Lenders in the subject's market are making loans on fourplexes at 70% of value. A typical loan of $350,000 carries 4.5% interest with level monthly payments of $1,750. Investors in these properties expect an 11% first-year cash-on-cash return. What overall capitalization rate is indicated by the band of investment?
Correct Answer
B) 7.50%
Annual debt service is $1,750 × 12 = $21,000, so the mortgage constant is $21,000 ÷ $350,000 = 6%. The band is then 0.70 × 6% + 0.30 × 11% = 7.50%. Using the 4.5% interest rate in place of the constant gives 6.45%. Averaging 6% and 11% without weights gives 8.50%, and swapping the weights gives 9.50%.
Why This Is the Correct Answer
Option B is correct because $1,750 × 12 = $21,000 of annual debt service, and $21,000 ÷ $350,000 = a 6% mortgage constant. The weighted rate is 0.70 × 6% + 0.30 × 11% = 4.20% + 3.30% = 7.50%. The constant is used because it captures the lender's full annual claim on income.
Why the Other Options Are Wrong
Option A: 6.45%
Option A, 6.45%, uses the note's interest rate: 0.70 × 4.5% + 0.30 × 11% = 6.45%. Interest alone leaves out the principal part of the payment, so the correct band is 0.70 × 6% + 0.30 × 11% = 7.50%.
Option C: 8.50%
Option C, 8.50%, averages the two rates: (6% + 11%) ÷ 2 = 8.50%. That treats debt and equity as equal shares when debt is 70% of value, so the correct band is 0.70 × 6% + 0.30 × 11% = 7.50%.
Option D: 9.50%
Option D, 9.50%, swaps the weights: 0.30 × 6% + 0.70 × 11% = 9.50%. The 70% share belongs to the debt, so the correct band is 0.70 × 6% + 0.30 × 11% = 7.50%.
Payment times twelve over loan
The constant comes from the payment, not the note's rate. Multiply the monthly payment by twelve and divide by the loan amount.
How to use: Use this formula whenever a band problem gives loan payment details instead of a constant.
Exam Tip
If a band problem gives an interest rate and a payment, derive the constant from the payment. On an amortizing loan, a calculated constant below the interest rate is a sign to recheck your work.
Common Mistakes to Avoid
- -Using the note's interest rate as the debt rate
- -Forgetting to annualize the monthly payment
- -Applying the debt weight to the equity rate
Concept Deep Dive
Analysis
The problem adds a step before the band because the debt rate is not given directly. The band uses the mortgage constant, which reflects both interest and principal, so it must be derived from the payment. Once the constant is known, each rate is weighted by its share of value.
Background Knowledge
The band of investment weights the capitalization rates for debt and equity by their typical shares of value. The debt rate is the mortgage constant, which is annual debt service divided by the loan principal and depends on the interest rate, payment frequency and term. It is normally higher than the interest rate on an amortizing loan.
Real-World Application
An appraiser in a thin sales market cross-checks a rate with current loan quotes. The lender gives a loan amount, rate and monthly payment, so the appraiser calculates the constant before weighting it. Equity evidence from comparable sales completes the band.
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