as372f11_utility

As372f11_utility - Investment Decision Under Uncertainty Lecture Notes for Actsc 372 Fall 2011 Ken Seng Tan Department of Statistics and Actuarial

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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 Neumann-Morgenstern (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 Neumann-Morgenstern Expected Utility Representation Given: a decision maker is facing two random prospects X and Y (with respective probability distribution) u(w ) is the decision-maker’s utility from outcome w Formally, a utility function is a mapping from the set of outcomes onto the real numbers. von Neumann-Morgenstern 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 50-50 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 risk-neutral; 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: Square-root 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.

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