Portfolio Risk and Return Calculation with Risk-Free Asset

This article explains the impact of a risk-free asset on portfolio risk and return. It includes calculations for five different portfolios with varying asset allocations and provides insights on the effects of diversification on portfolio risk and return.

Question 3
Part 1
A risk free asset has a standard deviation of urthermore the correlation between any risky asset and0. F ,
risk free asset is zero.
herefore if a risk free asset is added to a portfolio of two assets then the portfolio s standard deviationT , , ’
would be in a linear proportion to the risky asset s standard deviation ence the inclusion of a risk free’ . H ,
asset will have the effect of reducing the standard deviation.
σ2 = w12σ12 + w22σ22 +2w1w2σ1σ2ρ12
σ2= w12σ12 + w22*0 +2w1w2σ1*0…………………Add risk free asset where ρ12 = 0, σ2 = 0
σ2= w12σ12
σ2= (w1σ1)2
σ= w1σ1
Part 2
i) Portfolio 1- 30% A & 70% B
pected returnEx = Wara +Wbrb
=30%*10 +70%*15
=13.5%
Standard deviation σ2 w= 12σ12 w+ 22σ22 w+2 1w2σ1σ2ρ12
sqrt= (30%^2*20^2+70%^2*35^2+2*30%*70%*0.5*20*35)
=27.99%
ii) Portfolio 2- 50% A, 32.5% B, 17.5% C
e pected returnx = Wara +Wbrb +Wcrc

=50%*10 +32.5%*15+17.5%*20
=13.38%
Standard deviation
Standard deviation sqrt w= ( 12σ12 w+ 22σ22 w+ 32σ32 w+2 1w2σ1σ2ρ12 + w2 2w3σ2σ3ρ23 + w2 1w3σ1σ3ρ13)
sqrt= (50%^2*20^2+32.5%^2*35^2+17.5%^2*46^2+2*50%*20*32.5%*35*0.5+2*32.5%*35*17.5%*46*
0.3+2*17.5%*46*50%*20*0.15)
=22.07%
iii) Portfolio 3- 5% A, 75% B, 20% F
Standard deviation of risk free asset is zero
e pected returnx = Wara +Wbrb +Wcrc
=5%*10 +75%*15+20%*9.9
=13.73%
Standard deviation
sqrt= (5%^2*20^2+75%^2*35^2+2*5%*20*75%*35*0.5)
=26.76%
iv) Portfolio 4- 33.3% A, 33.3% B, 33.3% C
e pected returnx = Wara +Wbrb +Wcrc
=33.3%*10 +33.3%*15+33.3%*20
=15%
Standard deviation

sqrt= (33.3%^2*20^2+33.3%^2*35^2+33.3%^2*46^2+2*33.3%*20*33.3%*35*0.5+2*33.3%*35*33.3%*
46*0.3+2*33.3%*46*33.3%*20*0.15)
=25.13%
v) Portfolio 5- 25% A, 25% B, 25% C, 25% F
e pected returnx = Wara +Wbrb +Wcrc ++Wcrc
=25%*10 +25%*15+25%*20+25%*9.9
=13.73%
Standard deviation
sqrt= (25%^2*20^2+25%^2*35^2+25%^2*46^2+2*25%*20*25%*35*0.5+2*25%*35*25%*46*0.3+2*25
%*46*25%*20*0.15)
=18.85%
vi) Differences between portfolio 3,4 & 5
Portfolio Return Risk
3 13.73 26.76
4 15 25.13
5 13.73 18.85
or all portfolios diversification reduces the portfolio riskF 3 , .
ortfolio is heavily weighted towards Asset which has a high standard deviation owever the additionP 3 B . H ,
of the risk free asset has reduced the portfolio risk and return.
ortfolio is equally weighted to risky assets Diversification has reduced the portfolio risk however itsP 4 3 . ; ,
portfolio return is higher since there is no risk free asset .
ortfolio is equally weighted to risky assets and a risk free asset n comparison to portfolio it has aP 5 3 . I 4,
lowest return and risk due to inclusion of the risk free asset .
n summary inclusion of a risk free asset will reduce portfolio risk and returnI ,