The mathematical tool that displays most clearly a risk management strategy is a probability distribution. A bank will take the money it holds and invest it in various ways. It can hold currency, or loan the money to customers, or buy treasury bonds, or any number of other ways to try to make money. At the simplest level, a bank should put money into whatever has the best return. If a loan returns 8% while a treasury bond returns only 2%, it surely makes better sense to loan out money rather than buying treasury bonds. On the other hand, it happens all too often that borrowers default. The expected 8% return can turn into a 30% loss. Perhaps the greater safety of treasury bonds outweighs the lesser return. Of course, with enough loans, probably most of them will be paid back and only a few will default.
Somehow a bank needs to estimate with a reasonable degree of accuracy the probabilities of the various possible returns of the various possible investments. One source of risk is simple ignorance, the unknown unknowns. A bank needs to hire the experts who can make accurate estimates, avoid investments where such expertise is unavailable, and keep a watchful eye that the experts don't shy away from too many good new ideas or become overconfident about too many bad old ideas.
Given an understanding of the probabilities of various outcomes from individual investments, and of correlations among those outcomes, a bank can then evaluate proposed portfolios, can derives the probabilities of outcomes for various combinations of investments. Each potential portfolio will imply a cumulative probability distribution with a shape something like:
The most attractive sort of shift is a pure win:
Generally, of course, one is faced with more difficult decisions. A typical investment might add some potential for losses, but more likely will bring improved returns:
To regulate effectively the risk management practices of financial institutions, one must distinguish between the sorts of risks that can be taken:
- Investments can involve significant unknowable risks, e.g. because of possible political or technological shifts.
- Understanding risks, i.e. accurately estimating the probability of a proposed portfolio, can involve considerable expense that the institution might be unwilling or unable to incur.
- Even when accurate probability distributions are available, it is not clear or simple how to distinguish which distributions represent responsible risk and which constitute violations of the public trust.