The structural (Merton) model applied to a commercial mortgage, step by step, with the refinancing test that matters more and a grid by LTV and volatility.
Treat the property value as a random walk and the loan as a strike: the probability of default at maturity is N(−d2), with d2 = [ln(V/L) + (μ − σ²/2)T] ÷ σ√T. On an illustrative 60 per cent LTV, five-year interest-only loan with 15 per cent value volatility, the chance the building is worth less than the loan at maturity is 6.62 per cent. The chance it cannot be refinanced at 70 per cent LTV is 32.96 per cent, five times higher.
Worked in full in Real Estate Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
This is the structural, or Merton, view of a mortgage: the borrower holds the building and a right to hand it back, so the lender has written a put on the property with a strike equal to the loan. The model is crude, and every input is a judgement, but it does something no LTV grid does on its own: it turns a ratio into a probability, and shows how fast that probability moves with leverage, term and volatility.
| Input | Value |
|---|---|
| Property value today, V | 100,000,000 |
| Loan, interest-only, L | 60,000,000 |
| LTV at origination | 60% |
| Term to maturity, T | 5 years |
| Expected value growth, μ | 1.0% a year |
| Value volatility, σ (unsmoothed) | 15% a year |
| Maximum LTV of a refinancing lender | 70% |
Step 1: the distance to the strike. How far, in logarithms, the value sits above the loan.
ln(V ÷ L) = ln(100,000,000 ÷ 60,000,000) = 0.5108
Step 2: the drift. The median value grows at μ less half the variance, which is why volatility alone drags the median down.
(μ − σ²/2) × T = (0.01 − 0.01125) × 5 = −0.0062
Median value at year 5 = 99,376,949, although the mean is 105,127,110
Step 3: scale by the uncertainty over the term.
σ√T = 0.15 × √5 = 0.3354
d2 = (0.5108 − 0.0062) ÷ 0.3354 = 1.5044
Step 4: the probability. d2 is the number of standard deviations the value can fall before it reaches the loan. The default probability is the area of the normal curve beyond it.
PD = N(−1.5044) = 6.62% over five years, 1.36% a year
Excel: =NORM.S.DIST(-(LN(V/L)+(mu-sigma^2/2)*T)/(sigma*SQRT(T)),TRUE)
Annualised: 1 − (1 − 6.62%)1/5 = 1.36%
Step 5: the refinancing test. A loan rarely fails because the building is worth less than the debt. It fails because the next lender will not lend enough. At a 70 per cent maximum LTV, the 60,000,000 balance needs a value of 85,714,286 at maturity: a fall of only 14.3 per cent, against 40.0 for the repayment test.
ln(100,000,000 ÷ 85,714,286) = 0.1542; d2 = (0.1542 − 0.0062) ÷ 0.3354 = 0.4410
Probability of a refinancing gap = N(−0.4410) = 32.96%, 7.69% a year
Not every gap becomes a default: a sponsor may inject equity, as in paying down a refinancing gap. But it is the probability that the loan needs a negotiation at maturity, which is the number a credit committee should see first.
| LTV | σ 10% | σ 15% | σ 20% | Refi gap, σ 15% |
|---|---|---|---|---|
| 50% | 0.07% | 2.03% | 7.52% | 16.24% |
| 60% | 0.83% | 6.62% | 15.14% | 32.96% |
| 65% | 2.08% | 10.28% | 19.73% | 41.98% |
| 70% | 4.39% | 14.81% | 24.64% | 50.74% |
| 75% | 8.10% | 20.07% | 29.75% | 58.88% |
| 80% | 13.36% | 25.89% | 34.93% | 66.16% |
Default risk is convex in leverage. Moving from 60 to 75 per cent LTV multiplies the five-year PD by 3.0 at 15 per cent volatility, and the effect is far stronger when volatility is low: at 10 per cent, 0.83 becomes 8.10. Term works the same way. The 60 per cent loan has a 2.55 per cent PD over three years, 6.62 over five, 10.29 over seven and 14.67 over ten, because a longer term gives the value more room to wander below the strike.
Feeding the model appraisal volatility. Valuation indices are smoothed, and their volatility is often half or less of the true figure. At an illustrative 8 per cent the same 60 per cent loan shows a five-year PD of 0.12 per cent, roughly one fiftieth of the 6.62 per cent at 15 per cent, and the refinancing gap probability falls from 32.96 to 14.64. A model calibrated on smoothed returns will tell a lender that almost nothing can default; the method for recovering the true volatility is in how to unsmooth appraisal-based returns.
The model also has limits worth stating in the credit paper. It tests only the maturity date, so it misses defaults during the term when income falls below debt service. It treats value as a smooth walk with no jumps, so a tenant leaving overnight is not in it. And μ and σ are estimates for one building, not observed facts. Use it to rank loans and to show sensitivity, not to quote a PD to two decimals as if it were measured.
The book develops the loan as an option on the building and works the strike and the recovery on one office case; the free workbooks for the case carry that option table as live formulas. For the LTV a loan can be refinanced at, see what is a good LTV for commercial real estate.
An unsmoothed one. Appraisal indices understate volatility because valuations lag the market. In the example, 15 per cent gives a five-year PD of 6.62 per cent on a 60 per cent LTV loan; an illustrative 8 per cent appraisal-based figure gives 0.12 per cent, which is almost certainly too low. Show a range rather than one figure.
On value alone, yes, and the effect is convex: at 15 per cent volatility the five-year PD is 2.03 per cent at 50 per cent LTV, 6.62 at 60 and 20.07 at 75. But the model ignores income, so a low-LTV loan on a building losing its tenant can still default on debt service during the term.
Annual PD = 1 - (1 - cumulative PD) to the power 1/T. A 6.62 per cent five-year PD is 1.36 per cent a year; a 32.96 per cent probability of a refinancing gap over five years is 7.69 per cent a year. Dividing by the number of years gives a slightly different and less correct figure.
This article is one calculation from Real Estate Finance. 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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