gbf459 - mathematical derivatives.pdf

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

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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

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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

gbf459 - mathematical derivatives.pdf

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