What is the nominal annual rate of interest compounded quarterly if a loan of ​$23,000 is repaid in seven years by payments of ​$2000 made at the end of every six months?
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Part 1
The nominal annual rate of interest is enter your response here​% compounded quarterly.
​(Round to two decimal places as​ needed.)

To find the nominal annual rate of interest compounded quarterly, we'll use the formula for the present value of an annuity. In this case, we are given:

- Principal (Present Value), $ P = \$23,000 $
- Payment, $ PMT = \$2000 $
- Number of periods, $ n = 7 $ years, with semi-annual payments, so $ n = 7 \times 2 = 14 $ periods.
- Interest is compounded quarterly, and payments are made semi-annually.

Let $ r $ be the nominal annual interest rate compounded quarterly. The interest rate per period for the semi-annual payments is $ \frac{r}{2} $, and the quarterly interest rate is $ \frac{r}{4} $.

The formula for the present value of an annuity is:

\[
P = PMT \times \frac{1 - (1 + r/n)^{-nt}}{r/n}
\]

Since the interest is compounded quarterly, we need to account for that by modifying the formula slightly. The correct approach here is to equate the present value of the payments to the loan amount:

\[
23000 = 2000 \times \frac{1 - \left(1 + \frac{r}{2}\right)^{-14}}{\frac{r}{2}}
\]

This equation cannot be solved directly for $ r $ algebraically, so we'll solve it using numerical methods or a financial calculator. I will use numerical methods to approximate $ r $.The nominal annual rate of interest, compounded quarterly, is approximately **5.48%** (rounded to two decimal places).

Do you want any further details or have any questions?

Here are 8 related questions:
1. What is the effective annual rate corresponding to this nominal rate?
2. How does the compounding frequency affect the nominal interest rate?
3. What would be the nominal annual rate if payments were made quarterly instead?
4. How would the loan amount change if the nominal interest rate was 6% instead?
5. What would be the present value of the loan if the interest rate was compounded monthly?
6. How many payments would be needed to repay the loan if the interest rate was lower?
7. How would the payment amount change if the loan term was extended to 10 years?
8. What is the difference between nominal and effective interest rates?

**Tip:** The nominal rate is often used for quoting interest rates, but the effective rate gives a better picture of the actual cost of borrowing.

Learn how to calculate the nominal annual rate of interest compounded quarterly for a loan repayment scenario. Includes step-by-step calculation using the present value of annuity formula with detailed explanations. Suitable for advanced college-level students and professionals.

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