Chapter 6: Risky Securities and Utility Theory Topics 1. Principle of Expected Return 2. St. Petersburg Paradox 3. Utility Theory 4. Principle of Expected Utility 5. The Certainty Equivalent 6. Utility and Risk Attitudes 7. Measures of Risk Aversity
Background 1 1. A financial security is a document that records some entitlement or obligation and which can be traded for money. 2. E.g. stock, bonds, options, treasury bills etc. 3. The return and fair price or value of a riskless security are known with certainty 4. If the return of a (financial) security is risky, i.e. it is a random variable, then we ask how to value or price the (risky) security.
Background 2 1. To this end, consider a security which pays a single amount A. Since the security is risk-less its value is clearly S 0 = A. 2. If, however, A is a random variable then things change somewhat. To make things clear suppose that a (risky) security has value A 1 if some event E 1 occurs and A 2 otherwise. In this scenario the securities value is a discrete binomial random variable. Define probabilities P(E 1 ) = p; P(E 2 ) = 1 p.
Background 3 A sensible approach to pricing might be to take the value of the security as S 0 = pa 1 + (1 p)a 2, i.e. weighted sum of the possible payoff where the weights are the same as the probability of that event occuring. This is just pricing by expectation!
Principle of Expected Return 1. We have a risky security whose payoff is some random variable, say R. We assume that the distribution of R is known. 2. We then price the security by S 0 = E(R). 3. This way of pricing is called the Principle of Expected Return. 4. This method of pricing has some problems which we turn to now.
St. Petersburg Paradox 1 1. There is a problem with pricing with the Principle of Expected Return: 2. QUESTION: What is the fair price of a coin tossing game which pays an amount C t = 2 t 1 A if the first head appears on toss number t? 3. This is an example of a single pay-off at an unknown time (toss) in the future, with zero interest rates. 4. Payoff is time(toss)-dependent
St. Petersburg Paradox 2 1. The first t tosses must be TTT... TH which occurs with probability p t = 1/2 t. 2. The payoff for this example is C t = 2 t 1 A 3. By the Principle of Expected Return S 0 = E (C T ) = p t C t = t=1 = A 2 + A 2 + A 2 +... = 2 t 1 A 2 t = A 2 t=1 t=1 4. This appears nonsensical.
Utility Theory 1. Utility is a hypothetical quantity that is meant to measure how much satisfaction a sum of money is meant to give, 2. There is obviously more than one utility function. 3. There are a number of properties that utility functions are generally supposed to possess. 3.1 The law of non-satiety: you can never have too much money, i.e. U (x) > 0 3.2 The law of diminishing returns: an increment of wealth on a large fortune has less utility than an equal increment on a small fortune, i.e. U (x) < 0
Utility Theory 2 1. The law of diminishing returns may be interpreted as follows: If we thnk of U (x) as a measure of the satisfaction associated with a small increment in cash returns, U (x) < 0 means that the prospect of an extra dollar or two is less good if you are getting a lot of cash anyway. 2. The two conditions we supposed implies that U(x) is a monotonc increasing, concave function of wealth x.
Utility Theory 3 1. Since U is an increasing function, an infinitesimal increment in x leads to a positive increment du in utility. 2. Some people have argued that a good model for risk-averse investors is one for which the increment in utility du is proportional to the increment dx and inversely proportional to the resulting wealth x + W wher W denotes the individual s current wealth. 3. This ensures that an individual s satisfaction is determined by the proportion by which the investor s total wealth increases, rather than the actual value of the increase.
Utility Theory 4 This model implies that there exists a positive constant b such that du = bdx/(x + W ). The resulting ODE for U has solution U(x) = b ln( x + W W ) + a = b ln(1 + x W ) + a, where a is a constant of integration and U(0) = 0 means that a = 0
Principle of Expected Utility 1 1. Principle of Expected Utility is that an individual will aim to maximise expected utility rather than expected return. 2. The Principle of Expected Utility can be used to price securities.
Principle of Expected Utility 2 1. Suppose current purchase price of a security is S and that after its purchase the security will produce cash-flows with a combined value X 2. The net return from the security is X S and so the expected utility is E (U(X S)). 3. Is the investment better than doing nothing? If we do nothing then E (U(0)) = U(0). 4. According to the Principle of Expected Utility we must compare the expected utilities.
Principle of Expected Utility 3 1. Clearly, the purchase of the security is worthwhile if and only if E (U(X S)) > U(0). 2. it is not worthwhile if E (U(X S)) < U(0) 3. if E (U(X S)) = U(0) then neither alternative is preferrable. 4. Hence the root S 0 of the equation E (U(X S 0 )) = U(0) (1) is the price at which buying the security is neither advantageuos or disadvantageous. 5. The root S 0 of Equation (1) is clearly the maximum price at which the purchase of the security might be considered.
Principle of Expected Utility 4 Leave end of calculations as home work. [Do equations on black-board] 1. The price from the Principle Expected Utility is less than the price based on the Principle of Expected Return. 2. The discrepancy between the two prices is called the risk premium because it is a monetary measure of the extent to which the investors valuation of of the security is diminished due to the uncertainty in the reu=turn. In most cases, the risk premium is a decreasing function of the investor s wealth.
Principle of Expected Utility 5 Theorem Investment decisions based on the Principle of Expected Utility are unchanged by positive linear transformation. 1. Say that U(x) = a + bu(x) is a positive linear transformation of the function U(x) if a and b are constants with b > 0. 2. Let X 1 and X 2 be two random variables that represent net returns from two different investments. Also suppose that E (U(X 1 )) > E (U(X 2 )), i.e. choose investment 1 over investment 2 by the Principle of Expected Utility.
Principle of Expected Utility 6 1. Then E ( U(X 1 ) ) E ( U(X 2 ) ) = E (a + bu(x 1 )) E (a + bu(x 2 )) = b(e (U(X 1 )) E (U(X 2 ))) > 0 so E ( U(X 1 ) ) > E ( U(X 2 ) ) and the investor will make the same decision. NOTE: utility functions which are related by positive linear transformations are said to be equivalent
Principle of Expected Utility 8 Corollary Prices calculated using the principle of expected utility are unaffected by positive linear transformations.
Principle of Expected Utility 7 St. Petersburg Paradox To determine the maximum price S 0 that an investor should consider paying to play the game we need to solve the equation U(0) = E (U(C T S 0 )) = p t U(C T S 0 ) t=1
St Petersburg Paradox If we price via expected utility the fair price of the coin tossing game is finite.
The Certainty Equivalent 1 1. A security received as a gift may be valued differently from one which is purchased. 2. Consider the scenario: 2.1 A fixed amount of cash C 2.2 a risky security with net return X If the investor takes the cash the utility will be U(C) if she takes the risky security the expected utility is E(U(X )) According to the Principle of Expected Utility chose the security if E(U(X )) > U(C) and the cash if E(U(X )) < U(C)
The Certainty Equivalent 2 But if E(U(X )) = U(C) the cash and the security are equally attractive. THEREFORE the particular value C may be thought as the cash value of the security when received by the investor as a gift. The valuie of C which solves this equation is referred to as the certainty equivalent value of the security to the investor
The Certainty Equivalent 3 NOTE: the certainty equivalent may differ slightly from the previously introduced quantity S 0 which represented the maximum price at which the investor would consider purchasing the security. (In general, C is slightly higher than S 0 because the investor will be less concerned about the risks associated with a gift than a purchase.) The certainty equivalent is generally a very good approximation to the max. price S 0 the investor might consider paying. As it is usually easier to calculate than S 0, it is much more widely used.
Utility and Risk Attitudes 1 1. Since we have concave utility functions we always have U(E(X )) > E(U(X )) by Jensen s Inequality 2. This inequality is generally true for any random variable and any concave funcion, for e.g. U 3. The economic interpretation is that an investor with utility function U will prefer the certainty of receiving an amount of cash, here EX, to a risky security with return X.
Utility and Risk Attitudes 2 1. The risk premium is given by ρ = E(X ) U 1 (E(U(X ))) 2. It represents the amount an investor with utility function U must be compensated to accept the risky investment instead of a riskless investment with the same expected net return. 3. It s also the difference between the expected return price and certainty equivalent of the security. 4. Note that we have used that U is monotonic implies that U has a well-defined inverse 5. For a risk-averse investor the risk premium ρ is always positive
Utility and Risk Attitudes 3 Type U (x) ρ Risk Attitudes Risk averse < 0 (concave) ρ > 0 Prefers certainty to risk for the same expected return Risk neutral 0 (linear) ρ = 0 Indifferent between certainty and risk for the same expected return Risk loving > 0 (convex) ρ < 0 Prefers risk to certainty for (Black-board U plots) the same expected return.
Utility and Risk Attitudes 4 Note here that risk-neutral investors have linear utility functions U(x) = a + bx (equivalently U(x) = x. and the Principle of Expected Utility reduces to the Principle of Expected Return.
Blakcboard: Example
Measures of Risk Aversity 1 1. Risk premium ρ = E(X ) U 1 (E(U(X ))) 2. Absolute risk aversion 3. Relative risk aversion ρ abs = U (x) U (x) ρ rel = xu (x) U (x)
Measures of Risk Aversity 2 In all cases we have the classification 1. ρ > 0 (risk aversion) 2. ρ = 0 (risk neutrality) 3. ρ < 0 (risk loving)
Example: p84