Portfolio Analysis: Evaluating Risk, Return, and Diversification

This assignment provides a detailed analysis of portfolio risk and return, examining the impact of diversification and the inclusion of risk-free assets. It calculates the expected return and standard deviation for five different portfolios with varying asset allocations, including assets A, B, C, and a risk-free asset F. The analysis reveals that diversification generally reduces portfolio risk, while the inclusion of a risk-free asset further lowers both risk and return. The assignment compares and contrasts the risk-return profiles of the different portfolios, highlighting the trade-offs between risk and return based on asset allocation strategies, demonstrating how different weighting of assets impacts the overall portfolio characteristics.

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 ,