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.
The two moves, separately and together
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.
Exit value under each move on its own, and under both at once.
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.
Why the levered return reverses the sign
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.
Effect of each downside move on the levered return, in basis points.
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.
Where the difference actually matters
The two errors are not the same size, and they are not made in the same place.
On the return, the correction is 25 basis points on a downside that costs 877. A downside worth 877 basis points and one worth 852 are the same decision, taken by the same committee, with the same answer.
On the value, the correction is 314,523 on a base of 30,287,357, and it is the number a loan-to-value covenant is tested against. A value understated by five per cent can put a deal in breach on paper that is not in breach in fact.
Anyone modelling without debt — an open-ended fund, an insurer, a family holding — gets the sign wrong every time, because the effect that saves the combination is arithmetic and the effect that costs it is borrowed money.
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.
What to do with it
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.
The workbooks behind this article
Every figure above is a live formula in the free companion files for
Real Estate Financial Modeling. Each workbook ends with a Checks sheet
setting the printed figure beside the computed one. No account and no email address.
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