Lecture 6: Risk Sharing and Asset Pricing

1. Lecture 6: Risk Sharing and Asset Pricing • The following topics will be covered: • Pareto Efficient Risk Allocation • Defining Pareto Improvement and Pareto Efficient • Mutuality Principle • Optimal Risk Sharing • Asset Pricing • First Theorem of Welfare Economics • Equity Premium • CAPM • Interest rate determination L6: Risk Sharing and Asset Pricing

2. Pareto Efficient • Given a set of alternative allocations of say goods or income for a set of individuals, a movement from one allocation to another that can make at least one individual better off, without making any other individual worse off, is called a Pareto improvement. An allocation is Pareto efficient or Pareto optimal when no further Pareto improvements can be made. This term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution. L6: Risk Sharing and Asset Pricing

3. An Example of Optimal Risk Sharing • xs and xj have independent risks, distributed as (0, ½; 8000, ½) • Sempronius and Jacobus decide to form a joint venture, each taking care of a half of the pooled risk (fifty-fifty split) • Aj is less As • They have mean-variance preferences, thus L6: Risk Sharing and Asset Pricing

4. Example (Cont’d) • Suppose J will buy an 1% of the firm • The payment S ask for is: L6: Risk Sharing and Asset Pricing

5. Optimal Allocation L6: Risk Sharing and Asset Pricing

6. Pareto Efficient in a More General Setting • n risk-averse agents • Individual I has a twice differentiable increasing and concave utility function, ui, i=1, …, n • Agent I is endowed with wi(s) units of a single consumption good in state s • Agent I faces risk if there exists at least one pair of states of nature (s, s’) such that wi(s) not = wi(s’) • An allocation is characterized by c1(.), …, cn(.), where ci(s) is the consumption of agent i in state s • Edgeworth Box • Full insurance is not optimal. In other words, full insurance Pareto Efficient usually not feasible • Contract curve – Pareto efficient L6: Risk Sharing and Asset Pricing

7. Pareto Efficient Allocation L6: Risk Sharing and Asset Pricing

8. Solution L6: Risk Sharing and Asset Pricing

9. Mutuality Principle • The Principle: a necessary condition for an allocation of risk to be Pareto efficient is that whenever two states of nature s and s’ have the same level of aggregate wealth, z(s)=z(s’), then for each agent i consumption in state s must be the same as in state s’: ci(s)=ci(s’) for all agent i. • In other words, z is independent of the state of the nature • There is no aggregate uncertainty in the economy, i.e., no macroeonomic risk • It is a case of perfect insurance, see figure 10.3 on page 160 • This is more or less like the case with reinsurance and without catastrophic losses, all insurers share the risk and adequately diversify the risk • Is the insurance market achieving this principle? • However in reality, there are undiversifiable risks -- the economy faces random events that generate phases of recession or expansion • State price depends on z only. L6: Risk Sharing and Asset Pricing

10. Sharing of Macroeconomic Risk L6: Risk Sharing and Asset Pricing

11. Examples of Risk Sharing Rule (1) Risk neutral agent i=1: T1(c)  ∞. I.e., c1’(z)=1 and ci’(z)=0 if i not equal to 1. Agent 1 will bear all the risks. (2) All individuals have constant degree of risk tolerance (CARA) (3) HARA utility: Ti(c)=ti+αc See figure 10.4 L6: Risk Sharing and Asset Pricing

12. Aggregation of Preferences • The utility function of the central planner/representative agent is • The attitude towards risk of any economy implementing a Pareto-efficient allocation is to maximize Ev(z) by selecting z, the aggregate risk. • Risk tolerance of representative investor is L6: Risk Sharing and Asset Pricing

13. What is the life without the central planner? L6: Risk Sharing and Asset Pricing

14. General Solution • Figure 11.1 • AB is a curve for all the competitive equilibrium points, which satisfy the condition (1) v’(z)=µ(z)=π(z) • Combining all individual together, we have: L6: Risk Sharing and Asset Pricing

15. First Theorem of Welfare Economics • If there are two states with the same aggregate wealth, the competitive state prices must be the same, and agent I will consume the same amount of consumption good in these two states. • Individual consumption at equilibrium and competitive state prices depend upon the state only through the aggregate wealth available in the corresponding state. It implies that all the diversifiable risk is eliminated in equilibrium. • Competitive markets allocate the macroeconomic risk in a Pareto-efficient manner: Decentalized decision making yields efficient sharing. • The market in aggregation acts like the representative agent in the centralized economy. L6: Risk Sharing and Asset Pricing

16. Deriving Equity Premium L6: Risk Sharing and Asset Pricing

17. Gauging Equity Premium • The variance of the yearly growth rate of GDP per capita is around 0.0006 over the last century in the United States. Empirical feasible levels of relative risk aversion is between 1 and 4, then equity premium should be between 0.06% and 0.24% per year. • However, the observed equity premium has been equal to 6% over that period. L6: Risk Sharing and Asset Pricing

18. Pricing Individual Assets: Independent Asset Risk • We now examine the pricing of specific assets within the economy • The value of risky asset depends on the state of nature that will prevail at the end of the period • The simplest case: the risk is independent of the market z • The firm generate a value q with probability p and zero otherwise, i.e., (q,p; 0, 1-p). • In other words, whatever the GDP z, the firm generates a value q with probability p and zero otherwise. • When normalizing Eπ(z)=1, we have the expected value of the firm = qp. L6: Risk Sharing and Asset Pricing

19. Pricing Individual Assets: Correlated Asset Risk • What if asset return is correlated with the aggregate risk? • P(q)=Eq(z)π(z)=Eq(z)v’(z) • This is the asset-pricing formula common to all models with complete markets • Using first order Taylor Expansion, we have • γ is the relative risk aversion of the representative agent measured at the expected level of aggregate wealth • This is so called CAPM • It states that asset risk is measured by the covariance of the future asset value and the aggregate wealth in the economy, not the variance of q(z) L6: Risk Sharing and Asset Pricing

20. Risk Free Rate • No longer assume risk free rate to zero • Agent i is now endowed a fixed wealth of wi0 units of the consumption good at date 0, and with the state contingent claim for wi(s) unit at the second date in state s, for all s. • Aggregate wealth at date 0 is • The agent maximizes the discounted value of flow of EU over his lifetime: by choosing optimal ci0, ci(.) • Where ci0 is the consumption at date 0; ci(s) is consumption in date 1 in state s, and β is the rate of impatience • Subject to the constraint L6: Risk Sharing and Asset Pricing

21. Solution L6: Risk Sharing and Asset Pricing

22. Factors Affecting the Interest Rate • If x=0, use to denote the rate of pure preference for the present, then r=δ • Let B1=(1+r)-1 denote the price of a pure-discount bond. • If x=g>0, then B1<β, i.e., r>δ • Wealth effect • The price of zero-coupon bond can be written as: • The following factors matter: • x  lower the value of the debt • β ( increase the value of the debt • γ (constant risk aversion)  lower the value of the debt L6: Risk Sharing and Asset Pricing

23. Exercises • EGS, 10.1; 10.4; 10.5; 11.1; 11.5 L6: Risk Sharing and Asset Pricing