Sales of similar small apartment buildings indicate an overall rate of 6.7%. Lenders are quoting a 5.5% annual mortgage constant, and investors in these properties expect a 9.5% equity dividend rate. What loan-to-value ratio is consistent with these rates under the band of investment?
Correct Answer
B) 70.0%
Setting 6.7% = M × 5.5% + (1 − M) × 9.5% gives M = (9.5 − 6.7) ÷ (9.5 − 5.5) = 2.8 ÷ 4.0 = 70.0%. The 30.0% figure is the equity share, not the debt share. Dividing the overall rate by the equity rate gives 70.5%, and dividing the constant by the overall rate gives 82.1%, but neither weights the two rates.
Why This Is the Correct Answer
Option B is correct because 6.7% = M × 5.5% + (1 − M) × 9.5% rearranges to M = (9.5 − 6.7) ÷ (9.5 − 5.5) = 2.8 ÷ 4.0 = 70.0%. As a check, 0.70 × 5.5% + 0.30 × 9.5% = 3.85% + 2.85% = 6.70%. This debt share reconciles the band with the sales evidence.
Why the Other Options Are Wrong
Option A: 30.0%
Option A, 30.0%, is the equity share: 100% − 70.0% = 30.0%. Using 30.0% as debt gives 0.30 × 5.5% + 0.70 × 9.5% = 8.30%, which does not match the 6.7% overall rate.
Option C: 70.5%
Option C, 70.5%, divides the overall rate by the equity rate: 6.7% ÷ 9.5% = 70.5%. That shortcut does not solve the weighted formula, and a 70.5% debt share gives about 6.68% rather than 6.70%. The correct share is 2.8 ÷ 4.0 = 70.0%.
Option D: 82.1%
Option D, 82.1%, divides the constant by the overall rate: 5.5% ÷ 6.7% = 82.1%. That ignores the equity rate, and an 82.1% debt share gives about 6.22% rather than 6.70%. The correct share is 2.8 ÷ 4.0 = 70.0%.
Distance from equity over spread
The debt share equals how far the overall rate sits below the equity rate, divided by the gap between the equity rate and the constant. The closer the overall rate is to the constant, the more debt is implied.
How to use: Use this shortcut to solve for debt share, then confirm the answer by rebuilding the band.
Exam Tip
After solving for a share in a weighted formula, plug it back in to confirm that it reproduces the given rate. This catches both algebra errors and answers that report the wrong share.
Common Mistakes to Avoid
- -Reporting the equity share as the debt share
- -Dividing one rate by another instead of solving the weighted equation
- -Failing to check the answer by rebuilding the band
Concept Deep Dive
Analysis
The problem changes the unknown in the band formula from the overall rate to the debt share. You substitute the three known rates and solve for M. A quick check is that 6.7% sits between the two component rates, and the weights must reproduce it exactly.
Background Knowledge
The band of investment expresses the overall rate as RO = M × RM + (1 − M) × RE, where M is the debt proportion. With any three of the four terms known, the fourth can be found by algebra. Working the formula in reverse is a way to test whether financing assumptions are consistent with rates seen in sales.
Real-World Application
An appraiser sees sales at a rate lower than the band built on a lender's quoted terms. Solving for the implied loan-to-value ratio shows what financing would be needed to explain those sales. If that ratio is well above what lenders are offering, the appraiser revisits the inputs.
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The subject property has annual EGI of $92,000. A comparable with annual EGI of $100,000 sold for a cash-equivalent price of $850,000. Sales in the market area indicate the subject would sell at the same price-to-income ratio, with no other adjustments. What is the subject's indicated value, to the nearest dollar?
Next Question
Market data show that operating expenses, excluding debt service, run 38% of EGI for properties similar to the subject. The subject's annual EGI is $420,000. What is the projected annual NOI?