Certainty equivalents Risk premiums 19 Key concepts: Certainty Equivalent and Risk Premium Which is the amount of money that is equivalent in your mind to a given situation that involves uncertainty? Ex: for how much would you sell this lottery you own? Win 2000 with probability 0.5 Lose 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!!!! 20 1
EU vs. CE We can infer certainty equivalents... U(500)=0,65 The CE for the low risk stock must be only a little bit more than 500 The CE for the high risk stock must be less than 500 (EU=0.638) but not as little as 200 If we order by CE, the high risk stock has the lowest CE and is the lowest preferred 21 Risk Premiums (vs. CE vs. EU) Risk Premium = EMV Certainty Equivalent Ex: 680 300 =Risk Premium Graphical representation for risk premium: Note: EU(gamble)= =U(Certainty Equivalent) 22 2
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 positive CE, 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: 23 Four Steps in Finding the Gamble s Risk Premium 1. Find the EU for the gamble 2. 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 gamble 4. Subtract the certainty equivalent from the expected payoff to find the risk premium This is the difference between the expected value of the risky situation and the sure amount for which the risky situation would be traded 24 3
Example Using the utility function below, calculate the risk premium for the following gamble: Win 4000 with probability 0.4 Win 2000 with probability 0.2 Win 0 with probability 0.15 Lose 200 with probability 0.25 25 First step Expected Utility EU=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 26 4
Third step: EMV=0.4*4000 +0.2*2000 +0.15*0 +0.25* 200 =1500 Fourth step: Risk Premium= 1500 400 =1100 27 & Risk tolerance and the exponential utility function 28 5
Utility Function Assessment 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 preferences... The objective is to find a way to represent preferences that incorporates risk attitudes. A perfect representation is not necessary! 29 Utility Function Assessment Assessing a utility function is a matter of subjective judgement! Alternative approaches Two utility assessment approaches based on the CE concept: 1. Assessing using Certainty Equivalents 2. Assessing using Probabilities Assessment using a mathematical function 30 6
Utility Assessment using Certainty Equivalents The decision maker assesses several Certainty Equivalents A reference gamble for assessing a utility function your job is to find the CE so that you are indifferent to options A and B Get the first two points of your utility function by arbitrarily setting U(100)=1 and U(10)=0 Find B=30 U(30)=0.5*U(100)+0.5*U(10)=0.5 31 Utility Assessment using Certainty Equivalents Change the game for: Win 100 with probability 0.5 or win 30 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 Find B=18 U(18)=0.5*U(30)+0.5*U(10)=0.5*0.5+0.5*0=0.25 Iterate 32 7
And you will get... An appropriate representation of your utility function! 33 Utility Assessment using Probabilities The decision maker assesses the probability in the reference gamble to achieve indifference A reference gamble for assessing the utility of 65 using the probability equivalent method Find p Get the first two points of your utility function by arbitrarily setting U(100)=1 and U(10)=0 Compute: U(65)=p*U(100)+(1 p)*u(10)=p If you choose p=0.87, then U(65)=0.87 34 8
What if an individual distastes games? In the assessment of subjective probabilities, we have 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. 35 Assessment using a Mathematical Function Let us consider an exponential utility function: x R U ( x) = 1 e Concave risk averse preferences e = 2.71828 As x becomes large, U(x) approaches 1 U(o) equals 0 Uili Utility for negative x (being in debt) b) is negative R is a parameter that determines how risk averse is the utility function called risk tolerance (larger values of R make the utility function flatter; smaller values of R make a more curved utility function) How can R be determined? 36 9
Assessing your Risk Tolerance (I) An example to find R: Game Find the largest Y for which you would prefer: Win Y with probability 0.5 Lose Y/2 with probability 0.5 If Y=900, hence R=900, which would result in U ( x) = 1 e x 900 37 Assessing your Risk Tolerance (II) 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 0.4 Win 1000 with probability 0.4 Win 500 with probability 0.2 The expected utility is: EU=0.4*U(2000)+o.4*U(1000)+0.2*U(500)=0.7102 Which implies that: An approximation applies: x 900 0.7102 = 1 e CE = x = 1114.71 0.5( Variance) CE ExpectedValue RiskTolerance 2 0.5*(600) = 1300 900 38 10
Still on Risk Tolerance Individuals risk tolerance differs from corporate risk tolerance: Individuals risk tolerance depends on individuals risk attitude A 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 aversion (no matter how much wealth you have, you would view a particular gamble in the same way) this might not be reasonable Decreasing risk aversion: U ( x) = ln( x) 39 40 11
The Precision Tree software incorporates functionalities to deal with UTILITY 41 12