A property worth 50,000 can be purchased for 20% down and quarterly mortgage payments of $ 1000 for 25 years. What nominal rate of interest compounded monthly is charged?
The Correct Answer and Explanation is:
To find the nominal rate of interest compounded monthly, first identify key values from the problem:
- Property Value: $50,000
- Down Payment: 20% of $50,000 = $10,000
- Loan Amount (Principal): $50,000 − $10,000 = $40,000
- Quarterly Payment: $1,000
- Loan Term: 25 years = 100 quarters (since 4 quarters per year)
Step 1: Set Up the Present Value of an Annuity Formula
The present value of an ordinary annuity is:P=R⋅1−(1+i)−niP = R \cdot \frac{1 – (1 + i)^{-n}}{i}P=R⋅i1−(1+i)−n
Where:
- P=40,000P = 40,000P=40,000 (present value of the loan)
- R=1,000R = 1,000R=1,000 (quarterly payment)
- n=100n = 100n=100 (total number of payments)
- iii = unknown quarterly interest rate
Solve for iii numerically.
Step 2: Use a Financial Calculator or Iterative Method
Using a financial calculator or numerical solver:
- PV = 40,000
- PMT = −1,000
- N = 100
- FV = 0
Solving gives i≈0.006126i \approx 0.006126i≈0.006126 (quarterly interest rate ≈ 0.6126%)
Step 3: Convert to Nominal Annual Rate Compounded Monthly
Convert the quarterly rate to the nominal annual rate compounded monthly. This requires first converting the effective quarterly rate to a monthly rate:(1+iq)=(1+im)3⇒(1+0.006126)=(1+im)3(1 + i_q) = (1 + i_m)^{3} \Rightarrow (1 + 0.006126) = (1 + i_m)^3(1+iq)=(1+im)3⇒(1+0.006126)=(1+im)3
Solving:1+im=(1.006126)1/3≈1.002037⇒im≈0.002037 (monthly rate)1 + i_m = (1.006126)^{1/3} \approx 1.002037 \Rightarrow i_m \approx 0.002037 \text{ (monthly rate)}1+im=(1.006126)1/3≈1.002037⇒im≈0.002037 (monthly rate)
Nominal annual rate compounded monthly:r=im×12=0.002037×12=0.024444≈2.44%r = i_m \times 12 = 0.002037 \times 12 = 0.024444 \approx 2.44\%r=im×12=0.002037×12=0.024444≈2.44%
✅ Final Answer:
Nominal interest rate compounded monthly ≈ 2.44%
Explanation (300 words):
A real estate purchase includes a property valued at $50,000, with a 20% down payment reducing the actual borrowed amount to $40,000. The buyer agrees to repay the loan via quarterly installments of $1,000 over a period of 25 years, totaling 100 payments.
The present value of all these future payments must equal the amount borrowed. This relationship is captured by the annuity formula, which equates a stream of equal periodic payments to a lump-sum amount based on an interest rate and number of periods.
To solve for the quarterly interest rate, a numerical approach or financial calculator is applied. After determining the quarterly rate to be approximately 0.6126%, it must then be converted to a nominal annual interest rate compounded monthly, as requested.
This involves first finding the equivalent monthly rate using the compound interest relationship: a quarterly interest rate is equivalent to compounding the monthly rate three times. Solving for the monthly rate gives approximately 0.2037%. This is then scaled to a nominal annual rate by multiplying by 12, resulting in approximately 2.44%.
This rate represents the nominal interest rate compounded monthly that equates a $40,000 present value to 100 quarterly payments of $1,000. The process shows the difference between payment frequency (quarterly) and compounding frequency (monthly), which often occurs in financial contracts.
