Adding loans removes the risk specific to each borrower, not the risk they share, and the second sets a floor no loan count can beat.
Enough loans to remove the risk that is specific to each borrower, and no more, because the rest is shared. On illustrative inputs of a 2.0 per cent annual default rate, 40 per cent loss given default and 15 per cent asset correlation, about 75 equal loans bring the volatility of annual credit losses to 1.08 per cent, within 25 per cent of a 0.87 per cent floor that no number of loans can remove. Doubling to 150 loans only takes it to 0.98.
Worked in full in The Private Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A loan book loses money for two reasons. Some borrowers fail for reasons of their own: a lost contract, a fraud, a bad acquisition. Others fail because the economy turns and takes a cohort of borrowers with it. Adding loans dilutes the first kind of risk, because each borrower is a smaller slice of the fund. It does nothing to the second, because every new borrower is exposed to the same cycle. The question an investor asks a direct lending manager, how many positions are enough, is really a question about where the first kind of risk stops mattering.
Diversification also does not change the expected loss. A fund of 10 loans and a fund of 150 loans with the same default rate and recovery both expect to lose 0.80 per cent of principal a year. What changes is how widely the actual outcome scatters around that figure.
| Input | Value |
|---|---|
| Annual probability of default, each borrower | 2.0% |
| Loss given default (recovery of 60 cents) | 40% |
| Expected annual loss, any number of loans | 0.80% |
| Asset correlation between borrowers | 15% |
| Fund size, millions | 1000 |
The correlation is the share of each borrower's credit quality driven by one common factor, the economy. Bank capital rules use a range of 12 to 24 per cent for corporate borrowers; 15 per cent is an illustrative middle-market figure, not an estimate for any portfolio.
The standard tool is a one-factor model. In a given year the economy draws a value z; conditional on it, each borrower defaults independently with a probability that rises as z falls. The fund's loss rate is the loss given default times the share of borrowers that default.
Conditional default rate: p(z) = N( (N-1(PD) − √ρ × z) ÷ √(1 − ρ) )
Variance of the loss rate with n equal loans = LGD² × [ Var(p(z)) + (PD − PD² − Var(p(z))) ÷ n ]
The first term is the systematic floor; the second is the borrower-specific part, and it shrinks with 1/n.
Excel, conditional default rate for a scenario z in B2: =NORM.S.DIST((NORM.S.INV(PD)-SQRT(Rho)*B2)/SQRT(1-Rho),TRUE)
Var(p(z)) is found by averaging over the distribution of z. On these inputs it gives a floor of 0.87 per cent for the standard deviation of annual losses: the volatility an infinitely granular fund would still carry. The independent case, the one most quick estimates use, sets the correlation to zero and keeps only the second term.
| Loans | Position size | One default costs | Volatility, independent | Volatility, correlated | Borrower-specific share of variance | One-in-a-hundred bad year |
|---|---|---|---|---|---|---|
| 10 | 10.00% | 4.00% | 1.77% | 1.96% | 80% | 8.00% |
| 20 | 5.00% | 2.00% | 1.25% | 1.51% | 67% | 6.00% |
| 40 | 2.50% | 1.00% | 0.89% | 1.24% | 50% | 5.00% |
| 75 | 1.33% | 0.53% | 0.65% | 1.08% | 35% | 4.80% |
| 150 | 0.67% | 0.27% | 0.46% | 0.98% | 21% | 4.53% |
| Unlimited | 0 | 0 | 0 | 0.87% | 0% | 4.22% |
Going from 10 loans to 40 cuts the volatility of losses by 37 per cent. Going from 40 to 150 cuts it by only 20 per cent more, where the independent model promises 48. At 40 loans half of the remaining variance is still borrower-specific; at 75 it is about a third, and the volatility is within 25 per cent of the floor. Getting within 10 per cent of the floor takes 191 loans, which for most middle-market teams means smaller tickets in credits they know less well.
The tail tells the same story. On a 1,000 million fund, the one-in-a-hundred loss year is 80 million with 10 loans, 50 million with 40, 48 million with 75 and 45 million with 150. The last 75 loans buy 3 million of tail protection; the floor is 4.22 per cent whatever the count.
| Correlation | 20 loans | 40 loans | 150 loans | Floor | Loans to get within 10% of floor |
|---|---|---|---|---|---|
| 5% | 1.33% | 0.99% | 0.64% | 0.46% | 713 |
| 15% | 1.51% | 1.24% | 0.98% | 0.87% | 191 |
| 25% | 1.74% | 1.51% | 1.32% | 1.24% | 93 |
The more correlated the book, the fewer loans it takes to exhaust diversification, and the higher the floor once it has. A portfolio concentrated in one sector, one sponsor's deals or one vintage behaves like the bottom row: adding names helps little because they fail together. Sector and vintage spread do more for the floor than a longer list of similar borrowers.
Treating defaults as independent. It makes every extra loan look valuable: on the independent model, 150 loans give a volatility of 0.46 per cent, and the manager can argue the book is close to riskless around its expected loss. With 15 per cent correlation the true figure is 0.98 per cent, 2.2 times higher, and no amount of further origination closes the gap. The independent model is fine for the question "how big should one position be"; it is wrong for the question "how bad can a year get".
Position size is the other half of the answer. With 20 loans one default at 40 per cent loss costs 2.00 per cent of the fund, a quarter of an illustrative 8.0 per cent net return. With 40 loans it costs 1.00 per cent, 12.5 per cent of that return. Investors usually accept the second and question the first.
How expected loss feeds into a fund's net return is worked in how to calculate the loss rate on a private credit portfolio and how much leverage adds to a private credit fund's return. The book's fund economics, including what a single large loss costs a year of net output, are in the free workbooks for this book.
No. Expected loss is the default rate times the loss given default, whatever the number of loans: 2.0 per cent times 40 per cent is 0.80 per cent a year for 10 loans or 150. Diversification narrows the spread of outcomes around that figure. In the illustration the one-in-a-hundred loss year falls from 8.00 per cent of the fund with 10 loans to 4.80 per cent with 75.
Size it by what one default costs. With 40 per cent loss given default, a 5.00 per cent position that fails costs 2.00 per cent of the fund, a quarter of an illustrative 8.0 per cent net return. A 2.50 per cent position costs 1.00 per cent, 12.5 per cent of that return. A single-position limit is that trade-off written down.
Because correlation sets the floor that adding loans cannot breach. At 5 per cent asset correlation the floor on annual loss volatility is 0.46 per cent and it takes 713 loans to get within 10 per cent of it. At 25 per cent the floor is 1.24 per cent and 93 loans get there. A concentrated sector book is closer to the second case.
This article is one calculation from The Private Credit Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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