The issue is the notion of aggregation — we could possibly get a handle on credit in an agent-based model. I have discussed this topic before — for example, Hyman Minsky’s “Financial Instability Hypothesis” refers to “units,” which would be translated into “agents” in modern “agent-based modelling” terminology. Although I am pretty sure that I have discussed this before, I want to take a stab at the issues with credit modelling alone.

Why Does This Matter?

The bulk of this article is fairly abstract. I will now briefly outline why I believe this matters. Out of necessity, I will make assertions that are hopefully justified later. The main argument is that mainstream models — and any models that share key features with mainstream models — cannot properly model credit. The implications are as follows.

• Business cycles typically rise and fall with a credit cycle. (Admittedly, business cycles can happen for other reasons. For example, 2020 demonstrated that governments shut down activity, you can get a recession.) If the models cannot properly handle credit, they have a hard time offering useful insights into recession risks.

• Banking system behaviour revolves around the willingness to take credit risk. I have run into a number of debates about banking, such as banks as “intermediaries between savers and borrowers.” You cannot use mainstream models to make useful statements about banks if the models cannot handle credit.

The reason why the models have problems is that we cannot treat sectors of the economy as making “optimal” decisions as credit creates uncertainty that cannot be rammed into a mathematical optimisation problem. We can attempt to work around this in two ways.

1. We can drop the belief that sectors “optimise” their behaviour, and instead we just say that outcomes are just some “average behaviour” that we can try to estimate empirically. The problem with this approach in the area of credit is that behaviour changes: credit growth tends to grow rapidly until it doesn’t. The model offers no insight to when the change in behaviour happens.

2. We can build an agent-based model that replicates the point of view of individual economic actors in the real world. The agents face “uncertainty” in their decisions in that they are not privy to what entire sectors plan to do. The problem with this approach is that it is too flexible — you can build a model so that anything can happen.

Finally, the current crypto catastrophe is a wonderful example. It was abundantly clear to outsiders that the leveraged speculative cycle would blow up, and credit losses would lead to unwinding. However, the absence of a model means that there was no way to offer a quantitative prediction as to when the unwind would happen. This is yet another episode providing a counter-example to the assertion that economic theories must be embedded in a mathematical model.

I will now return to the justification for my statements about modelling credit.

Financial Modelling of Credit

From a pricing perspective, a bond with default risk is relatively straightforward to price (abstracting away from some technicals). The trick is to assume that we are risk neutral — the expected return on the bond (using the mathematical definition of “expectations”) equals that of a default risk-free bond. Although the exact calculation could be complicated if we try to model all the cash flows, we can approximate this well by having the bond spread equal the expected loss from default. E.g., if the expected loss was 1% of the market value of a one-year bond, the spread would probably be 1% (normally expressed as 100 basis points) over the risk-free curve. (I am just sticking with a one-year maturity for simplicity. For other maturities, we are looking at the average expected credit loss on an annual basis.)

(In the real world, we ought to take into account a premium for liquidity as well as financing differentials. These factors matter more for high quality bonds, as such concerns are a significant portion of spreads. Meanwhile, we need to model any embedded options in bonds, and so we end up with an option-adjusted spread, commonly abbreviated as OAS.)

For example, take a one-year bond where we believe that the probability of default is 2%. If a default happens, we also assume that the loss is 50% of the principal (or a 50% recovery rate). The expected loss is the probability of default times the loss, which equals 1% of capital. This implies a fair value spread of 100 basis points.

The assumed recovery rate is obviously important. If we assume a total loss (0% recovery), the expected loss doubles, and the fair value spread is 200 basis points (2%).

There are two (sensible) ways of using this theoretical approach.

1. I decide what the default probability is, as well as the recovery rate. This gives me a fair value spread, and I buy/sell based on market pricing relative to my view.

2. I get my hands on historical recovery rates (which vary by industry), and then back out an implied probability of default based on observed market pricing. I then go off and write a report on what is implied by market pricing.

All of the above is simplified, but is otherwise not controversial from the perspective of fixed income pricing. However, the important observation is that one is approaching this problem from a perspective of a bond portfolio manager, and we assume that the default/recovery probabilities exist “somewhere” independent of our investment decisions.

This perspective breaks down within economic models if we believe that the “actors” embedded in the model follow decision processes that match real world ones.

I will next run through issues that arise.

Expected Values Not Enough

If we look at the DSGE model literature, they feature extremely complex models which include random variables on continuous functions. It is clear that the full probability distributions would be extremely hard to compute. However, the complexity can be pushed aside by the focus on the vector of price variables. Behaviour is driven by the expected value of prices, and so we can abstract out the entire probability distribution by looking at the expected value.

This technique runs into problems when credit appears.

Let us imagine that we own a levered financial firm whose asset consists of a $100 million position in 1-year bond, financed with $90 million of debt, and $10 million of equity. (Yes, this is a silly investment strategy.) The spread of the bond over the financing is 50 basis points.

Let us imagine that the probability of default is 2%, with a 50% loss on default. This implies an expected loss of 1% of capital. Since the firm has $100 million in assets versus $90 million in liabilities, the expected value of equity is positive. Yay.

The problem appears when the owner of the firm ponders the full probability distribution more carefully. There is a 2% probability of the firm ending up with firm being wiped out, since the 50% recovery value is miles below the $90 million liability.

Whether or not a 2% probability of being wiped out or not is significant will depend upon whether the owner is feeling lucky. Furthermore, the probability of default could rise while the expected value of equity remains positive.

The effect of risk on behaviour will matter more than the expected value of assets.

Credit Risk Not “Independent Events”

The usual strategy for dealing with credit risk is rely on diversification. The hope is that defaults are somewhat independent. For banks in particular, many defaults are due to household-specific bad fortune (e.g., medical bills in the United States).

There are problems with this.

• Defaults will rise and fall with the business cycle. This is arguably not that hard to account for within a model, but it a major problem for risk management.

• The more difficult problem is that lender behaviour changes outcomes. For example, in the decades leading to the Financial Crisis, subprime lending was a growing profit centre for banks. Loosening lending standards buoyed house prices — which improved recovery values. This meant that the economic outcome validated loosened lending standards (a key observation of Minsky). Unfortunately, the process hit its limit, and the lack of lending standards led to default losses well beyond what lenders imagined.

On paper, we could imagine a model that allows for all the potential credit standards and the losses associated with them. Given the tendency of financial systems to evolve to avoid regulation, historical data are not useful for projecting potential losses.

Credit Often Bilateral

Although there are some semi-transparent markets in credit securities, credit transactions are typically bilateral. This means that there is no “market” to enter into “equilibrium.” Although we can try to model credit “as if” it were a public market, this downplays the importance of things like changing credit standards at large banks, which effectively are unilateral decisions.

Defaults Not Exactly “Random”

If we look at default events, they are not exactly “random”: the state of the borrower’s finances are determined by the state of the economy. As such, the default is either a deterministic side of effect of the evolution of the economy, or it was the result of a decision by either the borrower or lender.

Although having borrowers default because of the state of the economy might appear to be tractable, reality is more complicated. One of secrets of banking that loan officers do not like to publicise is the principle of “a rolling loan gathers no loss.” Although bankers are quite willing to be ruthless to weak borrowers who fall behind on their contractual obligations, lenders cannot be too aggressive in causing their own lending book to be filled up with defaulted loans. At some point, it is more attractive to roll over loans and hope that the borrowers get their act together at some point. Bailouts are a major point of controversy for a reason — the real world of credit relies on trust and relationships, and not just the “iron laws of the market.”

Finally, scam borrowers are a particular problem. Imagine a scam artist seeking a loan without any plausible business plan of paying it back. Although the credulous lender might view the default as being “random,” the scam artist knows with certainty that they will eventually not be paying their debts in full. Although this example might not seem that important, one need only reflect upon recent events in the crypto sector to see that these dynamics can grow.

Concluding Remarks

From the perspective of someone new to finance and economics, it probably seems unacceptable that the best forecast we can come up with is often along the lines of “the credit bubble will expand until it blows up.” In the absence of a time machine, getting a better forecast implies having a good mathematical model of the system. Unfortunately, credit does not sit well within such mathematical models.