Slides_IB_v_04 1 - Lehrstuhl für Bankwirtschaft

**Lehrstuhl** **für** **Bankwirtschaft** und Finanzdienstleistungen

Prof. Dr. Hans-Peter Burghof mit

Dipl. oec. Ulli Spankowski / Dipl. oec. Arne Breuer

Investment Banking and

Capital Markets

Monday 10 - 12 am

Exam: Last week of the semester - mid february

Guest lectures:

Rabea Bastges, Director, HSBC Trinkaus & Burkhardt AG,

„Introduction to Rating Advisory“, 24.11.08, 10 am - 13 pm !!!

Dirk Heß, Executive Director, Goldman Sachs International,

„Commodities as an Asset Class“, 08.12.08

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Table of Contents

What to expect of this lecture?

• Basic knowledge of theory and practice of investment banking

• Insight into the field of market microstructure

• Basic knowledge in trading theories

• Active trading experience by using a trading simulation software

• Fundamental theoretical models associated with banking and finance

• Fun, etc...

Table of Contents ⎜ Topic A.1 ⎜ Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

Table of Contents

A. Introduction to Investment Banking

A.1 The Investment Banking Environment

a. Universal Banking vs. Specialized Banking

b. Commercial Banking vs. Investment Banking

c. Definition of Investment Banking

d. A Basic Description of Investment Banking

A.2 Basic Principles of Capital Markets

a. Modern Portfolio Theory

b. The Capital Asset Pricing Model

Table of Contents ⎜ Topic A.1 ⎜ Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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Table of Contents

B. The Business of Investment Banking

B.1 Asset Management

a. Overview, Goals, and Asset Groups

b. Investment Styles and Asset Allocation

c. Performance Measures

d. Case Study

B.2 Trading Strategies Simulation

a. Trading Strategies

b. Introduction to TraderEx Software

Table of Contents ⎜ Topic A.1 ⎜ Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

Table of Contents

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Table of Contents

Table of Contents

B. The Business of Investment Banking

B.3 Capital Markets

a. Equity Markets

b. Debt Capital Markets

c. Derivative Markets

B.4 Structured Finance

B.5 Mergers & Aquisitions

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Table of Contents

A.1 The Investment Banking Environment

a. Universal Banking vs. Specialized Banking

b. Commercial Banking vs. Investment Banking

c. Definition of Investment Banking

d. A Basic Depiction of Investment Banking

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.1 The Investment Banking Environment

a) Universal Banking vs. Specialised Banking

The Universal Banking System

• Predominately present in Continental Europe

• In general, banks are allowed to offer all kinds of products to their

customers

• Banks offer a broad range of financial services e.g. deposit taking, real

estate and other forms of lending, foreign exchange (FX) trading, securities

trading, underwriting, portfolio management etc.

• Banks offer both financial and consultancy services; the principle of onebank-for-everything

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A.1 The Investment Banking Environment

The Universal Banking System

Advantages for the bank

• Detailed information about the

clients economic and business

activities

• Banking conditions are

tailored to the client

• Cross selling potential

• Competitive advantage due to

information efficiency about

clients

A.1 The Investment Banking Environment

The Specialised Banking System

Advantages for the client

• Individual customer service

• Clients can be assured that

the bank is very diplomatic

considering the disclosure of

the clients private information

• Implicit agreement between

bank and client

=> Banks tend to support

clients in distressed economic

situations

• Predominately present in the Anglo-Saxon countries and Japan

• Separation of commercial and investment banking

• Investment banking

- in the USA via investment banks (emerged by government regulations)

- in UK via merchant banks (emerged on a historical basis)

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A.1 The Investment Banking Environment

The Specialised Banking System

Emergence in the USA I

• 1933: Glass-Steagall-Act, Government regulation to separate commercial

and investment banking

• to moderate speculation

• to stabilize the financial system

and

• to prevent a banks’ conflict of interests

• The act was mainly triggered by the crash of the stock market and great

depression of the late 1920s

A.1 The Investment Banking Environment

The Specialised Banking System

Emergence in the USA I

• Regulators were afraid of

• the combination of a small group of banks

• high volatility at the stock markets

and

• the overall macroeconomic development

However:

• The development of the financial industry in the US, globalisation and

vertical integration lead to a slow but continuous maceration of the Glass-

Steagall-Rules

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A.1 The Investment Banking Environment

The Specialised Banking System

Emergence in the USA II

• After a continuous reduction of regulative restrictions the specialised

banking era ended 1999 with the Gramm-Leach-Bliley Act

• The act allowed US banks to offer the full range of financial products as

for instance credits, underwritings, structured finance products, deposit

taking, credit business

• It enabled financial institutions to do insurance broking, advisory

business, investment banking all in one company

• After Gramm-Leach-Bliley large financial holding companies emerged as

for instance JPMorgan Chase etc.

A.1 The Investment Banking Environment

The Specialised Banking System

Emergence in the United Kingdom

• Banks in UK developed to specialised institutions over the last two

centuries

• e.g. Barings and Schroders started to finance international merchant trade

in the 18th century and provided credit supply to European countries

• Their main activities at that time included corporate finance, issuance of

securities (bonds, stock, etc.) and principal investment projects

• The merchant banks’ capital structure was mainly relatively short in

equity capital which meant that they needed innovative ways to finance

their projects

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A.1 The Investment Banking Environment

The Specialised Banking System

Concluding remarks

Investment banking arose because of

• a declining attractiveness of commercial banking (smaller margins, larger

competition, etc.)

• a growing specialisation into some particular field of universal banks

• increasing legal regulations, which forced a separation of commercial and

investment banking

A.1 The Investment Banking Environment

b) Commercial Banking vs Investment Banking

Investors Banks Borrower

Investors:

Instrument:

Function:

Market Risk:

Commercial

Banking

Investment

Banking

Depositors Institutional Investors

Credits

Supervisor

Decision Maker

Securities

Analyst

Consultant

Taken by Bank Passed to market

Stability Change

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A.1 The Investment Banking Environment

b) Commercial Banking vs Investment Banking

The Downfall of Investment Banking

Zur Anzeige wird der QuickTime

Dekompressor „“

benötigt.

Bankrupt

A.1 The Investment Banking Environment

b) Commercial Banking vs Investment Banking

The Downfall of Investment Banking

• September 22nd ‘08: Goldman Sachs and Morgan Stanley resign from their

special status as investment banks

• Both banks received large short-term credits from the FED

• Stricter restrictions and controls from the SEC

• New sources for income via deposites

• Mitsubishi UFJ Financial Group plans to buy 20% of Morgan Stanley

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A.1 The Investment Banking Environment

c) Definition of Investment Banking

• It seems difficult to find an appropriate definition of investment

banking since investment banking covers a large variety of financial

services

• One of the best existing definitions of investment banking:

“Investment Banking is what Investment Banks do”

• Since this is quite a broad interpretation, the best way to define

investment banking is according to its areas of business

A.1 The Investment Banking Environment

c) Definition of Investment Banking

Investment Banking may include:

• International issuance of securities

• Special financial services (e.g. real estate, insurances, marketing

for private customers, etc.)

• Trading activity in various markets (fixed income, commodity and

proprietary trading, principal investment etc.)

• Activities in capital markets (e.g. corporate finance, M&A, IPO’s,

fund management, etc.)

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A.1 The Investment Banking Environment

d) The Basic Description of Investment Banking

A.1 The Investment Banking Environment

Literature:

• Liaw, K. Thomas (2006): The Business of Investment Banking,

Chapters 1 and 2.

• Hockmann, Heinz-Josef / Thießen, Friedrich (2007): Investment

Banking, Chapters 1.1 and 1.5.

• Achleitner, Ann-Kristin (2002): Handbuch Investment Banking, pp.

3-54.

• Rich, G. / Walter, C. (1993): The Future of Universal Banking, CATO

Journal.

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Investment Banking and

Capital Markets

Field-trip to the Stuttgart Stock Exchange

Possible Dates:

Friday 16th of January 2009, 14:00 - 15:30

Friday 23th of January 2009, 14:00 - 15:30

If you would like to join:

Register by clearly stating your NAME and EMAIL ADDRESS in the lists

Table of Contents

A.2 Basic Principles of Capital Markets

a. Modern Portfolio Theory

b. The Capital Asset Pricing Model

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory (MPT)

Introduction

• MPT bases on Harry M. Markowitz’s article “Portfolio

Selection”, first published in the Journal of Finance (1952)

• MPT states that it is not enough to consider the risk and return

pattern of a single asset

• Basic idea of portfolio selection: “Don’t put all eggs in one

basket”

• Reduction of idiosyncratic risk (unsystematic risk) via

diversification

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory (MPT)

Ideas of portfolio selection:

• Splitting an investment efficiently on various assets

• Diversification of a portfolio depends on the volatility of each

single asset but ALSO on the correlation of each assets’ risk and

return structure with other assets

• If single asset returns are not 100% positively correlated, risk

reduction in the portfolio is possible via diversification

• Risk reduction is possible via a simple split into equal units of

the investment into many assets (naïve diversification)

• Assets have to be split within the portfolio according to the

most efficient setting of risk and return (efficient frontier,

portfolio selection)

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory (MPT)

Model assumptions

• One period model

• Risk aversion of investors (concave risk utility function)

• Investors maximize their utility

• Returns are normally distributed (Gaussian distribution)

• Homogenous expectations of investors

• No risk free assets (preliminary)

• No transaction costs, no arbitrage

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) MPT

How are returns modelled in MPT?

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A.2 Basic Principles of Capital Markets

a) MPT

How are returns modelled in MPT?

A.2 Basic Principles of Capital Markets

a) MPT

Two different ways to calculate returns

• Discrete returns:

r

d,

t

K t − K

=

K

t−1

t−1

D

+

K

t−1

• Continuous returns:

t

K t + D

=

K

t−1

capital return + dividend return = general return

⎛ K

⎜

+ D ⎞

⎟

⎝ ⎠

t

−1

( K t + D t ) − ln K t 1

t t

rs, t =

ln⎜

ln

−

K ⎟ =

t−1

r d,t = discrete return in period t

K t = capital at the end of the period

K t-1 = capital at the beginning of the period

D t = accrued dividend per period

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) MPT

Risk and Return

• In MPT all assets are classified according to two criteria:

• Expected return E[r j], also know as µ

AND

• Expected variance of the return E[var(r j)],

also known as σ 2 , respective the standard deviation σ

• Markowitz defines the standard deviation (SD) of an expected

return as RISK

• This definition of risk is also know as volatility

• The return of an asset which bears a 20% SD is obviously more

risky than the return of another asset with 10% of SD

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) MPT

Risk, Return and an investor’s utility function

• An investor’s utility function of a portfolio depends solely on the

two variables µ p and σ p

• Remember: One of the assumptions of the model is risk aversion of

investors

• For this reason, it seems only logical that the return µ p is a desired

feature, while risk σ p is an undesired feature

• We are looking for an utility function which increases with the

desired feature and decreases with the undesired feature

U(RP ) = f (µ P ,σ P ) = µ P − a

2 •σ 2

P

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) MPT

Risk, Return and an Investor’s Utility Function

• This utility function points out the advantages of small risk and

the disadvantages of large risk

• a represents the risk-aversion-factor of the investor

• If a is small, the utility is larger for the investor, given a certain

return and risk of the portfolio

• This means that the investor does not attach too much

importance to risk

• If a becomes larger the investor cares more about risk and the

utility decreases

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Expected return, standard deviation, and variance

• p k = Probability of condition k to happen

• r k = Return of the asset in condition k

Expected return of an asset:

Variance of the assets return:

SD (volatility) of the assets return:

E(

r ) = µ =

i

K

∑

k =

p ⋅ r

Var(r) = σ 2 = p k

σ = σ 2

1

k

k

K

∑ (rk − µ)

k=1

2

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Covariance and Correlation

Covariance and correlation describe direction and strength of the

relation between the returns of two assets i and j

Covariance between the returns of assets i and j:

K

∑

cov(r i ,r j ) = σ ij = p k (r i,k − µ i,k )(r j,k − µ j,k )

k=1

Correlation between the returns of assets i and j:

Advantage of the correlation: Standardization between

−1≤ ρi, j ≤1

ρ ij = σ ij

σ iσ j

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Expected value of the portfolio return

• There are two ways to calculate the portfolio return

• Via the condition based portfolio returns:

K

∑

k =

with

E(

rP

) = µ P = pk

⋅ rP,

k rP,

k = ∑

1

i=

• Via the expected return of the assets:

E ( r ) = = x ⋅µ

P

N

µ P ∑

i=

1

i

i

with

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

N

∑

i=1

N

1

x i =1

x ⋅ r

i

i,

k

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Variance of the portfolio return

• Two ways to calculate the portfolio variance

• Via the condition based portfolio returns:

Var(r P ) = σ P

2

= pk (rP,k − µ P ) 2

K

∑

k=1

• Via the variance/covariance matrix of the asset returns:

Var(r P ) = σ P

N N

2

= xix jσ i, j

∑∑

i=1

j=1

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Naïve Diversification

• Suppose: Returns are normally distributed and stochastically independent

return

DiPo

2

• Diversification is possible if ρ < 1

3

5

4

1

risk

Various assets

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Naïve Diversification with uncorrelated returns

1

• Suppose: x and

i = σ ij = 0

N

•The variance is calculated according to the following formula:

σ P

N N

∑∑ N

N N

N

2 2 2 2

∑ σ i + ∑∑

xix jσ ij = ∑x

iσ

i

i=1 j=1

i=1

i=1 j=1

i ≠j

i=1

2 = xix jσ ij = x i

=

N

∑

i=1

⎛ 1

⎜

⎝ N

⎞

⎠

2

⎟ σ i

2 1 ⎡

= ⎢

N⎣

i=1

⎤

⎥ =

N ⎦

1

N

N 2

σ i

∑

•If more and more assets are added to the portfolio the variance becomes:

lim

N→∞ σ P

2 ⎛ 1

= lim⎜

N→∞⎝

N

2

⋅σ i

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

2

σ i

⎞

⎟ = 0

⎠

Naïve Diversification with positively correlated returns

• Usually asset returns (stock returns) are positively correlated, id est

• Calculating the variance under the premise of positive correlation brings:

N

2

σ P = ∑

i=1

⎛ 1

⎜

⎝ N

⎞

⎠

= 1 2

σ i +

N N −1

2

2

⎟ σ i +

⎛ 1

⎜

⎝ N

⎞

⎟

⎠

1 ⎛

⎜

⎝ N

⎞

N N

∑ ∑ ⎟ σ ij =

⎠

1 ⎛

⎜

N⎝

i=1 j=1

j≠i

1

σ ij =

N

• For N → ∞ the portfolio variance becomes

lim

N→∞ σ P

2 ( −σij ) + σ ij

N σ i

2 ⎡ 1

= lim

N→∞⎣

⎢

N

i=1

⎞

⎟ +

N ⎠

N 2

σ i

∑

N −1

N

N

N

∑∑

i=1

2

( σ i −σ ij)

+ σ ij = σ ij

⎤

⎦

⎥

j=1

j≠i

σ ij

N ⋅ (N −1)

σ ij > 0

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Comparing the effects of naïve diversification depending on the

assets’ correlations

σ P 2

2

σ i

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The most simple portfolio: Two assets

• Expected return of the portfolio:

µ P = x µ A+(1-x) µ B

• Variance of the portfolio:

σ 2 P = Var[R P]=x 2 σ 2 A+(1-x) 2 σ 2 B+2 x (1-x) σ A σ B ρ A,B

N

σ ij

σ ij

> 0

= 0

• The standard derivation of the portfolio is the square root of the

portfolio variance

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The most simple portfolio: Two assets

• The two asset portfolio depends on three variables only:

• The single risk of the assets (variance)

• The loading/weight of the two assets in the portfolio

• The correlation of the assets’ returns

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return of a two asset portfolio (example)

• Suppose there are only two assets in the capital market

• Asset 1

• Asset 2

• Suppose there are only three possible conditions to happen

• k 1 with probability p 1

• k 2 with probability p 2

• k 3 with probability p 3

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return of a two asset portfolio (example)

• For both assets the following information is given:

A 1

A 2

P 0j

100

200

Conditional

returns

Probabilities

k 1

p 1 : 0,3

130

190

k 2

p 2 : 0,5

90

210

k 3

p 3 : 0,2

70

240

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return of a two asset portfolio (example)

• Conditional returns: where Xij = return of

Rij =

asset j in condition i

X ij − P0 j

= X ij

−1

R 11 = (130-100)/100 = 30%

R 12 = -10%; R 13 = 40%; R 21 = -5%; R 22 = 5%; R 23 = 20%

• Expected returns are

P 0 j

E[R 1] = 0,3*0,3 + 0,5*(-0,1) + 0,2*0,4 = 0,12

E[R 2] = 0,05

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

P 0 j

i

R j = E[

R j ] = ∑ Rij

⋅pi

i=

1

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return of a two asset portfolio (example)

• Variances:

Var(

X ) = E

Var(

X ) = E

Var(

X ) =

2

[ ( X − E[

X ] ) ]

2 [ X ] − E[

X ]

• Var(R 1) = σ 1 2 = 0,**04**96 => σ1 = 0,2227

• Var(R 2) = σ 2 2 = 0,0075 => σ2 = 0,0866

i

∑

i=1

2

( R − R ) ⋅ q

ij

j

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return of a two asset portfolio (example)

µ

12%

5%

2

0,0866

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

2

i

1

0,2227

σ

100% of asset 1

100% of asset 2

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram of a two asset portfolio (example)

• But what happens if we change the portfolio loading?

• The following slides show the results of different portfolio loadings

with different correlations (e.g. 0,1; -0,5; -1)

• We will see that the correlation and the loading are important

factors

• The following portfolio loadings where used:

100/0; 90/10; 80/20; 70/30; 60/40; 50/50

0/100; 10/90...

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram with different correlations (example)

0,14

0,12

0,1

0,08

0,06

0,**04**

0,02

Risk-Return Diagram CC +0,1

0

0 0,05 0,1 0,15 0,2 0,25

Risk

Possible Portfolios

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram with different correlations (example)

0,14

0,12

0,1

0,08

0,06

0,**04**

0,02

Risk-Return Diagram Correlation Coef. -0,5

0

0 0,05 0,1 0,15 0,2 0,25

Risk

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram with different correlations (example)

0,14

0,12

0,1

0,08

0,06

0,**04**

0,02

Risk-Return Diagram CC -1

0

0 0,05 0,1 0,15 0,2 0,25

Risk

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

SD[P]

Possible Portfolios

53

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram with different correlations (example)

• The correlation between asset returns has a very important

influence on the portfolio risk-return pattern

• A perfect correlation of the assets return’s prohibits a risk

reduction

• A perfect negative correlation may lead to a complete risk

elimination depending on the portfolio loading

• In reality the correlation between two asset returns is usually

somewhere between -1 < ρ ij < 1

• Stock returns are usually positively correlated

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Risk and return diagram with different correlations

• Imperfect correlation may lead to a partial risk reduction,

depending the portfolio loading

µ

γ

c

b

A

a) ρ AB = 1

b) -1 < ρ AB < 1

c) ρ AB = -1

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

a

B

σ

55

56

28

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Efficient portfolios and efficient frontier

• Now that we created several portfolios we are searching for the

efficient portfolios

Definition of an efficient portfolio

• A portfolio P i is efficient, if there is no other portfolio P j which

dominates P i

• A portfolio P i with return R i dominates another portfolio P j with

return R j if µ i > µ j and σ i ≤ σ j

• Alternatively, µ i ≥ µ j and σ i < σ j

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Efficient portfolios and efficient frontier

0,14

0,12

0,1

0,08

0,06

0,**04**

0,02

Risk-Return Diagram Correlation Coef. -0,5

Efficient frontier

Unefficient portfolios

Minimum Variance Portfolio (MVP)

0

0 0,05 0,1 0,15 0,2 0,25

Risk

Efficient portfolios

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

SD[P]

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Efficient portfolios and efficient frontier

• All portfolios on the efficient frontier (also called efficient set) are

portfolios which are NOT dominated by any other portfolio

• Hence, they are desired/efficient in the sense of the model

• It depends on the individual risk aversion and hence the individual

risk utility function of each investor to choose his/her unique

portfolio loading

• An investor with the highest risk aversion will always choose the

Minimum-Variance-Portfolio (MVP)

• The MVP is characterized by a minimum deviation within the asset

returns and hence according to the definition it bears minimum risk

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Minimum Variance Portfolio

µ P = x µ A + (1 - x) µ B

σ 2 P = Var[R P] = x 2 σ 2 A + (1 - x) 2 σ 2 B + 2 x (1 - x) σ A σ B ρ A,B

δσ

δx

2

P 2

2 2

= ⋅ x ⋅σ

A + 2⋅

x ⋅σ

B − 2⋅σ

B + 2⋅σ

A ⋅σ

B ⋅ ρ A,

B − 4⋅

x

2 ⋅σ

⋅σ

⋅ ρ

• Consequently (if σ A ≠ σ B · ρ A,B)

xMVP

σ −σ

A ⋅σ

B ⋅ ρ A,

B

=

σ + σ − 2⋅σ ⋅σ

⋅ ρ

2

A

2

B

2

B

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A

B

A,

B

A

B

A,

B

59

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

Minimum Variance Portfolio

Assignment:

A portfolio manager may create a portfolio with the following two

assets. Asset A is an Asian-Stock-Index product and asset B is a

BRIC bond-index product. The expected return of asset A is 10%

(asset B 7% ) and the volatility of asset A is 20% (asset B 10%).

According to historical data, the correlation is expected to be 0,4.

How is the loading of the MVP?

What is the expected risk and return of the MVP?

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The Ideal Portfolio

• To select the ideal portfolio we need to know the individual risk-returnutility-function

of an investor

µ

Individual investors utility functions

Efficient set

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

σ

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The Ideal Portfolio

• Two asset case:

µ P (x1,x 2,...,x n) − a

2 ⋅σ 2

P (x1,x 2,...,x) ⇒ Max!

side condition

f (x) = x ⋅ µ A + (1− x)⋅ µ B

N

∑ xn =1

n=1

− a

2 ⋅ (x 2 2 2 2

⋅σ A + (1− x) ⋅σ B + 2⋅ x ⋅ (1− x)⋅σ A ⋅σ B ⋅ ρA,B )

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The Ideal Portfolio

• To discover the maximum we have to differentiate f(x)

• For f’(x) = 0 :

′

f (x) = µ A − µ B

2 2

−a(x σ A + (1− x)σ B + (1− 2x)σ Aσ Bρ A,B )

x =

µ A − µ B

a

σ A

2 + σ B

• f’’(x) < 0 , if σ A ≠ σ B and ρ A,B < 1

2

+ (σ B −σ Aσ Bρ A,B )

2 − 2σ Aσ Bρ A,B

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

The Ideal Portfolio - Assignments

A portfolio manager needs to create a portfolio for a new client

who has a risk aversion of a = 4. The manager can choose

between two assets A and B. The following information is being

provided to the manager: µ A = 10%, µ B = 5%, σ A = 20%,

σ B = 10%, ρ A,B = 0,3

Which is the ideal portfolio loading for the client?

Another client of the manager has already a portfolio set up.

The portfolio loading is 68,7% stocks and 31,3% bonds.

Additional portfolio information: µ A = 10%, µ B = 5%, σ A = 20%,

σ B = 6%, ρ A,B = 0,3

What is the client’s risk-aversion factor that the manager needs

to know to offer the client an additional product?

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

MPT - Risky and Risk-Free Assets

• So far, we were just considering risky assets

• Capital markets do not consist of only risky assets, but also of

risk-free assets, e.g. sovereign bonds, ...

• Now, we are also considering risk-free assets in the model

• A risk free asset can be described as:

E[r f ] = r f Var[r f ] = 0 cov[r f ] = 0

• A new, combined portfolio Q is created using both risky and riskfree

assets

• Return of the new portfolio Q is

r Q(w) = w r P + (1-w) r f

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

MPT - Risky and Risk-Free Assets

• Expected return of the combined portfolio:

µ Q = w µ P + (1-w) r f

• Expected variance of the combined portfolio:

σ 2 Q = w 2 σ 2 P + (1-w) 2 σ 2 0 + 2 w (1-w) σ P σ 0 ρ P,0

• We receive:

µ Q(w) = r f + w (µ P - r f)

σ Q(w) = w σ P => w = σ Q / σ P

• Introducing the excess return:

p = (µ P - R 0) / σ P

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

MPT - Risky and Risk-Free Assets

• Brings us to: µ Q = r f + p σ Q

µ

µ P1

µ T

µ MVP

µ P2

rf P 2

σ MVP

T

σ T

P 1

CML

Tobin Separation Theorem:

The structure of the ideal

portfolio is independent from

risky-asset portfolio

possibilities, if all investors have

homogenous expectations

σ

67

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A.2 Basic Principles of Capital Markets

a) Modern Portfolio Theory

MPT - Risky and Risk-Free Assets - Assignment

An investors’ utility function is given by

a 2

U ( RQ

) = f ( µ Q,

σ Q)

= µ Q(

w)

− ⋅σ

Q(

w)

2

What is the loading which maximizes the investors utility?

The market portfolio (tangential portfolio) has µ M = 10% and σ M =

20%. The risk free rate is given by an investment in a AAA

government bond with r f = 5%. Calculate the ideal asset

allocation for client A who has a risk aversion of a=1, client B

(a=1,25), client C (a=3), and client D (a=4)

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

A.2 The Investment Banking Environment

Literature:

• Markowitz, Harry M. (1952): Portfolio Selection, Journal of Finance,

Vol. 7, No. 1, 77-91, March 1952.

• Breuer, Wolfgang / Gürtler, Marc / Schuhmacher, Frank (20**04**):

Portfoliomanagement I, Chapters 1 & 4.

Table of Contents ⎜ Topic A.1 ⎜Topic A.2 ⎜ Topic B.1 ⎜ Topic B.2 ⎜ Topic B.3 ⎜ Topic B.4 ⎜ Topic B.5

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