Advanced Corporate Finance Lecture 2

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Lecture 2 Valuation and Risk
Anton Miglo Fall 2007
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Topics

• Recall valuation under certainty
• Risk and returns
• Expected utility
• Risk aversion
• Wealth certainty equivalent
• Stochastic dominance
• Mean-variance approach
• Applications

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Valuation Under Certainty
1. First step in finance is valuation assets,
securities, projects etc. 2. Net present value
(NPV) concept Relating the future to the
present.   Present value value today of a
future cash flow. Discount factor present
value of a 1 future payment. Discount rate
- interest rate used to compute present
values of future cash flows.
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Present Value (PV) Discount Factor (DF) Cash
Flow (CF)
r Discount/Interest rate t
Number of years until cash flow is realized
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A simple example of the steps involved in
calculating the present value of a risk-free
real-estate transaction Step 1 Forecast cash
flows           Cost of building
          Sale price in Year 1 Step 2
Estimate opportunity cost of capital
          If equally risky investments in the
capital market offer a return of 7, then cost
of capital 7
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Introduction to Risk

• Risk arises when future cash flows are uncertain.
  To deal with this uncertainty we must calculate
  Expected Utility
• Expected Utility is the utility a person expects
  to receive from
• his uncertain future cash flows
• e.g. Consider the following lottery
• Probability distribution of possible outcomes

Ci is a stochastic variable Expected Utility
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Example
Example with three probability distributions

• The probability distribution that leads to the
  highest utility depends
• on the utility function

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Example (Contd)
Assume the utility function is ln(x) The
expected utility for the three distributions is
thus
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Stochastic Variable Definitions
Stochastic Variable X 1. Cumulative distribution
function c.d.f.
For discrete variable
For continuous variable
2. Expected value
For discrete variable
For continuous variable
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Risk Aversion
The classical economic definition of risk
aversion is when an individual has a concave
utility function as shown below
This implies that the individuals marginal
utility of wealth diminishes as wealth increases
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A Graphical Illustration
Suppose a risk adverse person with a utility
function equal to ln(x) has the choice between
getting 50 for sure or being entered into a
lottery with a 50 chance of winning nothing and
a 50 chance of winning 100 (note the expected
value of the lottery is 50 also) what will this
person choose?
If the person takes the 50 his utility will be
ln(50) 3.91
If he takes the lottery his utility will be 0.5
ln(0) 0.5 ln(100) 2.3
Therefore the risk adverse person will take the
50 for sure. In fact, this result will hold
for all risk adverse people
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More Examples of Risk Aversion
1. Consider two lotteries a) 1000 with
probability p and 0 with probability (1-p) b) w
with probability 1 What is the relationship
between w and p assuming that these are equivalent
2. Assume u(0) 0 and u(1000) 1 By
definition
U(w) pu(1000) (1-p)u(0) p
w lt pu(w)1000
Important!
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Certainty Equivalent Wealth
In the last slide we learned that a risk averse
person will always prefer the expected value of
a lottery to the lottery itself. But how much
does a person value the lottery? Answer the
certainty equivalent wealth
Certainty equivalent wealth is the amount of
money a person would have to receive in order to
be indifferent between that sum of money and a
gamble
In our example recall that the utility of the
lottery was 2.3, therefore the person would have
to receive exp(2.3) 9.97 to make him
indifferent between that sum and the lottery
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Let be the stochastic outcome. Certainty
equivalent wealth is such that
Where
An individual is risk neutral if
Where
An individual is risk averse if
Finally an individual is risk loving if (this is
almost never the case)
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Consider the following lottery r1 with
probability p and r2 with probability 1 - p Let
u(w) aw b, where a, b gt0 Show that the
individual is risk-neutral
A risk-neutral utility function has the property
that marginal increases in wealth have the same
absolute effect on utility at all initial level
of wealth i.e. a risk neutral persons utility
will go up the same amount when given an extra
dollar whether his initial income is 5 or
1,000,000!
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Stochastic Dominance
So far we have developed ideas regarding the
preferences of risk averse individuals. We saw
that people who are risk averse prefer certain
wealth to a gamble, and that the more risk
averse someone is the less likely they are to
engage in a gamble. The notion of more risk
averse is hard to quantify however, and requires
precise utility functions, which in practice are
hard to calculate. This becomes problematic when
we attempt to analyze the behavior of risk averse
people when faced with a choice between two
gambles.
The idea of stochastic dominance however
eliminates the need to calculate utility
functions and sets hard and fast rules for
decision making under some conditions.
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First-order Stochastic Dominance
Stochastic variable X with cumulative probability
distribution (c.p.d.) F dominates stochastic
variable Y with c.p.d.G if
For all z
This inequality states that the probability
distribution of X is more heavily weighted
towards the higher values of z. In our case z
could represent corporate profits.
Important If the utility function is increasing
in wealth, then the individual will prefer F to
G.
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A Graphical Representation
Assume the Brown line is the cumulative
distribution function (CDF) of X (F(z)) and the
Black line is the distribution of Y (G(z)). X
stochastically dominates Y because the CDF of X
lies under the CDF of Y at all points
F(z)G(z), for all z
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A Numerical Example of First-order Stochastic
Dominance
Assume two random variable X Y with probability
distributions as follows
In this case z takes on three values 1, 2 and 3.
For X to stochastically dominate Y recall
For all z
Since F(z) is always less than or equal to G(z),
X dominates Y
F(z) is in read G(z) is in blue
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Second-order Stochastic Dominance
Stochastic variable X with cumulative
distribution function (CDF) F dominates
stochastic variable Y with CDF G if
For all z
This inequality implies that the distribution of
Y has more weight at lower values of wealth than
does X and this more than offsets the possible
higher weights of the distribution of Y at high
values of wealth
Important If the utility function is increasing
in wealth and it is concave, then the individual
will prefer F to G.
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A Graphical Representation
The blue curve is the density function of X The
red curve is the density function of Y
The blue curve is the cumulative distribution of
X The red curve is the cumulative distribution
of Y
X dominates Y because the yellow area in the
cumulative distribution graph is larger than the
blue area
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A Numerical Example of Second-order Stochastic
Dominance
Again z takes on three values 1, 2 and 3. For X
to stochastically dominate Y recall
For all z
Since ?F(z)-G(z)lt0 for all outcomes, X
stochastically dominates Y
F(z) is in read G(z) is in blue
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Increasing Risk
Consider stochastic variables X with distribution
function F(x) and Y with G(y), xb, yb. X
dominates Y in the increasing risk sense if
S(x)?(F(z)-G(z))dz lt 0, for all x
and S(b)0
Example
Result Eu(X)Eu(Y) for all u concave.
e.g. u(w) w1/2
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Mean Preserving Spread

• Assume we have two identical distributions
  as shown below, their means are 6 and their
  variances are 4

If we alter the green distribution by taking away
some of the weight in the center and adding it to
the tails in such a way as to keep the mean of
both distributions identical we get a Mean
Preserving Spread, and a risk averse person will
always choose the distribution with the lowest
spread. Graphically
The mean of the new distribution is still 6 but
the variance is now 9
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Relationship Between Different Stochastic
Dominance Relationships

• FOD implies SOD
• IR implies SOD
• Proof in class

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Mean-variance Criterion
In general this is not good, but sometime it is
possible, for example when utility is quadratic
or profits are normally distributed The mean
variance criterion takes note of the fact that
higher expected wealth is good, but higher
variance is not.
Let u(w) aw-(1/2)bw², where a,bgt0
Show that Eu(w)aµ-(1/2)b(µ²s²)
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Risk and Variance
Assume a utility function of u(w)ln w
EX EY 3 and VarX 4 lt VarY 5 Eu(X) 0.8 lt
Eu(Y) 1.42
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Finance Applications

• To invest or not to invest
• Which shares are better
• Simple portfolio selection problem

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To invest or not to invest?

• Assume an entrepreneur with initial wealth w
• 1. Expected utility function is "mean-variance"
  
• 2. Investment project cost B and profit
• 3. There is no inflation and discount rate is 0
• 4. NPV is positive ER gt B
• Under which conditions will the entrepreneur
  undertake this project?
• She will undertake the project if her expected
  utility from doing so
• is greater than her utility from not undertaking
  the project
• i.e. if
• 2. She will not undertake the project if the
  opposite is true
• i.e. if
• 3. After some simplification one can see that the
  entrepreneur will undertake the
• project if the following inequality holds

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This tells us that the higher is a (the marginal
increase in utility given higher income) and the
lower is K (the marginal decrease in utility
given higher uncertainty or variance) the more
likely the person is to undertake the project.
This illustrates that some positive NPV project
will be rejected if a persons risk aversion is
sufficiently high !!!
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Risk and Variance A Corporate Example
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Firm A share Expected EPS equals 5, standard
deviation 1.41
Firm B share Expected EPS equals 7 and deviation
2.82
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