A step-by-step return bridge gives a different answer for every order the drivers are switched on in, and the person drawing it chooses the order.
Switch each driver on one at a time, record the step in the IRR, repeat in every possible order, and report the average step for each driver. On an illustrative deal that turns a 5.00 per cent base return into 12.41 per cent, leverage is worth 3.95 points if it is switched on last and 1.00 if it goes first; only the average over all six orders, 2.50, is independent of the order, and the three averages add up exactly to the 7.415 points there are to explain.
Worked in full in Commercial Real Estate Investing by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Every investment report says what drove the return: so much from rental growth, so much from yield compression, so much from leverage. Almost none says in what order the drivers were switched on, and the order decides the answer. The drivers interact. Leverage gears the growth, growth raises the exit value that the yield is applied to. A step-by-step bridge allocates each interaction to whichever driver happens to come later, and the person drawing the bridge chooses the order.
| Input | Value |
|---|---|
| Purchase price | 50,000,000 |
| Net operating income, year 1 | 2,500,000 |
| Entry yield, and the base-case exit yield | 5.00% |
| Driver G: NOI growth a year | 2.5% |
| Driver Y: exit yield | 4.75% |
| Driver L: loan at 50% of price, interest only at 4.0% | 25,000,000 |
| Equity IRR, base case | 5.00% |
| Equity IRR, all three drivers on | 12.41% |
The exit is priced on year-six income. With no growth and no yield shift the property is sold for what it cost, and the unlevered IRR is simply the 5.00 per cent yield. There are 7.415 points to explain.
Three drivers, each on or off, make eight combinations. Every attribution, whatever the order, is built from these eight IRRs, so run them once and keep them.
| Drivers on | IRR |
|---|---|
| None | 5.00% |
| Growth | 7.50% |
| Exit yield | 5.93% |
| Leverage | 6.00% |
| Growth and exit yield | 8.46% |
| Growth and leverage | 10.71% |
| Exit yield and leverage | 7.80% |
| All three | 12.41% |
Step for a driver = IRR(drivers already on + this one) − IRR(drivers already on)
Attribution = average of that step over every order in which the drivers can be switched on
With three drivers there are 3! = 6 orders; with four, 24. In Excel, lay the eight IRRs out as a lookup keyed on the drivers that are on, then each order is three subtractions. The average is the Shapley value of the driver, the standard allocation from cooperative game theory: it treats the drivers symmetrically, adds up exactly, and does not depend on any order.
| Order | Growth | Exit yield | Leverage |
|---|---|---|---|
| Growth, yield, leverage | 2.50 | 0.96 | 3.95 |
| Growth, leverage, yield | 2.50 | 1.71 | 3.21 |
| Yield, growth, leverage | 2.53 | 0.93 | 3.95 |
| Yield, leverage, growth | 4.61 | 0.93 | 1.87 |
| Leverage, growth, yield | 4.71 | 1.71 | 1.00 |
| Leverage, yield, growth | 4.61 | 1.80 | 1.00 |
| Average | 3.576 | 1.342 | 2.497 |
Every row adds up to 7.41, so every row is a valid bridge. That is the problem. The first row, the one most reports draw because leverage feels like the last decision, credits leverage with 3.95 points, more than half the outperformance. The last two rows credit it with 1.00. The swing on leverage alone is 2.95 points, larger than the whole contribution of the exit yield.
The reason is visible in the arithmetic. Borrowing at 4.0 per cent against an unlevered 5.00 per cent return adds one point: half the capital earns a one-point spread, doubled by the gearing. Borrowing against an unlevered 8.46 per cent return, after growth and compression are already on, adds 3.95, because the spread being geared is now far wider. Neither number is wrong. They answer different questions, and the bridge does not say which.
| Driver | Minimum | Maximum | Average | Share |
|---|---|---|---|---|
| Rental growth | 2.50 | 4.71 | 3.576 | 48% |
| Leverage | 1.00 | 3.95 | 2.497 | 34% |
| Exit yield | 0.93 | 1.80 | 1.342 | 18% |
| Total | 7.415 | 100% |
Growth did nearly half the work. Leverage did a third, and the exit yield, the driver a manager has least control over and is most often credited for, less than a fifth.
The interaction terms are not lost in the average; they are shared. Growth and leverage together are worth more than the sum of each alone, and the average splits that extra equally between the two drivers that created it. A sequential bridge hands all of it to whichever comes second.
The common mistake is to present one sequential bridge as if it were the attribution. It is one of six here and one of twenty-four with four drivers, and the person choosing the order can move a driver's credit by almost three points. When a manager's fee or reputation rests on the split between skill (the business plan, the leasing) and market (the yield, the cost of debt), that choice is not neutral.
The second mistake is to attribute on unlevered returns and then add leverage as a separate line on top. That is simply the leverage-last order with another name, and it gives leverage the most it can ever be given.
Commercial Real Estate Investing runs this on four drivers and all twenty-four orders for a complete estate, and the free attribution workbook for that case computes every order live from sixteen runs of the model. How much of that leverage contribution reaches the manager as promote is a separate question, worked in does more gearing always raise the promote.
It credits each driver with its average marginal contribution across every order in which the drivers could be added. It adds up exactly to the total and does not depend on the order chosen. On an illustrative property it credits rental growth with 3.576 of 7.415 points of IRR, or 48 per cent.
Because leverage gears whatever spread the unlevered return already earns over the cost of debt. On an illustrative deal, borrowing at 4.0 per cent against a 5.00 per cent unlevered return adds 1.00 point; against an 8.46 per cent return, after growth and yield compression, it adds 3.95.
Two to the power of the number of drivers: every combination of drivers on and off. Three drivers need eight runs and give six orders; four drivers need sixteen runs and give twenty-four orders. Each order is then only subtractions between stored results.
Chapters 8 to 12 of Commercial Real Estate Investing attribute a complete estate's return to four drivers across all twenty-four orders; the free Attribution workbook computes each order live. 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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