The fund manager has 20 million euros to invest. He wants to buy some equities, which have an expected annual return of 14,5% and expected risk of 22,5%. To reduce the risk of the portfolio, he decided to add some fixed income securities. Bonds will have expected annual return of 3,5% and risk of 2,3% The correlation coefficient between two portfolios is -0,83. a. How much should be invested into equities and bonds in order to reduce overall risk level to minimum? b. What would be the expected combined return from both portfolios?

a.

Expected return of the equity Ea = 14.5%
Standard deviation of the equity Sa = 22.5%
Variance of the equity Va = 22.5² = 506.25%

Expected return of the fixed income Eb = 3.5%
Standard deviation of the fixed income Sb = 2.3%
Variance of the fixed income Vb = 2.3² = 5.29%

Let Wa be the weight in equities and Wb be the weight in fixed income
Wa + Wb = 1, Wb = 1 - Wa

Minimum variance portfolio Vp = Wa² x Va² + Wb² x Vb² + 2 x Wa x Wb x Sa x Sb x Corr.(a,b)

Vp = Wa² x 506.25 + (1 - Wa²) x 5.29 + 2 x Wa x (1 - Wa) x 22.5 x 2.3 x (-0.83)

Vp = Wa² x 506.25 + 5.29 - Wa² x 5.29 - 85.905 x Wa + 85.905 x Wa²

Vp = Wa² x 586.865 - 85.905 x Wa + 5.29

Taking the first derivative of the above equation,

0 = Wa x 1,173.73 - 85.905

Wa = 0.07319 or 7.319%

Wb = 100 - 7.319 = 92.681%

7.319% should be invested in equities and 92.681% should be invested in fixed income to reduce overall risk to minimum

b.

Expected return = Wa x Ea + Wb x Eb = 0.07319 x 14.5 + 0.92681 x 3.5 = 4.305%