Setting a Loan Yield for Default Risk and a Target Return
Summary
The document considers how to set a lending yield when a borrower has an estimated probability of default and the lender wants a specified expected return. Under the simplifying assumption that default destroys the entire principal, the borrower’s survival probability is multiplied by the repayment amount to express expected proceeds. The accepted answer emphasizes consistent units: probabilities and returns must be represented either as decimals or as percentages throughout the equation.
With decimal notation, the expected repayment must be equated to principal plus the desired profit, rather than to the profit alone. The source’s original setup mixes the target return with total repayment, and the accepted response’s displayed equation clarifies the unit convention but repeats that setup’s target expression. The framework is therefore a basic expected-value calculation, not a full credit-pricing model. It omits recovery values, timing, funding costs, uncertainty in default estimates, and risk compensation beyond the stated target return.
Key ideas
- Expected repayment depends on the probability that the borrower survives without default.
- Use a consistent decimal or percentage convention throughout the yield calculation.
- Distinguish total repayment from profit when defining the target return.
- Assuming total principal loss on default leaves out recovery, timing, and funding considerations.
- A default probability estimate alone does not capture every component of a credit spread.
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# Answer by Raskolnikov (score 0, accepted)
# 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:
- Risk (i.e. probability*) of a default to a particular borrower as P
- 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".
## Answer by Raskolnikov (score 0, accepted)
https://quant.stackexchange.com/a/37939
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.