Mathematical Tools for Asset Management MTH6113

Mathematical Tools for Asset Management, 2020 �1

Mathematical Tools for Asset Management MTH6113

Dr Kathrin GlauSchool of Mathematical Sciences

Semester B, 2021


OrganizationMathematical Tools for Asset Management


Semester B, 2021


Lectures and Tutorial style SeminarLectures (online)•Tuesday, 11:00 - 12:00•Tuesday, 17:00 - 18:00•Thursday, 11:00 - 12:00Seminar (online)•Thursday, 17:00 - 18:00 (tutorial style)

Office Hours•Thursday, 15:30-16:30 (see my Homepage for updates)

Contact by [email protected]

https://www.qmul.ac.uk/maths/profiles/glauk.html


Module Outline1. Efficient markets hypothesis (EMH)

2. Stochastic models of long-term behaviour of security prices

3. Investment risk and return

4. Mean-variance portfolio theory

5. Factor models of asset returns

6. Pricing

7. Utility theory

8. Behavioural finance


Detailed Module Outline (1/2)1. Efficient markets hypothesis (EMH)

• Various forms of the EMH• Comparison and consequences of each form of the hypothesis

2. Stochastic models of long-term behaviour of security prices• Continuous-time log-normal model• Autoregressive models• Alternative models• Simple calculations• Parameter estimation

3. Investment risk and return• Measures of investment risk• Comparison of investment opportunities using different risk measures• Assessment of risk• More on value-at-risk

4. Mean-variance portfolio theory• Mean-variance portfolio theory (Markowitz)• Risk and return of portfolio of many assets• Benefits of diversification• The risk-free asset (Tobin)


Detailed Module Outline (2/2)5. Factor models of asset returns

• Overview of multi-factor models• Single-index model• Types of risk• Calculations using various factor models

6. Pricing • Capital Asset Pricing Model (Sharpe and Lintner)• Arbitrage Pricing Theory (Ross)

7. Utility theory • Utility functions• Utility maximisation and the expected utility theorem• Simple investment allocation problems with finite number of possible outcomes• Portfolio optimisation problem• Stochastic dominance

8. Behavioural finance • Discussion of principal results


Expectations: What you’ll get

You will be safely guided to succeeding in the exam, if• you attend the lectures and follow my instructions,•participate in the seminars: use the occasion to ask questions! •work on all courseworks,• submit your quizzes

The contents is helpful to get your degree, but there is more to get• for your career, especially if you are interested in the financial sector,• the topic is how to manage assets. We will discuss crucial insights into asset management that are equally valid for all! You can even use the acquired knowledge for your own wealth creation!

You will be get even more out of it, by applying Matlab and using financial data! Then you actually can do your first step towards becoming

•Financial Analyst, a Quant, etc.•Data Analyst.


Expectations: Special!

This lecture is different! •Beware that this lecture is not pure maths!•This is applied maths!•Mathematics is here not the aim but the TOOL!

This has consequences: •We will introduce different financial concepts. •Financial concepts are no mathematical objects.•There comes into play the reality of financial markets, psychology, economic reasoning!

• If we derive for instance an equation, it is not about the equation itself. The actual meaning lies within its financial interpretation!


Expectations: Special!

Consequences on the presentation of the material:•Slides mainly for

•Financial concepts,•graphics of data

•Blackboard/ Whiteboard/ Visualiser mainly for•mathematical argumentation

Consequences assessing the material:•Financial data cannot be assessed with pencil and paper!•Use of Matlab to assess financial data: Matlab exercises.

Consequences for you in learning: • If you have no economic background: open yourself up to a different way of thinking! It is rewarding! •The economic concepts we are introducing are still very basic, so if you enter into the reasoning, you will succeed!•Use Matlab and data•How to work with slides: Many ways, for instance

•print them and make notes•use them on an ipad and make your notes electronically


Expectations: Matlab

Matlab in the Exam?• There will not be a Matlab exercise in the exam.

Why no Matlab Exercise in the Exam?• This is planned for future years.• Actuarials have discovered that the use of programming and data is essential to the

education and therefore it will become part of the exam in the future!Why should I care about Matlab Exercise if they are not part of the Exam?

• Matlab exercises are about data.• Interpretation of data is part of the exam. • The Matlab exercises prepare you to solve the exercises related to data in the

exam.

We offer Matlab exercises for you to • gain a better understanding of financial data! • gain an advantage on the job market! • get your first step towards getting a financial data analyst!


Expectations: Communication

The whole team is working extremely hard to ensure • that the contents is made accessible to you in the best possible way,

• responding to your queries in time,•caring for your individual needs.

How: •presentation of contents tailored to you! •different media in the lecture,•exercises,• sample exam,• tutorial style seminar,•please also ask questions during the seminar and in the forum!


Expectations: Communication

Efficient feedback: • Let us know what you like (-:• Let us know if you have problems:

•No panic & no worries: In almost all cases an easy solution can be found!

•Sometimes the solution will also require you to change habits (for instance a student in a previous year was not used to economic reasoning and preferred mathematical proofs. This is nothing the team can change. But you can change it, by giving it a try!).


Exercises andGraded Quizzes

•Please work on all exercises•Graded Quizzes:

• roughly every second week, • each counts 5% towards the final mark

•Final exam • in the exam period,• counts 75% towards the final mark

Three main types of questions:• Calculations, e.g. • Interpretations / Explanations, e.g.

does this plot confirm the Efficient Market Hypothesis and why?

• Matlab (not in the examination), e.g. plot the daily log-return of RDSA over the last 5 years

• Data (also in the examination)• e.g. 2 examples of plots, which ones is a realistic plot of the daily log-returns and why?

!(X1 + … + Xn ) for Xi ∼ #(μi, σ2i )


MaterialLecture:

• slides (main material)• lecture notes (additional source)• short lecture notes (supplementary, of previous years)

Exercises, seminar: • Exercise sheets• Written Solutions (not to feedback questions)• Matlab files

Exam: • Sample Exam• coursework exercises


LiteratureIntroduction to Macroeconomics:•Brealey, Myers, Allen

Principles of Corporate Finance, McGraw-Hill e.g. 12th edition: 2017

Focus on derivatives:• Hull - Options, Futures and other Derivatives,

Pearson, e.g. 9th edition: 2018

Risk management:•McNeil, Frey, Embrechts - Quantitative Risk Management, Princeton Univ. Press 2015


Literature (continued)Utility theory: •Cerny - Mathematical Techniques in Finance, Princeton Univ. Press, 2009

Portfolio theory:

• Capinski, Kopp - Portfolio Theory and Risk Management, Cambridge, 2014

• Grinold, Kahl - Active Portfolio Management, McGraw-Hill, 1999


Further Reading•Malkiel

A Random Walk Down Wall StreetNorton, e.g. 2016 (regularly revised)

Easy to read investment book; overview with many examples

Interesting links:•W. Sharpe: http://www.stanford.edu/~wfsharpe/mia/mia.htm•Reading list of Goldman Sachs: https://www.quantnet.com/threads/the-goldman-sachs-suggested-reading-list.4916

http://www.stanford.edu/~wfsharpe/mia/mia.htm

https://www.quantnet.com/threads/the-goldman-sachs-suggested-reading-list.4916

https://www.quantnet.com/threads/the-goldman-sachs-suggested-reading-list.4916


week 1 - IntroductionMathematical Tools for Asset Management


Semester B, 2021


0. Some Basics• Stock market basics

• Basic activities on financial markets• Some terminology• Basics in trading and risk management

• Working with data• Good practice• Matlab (see CW sheet and QMplus)• Further possibilities: python with pandas, R, Excel..

• Repetition of probability theory• Expectation value, Variance• Relation of random variables, Covariance, Independence


What is „the market“•Large amount of people trading (abstract and concrete) goods on an organised market (exchange)• e.g. corn, currencies, stocks, options

exchange

I would sell for £1.10

I would buy for £0.98




Hi, I’d like to sell for £1.00

exchange





Deal! I’ll buy it for £1.00

Great! There you are!

> current market price:£1.00


Basics on the marketOrder book: lists all current offers, e.g.• A would sell for £1.01, B for £1.10 and C for £1.20• A would buy for £0.9, D for £0.98, etc

Bid-Ask Spread• You can currently buy for £1.01, but sell only for £0.98

Balance of demand and supply• With people buying, the price increases:

Once A sold, you need £1.10 to buy it from B.


Daily Price and ReturnsExample: Apple Stocks (in USD)

Price

Returns

Zoomed in:

-4.5%(within 1 day)

-0.045

St

Rt

Data source: Yahoo Finance


Returns of a game of coin tosses as an illustration

• For each £1 deposited (=initial stock price), you may play the following game:• Each day a coin is tossed:

heads: 10% of your money is added to the accounttails: 10% is taken from your account

• You may withdraw your money whenever you likeExample run:

Day 0 1 2 3 4 5Coin - Tails Head Heads Tails HeadsReturn - -10 % +10 % +10 % -10 % +10 %Profit - -£0.10Deposit £1,00 £0.90


Returns of a game of coin tosses as an illustration

• For each £1 deposited (=initial stock price), you may play the following game:• Each day a coin is tossed:

heads: 10% of your money is added to the accounttails: 10% is taken from your account

• You may withdraw your money whenever you likeExample run:

Day 0 1 2 3 4 5Coin - Tails Head Heads Tails HeadsReturn - -10 % +10 % +10 % -10 % +10 %Profit - -£0.10 +£0.09 +£0.10 -£0.11 +£0.09Deposit £1,00 £0.90 £0.99 £1.09 £0.98 £1.08


Binomial model

£1

£1.10

£0.90

heads

heads

tails

tails

heads

tails

£1.21

£0.99

£0.99

£0.81

Note:Gain 10% & then lose 10%yields a small total loss of 1%. consider log-return instead:

i.e.

Then yields

Xt = . . .

St+ 1 = . . .

X0 = 10 % , X1 = −10 % ,

S2 = . . .


Binomial model

£1

£1.10

£0.90

heads

heads

tails

tails

heads

tails

£1.21

£0.99

£0.99

£0.81

Note:Gain 10% & then lose 10%yields a small total loss of 1%. consider log-return instead:

i.e.

Then yields

Xt = log (St+ 1 / St),

St+ 1 = St exp(Xt) .

X0 = 10 % , X1 = −10 % ,

S2 = S0 exp(−0.1)exp(0.1) = S0 .


Some more about the marketSo far: what is a market?Now: •What is traded on financial markets (security)?

• Equity: stocks/shares of a public company• Derivatives: Options, Futures, Swaps, e.g. Contracts for Difference (CFD)• Dept: Loans/Bonds (secured, unsecured; personal, commercial,

governmental)• Foreign currencies (FX)• …

•Who is trading and why?•How are they trading?•Some observed market mechanisms

And where is the mathematics?

buy?sell?

buy? sell?£14?

Deal!


Repetition: Risk-free interestLoans by other banks or stable governments considered risk-free

Example: £1,000 for one year at 2% nominal interest•Annual compounding:•Semi-annual compounding:•Quarterly compounding:•Monthly compounding:•Continuously compounding:

£1 000 × . . .£1 000 × . . .

£1 000 × . . .£1 000 × . . .

£1 000 × . . .

See FM1 for more details


Repetition: Risk-free interestLoans by other banks or stable governments considered risk-free

Example: £1,000 for one year at 2% nominal interest•Annual compounding:•Semi-annual compounding:•Quarterly compounding:•Monthly compounding:•Continuously compounding:

£1 000 × (1 + 0.02) = £1 020£1 000 × (1 + 0.01)2 = £1 020.10

£1 000 × (1 + 0.005)4 ≈ £1 020.15£1 000 × (1 + 0.02/12)12 ≈ £1 020.18

£1 000 × exp(0.02) ≈ £1 020.20

See FM1 for more details


Public companies (e.g. PLC)Company with shared ownership where shares are publicly traded1 share = owning 1 equal part of the company (example GE: ~1bn shares)

• proportional voting rights;• distribution of the profit: dividend;• limited liability (max. loss of the original investment).

Company’s advantage: raise capital by selling own shares

XY PLC

Shared ownership:XY plc = 100 shares:

1 2 3 4 5

100

6 7 8 9……

stockexchange

OK, deal

I’ll buy one shareof XY plc for £100

Shares publicly traded:

Profit is distributed to shareholders (dividend payment):

+£250

12

100

…£2.50

£2.50

£2.50

100

21…

Equal voting power(with certain exceptions)


Why do stocks even have a valueA. Intrinsic value given by the company’s value• own a share of a company

B. Cash-flow due to dividend payments• similar to interest payments

C. Commonly agreed value, reflected by stock price• can be significantly larger than the intrinsic values

(bubbles)• see cryptocurrency trading


How to make profit with stocks (1/3)A. Dividends paid yearly• Example: Apple in 2015: ≈ 2%• Stock price ~$105 (similar in Jan and Dec)

https://finance.yahoo.com/quote/AAPL/history?period1=1420070400&period2=1452384000&interval=1d&filter=history&frequency=1d• Dividend payment: 4x$0.52 = $2.08

https://investor.apple.com/investor-relations/financial-information/dividend-history/default.aspx• Compare: Fed-rate (Federal funds rate) in 2015

0.0%—0.5%

B. „Buy and hold“ - ignores temporary crisis and profit from the long-term growth of the market

C. Try to profit from short-term price changes

https://finance.yahoo.com/quote/AAPL/history?period1=1420070400&period2=1452384000&interval=1d&filter=history&frequency=1d

https://investor.apple.com/investor-relations/financial-information/dividend-history/default.aspx


How to make profit with stocks (2/3)B. „Buy and hold“ - ignores temporary crisis and profit from the long-

term growth of the market• Example: DAX (German stock market index; 30 public companies)

• Initial value (2010): 6k, current value: 11k• Each €1.000 broadly invested returned €1.800 (plus dividends)• Equivalent interest (continuously compound) 6.5%

C. Try to profit from short-term price changes

MTH6113: How to optimally design a portfolio

Mathematical Tools for Asset Management, 2020�34

How to make profit with stocks (3/3)C. Try to profit from short-term price changes

Example (completely random):

BP easyJet GE

a) Buy as price increases

b) Sell to realise profit

c) Buy again as price increases

d) Sell when price falls

Profit ~£80 per £1.000(within 3 months)


b) Sell to avoid further loss

c) Buy again as price increases

d) Sell to avoid loss


b) Sell to avoid loss

c) Buy again being optimistic

d) Sell in panicto avoid more loss

Profit ~£0

Hold, as we hope for an increase

Loss ~£200 per £1.000

see EMH


Some important market mechanisms: Markets react to news

Brexit referendum: 23 June 2016

see semi-strong EMHDax


Some important market mechanisms: Stocks don’t move independently

BP in 2018

Shell in 2018

Strong influence of the general market

Chapter 5:Factor models


Some important market mechanisms: Speculative bubbles exist frequently

•Dutch tulip mania (1600)•British South Sea Bubble (1700s)•Dot-com bubble (2000)•Bitcoin bubble (2017-2018):• 1 BTC in USD

Max value: $19.345

current value: $3.653


Leveraged tradingCapital: £100; Current stock price: £100Prediction: Tomorrows stock price: £110

•Trading with no leverage:• Buy one stock for £100, sell it tomorrow

• Ideal (i.e. stock sells for £110): Profit of £10• Risk (e.g. stock falls to £90): Loss of £10

•Leveraged (idealised)• Borrow £1k and buy 11 stocks, sell them tomorrow

• Ideal: Profit of £110• Risk: Loss of £110 (more than your capital) -> depth of £10 remain!

-> Higher profit for higher risk (not recommended for private investors)

£100 x1

£100 x11


HedgingYour trades can increase your risk, hoping for profit;Hedging: trades that reduce a particular risk, e.g.• airlines buy futures on kerosene to reduce their risk

of rising prices (hedge against rising of prices)• Also common: currencies, interest rates,

commodities• A bank selling options can buy the underlying

stock to reduce the impact of price changes• Hedge fonds originally tried to hedge the market

risk and wish to make steady profit (taking a large overall risk), on rising and falling markets

Maths: how to minimise risk?


DerivativesContract with a value deriving from an underlying entity

•Futures/Forwards: Agreed transaction at a fixed future time•Options: Granted possibility to buy/sell the underlying at a predefined time/time range•Swaps: exchange benefits of two instruments• e.g. exchange flexible interest payment by a fixed one

•… (see e.g. Hull)

Can be used to hedge risk or to use the leverage

FM2:How to pricethese derivatives


Who else is tradingPrivate investors:• Often buy and hold shares, exchange traded funds (ETF)

or bondsProfessional Funds (Mutual/private funds, hedge funds, …)• Active management, i.e. regular buying and (short) selling

(adjustment) of shares as well as complex derivatives• Risk management

Arbitrageurs (e.g. high frequency trading)• Exploit market inefficiencies, which then vanish• Transactions can happen within microseconds

(Not at all exclusive list)

Ch. 3-6: quantifying risk


Where may I need thisWorking in

•Banks: Barclays, HSBC• Investment companies: Goldman Sachs, BlackRock,…•Corporate Finance

• Take loan or sell shares to take investment?Reinvest profit or pay dividend?

•Consulting: KPMG, PwC• Risk Analysis; support with regulatory requirements

•Hedge Fund Management

Master suggested for financial mathematics jobs


week 1 - Lecture 2Mathematical Tools for Asset Management


Semester B, 2021


Outline

•Repetition from Probability Theory

•Working with Data

•Efficient Market Hypothesis


Repetition from probability theoryLet X be a discrete random variable with •possible values •and their possibilities

Expectation value:

Variance: Standard Derivation (SD):

x1, …, xn ∈ ℝp1 = P(X = x1), …, pn = P(X = xn ) ∈ [0,1]

!(X) = . . .

Var(X) = . . .σ = . . .


Repetition from probability theoryLet X be a discrete random variable with •possible values •and their possibilities

Expectation value:

Variance: Standard Derivation (SD):

x1, …, xn ∈ ℝp1 = P(X = x1), …, pn = P(X = xn ) ∈ [0,1]

!(X) =n

∑i= 1

xipi

Var(X) = !((X −!(X))2)σ = Var(X)


Example: Discrete random variableToss a fair coin, win £1 for heads and lose £1 for tailsRandom variable X: Profit in £

Values and probabilities:

Expectation value:

Variance & SD:

x1 = 1, p1 = 1/2, x2 = −1, p2 = 1/2

!(X) = . . . with (X −!(X))2 = . . .

Var(X) = . . .

σ = . . .


Example: Discrete random variableToss a fair coin, win £1 for heads and lose £1 for tailsRandom variable X: Profit in £

Values and probabilities:

Expectation value:

Variance & SD:

x1 = 1, p1 = 1/2, x2 = −1, p2 = 1/2

!(X) = x1p1 + x2p2 = 1/2 −1/2 = 0 with (X −!(X))2 = X2 :

Var(X) = !(X2) = x21 p1 + x2

2 p2 = 1.σ = Var(X) = 1


Continuous random variablesProbability Density Function (p.d.f.): where

Expectation value:

Variance:

Standard deviation:

fX : ℝ → [0,∞)

P(X ∈ (a, b)) = . . .

!(X) = . . .

Var(X) = . . .

σ = . . .


Continuous random variablesProbability Density Function (p.d.f.): where

Expectation value:

Variance:

Standard deviation:

fX : ℝ → [0,∞)

P(X ∈ (a, b)) = ∫b

afX(x) dx, a < b

!(X) = ∫ℝx f(x) dx

Var(X) = !((X −!(X))2)

σ = Var(X)


Example: Uniform distributionX distributed uniformly on the interval (-1,1), p.d.f.:

Expectation value:

Variance:

Standard deviation:

f(x) = {1/2, x ∈ (−1,1),0, else

!(X) = . . .

fulfills ∫ℝf(x) dx = 1

Var(X) = . . .

σ = . . .


Example: Uniform distributionX distributed uniformly on the interval (-1,1), p.d.f.:

Expectation value:

Variance:

Standard deviation:

f(x) = {1/2, x ∈ (−1,1),0, else

!(X) = ∫1

−1x 1/2 dx = 0.

fulfills ∫ℝf(x) dx = 1

Var(X) = ∫1

−1x2 1/2 dx = 1/3

σ = 1/ 3


Important distribution: GaussianGaussian distribution p.d.f.:#(μ, σ2)

f(x) = 12πσ2

exp (−(x −μ)2

2σ2 ) !(X) = μVar(X) = σ2

Exercise: Use the following identities to show the given expectation value and variance:

∫ℝx exp (−x2/2) dx = 0, ∫ℝ

x2 exp(−x2/2) dx = 2π .

μ = 0, σ = 1

μ = 0, σ = 2

μ = 3, σ = 1


Relation of several random variablesTwo random variables X, Y have a joint distribution (X,Y): Discrete: Values probabilities

Continuous: p.d.f. , s.t.

Independence: /

Covariance (measure for linear dependency):

Correlation: normed covariance

f(X,Y)(x, y) P(a ≤x ≤b, c ≤y ≤d ) = . . .

(xi, yj) P(X = xi, Y = yj)

P(X = xi, Y = yi) = . . .fX,Y(x, y) = . . .

Cov(X, Y) = . . .

X, Y independent ⇒ Cov(X, Y) = . . .Note: (reverse not true).

ρ(X, Y ) = . . .


Relation of several random variablesTwo random variables X, Y have a joint distribution (X,Y): Discrete: Values probabilities

Continuous: p.d.f. , s.t.

Independence:

Covariance (measure for linear dependency):

Correlation: normed covariance

f(X,Y)(x, y) P(a ≤x ≤b, c ≤y ≤d ) = ∫b

a ∫d

cf(X,Y )(x, y) dydx .

(xi, yj) P(X = xi, Y = yj)

P(X = xi, Y = yi) = P(Y = yi)P(X = xi)fX,Y(x, y) = fX(x) fY(y)

Cov(X, Y) = !((X −!(X))(Y −!(Y)))

X, Y independent ⇒ Cov(X, Y) = 0Note: (reverse not true).

ρ(X, Y ) = Cov(X, Y)/σXσY


Time seriesSeries of random variables •continuous: •discrete: (considered within this courseAutocorrelation for stationary series:(stationary: stochastic properties do not change when shifted in time)•Correlation between subsequent values: •Empirical mean: and•Empirical correlation:

(Xt)t

t ∈ 1,…, Nt ∈ ℝ

ρ(Xt, Xt+ 1)μ2 = . . .μ1 = . . .

ρ = . . .


Time seriesSeries of random variables •continuous: •discrete: (considered within this courseAutocorrelation for stationary series:(stationary: stochastic properties do not change when shifted in time)•Correlation between subsequent values: •Empirical mean: and•Empirical correlation:

(Xt)t

t ∈ 1,…, Nt ∈ ℝ

ρ(Xt, Xt+ 1)μ2 = 1/(N −1)∑N−1

t= 1 Xt+ 1μ1 = 1/(N −1)∑N−1t= 1 Xt

ρ =∑N−1

t= 1 (Xt −μ1)(Xt+ 1 −μ2)

∑N−1t= 1 (Xt −μ1)2 ⋅ ∑N−1

t= 1 (Xt+ 1 −μ2)2


Basics on asset returns•Consider time-series of asset values•Often daily values: with value on day t. •Daily return: i.e. relative profit in one day

•Log-returns:

•Aim: understand the stock market and investigate professional portfolio management

No advice for personal investments•Trading costs usually neglected during this course

(St)t

t ∈ 1,…, N St

St+ 1 = (1 + Rt)St

Rt = St+ 1/St −1

Xt = log(St+ 1/St)


Daily Price and ReturnsExample: Apple Stocks (in USD)

Price

Returns

Zoomed in:

-4.5%(within 1 day)

-0.045

St

Rt

Data source: Yahoo Finance


Working with data•Matlab is a common tool for mathematical research• Provided by QMUL -> see first coursework sheet

•Octave, Python, R, Excel are further possibilities• You may use them if you are comfortable with

them, however we will only consider Matlab


Why do we work with data?•Availability of huge amounts of data: „Big Data“•Huge potential to use data for benefit of the society• e.g. smart cities, optimal use of resources,

more precise risk evaluation➡ Active topic in research and industry

Financial Mathematics models the behaviour of markets• Data is the key to develop and test

models and hypotheses.


How do we work with data?•The whole story counts•No cherry picking. All results matter• Don’t just look at the results that confirm your

hypothesis•No cherry tree design• Don’t design your experiment to fit your hypothesis

•Think critical•Seek the „truth“•Don’t falsify data, or suppress relevant data•Transparency & reproducibility


Which data can we work with in financial mathematics?

Prices of •Financial indices: FTSE 100, Dow Jones, NIKKEI, ..•Stock prices: BP, GE, HSBC, Apple,…•Commodities: gold, silver,…•Currencies: EUR/GBP, CHF/EUR, GBP/USD,…•Derivative’s prices: European call/put, American call, …Related quantities:•Trade volume•Interest rates: LIBOR, EURIBORDerived quantities:•Daily returns•Volatility (SD of stock prices)


Where do we get the data?Data is power:financial data is valuable resource for many companies (e.g. Bloomberg)But, e.g. daily stock prices are available by Yahoo Finance:• https://finance.yahoo.com/• Search for a company, e.g. AAPL for Apple• Select the tab „Historical Data“• Select the desired time period, click „Apply“ and then „Download Data“• Open the obtained csv-file (comma separated values), e.g. with

MS Excel or LibreOffice Calc to inspect the data

Further possible data sources, e.g. Quandl, or the Google Dataset Search

https://finance.yahoo.com/

https://www.quandl.com/search?filters=%5B%22Free%22%5D

https://toolbox.google.com/datasetsearch


week 1 - Lecture 3 & seminarMathematical Tools for Asset Management


Semester B, 2021


Outline•The Efficient Market Hypotheses (EMH)• weak• semi-strong• strong form

•Criticism and use of EMH

•How to make money as one of the richest people on earth?


1. Efficient Market HypothesisMain question of portfolio analysis:•How to make money on trading stocks (with minimal risk)

Resulting questions:•What is the intrinsic value of an asset (is it over- or underpriced)?•How will the price behave in the near future?

Can we answer this?

Very first observation:Stock prices seem to behave randomly

-> can we still find patterns?

Apple in USD(two month period of daily prices)


Predicting future stock pricesIt is difficult to make predictions—especially about the future (old Danish proverb)

Assume you know future stock price movements•Are you the only one who knows?…


… most likely, you’re not!


There ain’t no such thing as free lunch

•If we expect a price rise tomorrow,•professional traders will buy the stock •and the price rises today:

Source: Brealey, Myers, Allen

If today’s market already reflects our prediction today, we call it an Efficient Market


No arbitrageOld story about a finance professor and his student:

A finance professor and his student go for a walk and find £50 on the pavement. When the student wants to pick up the note, his professor says: „Don’t try this, this cannot be. If there was a £50 note lying around, someone else would have already picked it up“

Note: While you might find £50 on the pavement, you can’t make a living by just looking for money on the street


First empirical test: Autocorrelation of log-returns

BP(correlation 0.0325)

easyJet(correlation 0.0290)

GE(correlation 0.0146)

Royal Dutch Shell(correlation -0.0052)

Correlationnot statistically

relevant

Can we predict tomorrow’s return?Consider autocorrelation, i.e. correlation of Xt and Xt−1

Data source:Yahoo finance


Different formulations of the Efficient Market Hypothesis (EMH)

Preliminary conclusion: Price changes cannot be predicted, i.e. markets are efficient

• Weak form of the EMH• Price reflects all the historical stock prices

• Semi-strong form of the EMH• Price reflects all public information

• Strong form of the EMH• Price reflects all public and private information


Weak form of the EMHInvestments based on past stock prices do not yield superior returns Technical Analysis: predict price movements based on past prices, e.g. „the trend is your friend“: invest in stocks when there is an upwards trend

Scientific studies:This yields less profit than „buy and hold“ when considering trading costs(See Part 2 of Malkiel for more examples)

drop, once we started investing

Upwards trend —> invest


Then how can we invest?A) Fundamental analysis: Model the intrinsic value of a company and invest in underrated stocks; then wait for the price to approach the intrinsic valueB) Quickly react to news with your investment strategy, e.g.:• announcements about the company / the market, e.g.

• higher profit then expected, new CEO, …• rumours, e.g.

• expected merger, expected contracts• related political events, e.g.

• Tax and tariffs; strike action, changed regulationsRecall our argumentation for the weak EMH, is it valid to gain profit based on public information?


How fast does the market react to news?

Source: Brealey, Myers, Allen Based on A. Keown, J. Pinkerton:

Merger Announcements and Insider Trading Activity.Journal of Finance 36, 1981, pp. 855—869)

Profit made before announcement:• Based on rumours (risky)• Based on insider information (illegal)

Little to none profit madeafter the announcement. -> efficient market

Rising price due to announced mergeron day 0:


Semi-strong form of the EMHInvestments based on any publicly available information do not yield superior returns

Assumes that the price adjusts immediately to new information, e.g. the announcement of quarterly earnings, dividends, new stocks

Public information: Anything that is publicly available and relatively easy to acquire (e.g. press releases, newspapers, financial magazines)Non-public Information: Insider trading - illegal in many cases!


Strong form of the EMHNobody can consequently beat the market with their investment

• Very strong and likely not fully valid• Requires a significant number of insiders to trade,

such that the price can reflect their private information(Though illegal, insider trading happens, see e.g. the case of the Galleon Group)

• Difficult to test, as private information is not available

https://en.wikipedia.org/wiki/Galleon_Group#Insider_trading_investigation


Consequence of the EMH:Random Walk Theory

Reminder: Coin tosses and the binomial model

£100

£103

heads

heads

tails

tails

heads

tails

£106.09

positive drift of 0.1%

£100.12

£100.12

£94.48

£97.2

Random outcome:(see Matlab files)

10 tosses 100 tosses 1000 tosses

Stock price Informationreflectedrandomly generated

ꔄ stock prices move randomly and independent of historic prices(i.e. a random walk)

(more sophisticated models considered in the next chapter)


Some empirical arguments concerning the EMH

1. Reminder: Lacking dependence of subsequent returns is a main argument for the weak EMH

BP(correlation

0.0325)The scatter plots shows that returns of subsequent days are uncorrelated. This underpins the hypothesis, that future price changes cannot be predicted based on historical prices.

GE(correlation

0.0146)


Source:Brealey, Myers, Allen

Some empirical arguments concerning the EMH

Can anybody beat the market?•Investment returns of mutual funds compared to the Wilshire 5000 Total Market Index:

In some years mutual funds outperform the market,but no fund does so consistently.Thus the market seems efficient.


CriticismNo clear answer to the validity of EMH:•Empirical studies both confirm and criticise the EMH•Arbitrageurs do gain from inefficiencies(see Pedersen - Efficiently Inefficient, Princeton, 2015)•Stock prices tend to under- or overreact to news•How do financial bubbles fit to the EMH?•Public information is not always free to use immediately. To collect and process business and market data is an effort that could be rewarded (fundamental analysis)•Strong EMH: Many examples of lucrative illegal insider trades were covered up, e.g. $50m-$100m by front-running: https://www.reuters.com/article/2014/01/14/us-usa-swaps-probe-idUSBREA0D04N20140114

https://www.reuters.com/article/2014/01/14/us-usa-swaps-probe-idUSBREA0D04N20140114


Repetition: Investment strategies

Sample Exam Question (1): Assume an investor believes in weak efficiency of markets. A. What type of financial strategy will they not look

for? B. Is there still a type of strategy that they might look

for in order to ”beat the market”?


Investment with weak EMH

If an investor believes in weak efficiency of markets, ...A. …they cannot expect to find a financial strategy

that predictably outperforms the market by seeking patterns in the time series of prices.

B. …they may still use knowledge, such as news and announcements other sources of knowledge, to find an investment strategy with superior returns.



Sample Exam Question (2): Assume an investor believes in semi-strong efficiency of markets. A. What type of financial strategy will they not look




Investment with semi-strong EMHIf an investor believes in semi-strong efficiency of markets, ...A. …they cannot expect to find a financial strategy

that predictably outperforms the market, neither by seeking patterns in the time series of prices, nor by investigating all publicly available information.

B. …they can still use special knowledge, such as insider information or other knowledge on firms capital structure or macroeconomics that is not publicly available, to find an investment strategy with superior returns.



Sample Exam Question (3): Assume an investor believes in strong efficiency of markets. A. What type of financial strategy will they not look




Investment with strong EMHIf an investor believes in strong efficiency of markets, ...A. …they cannot expect to find any financial strategy

that predictably outperforms the market, neither by seeking patterns in the time series of prices, nor by investigating all publicly available information, nor by using any other type of information.

B. Consequently, it is favourable to concentrate on shaping the risk-return profile they are seeking, and minimising the costs of his/her portfolio management.

See our subsequent chapters on risk and return and portfolio theory for more insight!


Repetition: EMHHypothesis: Market reacts immediately to news

• Weak form of the EMH• Price reflects all the historical stock prices

• Semi-strong form of the EMH• Price reflects all public information

• Strong form of the EMH• Price reflects all public and private information


Exam question 2018We read the following news: ”A new company called ’FutureGains’ has launched a new fund. The company uses market data and a new algorithm to find arbitrage possibilities.” The investors in FutureGains trust the strategy. Which form of the efficient market hypothesis do they believe is wrong?

• Weak form? (historical stock prices)• Semi-strong? (public information)• Strong? (all information)

Weak form: They use market data and new algorithms.


Warren Buffett

„Oracle of Omaha“ and 3rd richest person in the world

*1930


Warren Buffett’s investments•He is told to have made his first money buying six-packs of Coca-Cola for $0.25 and selling the bottles for $0.05 cents each.•Now he is one of the most successful investors

Buffet investment strategy, summarised fromhttps://www.investopedia.com/investing/warren-buffetts-investing-style-reviewed/1. Business Tenets2. Management Tenets3. Tenets in Financial Measures4. Value Tenets

„Buffett ignores short-term market volatility and focuses on long-term returns.“

https://www.investopedia.com/investing/warren-buffetts-investing-style-reviewed/


WB: Business Tenets”Buffett adamantly restricts himself to his ”circle of competence” – businesses he can understand and analyse.”Recipe:1. Analyse the business, not market/economy/investor

sentiment.2. Look for a consistent operating history.3. Seen 1. & 2., has the business favourable long-term

prospects?


WB: Management TenetsThis is considered to be the hardest part.1. ”You want to be greedy when others are fearful. You

want to be fearful when others are greedy. It’s that simple. . . . ” (Warren Buffett) In other words: Is the management long-term caring and investing enough or greedy and retaining too high profits?

2. Is management honest with shareholders and admits mistakes?

3. Is management trapped by trends or follow their decisions from reasonable investigations?


WB: Financial Measures and ValueTenets in Financial Measures 1. low leverage (debt/equity) and high profit margins (net profit/

revenue).2. what are the ”owner’s earnings,” so the cashflow available to

shareholders,3. the ”one-dollar premise,”: ”Within this gigantic auction arena, it

is our job to select businesses with economic characteristics allowing each dollar of retained earnings to be translated eventually into at least a dollar of market value.”http://www.berkshirehathaway.com/letters/1982.html

Value Tenets • ”Here, Buffett seeks to estimate a company’s intrinsic value.”

http://www.berkshirehathaway.com/letters/1982.html


Sample Exam Question (4)How does Warren Buffett’s investment style relate to EMHs?A. Does he seek for situations where weak EMH is

violated?B. Does he seek for situations where semi-strong EMH

is violated? C. Does he seek for situations where strong EMH is

violated?Explain your answer to your neighbour.


Sample Exam Question (4), cont’d His strategy violates the semi-strong form of efficient markets. Why do you think it might still work?

My idea:• He uses public information (available to everyone),• But collecting and interpreting this information is a

huge effort and not free (i.e. in practice only available to him)If his conclusions are correct, he can exploit them


Summarising the Efficient Market Hypothesis

• Weak form of the EMH• Price reflects all the historical stock prices • Collect information to outperform the market

• Semi-strong form of the EMH• Price reflects all public information • Only private/insider information can outperform the market

• Strong form of the EMH• Price reflects all public and private information • Nobody can consistently beat the market

Statistical evidence has been found pro and contra the EMH.


Further Reading on EMH• Ch. 13 of Brealey, Myers, Allen (Principles of Corporate Finance)• Malkiel (A Random Walk Down Wall Street)

Overview articles • E. F. Fama, “Efficient Capital Markets: A Review of Theory and

Empirical Work,” Journal of Finance 25 (1970), pp. 383–417. http://www.e-m-h.org/Fama70.pdf

• E. F. Fama, “Efficient Capital Markets: II,” Journal of Finance 46 (1991), pp. 1575–1617. http://www.jstor.org/stable/pdf/2328565.pdf

http://www.e-m-h.org/Fama70.pdf

http://www.jstor.org/stable/pdf/2328565.pdf

Mathematical Tools for Asset Management MTH6113

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