Download 1. How Capital Markets Work

International Financial Markets
1. How Capital Markets Work
Lecture Notes:
E-Mail:
Colloquium:
Prof. Dr. Rainer Maurer
www.rainer-maurer.de
[email protected]
Friday 17.15 - 18.45 (room W1.4.03)
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1. How Capital Markets Work
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.1.3. Investor and Saver Surplus
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
2. Questions for Review
Literature:1)
◆ Chapter 4, 25, Mankiw, N.G. (2001): Principles of Economics, Harcourt Coll. Publ., Orlando.
◆ Chapter 7, Mankiw, N.G. (2002): Macroeconomics, Worth Publishers, New York.
1) The
recommended literature typically includes more content than necessary for an understanding of this chapter. Relevant
for the examination is the content of this chapter as presented in the lectures.
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.1.1. Why People Save
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.1.1. Why People Save
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.1.1. Why People Save
➤ Why do people save?
■ Making savings means
◆ “consumption today” is postponed in favor of
◆ “consumption in the future”
■ Why are people willing to give up
“consumption today” in favor of
“consumption in the future”?
■ Because they receive interest payments for their savings.
■ The standard assumption is therefore that the willingness to save
depends positively on the interest rate:
Prof. Dr. Rainer Maurer
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The Slope of the Savings Curve
% 10%
Why do people
save more, when
they receive higher
interest payments?
8%
6%
4%
Savings = S(i)
Higher interest payments allow
for “higher consumption in the
future”. This compensates for
the “lower consumption today”.
2%
0%
0
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1. How Capital Markets Work
1.1.1. Why People Save
➤ How does this affect consumption of households?
➤ The relationship between “savings today” and “consumption
today” is inverse.
➤ The budget constraint of a household shows this. If we neglect
the necessity to pay taxes, the simplest form a budget constraint
is given by the equation:
Income = Savings + Consumption
Y
=
S
+
C
<=>
C
=
Y
–
S
<=>
C(i ) = [ Y
–
S(i ) ]
Prof. Dr. Rainer Maurer
+
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The Slope of the Savings Curve
% 10%
8%
As a consequence,
people consume
less, if the interest
rate is high.
6%
4%
2%
Consumption = C(i)
0%
0
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1. How Capital Markets Work
1.1.1. Why People Save
➤ How does an increase of permanent income affect the savings
function?
Y
=
S
+
C
➤ It must increase savings and/or consumption.
➤ Most likely is that it increases both savings and consumption at
the same time, because a permanent increase of income means
that higher income will also be available in future periods. So
people have no reason to postpone current consumption into the
future.
Prof. Dr. Rainer Maurer
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The Slope of the Savings Curve
% 10%
How does an increase
of permanent income
“y” change the
willingness to save?
8%
Savings = S(i, y1)
Savings = S(i, y2)
6%
4%
If the permanent income
of households y2 > y1
grows, households will
typically save more!
2%
0%
0
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The Slope of the Savings Curve
% 10%
Savings = S(i, y1)
8%
Savings = S(i, y2)
6%
Therefore household
permanent income “y” is
a shift parameter of the
savings function!
4%
2%
0%
0
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The Slope of the Savings Curve
% 10%
Savings = S(i, y2)
Savings = S(i, y1)
8%
6%
4%
If the permanent income
of households y2 < y1
decreases, households
will typically save less!
2%
0%
0
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1. How Capital Markets Work
1.1.1. Why People Save
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
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1. How Capital Markets Work
1.1.2. Why People Invest
➤ Why do people invest?
■ Investment means
◆ to spend money for “economic activities today”, which are
assumed to yield a “return in the future”
■ Investment projects can be ranked according to their expected
return.
■ This yields the following curve:
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The Slope of the Investment Curve
% 10% Investment volume of the first project
9%
8%
Expected return
7%
Available investment
projects depending on
their expected return
and investment volume
6%
5%
4%
3%
2%
1%
0%
0
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The Slope of the Investment Curve
% 10%
9%
Interest rate: 8%
8%
If the market interest rate
is 8%, only the first
investment project is
profitable! All other
investment projects are
not undertaken!
7%
6%
5%
4%
3%
2%
1%
0%
0
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The Slope of the Investment Curve
% 10%
9%
If the market interest rate
is 2%, only the first five
investment projects are
profitable!
8%
7%
6%
5%
4%
3%
Interest rate: 2%
2%
1%
0%
0
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If we addThe
the Slope
savings
to the curveCurve
of available
ofcurve
the Investment
investment projects we recognize, how many
% 10% investment projects savers are willing to finance:
9%
Savings = S(i)
8%
7%
6%
5%
4%
3%
2%
1%
0%
0
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If we addThe
the Slope
savings
to the curveCurve
of available
ofcurve
the Investment
investment projects we recognize, how many
% 10% investment projects saver are willing to finance:
9%
Savings = S(i)
Investor surplus
8%
7%
6%
Equilibrium interest rate
5%
4%
=> An exchange of savings at
the resulting equilibrium interest
rate is “mutual beneficial”!
3%
2%
Saver surplus
1%
0%
0
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The
Slope ofwe
the
Investment
Curve
In the
following,
will
for simplicity
approximate
the curve of investment projects with a straight line:
% 10%
9%
Savings = S(i)
8%
7%
6%
5%
4%
3%
Investment = I(i)
2%
1%
0%
0
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The
Slope ofwe
the
Investment
Curve
In the
following,
will
for simplicity
approximate
the curve of investment projects with a straight line:
% 10%
9%
Savings = S(i)
Investor surplus
8%
7%
6%
Market interest rate
5%
4%
3%
Investment = I(i)
2%
Saver surplus
1%
0%
0
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The Slope of the Investment Curve
% 10%
Contrary to the savings
curve, the investment
curve depends on the
negatively interest rate!
8%
6%
4%
The investment curve is
also influenced by shift
parameters, e.g. the
expected return of
investment projects, r1!
2%
Investment = I(i)
0%
0
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The Slope of the Investment Curve
% 10%
8%
6%
Investment = I(i,r2)
If firms expect on average a
higher return on investment
r1<r2 (e.g. because of an
expected higher demand for
their goods), the investment
curve shifts to the right!
4%
2%
Investment = I(i,r1)
0%
0
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The Slope of the Investment Curve
% 10%
If firms expect a lower
return on investment r1>r2
(e.g. because of a lower
demand for their goods),
they will typically want to
invest less.
8%
6%
4%
Investment = I(i,r1)
2%
Investment = I(i,r2)
0%
0
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The Capital Market
% 10%
9%
S(i,y)
8%
7%
Combination of
the savings
supply curve and
investment
demand curve
6%
i1* 5%
4%
Equilibrium
Interest
3% Rate
2%
I(i)
1%
0%
0
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S1*
€
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The Capital Market
% 10%
y1 < y2
9%
8%
S1(i,y1)
S2(i,y2)
7%
6%
i1* 5%
i2*
4%
3%
2%
I(i)
1%
0%
0
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S1* S2*
€
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The Capital Market
% 10%
r1 < r 2
9%
S(i)
8%
7%
i2*6%
i1*5%
4%
I2(i, r2)
3%
2%
I1(i , r1)
1%
0%
0
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S1* S2*
€
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
➤ Do you like risk?
➤ Experiment I:
What do you take (a) or (b)?
◆ (a) You receive 3 €.
◆ (b) You receive 3 €. You will get additional 2 € with a
probability of 50% and you will have to pay 3 € with a
probability of 50%.
Option (a):
Option (b):
EV: (0.5*(3+2) + 0.5*(3-3) = 2.5
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
➤ Do you like risk?
➤ Experiment II:
What do you take (a) or (b)?
◆ (a) You receive 3 €.
◆ (b) You receive 3 €. You will get additional 2 € with a
probability of 50% and you will have to pay 2 € with a
probability of 50%.
Option (a):
Option (b):
EV: (0.5*(3+2) + 0.5*(3-2) = 3
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
➤ Do you like risk?
➤ Experiment III:
What do you take (a) or (b)?
◆ (a) You receive 3 €.
◆ (b) You receive 3 €. You will get additional 7 € with a
probability of 50% and you will have to pay 1 € with a
probability of 50%.
Option (a):
Option (b):
EV: (0.5*(3+7) + 0.5*(3-1) = 6
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
➤ What does the experiment show?
■ Most people prefer a certain payment over a risky payment.
■ A risky payment is accepted only if it includes a premium,
which is “high enough”. In economics this premium is called
“risk premium”.
■ The magnitude of this “risk premium” individually differs from
person to person.
■ However, the existence of a risk premium shows that people
generally do not like risk: They are willing to accept risk only, if
they are compensated for the risk by a higher payment!
■ In economics we call this “being risk averse”.
Prof. Dr. Rainer Maurer
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8,0
Average Yields of Fixed Rate Securities with Time to Maturity
above 3 Years of Different German Issuers
%
Empirical example for
a risk premium:
Source: Deutsche Bundesbank
7,0
6,0
5,0
4,0
3,0
= Corporate Bonds
Industrieobligationen
Prof. Dr. Rainer Maurer
= Government Bonds
Bundeswertpapiere
2010-01
2009-07
2009-01
2008-07
2008-01
2007-07
2007-01
2006-07
2006-01
2005-07
2005-01
2004-07
2004-01
2003-07
2003-01
2002-07
2002-01
2001-07
2001-01
2000-07
2000-01
1999-07
1999-01
2,0
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
➤ Why do people demand a risk premium?
■ Our "self-experiment" and empirical data from financial markets
clearly show that people are risk averse and demand risk
premiums for risky investments.
■ Now the question is, why do people behave this way?
■ Is it "irrational fear" to be "risk averse" or can we explain it?
■ The next slides show the standard microeconomic explanation
for risk averse behavior.
◆ Standard microeconomics derives the explanation from a quite
plausible property of the utility function of people: Decreasing
marginal utility of consumption.
◆ The next slide gives an explanation of this property:
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
14
Utility from the Consumption of Cookies per Day
13
Utility =
U(Cookies)
12
11
10
9
8
…and so on
7
6
Utility of the 2nd
Cookie
5
4
Utility from
the Consumption of
16 Cookies
= 12 “Utils”
3
Utility of the 1st
Cookie
2
1
Quantity of Cookies (kg)
0
0
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12*50% + 4*50% = 8
1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
Experiment II: (a) You receive 3 € with a probability of 100%.
(b) You receive 3 €. You will get additional 2 €
Utility Units
with a probability of 50% and you will have
to pay 2 € with a probability of 50%.
Expected utility from the
uncertain payment is lower than
the expected utility from the
certain payment =>
A person with this utility function
will prefer the certain payment!
1
2
3
4
5
6
7
8
9
(a) 9,5 Expected Utility Units for the Certain Payment
(b) 8 Expected Utility Units for the Uncertain Payment
Prof. Dr. Rainer Maurer
Value of
Consumption
Goods (€)
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
Utility
Gain:
2
Utility
Loss:
5,5
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
Experiment II: (a) You receive 3 € with a probability of 100%.
(b) You receive 3 €. You will get additional 2 €
Utility Units
with a probability of 50% and you will have
to pay 2 € with a probability of 50%.
The reason for the lower
expected utility is the stronger
change of utility in case of a
loss compared to the case of a
gain, because of decreasing
marginal utility!
1
2
3
4
5
6
7
8
9
Income
Income
Loss of 2 € Gain of 2 €
(a) 9,5 Expected Utility Units for the Certain Payment
(b) 8 Expected Utility Units for the Uncertain Payment
Prof. Dr. Rainer Maurer
Value of
Consumption
Goods (€)
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
Utility
Gain: 4
Utility
Loss: 4
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
Experiment II: (a) You receive 3 € with a probability of 100%.
(b) You receive 3 €. You will get additional 2 €
Utility Units
with a probability of 50% and you will have
to pay 2 € with a probability of 50%.
In case of a utility function with
constant marginal utility, the utility
gain in case of an income gain would
be equal to the utility loss in case of
an income loss and hence expected
utility in case of a certain payment
would be equal to expected utility of
an uncertain payment!
1
2
3
4
5
6
7
8
9
Income
Income
Loss of 2 € Gain of 2 €
(a) 6 Expected Utility Units for the Certain Payment
(b) 6 Expected Utility Units for the Uncertain Payment
Prof. Dr. Rainer Maurer
Value of
Consumption
Goods (€)
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15*50% + 7*50% = 11
1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
Experiment III: (a) You receive 3 € with a probability of 100%.
(b) You receive 3 €. You will get additional 7 €
with a probability of 50% and you will have
Utility Units
to pay 1 € with a probability of 50%.
Utility Gain: 5,5
Expected utility from this
uncertain payment is higher than
the expected utility from the
certain payment =>
A person with this utility function
will prefer the uncertain payment!
Utility Loss: 2,5
1
2
3
4
5
6
7
8
9
10
Income
Income
Gain of 7 €
Loss of 1 €
(a) 9,5 Expected Utility Units for the Certain Payment
(b) 11 Expected Utility Units for the Uncertain Payment
Prof. Dr. Rainer Maurer
Value of
Consumption
Goods (€)
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1. How Capital Markets Work
1.2.1. Why People Don’t Like Risk
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1 0
-2
Risk Neutral Utility
Function
Utility Units
Normal Person
(Risk Averse)
Utility Function
Gambler (Risk Lover)
Utility Function
1
2
3
4
5
6
7
8
9
10
Value of
Consumption
Goods (€)
Since we know from experiments that most people are risk averse, we can
draw the conclusion that most people have a utility function with decreasing
marginal utility!
-44-
Prof. Dr. Rainer Maurer
1. How Capital Markets Work
1.2.2. How People Handle Risk
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ We have already seen, how normal people handle risk:
■ They demand a risk premium!
➤ Financial markets offer a possibility to eliminate risk:
■ Hedging!
➤ The following tables illustrate the principle of hedging based
on several numeric examples:
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1. How Capital Markets Work
1.2.2. How People Handle Risk
The Perfect Hedge
Stock
Raincoat Corporation
Sunglasses International
Portfolio: (R. & S.) / 2
Cloudy
13
-3
5,0
Sunny
-15
25
5,0
Rainy
17
-7
5,0
Mean
5,0
5,0
5,0
Variance Corr. Coeff.
203
-1,0
203
0
➤ This example shows:
■ If the return of one stock goes up exactly when the return of the
other stock goes down, a portfolio of both stocks completely
eliminates the risk!
■ Consequently, investing your money in a portfolio of both stocks
implies no risk, while investing your money in one of both
stocks only implies a lot of risk!
■ Note: In case of a perfect hedge, the correlation coefficient
equals exactly -1!
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1. How Capital Markets Work
1.2.2. How People Handle Risk
No Hedge
Stock
Raincoat Corporation
Umbrella Unlimited
Cloudy
15
15
Sunny
-15
-15
Rainy
20
20
Mean
6,7
6,7
Portfolio: (R. & S.) / 2
15
-15
20
6,7
Variance Corr. Coeff.
239
1,0
239
239
➤ This example shows:
■ If the return of one stock goes up exactly when the return of the
other stock goes up, a portfolio of both stocks does not affect
risk at all!
■ Consequently, investing your money in a portfolio of both stocks
implies the same risk, as investing your money in one of both
stocks only!
■ Note: In case of a no hedge, the correlation coefficient equals
exactly 1!
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.2. How People Handle Risk
The Miracle of
Hedging!
Normal Hedge
Stock
Raincoat Corporation
United Steel
Cloudy
10
-5
Sunny
-16
-10
Rainy
13
22
Mean
2,3
2,3
Portfolio: (R. & S.) / 2
2,5
-13
17,5
2,3
Variance Corr. Coeff.
170
0,7
198
155
➤ This example shows:
■ If the return of one stock goes up when the return of the other
stock goes up, but not by exactly the same degree, a portfolio of
both stocks can reduce risk somewhat but not completely
eliminate it!
■ Consequently, investing your money in a portfolio of both stocks
implies a lower risk, as investing your money in one of both
stocks only!
■ Note: In case of a normal hedge, the correlation coefficient lies
between 0 and 1!
Prof. Dr. Rainer Maurer
-50-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ In the real world, perfect hedges are as rare as no hedges!
➤ Fortunately, normal (imperfect) hedges are the rule, so that
investing in portfolios generally makes more sense than investing in
single stock!
➤ Why are stocks so often imperfect hedges?
■ On one hand, there are a lot of common economic factors that
effect all stocks in the same way, causing a positive correlation
of returns:
◆ The business cycle, prices of raw materials, wages, tax
reforms…
■ To the other hand, every firm has its own product markets and
these markets often react in a different way to these common
economic factors:
◆ For example, the Bicycle-Company profits from high consumer
confidence as well as the Snowboard-Company, but in summer
time the more so than in winter time and vice versa…
Prof. Dr. Rainer Maurer
-51-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ As the examples have shown, we can comfortably measure
the hedge quality of two kind of stocks by the correlation
coefficient.
➤ How do we compute the correlation coefficient?
Covariance (Stock A, Stock B)
Variance(S tock A) * Variance(S tock B) 
0, 5
Prof. Dr. Rainer Maurer
-52-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ How do we compute the variance?

Variance (Return Stock A)
 Mean Return A at Time j - Mean(All Returns A) 
2

➤ How do we compute the covariance?
Covariance (Return Stock A, Return Stock B, )
 Return A at Time j - Mean(All Returns A)  * 

 Mean 
 Return B at Time j - Mean(All Returns B) 

Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ Interpretation of the Correlation Coefficient:
Country
2002 2003 2004 2005 2006 2007 Correlation Coefficient
Raincoat Corp.
1
-1
1
-1
1
-1
Sunglasses International -1
1
-1
1
-1
1
-1
➤ A correlation coefficient of -1 indicates that the value of two
stocks moves through time with exactly opposite fluctuations:
■ If the stock of Raincoat Corp. displays a positive deviation from
its mean value, the stock of Sunglasses International displays a
negative deviation form its mean value.
■ If the stock of Raincoat Corp. displays a negative deviation from
its mean value, the stock of Sunglasses International displays a
positive deviation form its mean value.
Prof. Dr. Rainer Maurer
-54-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ Interpretation of the Correlation Coefficient:
Country
Raincoat Corp.
Umbrella Unlimited
2002 2003 2004 2005 2006 2007 Correlation Coefficient
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
➤ A correlation coefficient of 1 indicates that the value of two
stocks moves through time with exactly the same fluctuations:
■ If the stock of Raincoat Corp. displays a positive deviation from
its mean value, the stock of Umbrella Unlimited displays a
positive deviation form its mean value too.
■ If the stock of Raincoat Corp. displays a negative deviation from
its mean value, the stock of Umbrella Unlimited displays a
negative deviation form its mean value too.
Prof. Dr. Rainer Maurer
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1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ Interpretation of the Correlation Coefficient:
Country
2002 2003 2004 2005 2006 2007 Correlation Coefficient
Sunglasses International 1
-1
1
-1
1
-1
United Steel
1
-1
1
-1
-1
1
0,33
➤ A correlation coefficient between 0 and 1 indicates that the
value of two stocks moves through time with rather similar but
not exactly the same fluctuations:
■ If the stock of Sunglasses Int. displays a positive deviation from its
mean value, the stock of United Steel displays most of the time a
positive deviation form its mean value too – but not always.
■ If the stock of Sunglasses Int. displays a negative deviation from
its mean value, the stock of United Steel International displays
most of the time a negative deviation form its mean value too – but
not always.
-56-
Prof. Dr. Rainer Maurer
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ Now it’s up to you:
■ As portfolio manager you have to decide in the following cases,
whether to invest in single stock or in a portfolio.
■ What do you recommend?
Prof. Dr. Rainer Maurer
-57-
1. How Capital Markets Work
1.2.2. How People Handle Risk
Portfolio or not?
What do you recommend?
Stock
Pennylane Corp.
Mean
Mean-Deviation
Squared Mean-Deviation
2005
160
2006
130
2007
150
Galapagos International
Mean
Mean-Deviation
Squared Mean-Deviation
30
60
40
Variance
Corr. Coeff.
Portfolio: (P. & G.) / 2
Mean
Mean-Deviation
Squared Mean-Deviation
Variance = Mean( Squared Mean Deviation )
Covariance = Mean( (Mean Deviation Stock A) * (Mean Deviation Stock B) )
Prof. Dr. Rainer Maurer
Correlation Coefficient = Covariance / ( Variance * Variance)^0,5
-58-
1. How Capital Markets Work
1.2.2. How People Handle Risk
Portfolio or not?
What do you recommend?
Stock
Raincoat Corp.
Mean
Mean-Deviation
Squared Mean-Deviation
2005
110
2006
100
2007
60
Umbrella Unlimited
Mean
Mean-Deviation
Squared Mean-Deviation
100
90
50
Variance
Corr. Coeff.
Portfolio: (P. & G.) / 2
Mean
Mean-Deviation
Squared Mean-Deviation
Variance = Mean( Squared Mean Deviation )
Covariance = Mean( (Mean Deviation Stock A) * (Mean Deviation Stock B) )
Prof. Dr. Rainer Maurer
Correlation Coefficient = Covariance / ( Variance * Variance)^0,5
-59-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ The Market-Beta:
■ As already seen, it is almost impossible to find perfectly negatively
correlated stocks – so that all portfolios end up with a risk that cannot
be eliminated – the so called “market risk”.
■ The market risk is measured by the variance of the average return of the
market portfolio, i,e. the risk that cannot be eliminated by investing in
the market portfolio.
■ The tendency of the return of a stock to move with the average return of
the market portfolio is called its market-beta.
■ The market-beta is a measure of the relative volatility of a stock return
compared to the return of the total stock market (= a portfolio consisting
of all stocks minus the one stock whose beta is measured) as a whole.
◆ A beta of 1 means that a stock return moves exactly as the market does.
◆ A beta of 2 means that if a stock market return moves up by 10 %, the
stock return moves up by 20 %.
=> Adding a high (low) beta-stock to a portfolio means
increasing (reducing) the portfolio risk.
Prof. Dr. Rainer Maurer
-60-
1. How Capital Markets Work
1.2.2. How People Handle Risk
➤ How do we compute the market-beta?
Covariance (Return Market Portfolio, Return Stock A)
β
Variance(R eturn Market Portfolio)
➤ As we will see in chapter 2, the market-beta plays a central
role for the evaluation model of stocks – the so called CAPModel (Capital Asset Pricing Model).
➤ The following graph visualizes its interpretation.
Prof. Dr. Rainer Maurer
-61-
1. How Capital Markets Work
1.2.2. How People Handle Risk
50 Return of Stock A
β=1 => Stock A is as
volatile as the market.
40
30
=> Adding Stock A to the
market portfolio does not
change portfolio risk
“Average β-Stock“
20
10
Return of the Market Portfolio
0
-50
-40
-30
-20
β=1
-10
-10
0
10
20
30
40
50
-20
-30
-40
-50
Prof. Dr. Rainer Maurer
-62-
1. How Capital Markets Work
1.2.2. How People Handle Risk
50 Return of Stock A
β=0,5 => Stock A is less
volatile than the market.
“Low β-Stock“
40
β=0,5
30
=> Adding Stock A to the
market portfolio does
reduce portfolio risk
20
10
Return of the Market Portfolio
0
-50
-40
-30
-20
-10
-10
0
10
20
30
40
50
-20
-30
-40
-50
Prof. Dr. Rainer Maurer
-63-
1. How Capital Markets Work
1.2.2. How People Handle Risk
50 Return of Stock A
β=1,5 => Stock A is more
volatile than the market.
β=1,5
40
30
=> Adding Stock A to the
market portfolio does
increase portfolio risk
“High β-Stock“
20
10
Return of the Market Portfolio
0
-50
-40
-30
-20
-10
-10
0
10
20
30
40
50
-20
-30
-40
-50
Prof. Dr. Rainer Maurer
-64-
1. How Capital Markets Work
1.2.2. How People Handle Risk
Prof. Dr. Rainer Maurer
-65-
1. How Capital Markets Work
1.2.2. How People Handle Risk
Prof. Dr. Rainer Maurer
-66-
1. How Capital Markets Work
1.2.2. How People Handle Risk
Prof. Dr. Rainer Maurer
-67-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
-69-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
➤ As the next chapter will show, very different kind of
assets are traded on capital markets.
➤ Two technical procedures are important for the evaluation
of these different assets:
■
Discounted Cash-Flow Method
■
Internal Rate of Return Method
➤ Before we apply these procedures to the various types of
assets in the next chapter, we will analyze them in some
detail in the following:
➤ We start with the discounted cash-flow method:
Prof. Dr. Rainer Maurer
-70-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
➤ What do you prefer: 1 € today or 1 € in one year?
➤ The basic idea of the discounted cash-flow method is:
■
Determining the present value of a flow of future payments
– either from an investment project or a financial market
asset.
■
Technically, payments of different points in time are made
comparable by evaluating each payment with a time specific
discount factor and adding up these comparable payments to
the “present value” of the payment flow.
➤ The following examples show, how this works:
Prof. Dr. Rainer Maurer
-71-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
➤
Discounting a payment flow:
Cash Flow
Discount Rate: 5%
Present Values
Total Present Value
Discounting a Cash Flow
2008
2009
2010
2011
2012
0
100
100
100
100
*(1,05)^(-1) *(1,05)^(-2) *(1,05)^(-3) *(1,05)^(-4)
95,2
90,7
86,4
82,3
354,6
➤
This payment flow with equal annual payments of 100 per year clearly
shows that payments further in the future a more discounted than
payments closer to the present.
➤
A comparison with the following table shows that a lower discount rate
increases the present value:
Cash Flow
Discount Rate: 2%
Present Values
Total Present Value
Prof. Dr. Rainer Maurer
Discounting a Cash Flow
2008
2009
2010
2011
2012
0
100
100
100
100
*(1,02)^(-1) *(1,02)^(-2) *(1,02)^(-3) *(1,02)^(-4)
98,0
96,1
94,2
92,4
380,8
-73-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
➤ Problem of the discounting approach:
■
An appropriate discount rate has to be chosen!
➤ The appropriate discount rate should reflect the risk related to a
payment flow.
■
Uncertain payment flows, whose payments are based on
estimated forecasts only, should be discounted with a higher
discount rate than secure payment flows.
■
Hence, the discount rate should include an appropriate risk
premium.
➤ How can this be done?
Prof. Dr. Rainer Maurer
-74-
1. How Capital Markets Work
1.3.1. The Discounted Cash-Flow Method
➤ How to find the appropriate discount rate?
■
In many cases it is possible to find a market interest rate for a
payment flow of the same risk class:
◆ If the payment flow is nearly certain, the market interest rate
of a fixed rate security with the same risk structure, for
example a bond of a government with high creditworthiness,
should be chosen as discount rate.
■
However, for many uncertain payment flows, it is difficult to
find a market interest rate of the same risk class.
◆ If the payment flow is uncertain and based on a forecast (for
example the dividend payments from a stock company) a
market interest rates for exactly the same risk class are hard
to find.
◆ In this case, concepts like the Capital Asset Pricing Model
(CAPM) can to be employed to calculate an appropriate
discount rate. Chapter 2 will show, how this works.
Prof. Dr. Rainer Maurer
-75-
1. How Capital Markets Work
1.3.2. The Internal Rate of Return Method
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
-77-
1. How Capital Markets Work
1.3.2. The Internal Rate of Return Method
➤ The basic idea of the internal rate of return method is:
■
Very often one knows the market price of an asset or investment
project and its payment flow.
■
In such cases it is possible to determine the internal rate of return
of the asset or investment project.
■
Technically, this amounts to finding a discount rate that equals the
present value of the payment flow with the market price of the
asset.
➤ The following examples shows, how this works:
Prof. Dr. Rainer Maurer
-78-
1. How Capital Markets Work
1.3.2. The Internal Rate of Return Method
Calculation of the Internal Rate of Return
Market Price
of Asset
100
Periods
2009
t+1
Annual Payments :
102
!
102

=> Internal Rate of Return =
1  i 1
i=
2,00%
➤ If the payment flow consists of one period only, it is easy to
calculate the IRR (=internal rate of return) by hand:
102
100 
1 i
102
 1 i 
 1,02
100

i  1,02  1
Prof. Dr. Rainer Maurer
-79-
1. How Capital Markets Work
1.3.2. The Internal Rate of Return Method
Calculation of the Internal Rate of Return
Market Price
of Asset
100
Periods
2009
2010
2011
2012
2013
Annual Payments :
3
1
8
5
102
!
3

=> Internal Rate of Return =
➤
1
 i
i=
1

1
1
 i
2

8
1  i 
3

5
1  i 
4

102
1  i 5
3,78%
If the payment flow consists of more than one period, for example 5
periods, calculating the IRR implies solving a polynomial of degree 5.
This involves the following problem:
■
Formulas for analytical solutions exist only for polynomials lower
degree 4.
➤ Therefore, numerical solutions methods must be applied.
➤ One such method is available for Excel (the IKV() Function).
Prof. Dr. Rainer Maurer
-80-
1. How Capital Markets Work
1.3.2. The Internal Rate of Return Method
➤ Consequently, calculating the IRR typically implies the usage
of a computer.
➤ If the IRRs of several payment flows are calculated, it is in
principle possible to select the payment flow with the highest
IRR as the most profitable one.
➤ However, one has to take care of the risk implied by each
payment flow!
➤Since an uncertain payment flow is riskier than a certain
payment flow, the uncertain flow must offer a risk premium
(for people with normal utility function…)
➤One often used measure, which takes care of both return and
risk, is the so called Sharpe Ratio, which was proposed by
William F. Sharpe (1966).
Prof. Dr. Rainer Maurer
-82-
1. How Capital Markets Work
1.3.3. Risk and Return: The Sharpe Ratio
1. How Capital Markets Work
1.1. Supply and Demand on Capital Markets
1.1.1. Why People Save
1.1.2. Why People Invest
1.2. Capital Markets and Risk
1.2.1. Why People Don’t Like Risk
1.2.2. How People Handle Risk
1.3. Basic Evaluation Techniques for Capital Markets
1.3.1. The Discounted Cash-Flow Method
1.3.2. The Internal Rate of Return Method
1.3.3. Risk and Return: The Sharpe Ratio
Prof. Dr. Rainer Maurer
-83-
1. How Capital Markets Work
1.3.3. Risk and Return: The Sharpe Ratio
➤ The basic idea of the Sharpe Ratio is to determine the risk
premium paid per unit of risk.
➤ Therefore, the formula of the Sharpe Ratio relates the risk
premium of the internal rate of return of a specific investment j
(= the difference between this expected return, E(ij), and the
return of a risk free market interest rate, ro, e.g. a government
bond of high creditworthiness), E(ij)- ro, to the standard
deviation of the return of the specific investment
(= (variance(ij))^0,5 ):
E(i j )  ro
Risk Premium
Sharpe Ratio 

Risk
var( i j )
➤ The higher the Sharpe Ratio, the higher is the risk premium per
unit risk and hence the more attractive is the investment.
-84-
Prof. Dr. Rainer Maurer
1. How Capital Markets Work
1.3.3. Risk and Return: The Sharpe Ratio
➤ Which asset A or B offers the best relation between risk and
return? Calculate the Sharp Ratio for a risk free return of 2 %.
Computing the Sharpe Ratio: Asset A
Recession Normal
Return of Asset
-2%
3%
Mean Return
Mean Deviation
Squared Mean Deviation
Variance
Standard Deviation
Risk Free Return
2%
Sharpe Ratio
Prof. Dr. Rainer Maurer
Computing the Sharpe Ratio: Asset B
Recession Normal
Return of Asset
-4,0%
4,0%
Mean Return
Mean Deviation
Squared Mean Deviation
Variance
Standard Deviation
Risk Free Return
2%
Sharpe Ratio
Boom
8%
Boom
10,0%
-85-
1. How Capital Markets Work
1.3.3. Risk and Return: The Sharpe Ratio
➤ Problem with the Sharpe Ratio:
■
The standard deviation of an investment, i.e. the measure for its
risk, is typically not known, but has to be estimated based on
forecasted future returns. In our above example we simply
assumed to know them!
■
For financial market assets, e.g. the return of stocks, such kind
of forecasts are typically highly inaccurate.
■
Therefore, very often, the standard deviation of a stock is
estimated based on its past returns.
■
Even though such a calculation is easily done and seems to be
highly “accurate and plausible”, one has to be aware that the
application of such “historic” standard deviations for future
investment decisions, implies the “hidden assumption” that the
“future” will similar to the “past”.
■
For many kind of stocks, this assumption has proven wrong!
Prof. Dr. Rainer Maurer
-86-
Chapter 1: Questions
You should be able to answer the following questions at the
end of this chapter. If you have difficulties in answering a
question, discuss this question with me during or at the end of
the next lecture or attend my colloquium.
Prof. Dr. Rainer Maurer
-88-
Chapter 1: Questions for Review
Why can “saving” increase personal utility? Give a graphical and
verbal explanation.
2. How does saving behavior affects consumption behavior?
3. If saving increases utility, why do savers demand interest?
4. Why depends the willingness to save positively on the interest rate?
5. What is a “production function”?
6. Explain the motive for investment.
7. Why depends the willingness to invest negatively on the interest rate?
9. What insures the equality of saving and investment in a market
equilibrium? Explain your answer based on the a diagram of the
capital market.
10. How does an increase in savings supply affect the interest rate?
1.
Prof. Dr. Rainer Maurer
-89-
Chapter 1: Questions for Review
11. How does an increase in investment demand affect the interest rate?
12. Explain the meaning of “risk averse” and “risk neutral”.
13. Why are most people risk averse? Give a graphical and verbal
explanation.
14. What is a “risk premium”?
15. Why do people demand a risk premium?
16. What is “hedging”?
17. What property must two stocks have to be a “perfect hedge”?
18. Is it possible to hedge risk with two stocks, if there return is positively
correlated?
19. Why are perfect hedges rare?
20. What is a “normal hedge”?
21. How do you explain that so many stocks are “normal hedges”?
Prof. Dr. Rainer Maurer
-90-
Chapter 1: Questions for Review
22. Calculate the variances and the correlation coefficient of the following
stocks. Can they be used to hedge each other?
Stock
Stock A
Stock B
Sum / 2
2002
160
30
2003
130
60
2004
150
40
2005
80
110
2006
70
120
2007 Variance Corr. Coeff.
180
10
23. Calculate the variances and the correlation coefficient of the following
stocks. Can they be used to hedge each other?
Stock
Stock A
Stock B
Sum / 2
2002
110
100
2003
100
90
2004
60
50
2005
130
120
2006
110
100
2007 Variance Corr. Coeff.
170
160
24. What is the “market-beta”?
25. What is meant by “low beta stock” and “high beta stock”?
Prof. Dr. Rainer Maurer
-91-
Chapter 1: Questions for Review
26. Give a verbal explanation of the discounted cash flow and
the internal rate of return method.
Cash Flow
Discount Rate: 4%
Present Values
Total Present Value
27.
28.
29.
30.
Prof. Dr. Rainer Maurer
Discounting a Cash Flow
2008
2009
2010
0
100
100
2011
100
2012
100
Which criteria should an appropriate discount rate fulfill?
What is the definition of the Sharpe Ratio?
Give a verbal interpretation of the Sharpe Ratio.
What are the two major criteria for an investment decision?
-92-
Chapter 1: Questions for Review
31. What asset is the most attractive? Base your decision on the Sharpe
Ratio.
Computing the Sharpe Ratio: Asset C
Return of Asset
Mean Return
Mean Deviation
Squared Mean Deviation
Variance
Standard Deviation
Risk Free Return
Sharpe Ratio
Recession
-1,0%
Normal
4,0%
2%
Computing the Sharpe Ratio: Asset D
Recession Normal
Return of Asset
0,0%
3,5%
Mean Return
Mean Deviation
Squared Mean Deviation
Variance
Standard Deviation
Risk Free Return
3%
Sharpe Ratio
Prof. Dr. Rainer Maurer
Boom
10,0%
Boom
9,0%
-93-
Portfolio Theory
Prof. Dr. Rainer Maurer
-94-