# Heuristic (or algorithm) for calculating a risk premium, given a probability of default and a "minimum" profit margin (expressed as a yield)

Assuming that I have means of determining and calculating the following metrics:

1. Risk (i.e. probability*) of a default to a particular borrower as P
2. Profit margin of X%

The profit margin is taken to mean that "irrespective of defaults that might occur, in the long run, I expect to make X% for lending to this particular (class of) borrower.

Thinking it through (from first principles):

Expectation[given loans to borrower with P% of default at a rate of R%] = X%

For the sake of simplicity, lets assume that a default implies the entire lent out capital is lost, so then:

( (100 - P)/100 ) * (1+R) = X

We then trivially, solve for R.

Somehow, I think I've missed something. Can anyone shed some light on if this is a good (correct?) way to solve for R the interest rate to charge the borrower.

Note: I am aware that I'm using a slightly different definition of risk premium from that used in textbooks.

I'm using the frequentist interpretation of probability, where P denotes the number of occurrences (defaults) in a sequence of 100 "runs".

• I'd write either $(1-P)(1+R)=(1+X)$, either $\frac{100-P}{100}\frac{100+R}{100}=\frac{100+X}{100}$. Everything as perunages or everything as percentages. Jan 25, 2018 at 13:50
• Why do you feel that you've missed anything? Jan 25, 2018 at 21:54
• @Lipton I haven't done this sort of thing for a while - (I work in a different field now), so I wanted to double check with industry practitioners that my thinking through it was sound. Jan 26, 2018 at 10:49
• @Raskolnikov do you want to submit that as an answer? Jan 26, 2018 at 10:50

$$(1-P)(1+R)=1+X$$
$$\frac{100-P}{100}\frac{100+R}{100}=\frac{100+X}{100}$$