Statistics and Risk
Management
Ngai Hang Chan ( )
Department of Statistics
Risk Management Science Program
Chinese University of Hong Kong
Shatin, N.T., Hong Kong
[email protected]
Module One
Basics and Background
of Risk Management
Outlines
Concepts
RMSC4003
Homework 4
Due: November 13, 2014
1. Based on a one-factor model, consider a portfolio of two securities with
the following characteristics:
Security
A
B
Factor-sensitivity
.20
3.50
2
Non factor Risk (ei )
49
100
Proportion
.40
.60
(a) If the sta
RMSC 4003
Homework 3
Due: October 23, 2014
1. Find the tangency portfolio associated with the following securities
listed in Table 1. You may assume a risk-free rate of 4% and the
Table 1: Securities
Securities
1
2
3
4
5
Expected Return
15
11
10
9
7
Beta
RMSC4003
Homework 2
Due: October 9, 2014
1. The correlation between assets A and B is 0.1, and other data are
given in Table 1.
Table 1: Two Correlated Assets
Assets
A
B
10.0%
18.0%
15%
30%
(a) Find the proportions of A and (1) of B that dene a portfolio
RMSC4003
Homework 1
Due: September 18, 2014
1. Consider two portfolios A and B. Let the returns of portfolio A, rA be
normally distributed with mean 8% and standard deviation 10% and
let the corresponding returns for portfolio B be normally distributed
wi
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 10 Solution
LING Hok Kan, Brian
November 27, 2014
1
Examples on Itos Lemma
Example 1.1. The geometric mean reversion process Xt is dened as the solution of the stochastic
dierential equation
dXt
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 9: Its Lemma (Solution)
o
LING Hok Kan, Brian
November 13, 2014
In the following, Wt always denotes a standard Brownian motion.
1
Its Lemma
o
Denition 1.1. A diusion process is a stochastic proc
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 10
LING Hok Kan, Brian
November 27, 2014
1
Examples on Itos Lemma
Example 1.1. The geometric mean reversion process Xt is dened as the solution of the stochastic
dierential equation
dXt = K( log
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 9: Its Lemma
o
LING Hok Kan, Brian
November 13, 2014
In the following, Wt always denotes a standard Brownian motion.
1
Its Lemma
o
Denition 1.1. A diusion process is a stochastic process Xt sati
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 8 Solution
LING Hok Kan, Brian
November 6, 2014
1
Introduction
In the last two chapters, we would like to model asset dynamics and price options (or other derivatives). Recall that we are able t
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 8
LING Hok Kan, Brian
November 6, 2014
1
Introduction
In the last two chapters, we would like to model asset dynamics and price options (or other derivatives). Recall that we are able to price a
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 7 Solution
LING Hok Kan, Brian
November 6, 2014
Major criticism of CAPM theory: (1) assumptions about mean-variance analysis (everyone cares
about mean and variance only) and (2) the existence o
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 6 Solution
LING Hok Kan, Brian
October 16, 2014
1
Summary of Chapter 1
Typology of risks
Tracking error
Basis risk
2
Hedging with CAPM
This subsection summarizes what you learnt in RMSC2001 o
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 7
LING Hok Kan, Brian
October 31, 2014
Major criticism of CAPM theory: (1) assumptions about mean-variance analysis (everyone cares
about mean and variance only) and (2) the existence of the mar
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 3 Solution
LING Hok Kan, Brian
September 25, 2014
1
Portfolio Variance and Feasible Sets
1.1
Portfolio Variance
n
n
i=1 wi ri .
Example 1.1 (Portfolio Variance). Let rP =
n
Show that Var(rP ) =
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 6
LING Hok Kan, Brian
October 16, 2014
1
Summary of Chapter 1
Typology of risks
Tracking error
Basis risk
2
Hedging with CAPM
This subsection summarizes what you learnt in RMSC2001 on forward
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 3
LING Hok Kan, Brian
September 25, 2014
1
1.1
Portfolio Variance and Feasible Sets
Portfolio Variance
n
Example 1.1 (Portfolio Variance). Let rP =
n
i=1 wi ri .
n
Show that Var(rP ) =
wi wj ij
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 4 Solution
LING Hok Kan, Brian
September 30, 2014
This tutorial is for your reference on using R to solve Question 4 in Homework 2.
Example 0.1. (a) Using the data in the data le, calculate the
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 1
LING Hok Kan, Brian
September 11, 2014
1
Review of Basic Statistics and Probability Concepts
(1) Probability is a mathematical tool to quantify the uncertainty in the future.
(2) Risks and unc
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 2
LING Hok Kan, Brian
September 18, 2014
1
Asset Return
Using notations in lecture notes and assuming all vectors are column vectors:
(1) Simple return
(2) Log-return
(3) Portfolio return
Remark
RMSC 4003
Statistical Modeling in Financial Markets
Tutorial 1 Solution
LING Hok Kan, Brian
September 11, 2014
Example 0.1. Show that
(a) E(aX + b) = a E(X) + b.
(b) E(X + Y ) = E(X) + E(Y ).
(c) E(XY ) = E(X) E(Y ) if X and Y are independent.
Proof. (a)
RMSC4003
Statistical Modeling
in Financial Markets
Instructor: Professor Ngai Hang CHAN
Email:
[email protected]
Tel: 3943-8519
Office: Lady Shaw G18
Teaching Assistant:
Mr. LING, Hok Kan
[email protected]
Who should take this course: R
Chapter Two
Asset Management
Returns
rt = (Pt - Pt-1 + Dt)/Pt-1 , where Pt denotes the price of
the stock at the end of period t and Dt denotes the
dividends. WLOG, we take Dt = 0.
Since log(1+x) ~ x, when x is small, it follows that rt ~
log Pt log Pt-1
Chapter 6: Statistical Model
for Asset Dynamics
There are two classes of models in a one
period dynamic asset pricing:
Objective: To model the price of an
asset.
Discrete-time Binomial lattice
Continuous-time Wiener or Ito Processes
Discrete-time restrict
Chapter Three
Capital Asset Pricing Model
Perfect Market Conditions
Look for higher return, nonsatiation.
Look for lower risk, risk aversion.
Assets are infinitely divisible.
Taxes and transaction costs are negligible
All investors have the same one-perio
Chapter Four: Factor
Models
Introduction
CAPM is useful, but there are two drawbacks.
It depends on the knowledge of the mean
and the variance-covariance structure of the
return process. Suppose that we have a
portfolio consisting of N assets, then there
STATISTICAL MODELING IN FINANCIAL
MARKETS
Ngai Hang Chan
Department of Statistics
Chinese University of Hong Kong
Shatin, N.T.
Hong Kong
c N.H. Chan, 2010
Chapter 1
Introduction
1.1
Meaning of Risk Management
What exactly is risk management?
What is ris
Chapter 2
Asset Management
2.1
Introduction
Boradly speaking, asset management refers to the study of managing assets
in nance. For our purupose, we understand asset management mostly in
terms of portfolio selection. How to select an optimal portfolio has