Copyright © 2014 Pearson Education, Inc. All rights reserved. Copyright © 2014 Pearson Education, Inc. All rights reserved. 1
Teaching Staff for Summer Term 2014
Prof. Dr. Markus Glaser and Dipl. Finanzök. math. Peter Schmidt
Email: [email protected]
Time & Location:
Tuesday: 4-6 pm
Room: HGB, M 118
Prerequisite: Investition und Finanzierung
The first lecture will start on April 8th, 2014
Credits: 3 ECTS (ABWL)
Exam (1h): Do 17.07.2014, 15.30 - 16.30
Risk Management
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Timeline:
Date Session Content
08.04.2014 1 Introduction + Recap
15.04.2014 2 Financial Options 1
29.04.2014 3 Financial Options 2
06.05.2014 4 Option Valuation 1
13.05.2014 5 Option Valuation 2
20.05.2014 6 Tutorial 1
27.05.2014 7 Tutorial 2
03.06.2014 8 Insurance & Hedging 1
17.06.2014 9 Insurance & Hedging 2
24.06.2014 10 Tutorial 3
17.07.2014 Exam
2 Risk Management
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Lectures: Overview
Recap
1. Capital Markets and the Pricing of Risk
2. Optimal Portfolio Choice and the Capital Asset
Pricing Model
Financial Options
1. Option Basics
2. Option Payoffs at Expiration
3. Put-Call Parity
4. Factors Affecting Option Prices
5. Exercising Options Early
6. Options and Corporate Finance
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Option Valuation
1. The Binomial Option Pricing Model
2. Risk-Neutral Probabilities
3. The Black-Scholes Option Pricing Model
Insurance & Hedging
1. Insurance
2. Commodity Price Risk
3. Exchange Rate Risk
4. Interest Rate Risk
Readings:
B & D: Chapters 10, 11, 20, 21, & 30
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Risk Management
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Common Measures of Risk and Return
• Probability Distributions
• When an investment is risky, there are different returns it may earn. Each possible return has some likelihood of occurring. This information is summarized with a probability distribution, which assigns a probability, PR , that each possible return, R , will occur.
– Assume BFI stock currently trades for $100 per share. In one year, there is a 25% chance the share price will be $140, a 50% chance it will be $110, and a 25% chance it will be $80.
5 Risk Management
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Expected (Mean) Return
Calculated as a weighted average of the possible returns, where
the weights correspond to the probabilities.
Expected Return RRE R P R
25%( 0.20) 50%(0.10) 25%(0.40) 10% BFIE R
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Probability Distribution of Returns for BFI
7 Risk Management
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Variance
• The expected squared deviation from the mean
Standard Deviation
• The square root of the variance
• Both are measures of the risk of a probability distribution
( ) ( ) SD R Var R
2 2
( ) RR
Var R E R E R P R E R
8 Risk Management
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• For BFI, the variance and standard deviation are:
• In finance, the standard deviation of a return is also referred to as its volatility. The standard deviation is easier to interpret because it is in the same units as the returns themselves.
( ) ( ) 0.045 21.2% SD R Var R
2 2
2
25% ( 0.20 0.10) 50% (0.10 0.10)
25% (0.40 0.10) 0.045
BFIVar R
9 Risk Management
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The Expected Return of a Portfolio
Portfolio Weights
• The fraction of the total investment in the portfolio held in each
individual investment in the portfolio
– The portfolio weights must add up to 1.00 or 100%.
• Then, the return on the portfolio, Rp , is the weighted average of the
returns on the investments in the portfolio, where the weights
correspond to portfolio weights.
10
i
Value of investment
Total value of portfolio
ix
1 1 2 2 P n n i iiR x R x R x R x R
Risk Management
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The Expected Return of a Portfolio (cont'd)
• The expected return of a portfolio is the weighted average of the
expected returns of the investments within it
11
P i i i i i ii i iE R E x R E x R x E R
Risk Management
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Determining Covariance and Correlation
To find the risk of a portfolio, one must know the degree to which the
stocks’ returns move together.
Covariance
• The expected product of the deviations of two returns from their means
• Covariance between Returns Ri and Rj
• Estimate of the covariance from historical data
– If the covariance is positive, the two returns tend to move together.
– If the covariance is negative, the two returns tend to move in opposite directions.
12
( , ) [( [ ]) ( [ ])] i j i i j jCov R R E R E R R E R
, ,
1( , ) ( ) ( )
1
i j i t i j t jt
Cov R R R R R RT
Risk Management
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Determining Covariance and Correlation (cont'd)
Correlation
• A measure of the common risk shared by stocks that does not depend
on their volatility
• The correlation between two stocks will always be between –1 and +1.
13
( , )( , )
( ) ( )
i j
i j
i j
Cov R RCorr R R
SD R SD R
Risk Management
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Correlation
14 Risk Management
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Computing a Portfolio’s Variance and Volatility
• For a two-stock portfolio:
• The variance of a two-stock portfolio
15
1 1 2 2 1 1 2 2
1 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2
( ) ( , )
( , )
( , ) ( , ) ( , ) ( , )
P P PVar R Cov R R
Cov x R x R x R x R
x x Cov R R x x Cov R R x x Cov R R x x Cov R R
2 2
1 1 2 2 1 2 1 2( ) ( ) ( ) 2 ( , ) PVar R x Var R x Var R x x Cov R R
Risk Management
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Risk Versus Return: Choosing an Efficient Portfolio
Efficient Portfolios with Two Stocks
• Consider a portfolio of Intel and Coca-Cola
• Assume these two stocks are uncorrelated
16
Stock Expected Return Volatility
Intel 26% 50%
Coca-Cola 6% 25%
Risk Management
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Risk Versus Return: Choosing an Efficient Portfolio (cont'd)
Efficient Portfolios with Two Stocks (e.g. Intel and Coca-Cola)
17
Expected Returns and Volatility for Different Portfolios of Two Stocks
Risk Management
Calculation:
𝐸 𝑅40,60 = 0.40 ∗ 0.26 + 0.60 ∗ 0.06 = 0.14 = 14%.
𝑉𝑎𝑟 𝑅40,60 = 0.40² ∗ 0.502 + 0.60² ∗ 0.252 + 2 ∗ 0.40 ∗ 0.60 ∗ 0 ∗ 0.50 ∗ 0.25 = 0.0625 = 6.25%.
𝑆𝐴 𝑅40,60 = 𝑉𝑎𝑟 𝑅40,60 = 0.25 = 25%.
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Volatility Versus Expected Return for Portfolios of Intel and
Coca-Cola Stock
18 Risk Management
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Risk Versus Return: Choosing an Efficient Portfolio (cont'd)
Efficient Portfolios with Two Stocks
• Consider investing 100% in Coca-Cola stock. As shown in on the
previous slide, other portfolios—such as the portfolio with 20% in
Intel stock and 80% in Coca-Cola stock—make the investor better
off in two ways: It has a higher expected return, and it has lower
volatility. As a result, investing solely in Coca-Cola stock is
inefficient.
19 Risk Management
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Risk Versus Return: Choosing an Efficient Portfolio (cont'd)
Efficient Portfolios with Two Stocks
• Identifying Inefficient Portfolios
– In an inefficient portfolio, it is possible to find another portfolio
that is better in terms of both expected return and volatility.
• Identifying Efficient Portfolios
– In an efficient portfolio, there is no way to reduce the volatility of
the portfolio without lowering its expected return.
20 Risk Management
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Investing in Risk-Free Securities
Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the portfolio, while leaving the remaining fraction in risk-free Treasury bills.
• The expected return would be:
21
[ ] (1 ) [ ]
( [ ] )
xP f P
f P f
E R x r xE R
r x E R r
risk premium
Risk Management
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Investing in Risk-Free Securities (cont'd)
Risk Management
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Identifying the Tangent Portfolio
Sharpe Ratio
• Measures the ratio of reward-to-volatility provided by a portfolio
• The portfolio with the highest Sharpe ratio is the portfolio where the
line with the risk-free investment is tangent to the efficient frontier of
risky investments. The portfolio that generates this tangent line is
known as the tangent portfolio.
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[ ] Portfolio Excess ReturnSharpe Ratio
Portfolio Volatility ( )
P f
P
E R r
SD R
Risk Management
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The Tangent or Efficient Portfolio
24 Risk Management
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The Capital Asset Pricing Model
• The Capital Asset Pricing Model (CAPM) allows us to identify the
efficient portfolio of risky assets without having any knowledge of the
expected return of each security.
• Instead, the CAPM uses the optimal choices investors make to identify
the efficient portfolio as the market portfolio, the portfolio of all stocks
and securities in the market.
25 Risk Management
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The CAPM Assumptions (3 Main Assumptions)
• Assumption 1
– Investors can buy and sell all securities at competitive market prices (without
incurring taxes or transactions costs) and can borrow and lend at the risk-free
interest rate.
• Assumption 2
– Investors hold only efficient portfolios of traded securities—portfolios that yield the
maximum expected return for a given level of volatility.
• Assumption 3
– Investors have homogeneous expectations regarding the volatilities,
correlations, and expected returns of securities.
– Homogeneous Expectations
- All investors have the same estimates concerning future investments and
returns.
26 Risk Management
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Optimal Investing: The Capital Market Line
• When the CAPM assumptions hold, an optimal portfolio is a
combination of the risk-free investment and the market portfolio.
• When the tangent line goes through the market portfolio, it is called the
capital market line (CML).
27 Risk Management
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The Capital Market Line
28 Risk Management
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Market Risk and Beta
• Given an efficient market portfolio, the expected return of an investment is:
• The beta is defined as:
29
Risk premium for security
[ ] ( [ ] ) Mkt
i i f i Mkt f
i
E R r r E R r
Volatility of that is common with the market
i
( ) ( , ) ( , )
( ) ( )
i
Mkt i i Mkt i Mkti
Mkt Mkt
SD R Corr R R Cov R R
SD R Var R
Risk Management
(CAPM)
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The Security Market Line
• There is a linear relationship between a stock’s beta and its expected
return (See figure on next slide). The security market line (SML) is
graphed as the line through the risk-free investment and the market.
30 Risk Management
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The Capital Market Line and the Security Market Line
31 Risk Management
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The Capital Market Line and the Security Market Line, Panel (a)
32
(a) The CML
depicts portfolios
combining the risk-
free investment
and the efficient
portfolio, and
shows the highest
expected return
that we can attain
for each level of
volatility. According
to the CAPM, the
market portfolio is
on the CML and all
other stocks and
portfolios contain
diversifiable risk
and lie to the right
of the CML.
Risk Management
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The Capital Market Line and the Security Market Line, Panel (b)
33
(b) The SML shows
the expected return
for each security as
a function of its
beta with the
market. According
to the CAPM, the
market portfolio is
efficient, so all
stocks and
portfolios should lie
on the SML.
Risk Management
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The Security Market Line (cont'd)
The expected return of a portfolio
Beta of a portfolio
• The beta of a portfolio is the weighted average beta of the securities
in the portfolio.
34
, ( , ) ( , )
( ) ( ) ( )
i i MktiP Mkt i MktP i i ii i
Mkt Mkt Mkt
Cov x R RCov R R Cov R Rx x
Var R Var R Var R
)][(][ fMktPfP rRErRE
Risk Management
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Example
• Suppose the stock of the 3M Company (MMM) has a beta of 0.69
and the beta of Hewlett-Packard Co. (HPQ) stock is 1.77.
• Assume the risk-free interest rate is 5% and the expected return of
the market portfolio is 12%.
• What is the expected return of a portfolio of 40% of 3M stock
and 60% Hewlett-Packard stock, according to the CAPM?
Solution
35 Risk Management
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Summary of the Capital Asset Pricing Model
• The market portfolio is the efficient portfolio.
• The risk premium for any security i is proportional to its beta with the
market.
36
)][(][ fMktifi rRErRE
Risk Management
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