Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
1
Exercise 1 :
Q.1
Portfolio : Probability of a total loss
return of a combination of portfolios
Let P F (b1 , a1 ) and P F (b2 , a2 ) be 2 portfolios, r1 and r2 their respective returns. A
new portfolio, denoted PM is made of a mixture of the 2 previous
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Gram Matrix of a linearly dependent family of vectors
exercise 1
E
is a Vector Space and
a bilinear form on E. The aim of the exercise is to prove
the theorem :
If
(x1 , ., xp )
are linearly dependent then
(x1 , ., xp ) = 0
question 1
let
x = (x1 , ., x
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Proof and application of the Change of Time Formula
exercise 1
In this exercise, We only use the discrete model. The proof of the Change of time
formula for an exponential growth or decay :
unit
s
t
algebraic return
s
(1 + t )s = (1 + s )t
t
is aim of thi
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
4 December 2013
HomeWork chapter 4
Market without risk free asset
Partial correction
Mathematical Models
and Methods
in Asset Management
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
: Basic results about the Zero Model from Final Exam 2012
: Dec
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
10 December 2013
Correction of
Final Exam 2013
Mathematical Models
and Methods
in Asset Management
Notations : 5 points per question
Exercise 1 : Attributes of a Portfolio in a market with a risk free asset(8 q.)
Exercise 2 : Attributes after a small pert
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
NOM :
30 November 2013
Student ID :
Correction HomeWork
about chapter 3 : Eciency
Mathematical Models
and Methods
in Asset Management
Exercise 1 : Basic identities around the
and the CAPM
Exercise 2 : Impact of the risk free part on the CAPM identity
Exer
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
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CHAPTER
7.1
7
Eigenvalues
and
Eigenvectors
D
E
T
H
IG
R
ELEMENTARY PROPERTIES OF
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Chapitre 12
Linear Approximation, Multilinear
correlation coecient
12.1 Small introduction about Hilbert spaces
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes
the notion of Euclidean space. It extends the methods of ve
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Chapitre 2
The Portfolio Space
2.1 Computation of the basic attributes of a portfolio in a formal market
2.1.1 Description of a formal market
Denition 2.1 (risk free asset).
An asset is riskiness or risk free if its return over the period is perfectly kno
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
1
Chapitre 4
Estimation of the attributes related
to a portfolios
4.1 The Table of Data
4.1.1 The Data stream
The return of the asset number j at time ti will be denotes indiscriminately rti ,j or rj,ti
There are then, several ways to display the stream o
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
29
Chapitre 2
Eciency of a portfolio
2.1 The Mean Wealth Maximization Problem
Denition of an Ecient Portfolio, the Maximization Problem
Let (b,a) be the coordinates of a portfolio in the standard basis and (w,a) the coordinates of the same portfolio in In
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Chapitre 11
Bilinear Forms
11.1
Generalities on Bilinear forms
F is a vector space and B = (f1 , f2 , , fn ) a basis of this vector space. In all this
chapter C will be a Bilinear Form on this vector space.
11.1.1 Bilinear Form
Denition 11.1 (Bilinear For
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
1
Chapitre 3
Black'S Zero
(1972)
3.1 The Mean Wealth Maximization Problem
2
a is an ecient portfolio with an initial wealth w0 and a given variance 0 i:
a .e = w
0
EW1 (a ) EW1 (a) (a)/
V (a) = 2
0
We recall 2 basic rules in the search of the portfolio PF
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Chapitre 11
Bilinear Forms
11.1
Generalities on Bilinear forms
F is a vector space and B = (f1 , f2 , , fn ) a basis of this vector space. In all this
chapter C will be a Bilinear Form on this vector space.
11.1.1 Bilinear Form
Denition 11.1 (Bilinear For
Mathematical Models and Methods in Asset Management
MA 5623

Fall 2013
Chapitre 1
Computation of the returns
1.1
Computation with constant returns
1.1.1 Return of an investment over a given period
T is the length of the period.
t0 is the beginning of this period while t0 + T is its end.
Wt0 is the wealth invested at the begi