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Unformatted text preview: Investment Decision Under Uncertainty
Lecture Notes for Actsc 372  Fall 2011
Ken Seng Tan
Department of Statistics and Actuarial Science
University of Waterloo K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 1/31 Introduction
Recall (actsc 371) Net Present Value (NPV):
N P V ( r ) = C0 + C1
C2
+
+ ···
1 + r (1 + r)2 where r is the opportunity cost of capital
What is RISK?
In ﬁnance, risk is typically referred to the likelihood that
we will receive a return on an investment that is
different from the return we are expected to make.
variability of returns (downside risk vs upside
potential)
contingent events, catastrophic events, etc.
When faced with uncertainty of outcomes, how does a
rational individual make a decision?
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 2/31 Examples
If you have a number of investment opportunities
that could inﬂuence your wealth at the end of the year, can
you rank those opportunities (or state your preferences)?
Easy if the outcomes from all alternatives are certain!
prefer more to less
What if the cash ﬂows are random (i.e. uncertain)!
Example: What proportion of your wealth should be
invested in riskless asset and risky assets?
Example: Your friend is willing to invest in shares of RIM,
but you do not!
Example: Most of us are willing to insure against potential
losses even though the insurance premium is more than
the expected loss!
Example: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 3/31 Examples
You have a choice between two winning but
uncertain projects, which project should you select (ignore
initial cost)?
Project A pays either $1,000 or $5,000. Project B pays
either $1,500 at the same time project A pays $1,000,
or $5,500 when project A pays $5,000.
In another case, project B pays $5,500 when project A
pays $1,000, and it pays $1,500 when project A pays
$5000.
Example: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 4/31 Examples
A fair coin is tossed, if it’s “tail" you will
receive $10, otherwise you will receive $20.
Are you willing to gamble if each game costs
$20
$5
$15
What is the maximum cost of entry to deter you from
gambling?
Gambling Example: Your current wealth is $w and you are
facing a random loss of $X (a r.v.). What is the maximum
price (premium) you are willing to pay to insure the
potential loss completely?
Insurance Example: Answers to these questions lie in the decision maker’s risk
attitude
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 5/31 Topics Expected Value Principle
St. Petersburg Paradox
Expected Utility Principle
von NeumannMorgenstern (1944) Expected Utility
Representation
Insurance applications K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 6/31 Lottery/Gamble
A lottery or gamble is simply a probability distribution over a
given set of outcomes.
can be discrete or continuous
gamble ≡ lottery ≡ random prospect
≡ investment opportunity
outcome ≡ payoff ≡ prize Depending on the nature of the “lottery/gambling", the
outcomes need not always be expressed in terms of
monetary units.
But we are only interested in gambles with outcomes
that can be expressed in monetary units (e.g. wealth
level). K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 7/31 Example on Lottery/Gamble
A lottery, say P , has three possible outcomes: {0, 1, 2}. Probability of
0 occurring is 0.6, probability of 1 occurring is 0.3 and probability of 2
occurring is 0.1.
The lottery P can be represented as: P: p(0) = 0.6, p(1) = 0.3, p(2) = 0.1 where p(x) denotes the probability of outcome x occurring.
Note that p(x) = 0 ∀ x {0, 1, 2}
Expected value of the lottery = 0 × 0.6 + 1 × 0.3 + 2 × 0.1 = 0.5
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 8/31 Expected Value Principle
By this principle, a decision maker would choose an
investment that has the highest expected value (return)
Consider prospects A, B, C and D:
Investment A Investment B Investment C Investment D
payoff prob payoff prob payoff prob payoff prob
1
1
+4
1
+5
1
5
10
4
5
1
1
0
+10
2
5
1
2
+40
+20
4
5
1
+30
5
E (X )
4
5
8.75
14
If you are an expected value maximizer, which prospect
would you choose?
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 9/31 Expected Value Decision Maker
willing to pay up to E (X ) to participate in a gamble with
random payoff X
would be indifferent between assuming the random loss X
and paying amount E (X ) in order to be relieved of the
possible loss
In economics the expected value of a random prospect
(with monetary payments) is called the actuarial value of
the prospect.
Actuarial value of the gamble is
$15. Therefore the decision maker will pay up to $15 to
play this game.
Gambling Example Cont’d Is Expected Value Principle a reasonable criteria in
ranking gambles (opportunities)?
Do we adopt this principle?
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 10/31 Expected Value Principle & Insurance Application
A decision maker is considering insuring against an
accident. In all the cases below, assume the probability of
an accident is 0.01 and the probability of no accident is
0.99:
Possible Losses
Case No accident
Accident Expected Loss
A
0
1
0.01
B
0
1,000
10.00
C
0
1,000,000
10,000.00
Implications? K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 11/31 St. Petersburg Paradox
Suppose a fair coin is tossed until it comes up “tails". You
receive:
$2 if “tails" occurs on the opening toss, (with prob. 1/2)
$4 if “tails" occurs on the second toss, (with prob. 1/4)
$8 on the third toss,
$16 on the fourth toss, etc. . . .
How much are you willing to pay for this gamble?
What is its Expected Payoff?
This is the famous St. Petersburg Paradox, proposed by
Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli
in 1738. K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 12/31 St. Petersburg Paradox: D. Bernoulli’s Insights
What is important to investors is the utility derived from the
money received rather than the money itself.
Hypothesized that a person’s utility from money had the
form u(w) = log10 (w), where w stands for wealth.
This utility function has the characteristics that it increases
with wealth, but each extra dollar of wealth should
increase utility at a decreasing rate; i.e.
decreasing marginal utility of wealth.
u (w ) > 0, u (w ) < 0,
Compare log10 (10) and log10 (100)
people should rationally maximize expected utility rather
than expected monetary value.
∞
∞
k
E[u(X )] =
2−k log 2k = log 2
= log 4 = 0.602
2k
k =1 k =1 K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 13/31 Expected Utility Hypothesis
The expected utility “idea" was later formalized by John
Von Neumann and Oskar Morgenstern (1944).
The theory starts with the assumption that a decision
maker, when faced with two distributions of outcomes
affecting wealth, is able to express a preference for one of
the distributions or indifference between them.
Preference relation: “ ” is strictly preferred to
“ ∼ ” is indifferent to
“ ” is weakly preferred to; Furthermore, the preferences must satisfy certain
consistency requirements, known as the axioms of cardinal
utility.
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 14/31 von NeumannMorgenstern Expected Utility Representation Given:
a decision maker is facing two random prospects X
and Y (with respective probability distribution)
u(w ) is the decisionmaker’s utility from outcome w
Formally, a utility function is a mapping from the set
of outcomes onto the real numbers.
von NeumannMorgenstern Expected Utility Representation Rational individuals will choose among risky
alternatives as if they are maximizing the expected
value of utility (rather than the expected value of the
lottery)
such individual is referred to as the expected utility
maximizer ⇒
K.S. Tan/Actsc 372 F11 X Y if and only if E [u(X )] > E [u(Y )]
Investment Decision Under Uncertainty – p. 15/31 Properties of Utility Functions
The utility function has the property that for outcomes x
and y , x y iff u(x) > u(y )
i.e. u(x) is an increasing function in x
The utility units, which are called utiles, have no meaning,
what matter is the relative magnitude
Uniqueness of utility function u: Deﬁne a positive linear transformation as:
u∗ (x) = a · u(x) + b, a>0 What can you say about the preference ranking (based
on expected utility maximization) for both u(x) and
u∗ (x)? K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 16/31 Example Revisit
A: Payoffs of $50, $80, $100, with respective prob. 0.10, 0.45 & 0.45.
B: Payoffs of $0, $80, $100, with respective prob. 0.02, 0.45 & 0.53.
Suppose the utility of wealth is given by: u(w ) = √ w Expected utility of gamble A: Expected utility of gamble B: Hence Gamble B is preferred to Gamble A, using expected utility
criterion.
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 17/31 Example I
Gamble A: random payoff X with 5050 chance of receiving $2 and 0;
Gamble B: Receive $1 with certainty
Which gamble would you choose if your utility functions are: √
u1 (x) = x
u2 (x) = x2
u3 (x) = x Expected utility of gamble B is 1 under all three utility functions.
The expected utility of Gamble A: √
EA [u1 (X )] = 0.5 × 0 + 0.5 × 2 = 0.71
EA [u2 (X )] = 0.5 × 0 + 0.5 × 22 = 2
EA [u3 (X )] = 0.5 × 0 + 0.5 × 2 = 1 Conclusions?
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 18/31 Individual’s Attitude towards Risks (Risk Aversion)
An individual’s risk attitude (preference) can be classiﬁed
as either risk averse, risk seeking (risk loving) or risk neutral
Recall that actuarial value of a gamble X is the expected
value of the gamble.
one who prefers actuarial value with certainty is a risk
averter; i.e.
u(E(X )) > E[u(X )]
one who prefers the gamble is a risk lover (risk seeker) i.e.
u(E(X )) < E[u(X )] one who is indifferent is riskneutral; i.e.
u(E(X )) = E[u(X )]
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 19/31 Alternate Characterization of Risk Aversion
u (x) < 0 i.e u(x) is concave ⇒ risk averse
u (x) > 0 i.e. u(x) is convex ⇒ risk loving
u (x) = 0 i.e. u(x) is linear ⇒ risk neutral Theorem (Jensen’s Inequality): Suppose u (w ) ≤ 0 and X is a
random variable, then u(E[X ]) ≥ E[u(X )]. K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 20/31 Examples
Example I (Cont’d): u (x)
1
u (x)
2
u (x)
3 = 1 −1 / 2
2x For Gamble A with random payoff X :
> 0 and u (x) = − 1 x−3/2 < 0; ∀x > 0
1
4 = 2x > 0 and u (x) = 2 > 0; ∀x > 0
2
= 1 and u (x) = 0
3 Logarithmic Utility u(w ) = ln(w ),
w>0
1
u (w ) =
> 0,
w
1
u (w ) = − 2 < 0
w K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 21/31 Samples of Risk Averse Utility Functions
Quadratic Utility: u(w ) = w − αw 2 , for w < 1
, α > 0,
2α u (w ) = 1 − 2αw, u (w ) = −2α. note that: E[u(X )] = E[X − αX 2 ] = E[X ] − αE[X 2 ].
Exponential Utility: u(w ) = −e−αw ,
u (w ) = α e −αw ∀w and for ﬁxed α > 0 , 2 −αw . −αX u (w ) = −α e E[u(X )] = E[−e where MX (t) = E[etX ] ≡ m.g.f. = −M X (−α ) K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 22/31 An Example with Exponential Utility Function
A decision maker’s utility function is given by u(w ) = −e−5w . The decision
maker has two random economic prospects X and Y available. The
outcomes of these two prospects follow normal distribution such that: X ∼ N (5, 2) Y ∼ N (6, 2.5). and Which prospect will he prefer?
Solution:
First note that Y has higher expected return, but with higher std. dev.
22
If X ∼ N (µ, σ 2 ), then MX (t) = E[etX ] = eµt+σ t /2 K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 23/31 Samples of Risk Averse Utility Functions (Cont’d)
Fractional Power Utility u(w ) = w γ
u (w ) = γ w w > 0, 0 < γ < 1
γ −1 , u (w ) = γ (γ − 1)w γ −2 .
Power Utility wα − 1
, α < 1, α = 0,
α
u (w ) = w α−1 ,
u(w ) = w>0 u (w ) = (α − 1)w α−2 . K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 24/31 Connection Between EU Principle and EV Principle
Recall that Expected Value (EV) principle chooses a
prospect with higher expected value
If u(x) = a · x + b, a > 0, then E[u(X )] = a · E(X ) + b.
For prospects X and Y , we have E(X ) > E(Y ) ⇔ E[u(X )] > E[u(Y )] Therefore, decision maker who adopts EV Principle has a
linear utility function
decision maker is risk neutral
u(E(X )) = E[u(X )] K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 25/31 Insurance and Utility
Example: A decision maker, with an initial wealth $10,000, is facing a
loss of $3,600 with 10% probability, 0 otherwise. What is the maximum
premium he is willing to pay to completely insure against such loss if
√
his utility of wealth is u(w ) = w ?
Solution:
Possible losses?
E[loss] ?
Possible wealth level?
Expected utility of wealth with no insurance:
Expected utility of wealth with insurance:
Maximum premium he is willing to pay:
K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 26/31 Insurance Example: Squareroot Utility
√ A decision maker’s utility function is given by u(w ) = w . The decision
maker has wealth of w = 10 and faces a random loss X with a uniform
distribution on (0, 10). What is the maximum amount the decision maker
will pay for complete insurance against the random loss?
Solution: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 27/31 Insurance Example: Exponential Utility
A decision maker’s utility function is given by u(w ) = −e−αw . The decision
maker has initial wealth w0 and faces a random loss X . What is the
maximum amount the decision maker will pay for complete insurance
against the random loss?
Solution: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 28/31 Insurance Example: Quadratic Utility
A decision maker’s utility of wealth is given by u(w ) = w − 0.01w 2 w < 50 The decision maker will retain wealth of amount w0 with probability p and
suffer a ﬁnancial loss of amount c with probability 1 − p. Find the maximum
insurance premium that the decision maker will pay for complete insurance.
Solution: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 29/31 Risk Averse Decision Maker & Insurance Premium
A decision maker has initial wealth w0 , utility function u(w),
and faces a loss r.v. X . Let G denote the maximum insurance
premium she is willing to pay to completely insure her loss.
Then
G > µ = E[X ] if the decision maker is risk averse Proof: K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 30/31 Criticism of Expected Utility Theory Some experiments indicate that many people’s behavior in
some situation contradicts expected utility maximization
What is your utility function? K.S. Tan/Actsc 372 F11 Investment Decision Under Uncertainty – p. 31/31 ...
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This note was uploaded on 01/04/2012 for the course ACTSC 372 taught by Professor Maryhardy during the Fall '09 term at Waterloo.
 Fall '09
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