviation of each asset, and there is no diversification effect. However, only in this extreme case of a perfect positive correlation does that apply. We see this from the fact that any ρab < 1 would make the right hand side of Eq. (A.4) smaller than (A.6). Consider now a zero correlation between the two assets (ρab = 0 ). Equation (A.4) reduces to 2 σσ22=+ww2 1− σ 2 (A.8) paaa()b and 12/ 22 2 2 σσpa=+wwaa()1− σ b (A.9) 214 Appendix: The Mechanics of Mean-Variance Optimization σp is now smaller than waσa + (1 − wa)σb unless wa = 1 or 0, in which case the portfolio would be concentrated in either asset a or asset b, which would obviously exclude any diversification effect. Finally, let’s consider the case of a perfect negative correlation (ρab = − 1). As discussed previously, negative correlations between any two assets are hard to find over long time horizons, but frequently occur over shorter periods of time (e.g. in “flight to quality” situations). Substituting ρab = − 1 into Eq. (A.4) we get 2 σσ22=+ww2 12− σσ2 −−ww1 σ paaa()ba()aab which corresponds to 2 σσ2 =−ww()1− σ (A.10) pa aab or to 2 σσ2 =− ww+−()1 σ (A.11) pa aab hence σσ=−ww1− σ (A.12) paaa()b or σσ=−ww+−1 σ (A.13) paaa()b Either (A.12) or (A.13) is valid depending on which of the two is positive. In this case, it is (in principle) possible to entirely eliminate the risk of the portfolio. To find the wa (and, by implication, wb ) necessary for this, set (A.13) equal to zero and solve for wa to get σ w = b (A.14) a σσ+ ab The effect of diversification in the two assets case is illustrated in Fig. A.1, in stylized form. We saw that with a perfect positive correlation, both the risk and the return of the portfolio become linear combinations of the two assets in the portfolio. Appendix: The Mechanics of Mean-Variance Optimization 215 r = -1 = 0 = 0.5 = 1 r r r Fig. A.1 Correlation and diversification For correlations lower than one, however, the diversification effect is repre- sented by the non-linear combinations of assets a and b in Fig. A.1 and, in the extreme case of ρab = − 1, by the straight lines to the vertical axis. Comparing the linear combination of ρab = 1 with all the others, we see that diversification reduces the level of risk of the portfolio for a given level of return or, which is the equivalent, increases the level of return for a given level of risk. To find the minimum variance portfolio in the two-asset case, we differen- 222 2 2 tiate (A.4), that is, σσpa=+wwaa()12− σσba+−ww()1 aaσρbab , with respect to wa and set it equal to zero: ∂σ 2 p =−22wwσσ22++22σσ2 σρ −=40w σσρ ∂w aa babababaabab a Solving for wa, we get: σσ2 − σρ w = babab (A.15) a σσ22+−2σσρ ab abab σ b This is the general formula which, as we saw, becomes when ρab = − 1. σσab+ We see directly that for uncorrelated assets (ρab = 0), (A.15) is reduced to σ 2 w = b (A.16) a σσ22+ ab 216 Appendix: The Mechanics of Mean-Variance Optimization A Numerical Example A portfolio consists of an equity fund (asset a) and a high-yielding fixed income fund (asset b). The expected return of the equity fund is 12.5%, while that of the bond fund is 7.5%. The standard deviations of their returns are 18.0% and 12.5%, respectively. Table A.1 below shows the expected return and standard deviation of the portfolio in decimal form, using formulas (A.7), (A.9) and (A.15). As can be seen, the standard deviation of the portfolio falls the lower is the correlation, but this has no cost in terms of the return of the portfolio, which depends exclusively on the weighting of the two assets. Figure A.2 gives a graphical illustration. To calculate the weighting of each asset class required to minimize the stan- dard deviation of the portfolio we use formulas (A.14) and (A.15) to get the results set out in Table A.2. Table A.1 Standard deviation of portfolio returns with different levels of correlation Standard deviation of portfolio return wa wb Rp ρ = 1 ρ = 0.5 ρ = 0 ρ = −0.5 0.0 1.0 0.0750 0.1250 0.1250 0.1250 0.1250 0.1 0.9 0.0800 0.1305 0.1225 0.1139 0.1047 0.2 0.8 0.0850 0.1360 0.1220 0.1063 0.0878 0.3 0.7 0.0900 0.1415 0.1237 0.1028 0.0765 0.4 0.6 0.0950 0.1470 0.1273 0.1040 0.0735 0.5 0.5 0.1000 0.1525 0.1328 0.1096 0.0799 0.6 0.4 0.1050 0.1580 0.1399 0.1190 0.0936 0.7 0.3 0.1100 0.1635 0.1483 0.1315 0.1121 0.8 0.2 0.1150 0.1690 0.1580 0.1462 0.1333 0.9 0.1 0.1200 0.1745 0.1686 0.1625 0.1561 1.0 0.0 0.1250 0.1800 0.1800 0.1800 0.1800 0.2000 0.1800 ρ = 1 ρ = 0.5 ρ = 0 ρ = -0.5 0.1600 . 0.1400 0.1200 0.1000 folio St.D 0.0800 Port 0.0600 0.0400 0.0200 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Weight in asset 1 Fig. A.2 Portfolio volatility and asset correlation Appendix: The Mechanics of Mean-Variance Optimization 217 Table A.2 Minimum standard deviation portfolio Minimum St.D portfolio ρ = 0.5 ρ = 0 ρ = −0.5 wa 0.1714 0.3254 0.3811 wb 0.8286 0.6746 0.6189 Rp 0.0836 0.1028 0.1098 σp 0.122 0.1027 0.0734 Portfolios of n Risky Assets Generalizing to n assets, we can express the expected return on the portfolio as: n (A.17) Rwp = ∑ iiR i=1 n which is a simple weighted average of the return on each asset with ∑wi = 1.
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