Minimum variance portfolio mathematics

Minimum variance portfolio mathematics

Consider a portfolio of 2 mutual funds: long term debt securities (D) and sotck fund in equity (E). Debt Equity E(r) 8% 13% 12% 20% Cov(rD; rE) 72 D;E 0.3 weights wD wE = 1 wD We can compute the expected return on the portfolio P

E(rP) = wDE(rD) + wEE(rE); in our example we have

E(rP) = 0:08wD+ 0:13wE given that wD = 1 wE; E(rP) = 0:08(1 wE) + 0:13wE = 0:08 + 0:05wE; if we plot it we get 1 0.75 0.5 0.25 0 0.125 0.1 0.075 0.05 0.025 0 w_E E[r_P] w_E E[r_P]

The variance of the portfolio 2

P = w2D 2D+ wE2 E2 + 2wDwECov(rD;rE); in our example we have

2 P = 122wD2 + 202w2E+ 2wDwE72 2 P = 122(1 wE)2+ 202wE2 + 2(1 wE)wE72 2 P = 400w2E 144wE+ 144: This variance as a function of wE is

1 0.75 0.5 0.25 0 400 300 200 100 0 w_E var(r_P) w_E var(r_P)

Variance of a portfolio of two risky assets Assume a portfolio composed of two risky assets

rP = wDrD+ wErE The expected return is

E(rP) = wDE(rD) + wEE(rE); then, the variance of this portfolio will be

2 P = E [rP E[rP]]2 = E rP2] [E[rP] 2 = Eh(wDrD+ wErE)2 i [wDE(rD) + wEE(rE)]2 = = En(wDrD)2+ (wErE)2+ 2wDrDwErE h

(wDE(rD))2+ (wEE(rE))2+ 2wDwEE(rD)E(rE) io

= = w_D2E(r_D2) + w2_EE(r_E2) + 2wDwEE(rDrE) w2DE(rD)2 w2EE(rE)2 2wDwEE(rD)E(rE) = rearranging we have

= w_D2E(r_D2) w2_DE(rD)2+ w2EE(rE2) wE2E(rE)2+ 2wDwEE(rDrE) 2wDwEE(rD)E(rE) = = w2_D E(r2_D) E(rD)2 | {z } 2 D + w2_E E(r_E2) E(rE)2 | {z } 2 E

+ 2wDwE[E(rDrE) E(rD)E(rE)]

| {z }

Cov(rD;rE)

=

= w_D2 2_D+ w2_{E E}2 + 2wDwECov(rD; rE): Recall that the correlation coe¢ cient is

D;E=

Cov(rD; rE) D E

: then, we can express the portfolio variance as follows:

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= 1 ! Cov(r}D; rE) = D E; then the portfolio variance becomes

2

P = w2D 2D+ w2E E2 + 2wDwECov(rD;rE) = = w2_D 2_D+ w2_E 2_E + 2wDwE D E = = (wD D+ wE E)2

and P = wD D+ wE E :

If _{D;E= 0 ! Cov(r}D; rE) = 0; then the portfolio variance becomes 2 P = w2D 2D+ w2E E2 + 2wDwECov(rD;rE) = = w2_D 2_D+ w2_E 2_E + 0 = = w2_D 2_D+ w2_E 2_E that is, P = wD2 2D+ w2E 2E 1 2

If _D;E= _{1 ! Cov(r}D; rE) = D E; then the portfolio variance becomes 2

P = w2D 2D+ w2E E2 + 2wDwECov(rD;rE) = = w2_D 2_D+ w2_E 2_E 2wDwE D E = = (wD D wE E)2

and P = abs(wD D wE E): In this case, a perfectly hedging portfolio can be obtained by setting

P = abs(wD D wE E); that is, we are left with

P = wD D wE E ; and

P = wE E wD D : In general, the variance of the portfolio expressed as

2

P = w2D 2D+ wE2 E2 + 2wDwECov(rD;rE); if we replace wD = 1 wE; can be rewritten as follows:

2

P = 2D+ w2E 2D 2wE 2D + w2E 2E + 2wECov(rD;rE) 2wE2Cov(rD;rE):

If we plot the relationship between standard deviation of the portfolio ( P) and the propor-tion of wealth allocated to equity for alternative correlapropor-tion coe¢ cients, _D;E, we obtain

1 0.75 0.5 0.25 0 20 15 10 5 0 w_E sigma_P w_E sigma_P Solid line: _DE = 1 Dots line: _DE = 0 Circle line: _DE = 1

Notice that if all income is allocated to Debt (wE = 0) the volatility of the portfolio is that of Debt, whereas if all income is allocated to Equity (wE = 1); 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 wE and P:

When _DE = 1 (solid line), there is no room for reducing risk by diversi…cation. When DE = 0 (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 = 1 (circle line) in fact, portfolio volatility can be completely reduced. In our example, this would happen when wE is roughly around 0:4; and therefore wD is approximately 0:6: We will compute this optimal allocation later.

Computing the minimum variance portfolio Taking the formula of the variance of the portfolio

2

P = 2D+ w2E 2D 2wE 2D + w2E 2E + 2wECov(rD;rE) 2wE2Cov(rD;rE):

Which proportion of assets should we choose in order to minimize this variance? Derive 2_P with respect to wE d 2_P dwE = 2wE 2D 2 2D+ 2wE 2E+ 2Cov(rD;rE) 4wECov(rD;rE) = 0; that is, w = 2 D Cov(rD;rE) :

Notice that when _D;E = _{1 ! Cov(r}D; rE) = D E; this equation collapses to w_E = 2 D+ D E 2 D+ 2E+ 2 D E = D( D+ E) ( D + E)2 = D D+ E : In general, w_E = 2 D D;E D E 2 D+ 2E 2Cov(rD;rE) : If we apply this to the numbers in our example we obtain

2

P = 2D+ w2E 2D+ 2E 2Cov(rD;rE) 2 2D 2Cov(rD;rE) wE; the minimum variance is attained at

d 2_P dwE = 2wE 2D+ 2E 2 D;E D E 2 2D 2 D;E D E = 0; that is, w_E = 2 D D;E D E 2 D+ 2E 2 D;E D E = 144 240 D;E 544 480 _D;E = 9 15 _D;E 34 30 _D;E; then depending on _D;E we obtain di¤erent optimal allocations

D;E = 1 ! wE = 1:5 ! wE = 0 D;E = 0 ! wE = 26:47%

D;E= 1 ! wE = 37:5%

D;E= 0:9 ! wE = 64:28 ! wE = 0 D;E= 0:3 ! wE = 18%; then wD = 82% The risk-return tradeo¤

Now, we can put together all the relationships between risk and return, since E(rP) = 0:08 + 0:05wE;

and given the standard deviation in general

P = 2D+ wE2 2D+ 2E 2Cov(rD;rE) 2 2D Cov(rD;rE) wE

1 2 _;

20 15 10 5 0 0.125 0.1 0.075 0.05 0.025 0 sigma_P E[r_P] sigma_P E[r_P]

In the …gure, the gross solid line refers to the case _D;E = 1; the circle line is for _D;E = 1; the dotted line is for the case _D;E = 0; and …nally, the thin solid line refers to the numerical example _D;E= 0:3: