# Minimum variance portfolio mathematics

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1 Spring 6 Minimum variance portfolio mathematics Consider a portfolio of mutual funds: long term debt securities (D) and sotck fund in equity (E). Debt Equity E(r) 8% 3% % % Cov(r D ; r E ) 7 D;E.3 weights w D w E = w D We can compute the expected return on the portfolio P E(r P ) = w D E(r D ) + w E E(r E ); in our example we have given that w D = w E ; E(r P ) = :8w D + :3w E E(r P ) = :8( w E ) + :3w E = :8 + :5w E ; if we plot it we get E[r_P] The variance of the portfolio w_e P = wd D + we E + w D w E Cov(r D; r E ); in our example we have P = wd + we + w D w E 7 P = ( w E ) + we + ( w E )w E 7 P = 4wE 44w E + 44: This variance as a function of w E is

2 Spring 6 var(r_p) w_e Variance of a portfolio of two risky assets Assume a portfolio composed of two risky assets The expected return is then, the variance of this portfolio will be r P = w D r D + w E r E E(r P ) = w D E(r D ) + w E E(r E ); P = E [r P E[r P ]] = E rp ] [E[r P ] h = E (w D r D + w E r E ) i [w D E(r D ) + w E E(r E )] = = E n(w D r D ) + (w E r E ) + w D r D w E r E h io (w D E(r D )) + (w E E(r E )) + w D w E E(r D )E(r E ) = = w DE(r D) + w EE(r E) + w D w E E(r D r E ) w DE(r D ) w EE(r E ) w D w E E(r D )E(r E ) = rearranging we have = wde(r D) wde(r D ) + wee(r E) wee(r E ) + w D w E E(r D r E ) w D w E E(r D )E(r E ) = = wd E(r D ) E(r D ) + w E E(r {z } E ) E(r E ) + w D w E [E(r D r E ) E(r D )E(r E )] = {z } {z } D Cov(r D ;r E ) E Recall that the correlation coe cient is = w D D + w E E + w D w E Cov(r D ; r E ): D;E = Cov(r D; r E ) D E : then, we can express the portfolio variance as follows: P = w D D + w E E + w D w E Cov(r D ; r E ) = w D D + w E E + w D w E DE D E :

3 Spring 6 Relationship between correlation coe cients and portfolio variance Let s analyze the variance of the portfolio depending on the correlation coe cient of the assets. If D;E =! Cov(r D ; r E ) = D E ; then the portfolio variance becomes and P = w D D + w E E : P = w D D + w E E + w D w E Cov(r D; r E ) = = w D D + w E E + w D w E D E = = (w D D + w E E ) If D;E =! Cov(r D ; r E ) = ; then the portfolio variance becomes that is, P = wd D + w E E P = w D D + w E E + w D w E Cov(r D; r E ) = = w D D + w E E + = = w D D + w E E If D;E =! Cov(r D ; r E ) = D E ; then the portfolio variance becomes and P = abs(w D D by setting that is, we are left with and P = w D D + w E E + w D w E Cov(r D; r E ) = = w D D + w E E w D w E D E = = (w D D w E E ) w E E ): In this case, a perfectly hedging portfolio can be obtained P = abs(w D D w E E ); P = w D D w E E ; P = w E E w D D : In general, the variance of the portfolio expressed as P = w D D + w E E + w D w E Cov(r D; r E ); if we replace w D = w E ; can be rewritten as follows: P = D + w E D w E D + w E E + w E Cov(r D; r E ) w ECov(r D; r E ): If we plot the relationship between standard deviation of the portfolio ( P ) and the proportion of wealth allocated to equity for alternative correlation coe cients, D;E, we obtain 3

4 Spring 6 sigma_p Solid line: DE = Dots line: DE = Circle line: DE = w_e Notice that if all income is allocated to Debt (w E = ) the volatility of the portfolio is that of Debt, whereas if all income is allocated to Equity (w E = ); then the volatility of the portfolio is that of Equity. Then, depending on the correlation coe cient between these two assets we get di erent combinations between w E and P : When DE = (solid line), there is no room for reducing risk by diversi cation. When DE = (dotted line) some risk reduction is possible and this is shown in the shape of the curve. The highest risk reduction is achieved when DE = (circle line) in fact, portfolio volatility can be completely reduced. In our example, this would happen when w E is roughly around :4; and therefore w D is approximately :6: We will compute this optimal allocation later. Computing the minimum variance portfolio Taking the formula of the variance of the portfolio P = D + w E D w E D + w E E + w E Cov(r D; r E ) w ECov(r D; r E ): Which proportion of assets should we choose in order to minimize this variance? Derive P with respect to w E d P dw E = w E D D + w E E + Cov(r D; r E ) 4w E Cov(r D; r E ) = ; that is, w E = D Cov(r D; r E ) D + E Cov(r D; r E ) : 4

5 Spring 6 Notice that when D;E =! Cov(r D ; r E ) = D E ; this equation collapses to In general, w E = D + D E D + E + D E = D( D + E ) ( D + E ) = w E = D D;E D E D + E Cov(r D; r E ) : D D + E : If we apply this to the numbers in our example we obtain P = D + we D + E Cov(r D; r E ) D Cov(r D; r E ) w E ; the minimum variance is attained at that is, d P dw E = w E D + E D;E D E D D;E D E = ; we D D;E D E = D + = 44 4 D;E = 9 5 D;E ; E D;E D E D;E 34 3 D;E then depending on D;E we obtain di erent optimal allocations The risk-return tradeo D;E =! w E = :5! w E = D;E =! w E = 6:47% D;E =! w E = 37:5% D;E = :9! w E = 64:8! w E = D;E = :3! w E = 8%; then w D = 8% Now, we can put together all the relationships between risk and return, since and given the standard deviation in general E(r P ) = :8 + :5w E ; P = D + w E D + E Cov(r D; r E ) D Cov(r D; r E ) w E ; we could solve for the relationship between expected return and risk, the result is 5

6 Spring 6 E[r_P] sigma_p In the gure, the gross solid line refers to the case D;E = ; the circle line is for D;E = ; the dotted line is for the case D;E = ; and nally, the thin solid line refers to the numerical example D;E = :3: 6

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