1. Frank Cowell: Microeconomics Risk Taking MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Almost essential Risk Prerequisites November 2006

2. Frank Cowell: Microeconomics Economics of risk taking In the presentation Risk we examined the meaning of risk comparisons In the presentation Risk we examined the meaning of risk comparisonsRisk  in terms of individual utility  related to people’s wealth or income (ARA, RRA). In this presentation we put to this concept to work. In this presentation we put to this concept to work. We examine: We examine:  Trade under uncertainty  A model of asset-holding  The basis of insurance

3. Frank Cowell: Microeconomics Overview... Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Extending the exchange economy

4. Frank Cowell: Microeconomics Trade Consider trade in contingent goods Consider trade in contingent goods Requires contracts to be written ex ante. Requires contracts to be written ex ante. In principle we can just extend standard GE model. In principle we can just extend standard GE model. Use prices p i  : Use prices p i  :  price of good i to be delivered in state . We need to impose restrictions of vNM utility. We need to impose restrictions of vNM utility. An example: An example:  Two persons, with differing subjective probabilities  Two states-of the world  Alf has all endowment in state BLUE  Bill has all endowment in state RED

5. Frank Cowell: Microeconomics Contingent goods: equilibrium trade OaOa ObOb x RED a b x BLUE b a  RED – ____  BLUE  RED – ____  BLUE b b Contract curve   Certainty line for Alf  RED – ____  BLUE  RED – ____  BLUE a a   Alf's indifference curves   Certainty line for Bill   Bill's indifference curves   Endowment point   Equilibrium prices & allocation

6. Frank Cowell: Microeconomics Trade: problems Do all these markets exist? Do all these markets exist?  If there are  states-of-the-world... ...there are n  of contingent goods.  Could be a huge number Consider introduction of financial assets. Consider introduction of financial assets. Take a particularly simple form of asset: Take a particularly simple form of asset:  a “contingent security”  pays $1 if state  occurs. Can we use this to simplify the problem? Can we use this to simplify the problem?

7. Frank Cowell: Microeconomics Financial markets? The market for financial assets opens in the morning. The market for financial assets opens in the morning. Then the goods market is in the afternoon. Then the goods market is in the afternoon. We can use standard results to establish that there is a competitive equilibrium. We can use standard results to establish that there is a competitive equilibrium. Instead of n  markets we now have n+ . Instead of n  markets we now have n+ . But there is an informational difficulty But there is an informational difficulty  To do your financial shopping you need information about the afternoon  This means knowing the prices that there would be in each possible state of the world  Has the scale of the problem really been reduced?

8. Frank Cowell: Microeconomics Overview... Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Modelling the demand for financial assets

9. Frank Cowell: Microeconomics Individual optimisation A convenient way of breaking down the problem A convenient way of breaking down the problem A model of financial assets A model of financial assets Crucial feature #1: the timing Crucial feature #1: the timing  Financial shopping done in the “morning”  This determines wealth once state  is realised.  Goods shopping done in the “afternoon.”  We will focus on the “morning”. Crucial feature #2: nature of initial wealth Crucial feature #2: nature of initial wealth  Is it risk-free?  Is it stochastic? Examine both cases Examine both cases

10. Frank Cowell: Microeconomics Interpretation 1: portfolio problem You have a determinate (non-random) endowment y You have a determinate (non-random) endowment y You can keep it in one of two forms: You can keep it in one of two forms:  Money – perfectly riskless  Bonds – have rate of return r: you could gain or lose on each bond. If there are just two possible states-of-the-world: If there are just two possible states-of-the-world:  rº < 0 – corresponds to state BLUE  r' > 0 – corresponds to state RED Consider attainable set if you buy an amount  of bonds where 0 ≤  ≤ y Consider attainable set if you buy an amount  of bonds where 0 ≤  ≤ y  

11. Frank Cowell: Microeconomics Attainable set: safe and risky assets x BLUE x RED   P   P 0 y y _ _ _ A   Endowment   If all resources put into bonds   All these points belong to A   Can you sell bonds to others?   Can you borrow to buy bonds? unlikely to be points here   If loan shark is prepared to finance you [1+rº]y _ [1+r' ]y _ y+  r′, y+  r  _ _  +r′  y,  +r  y _ _

12. Frank Cowell: Microeconomics Interpretation 2: insurance problem You are endowed with a risky prospect You are endowed with a risky prospect  Value of wealth ex-ante is y 0.  There is a risk of loss L.  If loss occurs then wealth is y 0 – L. You can purchase insurance against this risk of loss You can purchase insurance against this risk of loss  Cost of insurance is .  In both states of the world ex-post wealth is y 0 – . Use the same type of diagram. Use the same type of diagram.

13. Frank Cowell: Microeconomics Attainable set: insurance x BLUE x RED   P y y _ _ _ A   Endowment   Full insurance at premium    All these points belong to A   Can you overinsure?   Can you bet on your loss? unlikely to be points here   P 0 y 0 – L y 0  L –  partial insurance

14. Frank Cowell: Microeconomics A more general model? We have considered only two assets We have considered only two assets Take the case where there are m assets (“bonds”) Take the case where there are m assets (“bonds”) Bond j has a rate of return r j, Bond j has a rate of return r j, Stochastic, but with known distribution. Stochastic, but with known distribution. Individual purchases an amount  j, Individual purchases an amount  j,

15. Frank Cowell: Microeconomics A 1 5 7 4 6 3 2 Consumer choice with a variety of financial assets x BLUE x RED   Payoff if all in cash   Payoff if all in bond 2   Payoff if all in bond 3, 4, 5,…   Possibilities from mixtures   Attainable set   The optimum 5 4 P* P*   only bonds 4 and 5 used at the optimum

16. Frank Cowell: Microeconomics Simplifying the financial asset problem If there is a large number of financial assets many may be redundant. If there is a large number of financial assets many may be redundant.  which are redundant depends on tastes…  … and on rates of return In the case of #  = 2, a maximum of two assets are used in the optimum. In the case of #  = 2, a maximum of two assets are used in the optimum. So the two-asset model of consumer optimum may be a useful parable. So the two-asset model of consumer optimum may be a useful parable. Let’s look a little closer. Let’s look a little closer.

17. Frank Cowell: Microeconomics Overview... Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Safe and risky assets  comparative statics

18. Frank Cowell: Microeconomics The portfolio problem We will look at the equilibrium of an individual risk-taker We will look at the equilibrium of an individual risk-taker Makes a choice between a safe and a risky asset. Makes a choice between a safe and a risky asset.  “money” – safe, but return is 0  “bonds”– return r could be > 0 or 0 or < 0 Diagrammatic approach uses the two-state case Diagrammatic approach uses the two-state case But in principle could have an arbitrary distribution of r… But in principle could have an arbitrary distribution of r…

19. Frank Cowell: Microeconomics Distribution of returns (general case) r f (r)   loss-making zone   the mean   plot density function of r ErEr

20. Frank Cowell: Microeconomics Problem and its solution Agent has a given initial wealth y. Agent has a given initial wealth y. If he purchases an amount  of bonds: If he purchases an amount  of bonds:  Final wealth then is y = y –  +  [1+r]  This becomes y = y +  r, a random variable The agent chooses  to maximise E u(y +  r) The agent chooses  to maximise E u(y +  r) FOC is E ( ru y (y +  * r) ) = 0 for an interior solution FOC is E ( ru y (y +  * r) ) = 0 for an interior solution  where u y () =  u() /  y   * is the utility-maximising value of . But corner solutions may also make sense... But corner solutions may also make sense...     

21. Frank Cowell: Microeconomics A Consumer choice: safe and risky assets x BLUE x RED y y _ _ P* P* P 0     Attainable set, portfolio problem. _   P   Equilibrium -- playing safe   Equilibrium - "plunging"   Equilibrium - mixed portfolio

22. Frank Cowell: Microeconomics Results (1) Will the agent take a risk? Will the agent take a risk? Can we rule out playing safe? Can we rule out playing safe? Consider utility in the neighbourhood of  = 0 Consider utility in the neighbourhood of  = 0  E u(y +  r)   E u(y +  r)  ———— | = u y (y ) E r ———— | = u y (y ) E r  |   |  u y is positive. u y is positive. So, if expected return on bonds is positive, agent will increase utility by moving away from  = 0. So, if expected return on bonds is positive, agent will increase utility by moving away from  = 0.  

23. Frank Cowell: Microeconomics Results (2) Take the FOC for an interior solution. Take the FOC for an interior solution. Examine the effect on  * of changing a parameter. Examine the effect on  * of changing a parameter. For example differentiate E ( ru y (y +  * r) ) = 0 w.r.t. y For example differentiate E ( ru y (y +  * r) ) = 0 w.r.t. y E ( ru yy (y +  * r) ) + E ( r 2 u yy (y +  * r) )  * /  y = 0 E ( ru yy (y +  * r) ) + E ( r 2 u yy (y +  * r) )  * /  y = 0  * – E (ru yy (y +  * r))  * – E (ru yy (y +  * r)) —— = ———————— —— = ————————  y E (r 2 u yy (y +  * r))  y E (r 2 u yy (y +  * r)) Denominator is unambiguously negative Denominator is unambiguously negative What of numerator? What of numerator?      

24. Frank Cowell: Microeconomics Risk aversion and wealth To resolve ambiguity we need more structure. To resolve ambiguity we need more structure. Assume Decreasing ARA Assume Decreasing ARA Theorem: If an individual has a vNM utility function with DARA and holds a positive amount of the risky asset then the amount invested in the risky asset will increase as initial wealth increases Theorem: If an individual has a vNM utility function with DARA and holds a positive amount of the risky asset then the amount invested in the risky asset will increase as initial wealth increases

25. Frank Cowell: Microeconomics A   Attainable set, portfolio problem. An increase in endowment P* P* x BLUE x RED y y _ _ P ** oo oo y+  _ _       DARA Preferences   Equilibrium   Increase in endowment   Locus of constant    New equilibrium try same method with a change in distribution

26. Frank Cowell: Microeconomics A rightward shift r f (r)    original density function   original mean   shift distribution by    will this change increase risk taking?

27. Frank Cowell: Microeconomics x RED A A rightward shift in the distribution x BLUE y y _ _ P ** P* P*   P 0   oo oo   Attainable set, portfolio problem. _   P   DARA Preferences   Equilibrium   Change in distribution   Locus of constant    New equilibrium What if the distribution “spreads out”?

28. Frank Cowell: Microeconomics A An increase in spread x BLUE x RED y y _ _ P* P*   P 0     Attainable set, portfolio problem. _   P   Preferences and equilibrium   Increase r′, reduce r  y+   r′, y+   r  _ _   P * stays put   So  must have reduced.   You don’t need DARA for this

29. Frank Cowell: Microeconomics Risk-taking results: summary If the expected return to risk-taking is positive, then the individual takes a risk If the expected return to risk-taking is positive, then the individual takes a risk If the distribution “spreads out” then risk taking reduces. If the distribution “spreads out” then risk taking reduces. Given DARA, if wealth increases then risk-taking increases. Given DARA, if wealth increases then risk-taking increases. Given DARA, if the distribution “shifts right” then risk-taking increases. Given DARA, if the distribution “shifts right” then risk-taking increases.