FRE6083, Midterm Examination, Monday October 21 2013
2:00pm-4:30pm
1. Number of pages including this one: 2
2. For this examination, you may only use a 2-page cheat sheet. No other notes, books or electronic
devices may be used. Cell phones may not be use

Value at Risk and Expected Tail Loss Conditional value at risk (CVaR) is also referred to as expected tail loss (ETL) [see Acerbi and Tasche (2001)]. It has gained significant importance in the risk management field and is our choice of downside risk meas

CALIBRATING AND SIMULATING COPULA FUNCTIONS: AN APPLICATION TO THE ITALIAN STOCK MARKET Claudio Romano1 Abstract
Copula functions are always more used in financial applications to determine the dependence structure of the asset returns in a portfolio. Emp

Order Statistics Let X1, X 2 , , X n be a random sample from the density fX (x ) and suppose that we want the joint density of Y1,Y2 , ,Yn where the Yi are the X i arranged in order of magnitude so that Y1 Y2 Yn . In what follows, we develop the expressio

Midterm Review
1. Let
fX|Y (x|y) = yexy f or x > 0, 0 < y < 1
and assume that the distribution of Y is uniform on (0, 1). Calculate
1.The marginal density of X, fX (x),
2.P [XY > 1],
3.E[X|Y ],
4.V ar[X|Y ].
2. Consider a standard normal random variable X

Continuous Optimization Problems Mathematical optimization techniques are crucial in the design and construction of portfolios of financial assets. It is assumed that you have taken an introductory course in linear and nonlinear programming such as MA 614

FRE 6083 Homework 5 Question 1 Is the mid-range, 1/2( + ) a statistic?
Question 2 Show that the average of the rst two observations in a random sample is unbiased but not consistent, in estimating a population mean. Question 3 Show that if is an unbiased

Solutions Question 1 Similar to the example in the notes, diering only in the form of the starting pdf. ( )= ( )= (
2
)= 1 2 ()
(
)=
( ) 1/ (1 + )
(
)
The pdf is obtained from the distribution function by dierentiation, ( )= ( )= () ( ) 1 = for 0
2
()

FRE 6083 Homework 4 Question 1 Scaling Normal VaR with independent and with autocorrelated returns. A portfolio has daily returns, discounted to today, that are normally with zero expectation and standard deviation 1.5%. (a) Calculate the 1% 1-day VaR. (b

MONTE CARLO SIMULATION OF CORRELATED, LOGNORMALLY DISTRIBUTED ASSET PRICES
Dynamic investment strategies based on continuous time stochastic models of asset prices frequently require that the portfolio manager simulate the portfolio process using the Mont

ESTIMATING VALUE-AT-RISK WITH A PRECISION MEASURE BY COMBINING KERNEL ESTIMATION WITH HISTORICAL SIMULATION
J. S. Butler Professor of Economics Department of Economics and Business Administration Vanderbilt University, Nashville, TN, USA
Barry Schachter M

Lecture 3 Univariate and Bivariate Probability
Agenda
Simulation of Exponential Random Variable Simulation of Uniform Random Variable Simulation of Gumbel Random Variable Non-Parametric Density Models Bivariate Continuous Distributions Multivariate Contin

FRE 6083 Homework 1 Question 1 Suppose that a number is to be selected from the real line , and let , , and be the events represented by the following subsets of , where the notation cfw_ : denotes the set ocontaining every point for which the proporty p

Set Theory Review
Definitions Set Operations
FRE 6083 Quantitative Methods in Finance - Copyright F. Novomestky 2007
1
Set Theory - Definitions
Set = a collection of objects Symbols for sets
Upper case block letters A, B, C, E,F, M, S Upper case italic le

Random Variables, Densities, Distributions
This supplement provides a catalog of some of the most widely used random variables in financial risk modeling. Extensive use of the Dirac delta function, unit step function and ramp function. Additional comments

Gamma Function Notes The gamma function is widely used in applied mathematics and probability. An discussion of this function provides a review of calculus and an introduction to numerical methods. The gamma function, denoted by (x ) , is defined for x >

Function of One Random Variable
Let X be a continuous RV with density function differentiable function
fX ( x) .
Suppose that there is a Suppose that for a given Y The
Y = g ( X ) of the random variable.
there are n unique roots
X 1 , X 2 , X n
such that

The Concept of a Random Variable
Random variables; distributions; densities Examples of distributions and density functions Conditional distributions and densities
FE 6083 Quantitative Methods in Finance - Copyright F. Novomestky 2007
1
Random Variables;