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    الرياضيات

    CHAPTER 1. LINEAR ALGEBRA 13
    1.8 Rank
    Definition 1.8.1
    The row space of an m  n matrix A is the vector subspace of <n generated by
    the m rows of A.
    The row rank of a matrix is the dimension of its row space.
    The column space of an m  n matrix A is the vector subspace of <m generated
    by the n columns of A.
    The column rank of a matrix is the dimension of its column space.
    Theorem 1.8.1 The row space and the column space of any matrix have the same
    dimension.
    Proof The idea of the proof is that performing elementary row operations on a
    matrix does not change either the row rank or the column rank of the matrix.
    Using a procedure similar to Gaussian elimination, every matrix can be reduced to
    a matrix in reduced row echelon form (a partitioned matrix with an identity matrix
    in the top left corner, anything in the top right corner, and zeroes in the bottom left
    and bottom right corner).
    By inspection, it is clear that the row rank and column rank of such a matrix are
    equal to each other and to the dimension of the identity matrix in the top left
    corner.
    In fact, elementary row operations do not even change the row space of the matrix.
    They clearly do change the column space of a matrix, but not the column rank as
    we shall now see.
    If A and B are row equivalent matrices, then the equations Ax = 0 and Bx = 0
    have the same solution space.
    If a subset of columns of A are linearly dependent, then the solution space does
    contain a vector in which the corresponding entries are nonzero and all other entries
    are zero.
    Similarly, if a subset of columns of A are linearly independent, then the solution
    space does not contain a vector in which the corresponding entries are nonzero
    and all other entries are zero.
    The first result implies that the corresponding columns or B are also linearly dependent.
    The second result implies that the corresponding columns of B are also linearly
    independent.
    It follows that the dimension of the column space is the same for both matrices.
    Q.E.D.
    Revised: December 2, 1998

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    الرياضيات2

    14 1.9. EIGENVALUES AND EIGENVECTORS
    Definition 1.8.2 rank
    Definition 1.8.3 solution space, null space or kernel
    Theorem 1.8.2 dimension of row space + dimension of null space = number of
    columns
    The solution space of the system means the solution space of the homogenous
    equation Ax = 0.
    The non-homogenous equation Ax = b may or may not have solutions.
    System is consistent iff rhs is in column space of A and there is a solution.
    Such a solution is called a particular solution.
    A general solution is obtained by adding to some particular solution a generic
    element of the solution space.
    Previously, solving a system of linear equations was something we only did with
    non-singular square systems.
    Now, we can solve any system by describing the solution space.
    1.9 Eigenvalues and Eigenvectors
    Definition 1.9.1 eigenvalues and eigenvectors and -eigenspaces
    Compute eigenvalues using det (A

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    المصفوفات

    CHAPTER 1. LINEAR ALGEBRA 15

    1.10 Quadratic Forms

    A quadratic form is

    1.11 Symmetric Matrices

    Symmetric matrices have a number of special properties

    1.12 Definite Matrices

    Definition 1.12.1 An n  n square matrix A is said to be

    positive definite () x>Ax > 0 8x 2 <n; x 6= 0

    positive semi-definite () x>Ax  0 8x 2 <n

    negative definite () x>Ax < 0 8x 2 <n; x 6= 0

    negative semi-definite () x>Ax  0 8x 2 <n

    Some texts may require that the matrix also be symmetric, but this is not essential

    and sometimes looking at the definiteness of non-symmetric matrices is relevant.

    If P is an invertible n  n square matrix and A is any n  n square matrix, then

    A is positive/negative (semi-)definite if and only if P

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    الاحتمالية

    PORTFOLIO THEORY

    6.1 Introduction

    Portfolio theory is an important topic in the theory of choice under uncertainty. It

    deals with the problem facing an investor who must decide how to distribute an

    initial wealth of, say, W0 among a number of single-period investment opportunities,

    called securities or assets.

    The choice of portfolio will depend on both the investor’s preferences and his

    beliefs about the uncertain payoffs of the various securities. A mutual fund is just

    a special type of (managed) portfolio.

    The chapter begins by considering some issues of definition and measurement.

    Section 6.3 then looks at the portfolio choice problem in a general expected utility

    context. Section 6.4 considers the same problem from a mean-variance perspective.

    This leads on to a discussion of the properties of equilibrium security returns

    in Section 6.5.

    6.2 Notation and preliminaries

    6.2.1 Measuring rates of return

    Good background reading for this section is ?.

    A rate of interest (growth, inflation, &c.) is not properly defined unless we state

    the time period to which it applies and the method of compounding to be used.

    2% per annum is very different from 2% per month.

    Table 6.1 illustrates what happens to £100 invested at 10% per annum as we

    change the interval of compounding. The final calculation in the table uses the

    Revised: December 2, 1998

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    3

    106 6.2. NOTATION AND PRELIMINARIES
    Compounded
    Annually £100 ! £110
    Semi-annually £100 ! £100  (1:05)2 = $110:25
    Quarterly £100 ! £100  (1:025)4 = $110:381. . .
    Monthly £100 ! £100  1 + :10
    12 12
    = $110:471. . .
    Weekly £100 ! £100  1 + :10
    52 52
    = $110:506. . .
    Daily £100 ! £100  1 + :10
    365365
    = $110:515. . .
    Continuously £100 ! £100  e0:10 = $110:517. . .
    Table 6.1: The effect of an interest rate of 10% per annum at different frequencies
    of compounding.
    fact that
    lim
    n!11 +
    r
    nn
    = er
    where e  2:7182. . .
    This is sometimes used as the definition of e but others prefer to start with
    er  1 + r +
    r2
    2!
    +
    r3
    3!
    + : : : =
    1Xj=0
    rj
    j!
    where n!  n (n

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    Definition 6.2.1 w is said to be a unit cost or normal portfolio if its weights sum
    to 1 (w>1 = 1).
    The portfolio held by an investor with initial wealth W0 can be thought of either
    as a w with w>1 = W0 or as the corresponding normal portfolio, 1
    W0
    w. It will
    hopefully be clear from the context which meaning of ‘portfolio’ is intended.
    Definition 6.2.2 w is said to be a zero cost or hedge portfolio if its weights sum
    to 0 (w>1 = 0).
    The vector of net trades carried out by an investor moving from the portfolio w0
    to the portfolio w1 can be thought of as the hedge portfolio w1

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    6.3 The Single-period Portfolio Choice Problem
    6.3.1 The canonical portfolio problem
    Unless otherwise stated, we assume that individuals:
    1. have von Neumann-Morgenstern (VNM) utilities:
    i.e. preferences have the expected utility representation:
    v(~z) = E[u(~z)]
    = Z u(z)dF~z(z)
    where v is the utility function for random variables (gambles, lotteries)
    and u is the utility function for sure things.
    2. prefer more to less
    i.e. u is increasing:
    u 0(z) > 0 8z (6.3.1)
    3. are (strictly) risk-averse
    i.e. u is strictly concave:
    u 00(z) < 0 8z (6.3.2)
    Date 0 investment:
     wj (pounds) in jth risky asset, j = 1; : : : ;N
     (W0

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    6.3 The Single-period Portfolio Choice Problem
    6.3.1 The canonical portfolio problem
    Unless otherwise stated, we assume that individuals:
    1. have von Neumann-Morgenstern (VNM) utilities:
    i.e. preferences have the expected utility representation:
    v(~z) = E[u(~z)]
    = Z u(z)dF~z(z)
    where v is the utility function for random variables (gambles, lotteries)
    and u is the utility function for sure things.
    2. prefer more to less
    i.e. u is increasing:
    u 0(z) > 0 8z (6.3.1)
    3. are (strictly) risk-averse
    i.e. u is strictly concave:
    u 00(z) < 0 8z (6.3.2)
    Date 0 investment:
     wj (pounds) in jth risky asset, j = 1; : : : ;N
     (W0

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    0. Since we have assumed that investor
    behaviour is risk averse, A is a negative definite matrix and, by Theorem 3.2.4,
    f is a strictly concave (and hence strictly quasiconcave) function. Thus, under
    the present assumptions, Theorems 3.3.3 and 3.3.4 guarantee that the first order
    conditions have a unique solution. The trivial case in which the random returns
    are not really random at all can be ignored.
    The rest of this section should be omitted until I figure out what is going on.
    Another way of writing (6.3.3) is:
    E[u0( )~rj ] = E[u0( ~W
    )]rf 8j; (6.3.5)
    or
    Cov hu0( ); ~rji+ E[u0( )]E[~rj ] = E[u0( )]rf 8j; (6.3.6)
    or
    E[~rj

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