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7/27/2019 GBF459 - Mathematical Derivatives.pdf
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Investment Analysis
GBF459
Short notes on Total and Partial Derivatives
Dr. Dimitris A. Tsouknidis
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Rate of Change for a Straight Line
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
The fraction
measures the
rate of change in
y=0.5+0.5x as x
changes. What if
we tried the
same idea with
y=ln(x)?
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Derivatives
More generally we define the derivative of f at xby
when the limit exists. The derivate may be differentiated as well, resulting in thesecond derivative of f:
and we can continue differentiating to higher and higher orders, the nth
derivative being written
3
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Derivatives of Polynomials
Derivatives of polynomial functions are given by a simple formula
(more about that later) and because
we can differentiate any polynomial function, for example
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The Derivative of the Logarithm Function
To find the derivative of the logarithm function, lets apply the definition
But recall If we take the logarithm of both sides
so the limit above equals 1.
The conclusion is
5
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The Chain Rule
We still havent figured out the derivative of the exponential function. For that we
need a couple of tricks. The first is the Chain Rule.
For example
If we now apply the Chain Rule to the equation
by differentiating both sides, we get
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Product and quotient rules
For example
For example
if
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Convexityln(x)
-5
-4
-3
-2
-1
0
1
2
0 1 2 3 4 5 6
exp(x)
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Were back where we started, and if we differentiate the two functions
one more we get
Which means that they, respectively, are concave and convex.
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And finally
We wrote down the Chain Rule
which was used to find the derivatives of some important functions:
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Problem
Consider three stocks: A, B and C. Stock A has an expected
return of 0.01 and a standard deviation of 0.20. Stock B 0.02
and 0.3 and stock C 0.03 and 0.4 respectively. The
covariances between the three stocks are:
, = . , , = .,, = . .a. Find the first order conditions using Lagrange multiplier method to
define the minimum variance portfolio.
b. Find the first order conditions using Lagrange multiplier method to
define the minimum variance portfolio if the investor wants to
achieve a return of 20%.
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Problem
A. Objective: Find the minimum of:min
=0.2 + 0.3
+ 0.4 + 2 0.012 + 2 0.024 + 2
0.032
. . : + + = 1
The Lagrange function combines the original function and the constraints as follows:
= 0.2 + 0.3
+ 0.4 + 2 0.012 + 2 0.024 + 2 0.032
+ ( 1 )
Finding the first order conditions:
= 2 0.2 + 2 0.012 + 2 0.024 = 0
= 2 0.3 + 2 0.012 + 2 0.032 = 0
= 2 0.4 + 2 0.024 + 2 0.032 = 0
= 1 = 0
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Problem 3 Objective: Find the minimum of:
min = 0.2
+ 0.3 + 0.4
+ 2 0.012 + 2 0.024 + 2
0.032
..:0.01 + 0.02 + 0.03 = 0.20 + + = 1
The Lagrange function combines the original function and the constraints as
follows:
= 0.2 + 0.3
+ 0.4 + 2 0.012 + 2 0.024 + 2
0.032
+ 0.20 0.01 0.02 0.03 + ( 1 )
Finding the first order conditions:
= 2 0.2 + 2 0.012 + 2 0.024 0.01 = 0
= 2 0.3 + 2 0.012 + 2 0.032 0.02 = 0
= 2 0.4 + 2 0.024 + 2 0.032 0.03 = 0
= 0.20 0.01 0.02 0.03 = 0
= 1 = 0
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See also
https://www.khanacademy.org/math/calculus
/partial_derivatives_topic/partial_derivatives/
v/partial-derivatives