The rule that correlated stresses compound is a statement about leverage, not about correlation.
On value, no. A rental fall and a yield widening applied together cost 5,906,035 against 6,220,557 for the two applied separately and added — the combination is 314,523 less bad than the sum, about five per cent of the loss. Value is income over yield, so the moves are multiplicative and each shrinks the base the other acts on. On the levered return the sign reverses: the same pair costs 877 basis points against a sum of 852.
Worked in full in Real Estate Financial Modeling by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A recession downside does not move one assumption. It moves several, and on a standing investment the pair that drives the value is the income and the exit yield. Take a worked case with a base exit value of 30,287,357 and apply the downside's two moves — income nine per cent lower, the exit yield 75 basis points wider — first one at a time, then together.
| Case | Value | Change from base |
|---|---|---|
| Base | 30,287,357 | — |
| Income falls, yield unchanged | 27,561,494 | −2,725,862 |
| Yield widens, income unchanged | 26,792,662 | −3,494,695 |
| The two added together | — | −6,220,557 |
| The two together, actually | 24,381,322 | −5,906,035 |
The two single moves sum to a loss of 6,220,557. The combination costs 5,906,035. The difference of 314,523 is not an artefact of this example and it admits no exception. Value is income divided by yield, so the two moves are multiplicative, and each one shrinks the base the other acts on. The cross term is exactly the income fall times the yield widening, over one plus the yield widening, times the value — about five per cent of the total loss here. On value, the combination is never worse than the sum. It is always better, and by an amount that can be written down in one term.
The received rule says a sensitivity table cannot describe a recession because the combination is worse than the sum of the individual moves. On value that is not merely imprecise, it is the wrong sign. An ungeared investor who adds two downside effects together is booking a loss that is not there, and inside a covenant test — where the number that binds is the value, not the return — a five per cent overstatement is not nothing.
Run the same two moves through the same cash flow with the debt left in, and measure the effect on the equity return rather than on the value.
| Move | Cost, basis points |
|---|---|
| Income falls, yield unchanged | 426 |
| Yield widens, income unchanged | 426 |
| The two added together | 852 |
| The two together, actually | 877 |
Together the two moves cost 877 basis points against a sum of 852. The debt is what does it. A fixed interest charge and a fixed loan repayment sit between the asset and the equity, and they turn a proportional loss into a leveraged one. The leverage acts on the second move as well as on the first, so the interaction that reduces the loss on value increases it on the return.
Which means the sign is a function of gearing, not of correlation. Below a loan-to-value of about 19.6 per cent the combination is less bad than the sum, exactly as the value arithmetic requires. Above it, worse. A deal at 55 per cent sits well past that line, so the conventional claim holds for it — by 25 basis points, not by a wide margin, and because of the capital structure rather than because the variables move together.
The two errors are not the same size, and they are not made in the same place.
None of this touches the practical advice that surrounds the rule, which is correct. A downside must be built as a coherent story in which every variable moves together, driven by one scenario switch, and the individual assumptions must not be softened because they look harsh in isolation. Treating the variables as independent and multiplying probabilities is the serious error, and it understates the downside every time. The point of sizing the cross term is not to argue with that.
Compute the combined case in the model, from a single scenario column, and never by adding the results of single-variable runs. Where the output is a value — a covenant test, a valuation, a lender's advance — the addition overstates the loss, and the overstatement is the cross term, which can be quantified in one line rather than argued about. Where the output is a levered return, the addition understates it, by an amount that grows with gearing and that at moderate leverage is smaller than the precision the recommendation is written to. Report the combination. Keep the single moves in the tornado, where they belong, as a ranking of drivers and not as an addition.
Not on value. Income nine per cent lower costs 2,725,862 and a yield 75 basis points wider costs 3,494,695, summing to 6,220,557, but applied together they cost 5,906,035. The gap of 314,523 is the cross term: value is income divided by yield, so each move shrinks the base the other acts on.
At a loan-to-value of about 19.6 per cent. Below that the combination is less bad than the sum, as the value arithmetic requires. Above it the fixed interest charge and fixed loan repayment leverage the second move as well as the first, and at 55 per cent gearing the pair costs 877 basis points of return against a sum of 852.
No. Added single moves overstate the value loss by the cross term — 314,523, or about five per cent, in this case — and understate the levered return loss by 25 basis points. Build the downside as one coherent column of assumptions and read the combined result from the model.
This article is one calculation from Real Estate Financial Modeling. 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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