σ2 = wa2σa2 + wb2σb2 + 2?wa?wb?σa?σb?ρ
substitute weight
σ2 = wa2σa2 + (1-wa)2σb2 + 2?wa?(1-wa)?σa?σb?ρ
add parameters:
σ=0.1
ρ=0.0
σa=0.1
σb=0.2
0.12 = wa2?0.12 + (1-wa)2?0.22 + 0
get two solutions:
wa = 0.6 : wb = 0.4 & wa = 1.0 & wb = 0.0
By eliminating unsuitable solution (bottom wing of curve because return lower there r=5%) we get portfolio weights wa = 0.6 and wb = 0.4
Return is:
rp = wa?ra + wb?rb
since individual assets returns are given:
ra = 0.05
rb = 0.10
rp = 0.05?wa + 0.1?wb = 0.05?0.6 + 0.1?0.4 = 0.07
Answer: Expected portfolio return is 7% (with risk = 10%, asset A weight = 0.6 and asset B weight = 0.4)
Best strategy depends on market conditions and trader' risk preferences.
Formal Modern Portfolio Theory would suggest (assuming risk free rate and returns are σf=0 & rf=0) best investment strategy as defined by SML line (tangent to efficient frontier):
Find equation for SML going through σf=0 & rf=0 , efficient frontier, derivative, etc. ending with SML E[r]= σ/√2
wich touches (tangent) efficient frontier ((3/50) + 0.02√(125σ2-1)) at point with r=1/15≈0.066(6) and σ=(√2)/15≈0.0942809
asset weights at this point will be:
r = 1/15 = wa?ra + wb?rb = 0.05wa + 0.1(1-wa)
wa=2/3 & wb=1/3 or in monetary terms: 6'666.6(6) should be invested in stock A and 3'333.3(3) in stock B
in more realistic case say risk "free" asset having rf=1% and σf=3%
solution (tangent point) to efficient frontier would be: r≈0.0652266 & σ≈0.0924465 with asset A weight wa≈0.695468 and asset B weight wb≈0.304532 or $6'954.68 should be invested in stock A and $3'045.32 in stock B
A market has just two stocks, A and B.
Stock A has an average return 5 % and risk 10%.
Stock B has an average return 10% and risk 20%.
The returns of the two stocks are independent.
A trader sets up a portfolio with a risk of 10% and the highest possible mean return.
Calculate the mean return of this portfolio and prove that this is the highest mean return you can get with a risk of 10%
If you have $ 10,000 to invest what is the best investment strategy.
