Practical Tutorial 1
Dr R Kufakunesu
February 11, 2016
Chapter 6: Introduction to Portfolio Theory.
(1) Given that X and Y are random variables on the same set. Prove that
corr(X, Y ) 1. ( Hint: Suppose that V ar(X + Y ) = .)
Suggested Solution
Proof. L
WTW 354 Tutorial Test 4A MAX 10
r Surname: E’IWUIW Initials:TH E Student 110.:
1. (5 marks) Let A be a uniformly distributed random variable on the interval (1, 2), and B on
the interval (1, 3). (This means that the each of the density functions are const
VVTW 354 Tutorial Test 5A MAX 10
Surname: golutlml’s Initialszhl H E Student no.1
l. (2 marks) The logwoptirnal formula for the price of an asset with random payoff d, is
d
Pi:lE—z, '=l,2,.,n
( R. z
where R* denotes the total return of the log~optimal
Financial Engineering
Extra notes ll
1. Stochastic Differential Eguations
The Product Rule
Just like in Newtonian Calculus, Stochastic calculus has its own product rule for calculating differentials.
Remember in Newtonian Calculus we had:
d
50mm»: f'(x)g(
Memo
UNIVERSITY OF PRETORIA
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING
EXAMINATION : MAY 2010
, TIME : 2 hours
MARKS : 60
SURNAME:
FIRST NAMES:
STUDENT NUMBER:
SIGNATURE:
PHONE NUMBER:
Exte
MODERN FINANCE vs. BEHAVIOURAL FINANCE:
AN OVERVIEW OF KEY CONCEPTS AND MAJOR ARGUMENTS
PANAGIOTIS ANDRIKOPOULOSa
Leicester Business School
De Montfort University
Abstract
Modern Finance has dominated the area of financial economics for at least four deca
Investment Science
Remark. The following Notes facilitates the understanding of the textbook by David
Luenberger . Investment Science, CH 69. Oxford University Press, 1998, and is in no
way a replacement of it.
6.1 Modern Portfolio Theory (MPT)
(i)Financ
making actuaries less human
lessons from behavioural finance
presented to the Staple Inn Actuarial Society, 18 January 2000
by Nigel Taylor
Summary
Policyholders of insurance companies and beneficiaries of pension schemes whether they know it
or not are p
THEME 7 : MODELS OF SECURITY PRICES AND THE WILKIE
MODEL.
1. Introduction : Autoregressive model
Autoregressive models are discrete time models, in contrast to the continuoustime
lognormal model.
However, we can always consider the lognormal model in d
WTW 354
STOCHASTIC DOMINANCE AND BEHAVIOURAL
FINANCE. Updated by Dr R.K., 2014.
1. Introduction
Up to now we have assumed that investors make decisions in accordance with the
maximization of expected utility of wealth.
The meanvariance approach where cho
Risk Measures.
Last modified April 11, 2014 by Dr R. Kufakunesu.
In financial mathematics return is almost always measured by expected return.
However, there are many possible interpretations and ways of measuring investment risk, of which variance
is jus
Chapter 7
The Capital Asset Pricing Model
Chapter 7 p. 1/?
The Capital Asset Pricing Model
The CAPM is a pricing model
Chapter 7 p. 2/?
The Capital Asset Pricing Model
The CAPM is a pricing model
 how to determine the correct (fair,
arbitragefree) price
I"? 97 0
VVTW 354 Practical Test 2
Surname:_.—_ Initiais: Student number:
(1) Censider a market where there are two risky assets A and B and a risk free asset.
Both rislqr assets have the same market capitalization. .
Assume that ail the assumptions of
WTW354 FINANCIAL ENGINEERING I Class test 1B
Surname and initials
Student number
1. Consider a portfolio consisting of two independent assets with the same mean rate
of return 'F. The variances are 0.25 and 0.36 respectively. Short selling is allowed.
WT‘W 3554 Tutorial Test SB . MAX 10
Surname: I Initials: Student no: A
1. (2 marks) Deduce the log—optimal pricing formula B : ELIE3%) from the portfolio choice
solution equations E(U’(23*)di) 2 AH. State the meanings of all symbols used.
R: Ram etc (15
M W o
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA
DEPT VVISKUNDE EN TOEGEPASTE W ISKUNDE
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING
SEMESTER TEST 1
24 March 2011
TIME: 90 min
MARKS: 45
VAN/SURNAME:
VOORNAME / FIRS
Memo,
UNIVERSITY OF PRETORIA
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING
SEMESTER TEST 1 : 15 March 2010
TIME : 90 min
MARKS : 45
SURNAME:
FIRST NAMES:
STUDENT NUMBER:
SIGNATURE:
PHONE NUNIBER:
W .
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA
DEPT VVISKUNDE EN TOEGEPASTE WISKUNDE
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
‘WTW 354 : FINANCIAL ENGINEERING
SEMESTER TEST 2
3 May 2011
TIME: 90 min
MARKS: 35
VAN/SURNAME:
VOORNAME/FIRST NAMES:
Question 1
In a certain “short selling” transaction you borrow 1000 shares from your broker at a price of X each and
sell the shares immediately. A year later you buy shares at Y each, and give the shares to your broker
to close the loan. The broker requi

Note: This is the discounted qexpected value of a security that pays out "1" in state
uuud=uudu=uduu=duuu only, and "0" in all other states. This is the same as the discounted qprobability of that state. Now reference the binomial distribution, or coun
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA
DEPT WISKUNDE EN TOEGEPASTE WISKUNDE
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING
SEMESTER TEST 1
24 March 2011
TIME: 90 min
MARKS: 45
VAN/SURNAME:
VOORNAME/FIRST NAMES:
STU
Mam/LO _
. UNIVERSITY OF PRETORIA
FACULTY OF SCIENCE ' _
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING '
_ Semester Test 2: May 2010
TIME : 90 minutes
MARKS : 35
SURNAME:
FIRST NAMES: _
STUDENT NUMBER:
SIGNATURE:
PHONE
Z Qadrﬁ CAL‘ZB GD
(1)
(1 1) Explain what as meant by the multifactor model. You should deﬁne any notation you
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WTW354 FINANCIAL ENGINEERING Class test 1A
Surname and initials
Student number
1. Consider a portfolio consisting of two assets.
of 10% and a variance of 0.16,
and a variance of 0.25. The co
The ﬁrst asset has a mean rate of return
while the second asse
WTW354 FINANCIAL ENGINEERING Class test 10
Surname and iniiiais —
Sindeni number —
1. Consider a portfolio consisting of two independent assets with the same standard
deviation 0. The mean rates of return are 0 25 and 0. 36 respectively.
(i) Represent the
Chapter 6
Mean variance portfolio theory
Chapter 6 p. 1/1
The Feasible Set
Chapter 6 p. 2/1
The Feasible Set
The set of points in the r plane
corresponding to portfolios, made up of weighted
combinations of assets, is called the feasible set.
Chapter 6 p.
Chapter 6:
Meanvariance portfolio theory
Chapter 6:Meanvariance portfolio theory p. 1/1
Combining two assets in a portfolio
Chapter 6:Meanvariance portfolio theory p. 2/1
Combining two assets in a portfolio
Consider two assets with mean returns r1 and
Chapter 6
Mean variance portfolio theory
Chapter 6 p. 1/1
The Markowitz problem
A risk averse, nonsatiated investor would like to
invest in a portfolio with a certain reward, say
r = 14%.
Of all the possible portfolios in the feasible
region, he would ch
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA
DEPT WISKUNDE EN TOEGEPASTE WISKUNDE
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354 : FINANCIAL ENGINEERING MEMO
SEMESTER TEST 1
5 April 2013
TIME: 90 min
MARKS: 45
VAN/SURNAME:
VOORNAME/FIRST NAMES:
Semester Test 1
25 February 2016
UNIVERSITY OF PRETORIA
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 354: FINANCIAL ENGINEERING MEMO
External Internal Examiner:
Internal Examiner:
Prof K Jordaan (University of Pretoria)
Dr R Kufakunesu (Universit