Microeconomics Principles and Risk Analysis

Prerequisites Almost essential Risk Risk Taking MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

Overview... Risk Taking Trade and equilibrium Extending the exchange economy Individual optimisation The portfolio problem

Trade • Consider trade in contingent goods • Requires contracts to be written ex ante. • In principle we can just extend standard GE model. • Use prices piw: • price of good i to be delivered in state w. • We need to impose restrictions of vNM utility. • 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

a pRED – ____ pBLUE a b pRED – ____ pBLUE b a b b a xBLUE xBLUE xRED xRED Contingent goods: equilibrium trade • Certainty line for Alf • • Alf's indifference curves Ob • Certainty line for Bill • Bill's indifference curves • Endowment point • Equilibrium prices & allocation • Contract curve Oa

Trade: problems • 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. • Take a particularly simple form of asset: • a “contingent security” • pays $1 if state w occurs. • Can we use this to simplify the problem?

Financial markets? • The market for financial assets opens in the morning. • Then the goods market is in the afternoon. • We can use standard results to establish that there is a competitive equilibrium. • Instead of n markets we now have n+. • 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?

Overview... Risk Taking Trade and equilibrium Modelling the demand for financial assets Individual optimisation The portfolio problem

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

Interpretation 1: portfolio problem  • You have a determinate (non-random) endowment y • 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: • rº < 0 – corresponds to state BLUE • r' > 0 – corresponds to state RED • Consider attainable set if you buy an amount b of bonds where 0 ≤ b ≤ y 

_ _ y+br′, y+br _ _ [1+r′ ]y, [1+r]y Attainable set: safe and risky assets • Endowment xBLUE • 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 _ _ • P y • P0 unlikely to be points here _ [1+rº]y A xRED _ _ [1+r' ]y y

Interpretation 2: insurance problem • You are endowed with a risky prospect • Value of wealth ex-ante is y0 . • There is a risk of loss L. • If loss occurs then wealth is y0 – L. • You can purchase insurance against this risk of loss • Cost of insurance is k. • In both states of the world ex-post wealth is y0 – k. • Use the same type of diagram.

Attainable set: insurance • Endowment xBLUE • Full insurance at premium k • All these points belong to A • Can you overinsure? • Can you bet on your loss? unlikely to be points here _ _ partial insurance • P y L – k unlikely to be points here • P0 y0– L A k xRED _ y0 y

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

A Consumer choice with a variety of financial assets • Payoff if all in cash • Payoff if all in bond 2 • Payoff if all in bond 3, 4, 5,… • Possibilities from mixtures xBLUE • Attainable set • The optimum 1 2 • only bonds 4 and 5 used at the optimum 3 4 4 P* 5 5 6 xRED 7

Simplifying the financial asset problem • 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 #W = 2, a maximum of two assets are used in the optimum. • So the two-asset model of consumer optimum may be a useful parable. • Let’s look a little closer.

Overview... Risk Taking Trade and equilibrium Safe and risky assets  comparative statics Individual optimisation The portfolio problem

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

Distribution of returns (general case) • plot density function of r f (r) • loss-making zone • the mean r Er

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

_ _ • P y A _ y Consumer choice: safe and risky assets • Attainable set, portfolio problem. xBLUE • Equilibrium -- playing safe • Equilibrium - "plunging" • Equilibrium - mixed portfolio P* P0 xRED

Results (1) • Will the agent take a risk? • Can we rule out playing safe? • Consider utility in the neighbourhood of b= 0 • Eu(y + br) | ———— | = uy(y )Er b|b=0 • uy is positive. • So, if expected return on bonds is positive, agent will increase utility by moving away from b= 0.  

Results (2) • Take the FOC for an interior solution. • Examine the effect on b* of changing a parameter. • For example differentiate E(ruy(y + b*r)) = 0 w.r.t.y • E(ruyy(y + b*r)) + E(r2 uyy(y + b*r))b*/y = 0 • b*– E(ruyy(y + b*r)) —— = ———————— y E(r2 uyy(y + b*r)) • Denominator is unambiguously negative • What of numerator?        

Risk aversion and wealth • To resolve ambiguity we need more structure. • 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

_ _ y+d y+d _ y A _ y An increase in endowment • Attainable set, portfolio problem. xBLUE • DARA Preferences • Equilibrium • Increase in endowment • Locus of constant b • New equilibrium P** P* try same method with a change in distribution xRED

A rightward shift • original density function f (r) • original mean • shift distribution by t • will this change increase risk taking? r t

_ _ • P y A _ y A rightward shift in the distribution • Attainable set, portfolio problem. xBLUE • DARA Preferences • Equilibrium • Change in distribution • Locus of constant b • New equilibrium P* P** What if the distribution “spreads out”? P0 xRED

_ _ _ _ y+b*r′, y+b*r • P y A _ y An increase in spread • Attainable set, portfolio problem. xBLUE • Preferences and equilibrium • Increase r′, reduce r P* • P* stays put • So b must have reduced. • You don’t need DARA for this P0 xRED

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

Microeconomics Principles and Risk Analysis