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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".

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  • $\begingroup$ 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. $\endgroup$
    – Raskolnikov
    Commented Jan 25, 2018 at 13:50
  • $\begingroup$ Why do you feel that you've missed anything? $\endgroup$
    – Lipton
    Commented Jan 25, 2018 at 21:54
  • $\begingroup$ @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. $\endgroup$
    – Homunculus Reticulli
    Commented Jan 26, 2018 at 10:49
  • $\begingroup$ @Raskolnikov do you want to submit that as an answer? $\endgroup$
    – Homunculus Reticulli
    Commented Jan 26, 2018 at 10:50

1 Answer 1

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I think your idea is right. But I would just stay consistent with the representation of the numbers and not mix percentages and perunages. Either write

$$(1-P)(1+R)=1+X$$

in which case the numbers are perunages. Either you write

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

and all the numbers are interpreted as percentages. It's really not a fundamental distinction and of course solving using one of the equations, you can always rapidly get the values of the other by multiplying/dividing by 100.

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