7) Uncertainty Flashcards
How does no arbitrage reflect optimality?
-the same good must sell at the same price at different locations
-£1 today = Expected present discounted value of investing £1 (if it is true then you have optimized)
^if this holds, no arbitrage and resources are being used optimally
Explain 1 = E(M (1+ri))
-no arbitrage pricing
-if exceeds equation, there is arbitrage
What is M?
-stochastic discount factor
-measures time preference and risk preference
Explain risk premium?
-paid to hold risky investments
E(ri) - r / 1+r = -cov(M , 1+ri)
Explain how covariance links to risk premium?
-if negative covariance, investors dislike assets that do not do well in recessions so demand a positive risk premium to hold stock
How to calculate covariance?
E(xy) - E(x)E(y)
Risk pooling?
-sum all income and use weight of utilities then redistribute
Risk pooling eq?
(c i1/ cj1) =( ki2 /kj2) =( λ / 1-λ)
rate of MU is equal at all times
and if utility function is the same
If a change in relative MU , then what happens to ratio of weight of utility?
-changes
In the eq c1 + k2/1+r = y , why do we divide by 1+r
just gives you this value of k2 in the present
When marginal utility is equal what is Beta (discount factor) for optimality?
-1/1+r
When calculating discount factor what do we do?
-take derivatives and manipulate so one side is 1
Explain intuitively how discount factor effects optimality?
keeping £1 as good as investing £1
What is optimum for insurance?
-marginal cost = marginal benefit
What are actuary fair prices for insurance?
-price = probability
When insurance market functions optimally what is condition (actuary fair insurance)?
-marginal consumption of c1 is equal to discounted marginal utility in high and low state in future
as probability cancels out and they all equal 1
Expected return of risky asset?
-return from risk free asset + risk premium
Expected inflation?
π^e
we divide by 1+ π^e
If optimising for asset pricing condition?
-cannot invest more/less and increase utility
How to get risk premia condition from asset pricing condition (no arbitrage)?
-use covariance definition
-make 1 = E(X)E(Y) +COV(X,Y)
-use risk free asset , as cov = 0 as risk free return is a constant
we get E(M) = 1/1+r then sub in to 1 = E(X)E(Y) …. then rearrange for risk premia
