Leverage multiplies value changes by a fixed factor and income by a spread that can turn negative. Both are one line of arithmetic.
At 70 per cent loan-to-value, every move in property value reaches the equity multiplied by 1 / (1 − LTV), or 3.33 times. A 10 per cent rise in value is a 33.3 per cent gain on the equity, and a 10 per cent fall is a 33.3 per cent loss. Income is amplified too, but only by the spread between the cap rate and the cost of debt, which can be negative.
Worked in full in Private Equity Real Estate by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Private Equity Real Estate uses one example to show what leverage does: a property bought at 70 per cent LTV whose value then moves by 10 per cent. The figure the chapter gives, roughly 33 per cent on the equity, is right; the capital stack workbook that accompanies the book sets out the working that reaches it. Here the same case is put in dollars, with an illustrative income line added so that the cash return can be seen beside the value effect.
| Input | Value |
|---|---|
| Purchase price | $100M |
| Loan to value | 70% |
| Debt / equity | $70M / $30M |
| Change in property value | +/- 10% |
| Cap rate (NOI of $6.0M) | 6.00% |
| Interest rate, interest only | 5.00% |
The debt does not move with the property. Every dollar of value change therefore lands on the equity, which is a smaller base.
Equity after the move = value × (1 + move) − debt = $110M − $70M = $40M
Change in equity = $40M / $30M − 1 = 33.3%
In general: equity change = property change / (1 − LTV) = 10% / 0.30 = 33.3%
Excel: =Move/(1-LTV)
On the way down the same arithmetic runs in reverse: value falls to $90M, equity to $20M, a loss of 33.3 per cent. For a fixed loan balance the amplification is exactly symmetric. That is the point the chapter makes: higher leverage amplifies both directions.
| LTV | Amplification | Equity on +10% | Equity on -10% | Fall that breaches 90% LTV | Fall that wipes out equity |
|---|---|---|---|---|---|
| 0% | 1.00x | +10.0% | -10.0% | n/a | -100% |
| 50% | 2.00x | +20.0% | -20.0% | -44.4% | -50% |
| 60% | 2.50x | +25.0% | -25.0% | -33.3% | -40% |
| 70% | 3.33x | +33.3% | -33.3% | -22.2% | -30% |
| 75% | 4.00x | +40.0% | -40.0% | -16.7% | -25% |
| 80% | 5.00x | +50.0% | -50.0% | -11.1% | -20% |
| 85% | 6.67x | +66.7% | -66.7% | -5.6% | -15% |
The amplification is not linear in LTV. Going from 50 to 70 per cent raises it from 2.00x to 3.33x; going from 70 to 85 per cent doubles it again, to 6.67x. At 85 per cent a 15 per cent fall in value erases all the equity, and a 5.6 per cent fall already takes the loan through a 90 per cent LTV covenant. The fall that breaches a covenant is LTV / covenant − 1: at 70 per cent and a 90 per cent covenant, -22.2 per cent.
The value example says nothing about cash flow. Leverage amplifies income only to the extent that the property yields more than the debt costs.
Levered cash yield = (cap rate − LTV × interest rate) / (1 − LTV)
At 6.00% and 5.00%: (6.00% − 0.70 × 5.00%) / 0.30 = 8.33%
Excel: =(Cap-LTV*Rate)/(1-LTV)
| Interest rate | Spread to cap rate | Interest | Cash to equity | Levered yield |
|---|---|---|---|---|
| 4.50% | +150bps | $3.15M | $2.85M | 9.50% |
| 5.00% | +100bps | $3.50M | $2.50M | 8.33% |
| 6.00% | 0bps | $4.20M | $1.80M | 6.00% |
| 7.00% | -100bps | $4.90M | $1.10M | 3.67% |
When the debt costs exactly the cap rate, leverage adds nothing to the cash yield: 6.00 per cent levered or unlevered. Above it, leverage is negative and the equity earns less than an all-cash buyer, while still carrying the 3.33x value risk. It is the position a floating-rate buyer reaches as soon as the all-in rate rises past the going-in cap rate.
Over one year at 5.00 per cent interest, the equity return is (cap rate + value change − LTV × rate) / (1 − LTV). With value up 10 per cent it is 41.7 per cent, against 16.0 per cent unlevered. With value flat it is 8.3 per cent. With value down 10 per cent it is -25.0 per cent, against -4.0 per cent unlevered. The income cushion makes the outcome slightly asymmetric, but only slightly: a fall of just 2.5 per cent in value is enough to wipe out a year of levered income.
The usual mistake is to quote the upside as the reason for leverage and to model the downside as a lower IRR, rather than as a share of equity lost. At 70 per cent LTV the right sentence is that the equity is a 3.33x bet on the property's value, which disappears at a 30 per cent fall and breaches a 90 per cent covenant at a 22.2 per cent fall. The second mistake is to assume leverage always adds return. It adds return on income only while the cap rate exceeds the cost of debt, and the sign of that spread can change after closing.
The full leverage ladder and both of the book's capital stacks are live in the free workbook for this case. Leverage also changes how a manager is paid: see does more gearing always raise the promote.
For a change in value, equity change equals property change divided by (1 minus LTV). For income, levered cash yield equals (cap rate minus LTV times interest rate) divided by (1 minus LTV). At 70 per cent LTV, a 6.00 per cent cap rate and 5.00 per cent debt, that is a 3.33x value multiplier and an 8.33 per cent cash yield.
When the cost of debt exceeds the property's cap rate. At 70 per cent LTV on a 6.00 per cent cap rate, debt at 7.00 per cent cuts the cash yield on equity to 3.67 per cent, below the unlevered 6.00, while the equity still carries 3.33 times the value risk.
By (1 minus LTV). At 70 per cent LTV the equity is gone after a 30 per cent fall; at 85 per cent after 15 per cent. A covenant breaks sooner: at a 90 per cent LTV covenant the 70 per cent loan breaches after a 22.2 per cent fall.
The leverage arithmetic is worked in the Chapter 21 capital stack workbook of Private Equity Real Estate. 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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