Testing Whether a Bank Can Profitably Pool Borrower Risks
Summary
The document analyzes a bank that cannot distinguish high-risk from low-risk borrowers and considers whether a single loan rate can attract both groups while covering expected repayment losses. It defines each group’s population weight, default probability, and reservation rate, assumes zero recovery and no competition, and calculates expected profit from the pooled borrower mix. The break-even rate is found by dividing principal by the weighted probability of repayment and subtracting one.
For the stated assumptions, the answer reports a break-even rate of 13.96%. Because this exceeds the low-risk borrowers’ reservation rate, that rate would not attract the full pool, so the calculation does not establish a viable pooling equilibrium. The notional amount is immaterial in the setup, while capital costs and other lending frictions are excluded. The conclusion is therefore specific to the simplified assumptions and does not model strategic selection beyond the stated willingness-to-pay limits.
Key ideas
- A pooled loan rate attracts both borrower types only if it is within the low-risk group’s reservation limit.
- Expected repayment depends on the population-weighted survival probability of the borrower pool.
- Under zero recovery, the break-even rate is the reciprocal of the weighted repayment probability minus one.
- The calculated break-even rate exceeds the low-risk group’s stated willingness to pay, undermining pooling under these assumptions.
- Competition, capital costs, and other lending considerations are omitted.
Tags
Full text
# pooling equilibrium
# pooling equilibrium
I was hoping for some help on how to answer a question about pooling equilibrium. Suppose a bank wants to give loans of 1 million dollars to people, but it cannot differentiate between high risk borrowers and low risk borrowers. The high risk borrowers make up 15% of the population and have a default rate of 25% and would be willing to pay a max interest rate of 30%. The low risk borrowers make up 85% of the population and have a default rate of 10%, and would be willing to pay a max interest rate of 13.5%. In this scenario, would the bank be willing to pool together the risk, and if so what interest should it charge on the loans ? Thank you
## Answer by Kermittfrog (score 1)
https://quant.stackexchange.com/a/68917
I'd argue as follows. Let there be two borrower groups, $l$ow risks and $h$igh risks. The population fractions are $w_l$ for the low risks and $1-w_l$ for the high risks. Their default probabilities are $l$ and $h$, respectively. No recovery. Their reservation interest rates are $r_l$ and $r_h$. We assume one bank (no competition), so any interest rate at or below $r_l$ will attract both, high and low risk debtors, at fractions equal to their population share. The total notional to be invested is irrelevant. We do not consider cost of capital etc.
From the institution’s perspective, an offered rate $r$ will pool the borrowers if $r\leq r_l$ and it will be beneficial if the bank's expected profit $\pi$ is positive (or at least not negative)
$$ \begin{align} 0\stackrel{!}{\leq} \pi & \equiv -1+w_l \left[0\times l + (1+r)(1-l)\right] + (1-w_l)\left[0\times h + (1+r)(1-h)\right]\\ &=-1+(1+r)\left[w_l (1-l) + (1-w_l) (1-h)\right]\\ \Rightarrow\quad r&\geq \frac{1}{w_l (1-l) + (1-w_l) (1-h)}-1 \end{align} $$
which, in your case, equals 13.96% (NB: not 13.69% as in your comment). This rate will not be accepted by the low risks.
Does that make sense?
NB: The bad risk group does not accept a rate that can sustain its own PD. That seems odd ;-)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.