Review Session Prof. Manuela Pedio 20135 Theory of Finance 12 October 2018
Three most common utility functions (1/3) We typically assume that investors are non satiated (they always prefer more to less) and risk averse Three most common VNM felicity functions of wealth that respect these two assumptions Negative exponential utility which is always positive when, ensuring that non satiation is respected which is always negative => concavity implies risk-averse investors ARA(W)= (IRRA) RRA(W)=ARA(W)W= 2
Three most common utility functions (2/3) Power utility wealth is positive which is always positive since which is always negative => concavity implies risk-averse investors ARA(W)= (DARA), RRA(W)= (CRRA) Very plausible behavior that implies that the weights assigned to the risky and the risk-free asset do not depend on wealth 3
Three most common utility functions (3/3) Quadratic utility in order to ensure positivity (and hence non-satiation) we need where is called bliss point which is always negative => concavity implies risk-averse investors ARA(W)= (IARA), RRA(W)= (IRRA) Quadratic function is one (not the only) of the potential foundations of the mean-variance approach 4
Minimum odds and their meaning We know that approximately (i.e., for small enough bets) the following relationship holds Think about what this means: for a given ABSOLUTE size (h) of the bet, a CARA individual would always require the same minimum odd to enter into the bet, whatever the size of her wealth This means that given the RELATIVE size of a bet (with respect to the individual s wealth) a CRRA individual will always require the same minimum odd to enter into the bet 5
Certainty equivalent: an example Suppose that Mary, who is characterized by a power utility function with (i.e., logarithmic, ) is facing a fair bet in which she can either loose or win 100 euros with the same probability. Her wealth is equal to Euros 1000. Compute the certain equivalent The certain equivalent is the CERTAIN amount that makes her indifferent between the amount itself and the bet Therefore, in this case we need to solve 077=>. Note: I changed the size of the bet as the previous result obtained in class was really sensitive to rounding 6
An example on background risk (1/2) The presence of non-tradable, risky labor income modifies the standard mean-variance closed form result as follows Let s see this in action: consider the case of Mary, currently unemployed, that is investing only in two equity indices: E R R 1.5% 0.5% ; Σ 4.5 0 0 2.5 She has κ 0.1 and the risk free rate is R 0.1% Suppose that she finds a job in the energy sector; her new labor income process implies the following covariances with equity index returns: Cov Y,R =6 Cov Y,R =2. 7
An example on background risk (2/2) Her optimal risky portfolio before finding a job was ω ω 1 0.1 0.05 0 0 0.16 Σ E R R ι 1.5 0.1 0.5 0.1 1 0.1 0.05 0 0 0.16 1.4 0.4 0.69 0.64 To compute her new optimal portfolio once she finds a job we need to compute Σ σ that is 0.05 0 0 0.16 6 2 The new weights are obtained by subtracting this vector from the original weights Probably also the weight invested in cash will change 8
Appendix: Cass-Stiglitz and quadratic utility A quadratic utility function is a special case of where A=1, and C=1 However, for Cass-Stiglitz theorem to apply we need the following condition to be satisfied :, and (the alternative set of conditions is never met as B is negative) This latter condition means that BELOW THE BLISS POINT, i.e., we are We cannot say that Cass-Stiglitz holds in general under quadratic utility, but we can apply it provided that we stay below the bliss point for all levels of wealth, initial and final 9