Home control work
Consider a virtual portfolio. Your portfolio consists of two assets: I and II.
These assets are characterized by the following parameters
Asset I
Total value of portfolio, $ mln
Volatility of an asset, annual
Correlation coefficient
Expected return, annual
Asset II
10
0,04
0,12
0,23
0,06
0,15
1) Write down a covariance matrix 
𝜎1 2 𝜎1,2
0,042 𝜎1,2
=
[
]
[
]
𝜎2,1 0,12
𝜎2,1 𝜎2 2
𝜎1,2 = 𝜎2,1 = 𝜎1 × 𝜎2 × 𝜌 = 0,04 × 0,12 × 0,23 = 0,0011;
0,042
[
𝜎2,1
𝜎1,2
0,0016 0,0011
]
]=[
0,0011 0,0144
0,12
 x1 
 , where x1 and x2 are
x
 2
2) Determine the vector of initial investments x  
investments in the Asset I an Asset II correspondingly.
To obtain minimal level of risk, we can find initial investment in the following
way:
𝑥1 = 𝑊1 × 𝑊; 𝑥2 = 𝑊2 × 𝑊
𝑊1 =
𝜎1 2 −𝜎1 𝜎2 𝜌
𝜎1 2 +𝜎2 2 −2𝜎1 𝜎2 𝜌
; 𝑊2 = 1 − 𝑊1
𝜎1 2 − 𝜎1 𝜎2 𝜌
𝑊1 = 2
; 𝑊2 = 1 − 𝑊1
𝜎1 + 𝜎2 2 − 2𝜎1 𝜎2 𝜌
0,042 − 0,04 × 0,12 × 0,23
𝑊1 =
= 0,036 ;
0,042 + 0,122 − 2 × 0,04 × 0,12 × 0,23
𝑊2 = 1 − 0,036 = 0,964
𝑥1 = $0,36 𝑚𝑙𝑛 ; 𝑥2 = $9,64 𝑚𝑙𝑛
3) Calculate the portfolio variance  p W as a product xT  x
2
𝑥⃗ ∑ 𝑥⃗ = [𝑥1
𝑇
2
𝜎1 2
𝑥2 ] [
𝜎2,1
𝜎1,2 𝑥1
] [ ];
𝜎2 2 𝑥2
0,042 0,0011 0,36
𝑥⃗ 𝑇 ∑ 𝑥⃗ = [0,36 9,64] [
];
][
0,0011 0,122 9,64
0,000576 + 0,010604
[0,36 9,64] [
] = $0,0040248 𝑚𝑙𝑛 + $1,34200368 𝑚𝑙𝑛
0,000396 + 0,138816
= 1,346028 𝑚𝑙𝑛
Volatility is √1,346028 = $1,1602 mln
If 𝛼 = 1,65 , than:
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑉𝐴𝑅 = 𝛼 × √𝑥⃗ 𝑇 ∑ 𝑥⃗ = 1,65 × 1,1602 = $1,9143 𝑚𝑙𝑛
4) Calculate portfolio VAR (diversified)
2
𝐷𝑖𝑣𝑉𝐴𝑅𝑃 = 𝛼𝜎𝑝 𝑊 = 1,65√𝑤1 2 𝜎1 2 + 𝑤2 2 𝜎2 + 2𝑤1 𝑤2 𝜎1 𝜎2 𝜌
= 1,65 × 0,11602 × 10 = $1,9143 𝑚𝑙𝑛
5) Calculate the sum of two assets’ VARs (undiversified VAR for portfolio)
𝑉𝐴𝑅1
1,65 × 0,04 × 0,36
$0,0238 𝑚𝑙𝑛
[
]=[
]=[
]
𝑉𝐴𝑅2
1,65 × 0,12 × 9,64
$1,9087 𝑚𝑙𝑛
Undiversified VAR = $1,9325 mln
6) Calculate vector of Betas as   W 
𝛽=𝑊×
x
xT  x
∑ 𝑥⃗
1
0,01118
10 × 0,0083
0,0831
=
10
×
[
]
×
=
[
]
=
[
]
0,139212
10 × 0,1034
1,0342
𝑥⃗ 𝑇 ∑ 𝑥⃗
1,346028
7) Now you can see, which portfolio position should be increased and which
should be decreased to improve your portfolio risk characteristics. Your aim is to
create a portfolio with equal betas for both components – this portfolio will have
the least VAR.
Use Excel to carry out few steps of optimization to obtain components’ betas,
different no more than in the second decimal digit.
8) Present your results in form of the following Table 1. Risk-minimizing
position
Risk-minimizing position
Asset
I
II
Total
Diversified VAR
Undiversified VAR
Standard deviation
Original position
3,6%
96,4%
100%
$1,9143 mln
$1,9325 mln
$1,1602 mln
Beta
0,0831
1,0342
Final position, %
96,4%
36%
100%
$0,6563 mln
$0,7075 mln
$0,3978 mln
Beta
0,99
0,99
9) Now take into account both risks and returns of your portfolio. That means you
Ep
should maximize the Sharpe ratio SR p 
for your portfolio. In the point of
p
optimum the ratio
Ei
i
for both portfolio components should be equal to each other.
Start from the Risk-minimizing position and carry out few steps of optimization
to obtain Sharpe ratios different no more than in the second decimal digit.
In the point of optimum calculate the diversified VAR, expected return and Sharpe
ratio for your final portfolio.
10) Present your results in form of the following Table 2. Risk and returnoptimizing position
Risk and return-optimizing position
Asset
Expected
return
Original
position, %
Beta
3,6%
96,4%
100
$1,9143 mln
Ei
I
II
Total
Diversified
VAR
Standard
deviation
Expected
return for
portfolio
Sharpe
ratio for
portfolio
0,06
0,15
$1,1602 mln
3,6%*0,06+
96,4%*0,15
=0,1468
0,1265
i
Ratio
Ei i
Final
position, %
Beta
0,0831
1,0342
0,7220
0,1450
80,1%
19,9%
100
0,77
1,9226
i
Ratio
Ei i
0,078
0,078
$0,7284
mln
$0,4414
mln
80,1%*0,06+
19,9%*0,15=
0,0779
0,1765
In optimal portfolio, the ratio of all expected returns to marginal VARs or betas must be equal.
Ei i must be constant.