I think it is not clear to all readers why I assert that in the simplest possible model of financial markets all agents will invest proportionally in all tranches of all CDOs. I’m not sure I can present even the simplest model in plain ascii. Also the model is very simple and you will have to trust me (or not) when I claim that the results carry over to standard simple models. There are two key assumptions – 1) all assets are traded and 2) there are no transactions costs. Assumption 1 implies that agents don’t have risks from some source other than asset returns to hedge. In the real world, clearly farmers will have different positions in grain futures than non farmers. Assumption 2 implies that agents will take tiny positions, since there is no problem with odd lot fees or anything like that.
Simple math after the jump.
update: Down in a thread I made a big deal out of someone making an assumption without stating it, so I really should have made my assumptions more explicit (I considered them implied by “standard simple model”).
First I am assuming all agents have rational expectations so they know the conditional joint distribution of all variables conditional on their information.
Second I am assuming that they all have the same information so they agree on the contitional joint distribution of all variables.
Third I assume that there is a maximum possible expected utility for each agent, that is that no agent can achieve any level of expected utility up to infinity. Basically, I assume there are no riskless arbitrage opportunities.
Just to go on and address a point raised in comments a complete model would include a market price for the risky assets in period 1 say p_i. Each risky asset would turn into a stochastic amount of consumption good in period two — say v_i. Then 1+r_i = (v_i/p_i). To close the model, p_i would adjust until demand for the risky assets was equal to supply. This (with the assumptions of symmetric information and rational expectations) implies that there are no riskless arbitrage opportunities.
It also means that there is no excess demand for any risky security in equilibrium. Even if investors really want to hold lots of AAA debt instruments, there will be no shortage of AAA debt instruments, then the price of AAA debt instruments will rise (and the yield decline) until investors want no more than are supplied. In that equilibrium, no profits can be obtained by pooling and tranching and slicing and dicing assets.
End update except for the words and colons “proposition:” “comment:” and “proof:”.
So agents life two periods. There is a risk free asset (Treasury inflation protected securities or TIPS) which pays a return r. There are risky assets which pay r_i (these include ordinary t-bills which are risky because of not completely predictable inflation). Agents have wealth in period 1, they invest and then consume in period two.
Let’s say agent j has a have CRRA utility function with parameter a_j so utility is given by -exp(-a_jC_j) that is –e^(-a_jC_j) where C_j is consumption of agent j equal to agent j’s initial wealth plus the return agent j gets from investing. Agent j’s demand for the risk free asset is d_j. Agent j’s demand for asset i is d_ji.
d_j + sum_i (d_ji )= W_j, that is agent j’s initial wealth. C_j = d_j(1+r) + sum_i (d_ji(1+r_i)). It will help to write C_j = W_j(1+r) + sum_i (d_ji(r_i-r)) that is plug the period 1 budget constraint into the equation for C_j.
Proposition: Then d_ji = is b_i/a_j where b_i is some function of the the joint distribution of the risky returns.
Comment: Oh so note that the ratio of demand for asset i by agent j and agent k is equal to a_k/a_j and is the same for all risky assets. If a_j > a_k then agent j will buy less of each risky asset. However, it is not true that agent j will buy the less risky of the risky assets and agent k won’t.
Proof: This is basic micro. Consider a feasible change in agent j’s portfolio where d_j is decreased by x and d_ij is increased by x. For the optimal portfolio, the derivative of expected utility is zero at x=0. That is
0 = E((a_j exp(-a_jC_j))(r_i-r)) that is the derivative of expected utility is the expected value of the product of the marginal utility of consumption and the derivative of consumption (both of which are stochastic)
That is 0 = Exp(-a_j(w_j(1+r)) (a_j)E( exp-(a_j sum_i ((dji)(r_i-r))(r_i-r))
Dividing both sides by a constant gives
0 = exp(-(a_j sum_i(dji(ri-r))(r_i-r) = exp(-sum_i((a_jdji)(ri-r))(r_i-r)
Notice that the only part of the first order condition which depends on j (the agent) is a_jd_ji so to make the condition hold for all agents it is necessary that d_ji is of the form b_i/a_j so the first order conditions become
0 = exp(-sum_i((b_i)(ri-r))(r_i-r)
b_i is the solution to this equation.
If agents all have constant absolute risk aversion, then their demand for all risky assets is proportional. Agent j’s demand is a constant (whichdepends on the joint distribution of the returns) divided by agent j’s coefficient of risk aversion.
This means that agent j will buy equal amounts of all tranches of a CDO undoing the tranching.
A similar result holds for constant relative risk aversion. In that case, demand for risky assets is proportional to wealth divided by the coefficient of relative risk aversion.
The world lasts longer than two periods. The result carries over to optimal investment in continuous time (I won’t show the proof – trust me or don’t trust me).
In the standard simple models of asset demand, the profitability and existence of tranching is not explained by differences in risk aversion across agents. The ratio of demand for the safest tranche (which bears approximately only inflation risk) and the riskiest tranche is the same for extremely risk averse and less risk averse agents.