Asset Pricing Theory – Assignment (by two)
Due date: 11:59 pm, November 15th 2024
1 CAPM in a CARA-Normal Setup
Consider an economy with 2 dates t = 0 and t = 1
Financial markets:
● At t = 0, K risky securities can be traded at price pk , as well as a risk-free security, at price pf .
● At t = 1, risky security k pays of a random amount per share equal to ˜(a)k ~ N(μk , σk(2)), with Var[˜(a)] the variance-covariance matrix which is invertible by assumption. The risk-free security always pays 1 per share.
● Denote p, = (p1, . . . , pK ) the vector of prices, pf = , and ˜(a), = (˜(a)1 , . . . , ˜(a)K ) the vector of random pay-ofs
Investors:
● I investors can trade on financial markets
● At t = 0, investor i ∈ {1, . . . , I} already holds a portfolio of risky securities zi (0) = (z1(i)(0), . . . , zK(i) (0)) (no risk-free security).
● Each investor will trade to obtain a new portfolio of risky securities, (zi ), = (z1(i) , . . . , zK(i)),
and risk-free security zf(i) .
● At t = 1, on top of the portfolio’s pay-of, investor i receives a random wage ξi.
● All investors agrees about the pay-ofs distributions
● For a risky pay-of ˜(c)i at t = 1, investor i draws at t = 0 an expected utility, E[ui (˜(c)i )], with ui (x) = -τi exp(-x/τi ). (τi = risk tolerance = 1/risk aversion).
Questions:
1. For given prices, compute the initial wealth of investor i.
2. Define the aggregate supply of security k , Sk , for all k ∈ {1, . . . , K}.
3. Give the market equilibrium conditions for all financial markets (risk-free security in- cluded).
4. Give the investor i budget constraint at t = 0.
5. Show that investor i objective function, is equivalent to the following mean-variance criterion:
, avec Izi -
6. Determine investor i demand function zi (p).
7. Determine the aggregate demand function z(p) (we will denote τ =εi τi ).
8. Determine the equilibrium vector of prices, p* .
9. Consider the case where there is no random wage (ξi = 0 8i). The objective is to study the risk premium of risky securities.
● Denote ZM = (S1 , S2, . . . , SK ) the market portfolio, pM(*) = (p* )I ZM its price, and˜(a)M =˜(a)I ZM its pay-of.
(i) W˜rite the risk premium of security k , as a of the covariance between
Write the relation between the market portfolio risk premium,
the variance of its pay-of, Var[˜(a)M ].
Write the relation between the risk premium of security k , , and the market portfolio risk premium,
● Let ˜(r)k = - 1 and - 1, the returns of security k and the market
portfolio. Write the risk premium of security k , E[˜(r)k ] - rf as a function of the market portfolio risk premium, E[˜(r)M ] - rf .
2 Arrow-Debreu securities in a CARA-Normal Setup
Consider an economy with two date t = 0 and t = 1. In this economy there is a risky security that pays-of a random amount per share˜(a) ~ N(µ, σ2 ) at t = 1. The asset aggregate supply is Q.
At t = 0 investors pick their portfolios. A portfolio is made of the risky asset and the risk-free asset that pays-of 1 per share at t = 1. The risk-free rate is rf .
Investor i draws some expected utility from her final wealth, ˜(y)i a、t = 1. It is computed as E[ui (yi )] where,
We will denote τ =Σi τi.
1. Show that the equilibrium price of the risky asset is
From now on, we reconsider the problem by an approach 、a la Arrow-Debreu. We assume that at t = 1, there are an infinite number of states of the world. Each state is indexed by the realization of the random variable˜(a) . State of the world a occurs with probability φ(a).da where
In state a, each agent i is endowed with wealth (or consumption units) wi (a) such that
The Arrow-Debreu security indexed by a pays-of one unit of wealth per share in state a, and zero otherwise. Its price (infinitesimal) is denoted q(a).da. We will take the Arrow- Debreu security indexed by a = 0 as the num/eraire, that is q(0) = 1. Function q is assumed integrable.
2. Explain why agent i Lagrangian to be optimized is
3. For an integrable function f we can define the following derivative
Show that the first order conditions of the maximization program, max{yi (a)}a∈R Li , imply
4. Show that the equilibrium prices are
5. Show that the risk free rate is defined as follows
6. Finally, show that a security that pays-of ˜(a) per share at t = 1 has a price equal to