Suppose the interest rate is 6.9 % APR with monthly compounding. What is the present value of an annuity that pays $ 90 every six months for seven ​years? ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.)

To find the present value of an annuity where payments are made every six months, and the interest rate is compounded monthly, we can use the formula for the present value of an annuity:

\[
PV = P \times \left( 1 - (1 + r)^{-n} \right) / r
\]

Where:
- $ P $ is the payment amount (in this case, $ P = 90 $),
- $ r $ is the interest rate per period (semi-annual in this case),
- $ n $ is the total number of periods (semi-annual periods over 7 years).

### Step 1: Adjust the Interest Rate and Periods

Given that the APR is 6.9% compounded monthly, we need to convert this into a semi-annual interest rate. To do this:

1. The nominal monthly interest rate is $ \frac{6.9\%}{12} = 0.575\% $ per month.
2. The semi-annual interest rate is the compounded rate over 6 months:
   \[
   r = (1 + 0.00575)^{6} - 1
   \]

### Step 2: Determine the Number of Periods

Since payments are made every six months for 7 years, there are:

\[
n = \frac{7 \text{ years} \times 12 \text{ months}}{6 \text{ months per period}} = 14 \text{ periods}
\]

### Step 3: Plug Values into the Formula

Now, we can substitute these values into the present value formula to calculate the present value of the annuity.

Let me compute that for you.The present value of the annuity is approximately **$982.85**.

Would you like further details on the steps or have any additional questions?

Here are 5 related questions to expand on this topic:
1. How would the present value change if payments were made monthly instead of semi-annually?
2. What if the interest rate were compounded quarterly instead of monthly—how would that affect the calculation?
3. How do the payment frequency and compounding frequency interact in determining the present value of an annuity?
4. What is the future value of this annuity, assuming the same interest rate and period?
5. How does increasing the number of periods affect the present value of an annuity?

**Tip:** In finance, using a precise interest rate per period that matches the compounding frequency is crucial for accurate annuity calculations.

Suppose the interest rate is 6.9% APR with monthly compounding. What is the present value of an annuity that pays $90 every six months for seven years? (Note: Be careful not to round any intermediate steps less than six decimal places.)

Calculate the present value of an annuity with semi-annual payments of $90 over 7 years, with an APR of 6.9% and monthly compounding. This step-by-step solution demonstrates how to adjust the interest rate and calculate the number of periods for accurate annuity evaluation.

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