An overall cap rate extracted from a sale is 7.2%. The buyer financed 75% at a 6.9% constant. The equity dividend rate implied for the 25% equity share is about:
Correct Answer
A) 8.1%, the leverage-adjusted residual
Why this is correct: The overall cap rate (R) is a weighted average of the mortgage constant (M) and equity dividend rate (E). Given R = 7.2%, loan-to-value ratio = 75%, and M = 6.9%, solve: 0.072 = (0.75 × 0.069) + (0.25 × E). First, 0.75 × 0.069 = 0.05175. Then, 0.072 - 0.05175 = 0.02025. Finally, E = 0.02025 ÷ 0.25 = 0.081 or 8.1%. Why the other choices are wrong: "7.05%, splitting the difference" is an incorrect average. "6.9% exactly, matching the loan's constant" ignores that equity typically requires a higher return. "28.8%, the rate times leverage" misapplies the calculation. Exam tip: Use the band of investment formula: R = (LTV × M) + (Equity % × E). Solve for E when other variables are known.
Why This Is the Correct Answer
Solving the band of investment for the unknown component gives the answer directly. The debt contribution is 0.75 times 0.069, which is 0.05175. Subtracting that from the overall rate of 0.072 leaves 0.02025 as the equity contribution, and dividing by the 0.25 equity share gives 0.081, or 8.1 percent. The result also passes a sanity check, since equity stands behind debt in the capital stack and therefore demands the higher return, which is why positive leverage exists at all. When the equity dividend rate solves to less than the mortgage constant, the property carries negative leverage and that finding deserves comment in the report.
Why the Other Options Are Wrong
Option B: 7.05%, splitting the difference
Averaging the overall rate and the mortgage constant produces a number with no meaning in the band of investment, since the formula weights components by their share of value rather than splitting differences. The two rates being averaged are not two components of anything; one of them is the result the components produce. The option substitutes arithmetic convenience for the weighted relationship.
Option C: 6.9% exactly, matching the loan's constant
Setting the equity dividend rate equal to the mortgage constant would make the overall rate equal to 6.9 percent as well, since a weighted average of two identical numbers is that number. The stem gives an overall rate of 7.2 percent, so the equity component must exceed the debt component. It also ignores the economic reality that the subordinate position demands the higher return.
Option D: 28.8%, the rate times leverage
Multiplying the overall rate by leverage produces a figure with no place in the formula and lands far outside any plausible equity dividend rate for a stabilized property. An equity return near 29 percent would signal extraordinary risk or an error in the inputs. Checking an answer against the range of realistic outcomes eliminates it before any arithmetic is needed.
Weights Must Sum To One
Two slices of the same pie, seventy-five and twenty-five, each earning its own rate. Multiply each slice by its rate, add them, and you get the whole pie's rate. Given any three of the four numbers, the fourth is simple subtraction and division.
How to use: Write the formula with all four terms before substituting, then fill in what the stem gives and solve for the blank. Check the result against the expectation that equity earns more than debt on a positively leveraged property. If it does not, say so rather than assuming an arithmetic error.
Exam Tip
The mortgage constant is not the interest rate. On any amortizing loan the constant is higher, and stems often supply the interest rate hoping you will use it in the constant's place.
Common Mistakes to Avoid
- -Substituting the mortgage interest rate for the mortgage constant on an amortizing loan
- -Confusing the equity dividend rate with the equity yield rate over a holding period
- -Failing to notice that a solved equity rate below the mortgage constant indicates negative leverage
Concept Deep Dive
Analysis
This item tests the band of investment, which treats the overall capitalization rate as a weighted average of the returns demanded by the two capital positions in a property. Debt requires the mortgage capitalization rate, usually called the mortgage constant, which is annual debt service divided by the loan amount and therefore includes both interest and principal amortization. Equity requires the equity dividend rate, sometimes called the cash-on-cash return, which is annual pre-tax cash flow divided by the equity invested. Weight each by its share of value and the components sum to the overall rate: R equals the loan-to-value ratio times the mortgage constant, plus the equity ratio times the equity dividend rate. The formula is usually presented as a way to build a rate from its parts, and this item runs it backward, giving the overall rate and the debt component and asking for the equity residual.
Background Knowledge
You need the band of investment relationship, that the overall rate equals the loan-to-value ratio times the mortgage capitalization rate plus the equity ratio times the equity dividend rate, and the ability to solve it for any single unknown. You should know that the mortgage constant is annual debt service divided by loan amount and therefore exceeds the interest rate whenever the loan amortizes, and that the equity dividend rate is annual pre-tax cash flow divided by equity. You also need to distinguish the equity dividend rate, a single-year cash-on-cash measure, from the equity yield rate, which is a multi-year internal rate of return including reversion.
Real-World Application
Extracting a 7.2 percent overall rate from a comparable sale, the appraiser learns the buyer financed 75 percent with a loan carrying a 6.9 percent constant. Solving the band of investment yields an implied equity dividend rate of 8.1 percent, which the appraiser compares against equity returns reported by investors in that market as a test of whether the extracted overall rate is reasonable.
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