Basic Mathematics for Portfolio Management. Statistics Variables x, y, z Constants a, b Observations...
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Transcript of Basic Mathematics for Portfolio Management. Statistics Variables x, y, z Constants a, b Observations...
![Page 1: Basic Mathematics for Portfolio Management. Statistics Variables x, y, z Constants a, b Observations {x n, y n |n=1,…N} Mean.](https://reader037.fdocuments.net/reader037/viewer/2022110323/56649d775503460f94a58673/html5/thumbnails/1.jpg)
Basic Mathematicsfor
Portfolio Management
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Statistics
• Variables x, y, z
• Constants a, b
• Observations {xn, yn |n=1,…N}
• Mean
mean or expected value of x E x x
E a x b y a x b y
1
1 N
nn
x xN
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Variance
22x Var x E x x
2 2xVar a x a
22
1
1 N
x nn
x xN
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Covariance
,xy Cov x y E x x y y
xy yx
2, xCov x x
, xz yzCov a x b y z a b
2 2 2 2
,
2x y xy
Var a x b y Cov a x b y a x b y
a b ab
1
1 N
xy n nn
x x y yN
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Regression
• This is an example of a linear model.• The model is unbiased if:
• How do we choose the best {a,b}?
ˆn n n n ny a b x y
1
0N
nn
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Regression
• The best estimates for {a,b} should minimize the sum of squared errors:
• This corresponds to minimizing:
• Which leads to two equations:
2
1
ˆ ˆ2N
n n nn
y y y
22
1 1
ˆN N
n n nn n
ESS y y
1
1
ˆˆ 0
ˆˆ 0
Nn
n nn
Nn
n nn
yy y
a
yy y
b
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Regression
• The first equation leads to an unbiased model.
• We can use this to solve for a:
1
ˆˆ1 0
Nn
n nn
yy y
a
n n ny a b x
y a b x
a y b x
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Regression
• The second equation relates the coefficient b to the sample covariance of x and y:
• Putting this together:
• The model is the best linear unbiased estimate (BLUE), given only the sample data.
1
ˆˆ 0
,
Nn
n n n nn
yx y y x
b
Cov x yb
Var x
,
n n n
Cov x yy y x x
Var x
y
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BLUE of y conditional on x
• More generally, the BLUE has the form:
• You can see how this directly relates to our prior regression result.
• We will apply this result more generally than just in the regression context. For example, our estimates of variances and covariances may improve upon sample estimates.
1| ,E y x E y Cov x y Var x x E x
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Linear Algebra• Vectors and Matrices (We will denote in bold)
• Transpose
• Dimensions– (Rows x Columns)– Keep track of these
• Addition and Multiplication– Can only add matrices of identical dimension– For multiplication,
1
2
N
h
h
h
h
1 2, ,TNh h hh
, , ,N M M K N K
Tkn nkX X
11 12 13 1
21 22
1
N
T
K KN
X X X X
X X
X X
X
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Matrix Multiplication: AxB
• Move across A and down B
11 12 13 1 1 11
221 22
11
N
N n nn
T
N
Kn nNK KNn
X X X X h X h
hX X
X hhX X
X h
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Hint 1
• Keep track of the size of the matrices, to make sure the algebra makes sense:
1
2
1
1 2N
TP P P P P n
n
N
r
rh h h N h n r
r
h r
122
11 1
2 2
21
21
11
Nn P
n BPN
B
B B NPN N Nn P
n B
V h n
h
h N V h n
V h
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Matrix Inverse
1
1
1 1 1
A x y
x A y
A A I
A B B A
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Using Linear Algebra
• Portfolio Variance21 12 1
221 2 2
21
N
N
N N
V
1
2
N
h
h
h
h
21 1 2 2
2 2 2 21 1 1 2 12 1 3 132 2
P N N
N N
T
Var h r h r h r
h h h h h h
h V h
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Multivariate Regression
• The multivariate linear model is:
• We can include an intercept as a column of X.• To generalize, we minimize the weighted sum of squared
errors:
• Our resulting estimates are:
Cov
y X f ε
Ω ε
1T ε Ω ε
11 1T T f X Ω X X Ω y
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Basic Utility Function
• Mean/Variance
• What are the dimensions of U?
2P P
T T
U f
h f h V h
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Portfolio Optimization
• Choose portfolio h to maximize U.• What does that mean? We must take the
derivative of U with respect to each of the N elements of h. That leads to N equations in N unknowns.
0n
U
h
0T
U
h
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More Hints
• Hint 2: Try out 2x2 or 3x3 examples.
• Hint 3: To start, try working with individual elements. For example, I will use shorthand like:
– This says to take derivatives with respect to each element of h. This should lead to N separate equations.
– Example:
T
h
2TT
h V h V h
h
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