A First Year (graduate) Microeconomics Lesson
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.
OK, so what if there is a demand to hold all assets in equal proportion, but borrowers are only able to issue a different distribution of liabilities. Say 2 AA securities for every AAA security. Wouldn’t there be profit opportunities for the financial system to synthetically create enough AAA securities to meet the symmetric demand?
*** I’m not sure I can present even the simplest model in plain ascii.***
I’m not sure either
***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.***
Wealth and demand have the same units? I thought demand and supply usually had dimensions of cost/unit? But I’m willing to believe that somewhere there is a constant K — omitted for simplicity — that coverts demand to Doubloons or whatever wealth is measured in. I only ask in case my lack of understanding stems from something deeper.
***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.***
This is an assumption, not a conclusion? Why would I assume that? If true, would it not make tranching pretty much pointless? What if j is restricted by law or policy to only the (purportedly) less risky, AAA rated assets? And what if k is Nassim Nicholas Taleb who is probably looking only for very safe or very risky investments and is uninterested in anything in between? Clearly I’m lost at this point.
***This is basic micro.***
Clearly, I am much too dumb to pass even basic Microeconomics.
RSJ: In theory the prices of assets will change to equate demand and supply. I have updated the post and also discuss the question when replying to VtCodger below. I honestly don’t understand where this idea that there was un-met demand for AAA assets comes from. AAA rated corporate bonds paid positive nominal yields. Their price could have risen to equate demand and supply.
I think the issue is that investord demanded moderately high returns on AAA rated instruments. This was demanding something that could only be produced by inventing riskier AAA rated instruments (that is by destroying the ratings system). This demand (in the sence of I demand that you do the impossible not I order so and so many shares of BP) put intollerable pressure on the ratings agencies and investment banks — tempting them to profit from selling illusions to people who refused to accept reality.
Demand and wealth are both in the same units called wealth (in a richer model this is first period consumption good). The constant suppressed as I describe the asset as a return 1+r_i, supprssing the first period price of the asset. The asset becomes v_i of consumption good in the second period and 1+r_i=v_i/p_i. Even before reading your comment, I realised I should have written this out a bit more.
The statement is a conclusion not an assumption. It was stated before the proof. I should have used proposition proof notation. I will go and do that.
Clearly, I am much to incoherent to teach basic (graduate) micro.
Thanks for both of you for dealing with the math in plain ascii and making useful comments.
“I honestly don’t understand where this idea that there was un-met demand for AAA assets comes from.”
Let me try to list some factors:
* Trade deficit means that China and others remove a few trillion of AAA assets from the market, but they do not have symmetric demand for the other assets.
* Growing indebtedness means that the financial position of borrowers deteriorates (e.g. subprime), so less AAA rated debt is being issued (e.g. sub-prime borrowing)
* Growing assets (e.g. growing wealth inequality) means more demand for all assets, which would be met with a growing supply of less safe assets
* Banks require AAA assets to hold on the asset side of the balance sheet when they take on leverage. Growing leverage in the financial sector also sucks out more AAA assets.
* Some funds require AAA assets as part of their charter.
Frank Portnoy and others on the sell side have noted that there was a large unmet demand for AAA assets.
“In theory the prices of assets will change to equate demand and supply. “
This would allow for arbitrage, right? The price cannot both equate supply and demand and also be equal to the expected return (which is not a function of the lender’s risk aversion or the lender’s total wealth). If you are talking about bananas, then I see your point, but for financial assets, I don’t see how safer assets can have a lower risk-adjusted yield than less safe assets. The financial system will intermediate this just as they do with maturity.
None of this is to say that your other points about fraud, regulatory arbitrage, etc, are not valid.
To put it more succinctly, even without the fraudulent aspects, there would still be a large demand for credit enhancement anytime the psychological biases (otherwise known as “utility”) of households are such that they are willing to accept a lower risk-adjusted return for a safer asset. Then arbitrageurs in the financial system buy the risky asset and sell the safe asset, and they keep expanding their balance sheet until price is no longer distorted by the psychological biases — i.e. no free money according to the model.
When the crisis comes the financial system finds its balance sheet filled with junk.
The domand equals supply condition and the condition that there are no extraordinary risk adjusted returns are the same condition. What should happen in theory if there is excess demand for safe assets is that the returns on safe assets should decline compared to the returns on risky assets.
Notice I said returns not risk adjusted returns. According to the model (which is an efficient markets model) correctly risk adjusted returns are equal on all assets. However, this is just a tautology. There is nothing that says that the appropriate risk adjustment gives a constant quality premium or is given by a simple formula of variances and covariances of returns.
In any case, there are standard finance models in which financial markets clear for any assumptions at all about asset supply. These models obtain market clearing without any opportunities for extraordinary risk adjusted returns. In fact, market clearing and no opportunities for extraordinary risk adjusted returns are the same thing. There is no case in which there can be excess demand for an asset because not enough is supplied.
In your explanation of the demand for AAA assets, the last two partial explanations refer to regulations and charters which explicitly refer to the rating. In my opinion, they are the explanation of the profitability of pooling and tranching. They don’t appear in the simple model. Aside from that, shifts in demand for and supply of AAA rated assets compared to other assets should have caused a change in the relative price (the appropriate risk adjustment to returns) but should not cause gains from pooling and tranching.
Basically the EMH states that unless you have information unavailable to the public or some risk from non traded assets (like your labor income) that you want to hedge, you should buy and hold a mix of the market portfolio and a position in the risk free asset (which is often a short position which is possible via a repo or brokerage accont). This means that it is inconsistent with profits from pooling and tranching. Beh so what, who ever believed the EMH? I’d say obviously the people who bought AAA rated CDO tranches certainly believe in the EMH. They thought they were beating the market and earning extraordinary risk adjusted returns.
I don’t see what you mean by a “lower risk-adjusted return for a safer asset.” Basically I don’t see how the statement is meaningful, because risk adjustment reflects investors desire for a safe asset. In any case, extremely risk averse investors are not an exception to the simple mathematical result (based on very strong assumptions) which I proved. In market clearing (hence no extraordinary risk adjusted returns available) equilibrium, a very cowardly investor will buy only the risk free asset. One who is slightly less cowardly will divide her wealth between the risk free asset and a tiny bit of the market portfolio of riksy assets. She would buy very little of each, but buy the same tiny proportion of the stock of each risky asset on the market. She will buy the same tiny proportion of the total offer of each tranche of a CDO undoing the tranching.
A very bold investor will short the risk free rate and buy leveraged market portfolio of risky assets.
All investors will undo tranching in their portfolio. Given my strong assumptions, this is simply a mathematical result. A helicopter drop of risky assets or extremely risk averse investors (with parachutes) would change equilibrium prices, but wouldn’t invalidate the result.
Thanks for engaging and thanks for your time and thoughtfulness.
“Basically I don’t see how the statement is meaningful, because risk adjustment reflects investors desire for a safe asset. “
OK, with probability p, the bond will pay x. The expected return is the expectation of x, and is dependent only on the borrower’s financial prospects.
The expected *utility* of a household buying the bond will be a function of the distribution, marginal utility of consumption, time horizon, risk aversion, target wealth level, tax policy, etc. Maybe some investors are really sensitive to the variance, while others are sensitive to the variance and the third moment — who knows?
In any case, when the repayment characteristics of borrowers is different from the the types of debt that households want to hold, does price adjust, or does quantity supplied adjust?
If there were no financial system, and households were the only actors, then only price would adjust. Risky borrowers would pay an excessive premium, and safer borrowers would receive an excessive discount. Or the other way around — it depends on how the subjective utility of households corresponds to the financial position of borrowers.
But as soon as there is a complete market with financial intermediaries, then these institutions do not price debt based on expected utility, but based on expected return. Banks have no notion of utility. They can expand or contract their balance sheet, decreasing the number of risky assets sold to households by taking some of those assets onto their own balance sheet and issuing safer debt to households. Even without the endemic fraud, they would “in theory” be able to convert X risky assets into kX safe assets, and in the process, the supply of risky assets would decrease and the supply of safe assets would increase so that quantity, rather than price, adjusts. As long as k is non-zero, this process can result in the supply of debt being exactly in the proportions that households want to hold, with no price adjustments necessary — assuming leverage requirements can be ignored 🙂
So when Frank Portnoy said that there was an excess need for AAA debt, it was understood that yields would not fall to zero, because as soon as households were willing to pay a premium for AAA debt, then the financial system would create AAA debt and sell it to households, pocketing the premium. The fact that they got greedy and decided that they could convert 100 risky bonds into 95 safe ones, instead of 20 safe ones, is a separate issue.
And this is the only rationale for the financial system to exist. According to the model in the post, no one would buy bank debt, because everyone would “untranche” that debt and directly lend to borrowers.