concepts: Certainty Premium - ULisboa II7 B.pdf · 5/19/2010 3 yRisk averse individual: the...

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Certainty equivalentsRisk premiums

Mónica Oliveira, RPEM 2009/201019

Key concepts: Certainty Equivalent and Risk Premium

Which is the amount of money that is equivalent in your y q ymind to a given situation that involves uncertainty?Ex: for how much would you sell this lottery you own?

Win 2000€ with probability 0.5Lose 20€ with probability 0.5

300€: that amount would be the Certainty Equivalent

the gamble will be equal in your mind to a sure 300€!

This amount is the CERTAINTY EQUIVALENT!Ranking alternatives by certainty equivalents is the same as ranking them by their expected utilities!!!!

20Mónica Oliveira, RPEM 2009/2010

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We can infer certainty equivalents...

EU vs. CE

U(500)=0,65 The CE for the low risk stock must be only a little bit more than 500than 500The CE for the high risk stock must be less than 500 (EU=0.638) but not as little as 200If we order by CE, the high risk stock has the lowest CE and is the lowest preferred


Risk Premiums (vs. CE vs. EU)Risk Premium = EMV ‐ Certainty EquivalentRisk Premium EMV Certainty Equivalent

Ex: 680€‐300€=Risk Premium

Graphical representation for risk premium:

Note:


Note:EU(gamble)=

=U(Certainty Equivalent)

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Risk averse individual: the horizontal EU line reaches Risk averse individual: the horizontal EU line reaches the concave utility curve before it reaches the vertical line that corresponds to the expected value Risk premium is positiveCE, EU and Risk premium depend on two factors:

Decision maker’s utility function Probability distribution for the payoffs

If the CE for a gamble is assessed directly, finding the risk premium is straightforward; if not.... Four steps:


Four Steps in Finding the Gamble’s Risk Premium1. Find the EU for the gamble1. Find the EU for the gamble2. Find the Certainty Equivalent, or the sure amount

that has the utility value equal to the EU that was found in 1.

3. Calculate the EMV for the gamble4. Subtract the certainty equivalent from the expected 4 y q p

payoff to find the risk premiumThis is the difference between the expected value of the risky situation and the sure amount for which the risky situation would be traded


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ExampleUsing the utility function below, calculate the risk premium for the following gamble:

Win 4000€ with probability 0.4Win 2000€ with probability 0.2Win 0€ with probability 0.15Lose 200€ with probability 0.25


First step Expected UtilityFirst step Expected UtilityEU=0.4*U(4000€)+0.2*U(2000€)+0.15*U(0€)+0.25*U(‐200€)=0.72

Second step Finding the certainty equivalent for EU=0.72


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Third step:EMV=0.4*4000€+0.2*2000€+0.15*0€+0.25*‐200€=1500€

Fourth step:Fourth step:Risk Premium= 1500€‐400€=1100€


& Risk tolerance and the exponential utility function


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Utility Function AssessmentObjective in decision analysis of making a better Objective in decision analysis of making a better decision need to:

Construct a model or representation of decision, which might need assessing a utility function.When we assess a utility function, we are constructing a mathematical model or representation of preferencespreferences...The objective is to find a way to represent preferences that incorporates risk attitudes. A perfect representation is not necessary!


Utility Function AssessmentAssessing a utility function is a matter of subjective g y jjudgement!

Two utility assessment approaches based on the CE concept:

Alternative approaches

concept:1. Assessing using Certainty Equivalents2. Assessing using ProbabilitiesAssessment using a mathematical function


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Utility Assessment using Certainty Equivalents

The decision maker assesses several Certainty The decision maker assesses several Certainty Equivalents

A ‘reference gamble’ for assessinga utility function your job is to findthe CE so that you are indifferentto options A and B

Get the first two points of your utility function by arbitrarily setting U(100)=1 and U(10)=0Find B=30 U(30)=0.5*U(100)+0.5*U(10)=0.5


Utility Assessment using Certainty Equivalents

Change the game for: Win 100 with probability 0.5 or win g g p y 530 with probability 0.5, and find another point…

Find B=50 U(50)=0.5*U(100)+0.5*U(30)=0.5*1+0.5*0.5=0.75

Change the game for: Win 30 with probability 0.5 or win 10 with probability 0 5 and find another point with probability 0.5, and find another point…

Find B=18 U(18)=0.5*U(30)+0.5*U(10)=0.5*0.5+0.5*0=0.25

Iterate…


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And you will get...


An appropriate representation of your utility function!

Utility Assessment using Probabilities

The decision maker assesses the probability in the reference bl hi i diffgamble to achieve indifference

A ‘reference gamble’ for assessingthe utility of 65 using theprobability‐equivalent method

Find pGet the first two points of your utility function by arbitrarily setting U(100)=1 and U(10)=0Compute:

U(65)=p*U(100)+(1‐p)*U(10)=pIf you choose p=0.87, then U(65)=0.87


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What if an individual distastes games?In the assessment of subjective probabilities, we have j p ,framed utility assessment in terms of gambles and lotteries... BUT...For many individuals this evokes images of carnival games or gambling in a casino, images that may seem irrelevant to the decision at hand or even distasteful. Which alternatives?

An alternative is to think about investments that are risky!Change the gamble to whether you would make a particular investment.


Assessment using a Mathematical FunctionLet us consider an exponential utility function:

Concave risk averse preferencese = 2.71828As x becomes large, U(x) approaches 1U(o) equals 0U ili f i (b i i d b ) i i

Rx

exU−

−=1)(

Utility for negative x (being in debt) is negativeR is a parameter that determines how risk averse is the utility function called risk tolerance (larger values of Rmake the utility function flatter; smaller values of R make a more curved utility function)

How can R bedetermined?


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Assessing your Risk Tolerance (I)An example to find R: An example to find R: Game –Find the largest Y for which you would prefer:

Win Y€ with probability 0.5Lose Y/2€ with probability 0.5

If Y=900€, hence R=900€, which would result in 9001)(

x

exU−

−=37


Assessing your Risk Tolerance (II)Once you have R and an exponential utility function, it Once you have R and an exponential utility function, it is easy to find the CE, e.g., if you have:Win 2000€ with probability 0.4Win 1000€ with probability 0.4Win 500€ with probability 0.2

The expected utility is:EU=0 4*U(2000)+o 4*U(1000)+0 2*U(500)=0 7102EU=0.4 U(2000)+o.4 U(1000)+0.2 U(500)=0.7102

Which implies that:

71.111417102.0 900 ==⇔−=−

xCEex

An approximation applies:

900)600(*5.01300

)(5.0

2

−=

−≈nceRiskTolera

VariancelueExpectedVaCE


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Still on Risk ToleranceIndividuals’ risk tolerance differs from corporate risk tolerance:

Individuals’ risk tolerance depends on individuals risk attitudeA board of directors might adopt a decision‐making attitude based on corporate goals and acceptable risk levels for the corporation

Howard (1988) suggests guidelines for determining a corporation’s risk tolerance in terms of total sales, net income, or equity. Reasonable risk have been 6.4% of total sales, 1.24 times net income, or 15,7% of equity (values based on consultancy)

In the exponential function, it is assumed a constant risk p ,aversion (no matter how much wealth you have, you would view a particular gamble in the same way) this might not be reasonableDecreasing risk aversion: )ln()( xxU =



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The Precision Tree software incorporates functionalities to deal with UTILITYdeal with UTILITY


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