Suppose the interest rate is 

6.9 %6.9% 

APR with monthly compounding. What is the present value of an annuity that pays 

$ 90$90 

every 

sixsix 

months for 

sevenseven 

​years? 

​(*Note*​: 

Be careful not to round any intermediate steps less than six decimal​ places.)

To find the present value of an annuity, we use the formula for the present value of an ordinary annuity:

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

Where:
- $ PV $ is the present value of the annuity.
- $ P $ is the payment per period.
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

### Step 1: Determine the payment per period.
Here, the payment $ P = 90 $ dollars.

### Step 2: Adjust the interest rate for semiannual payments.
Since the interest rate is **6.9% APR** (Annual Percentage Rate) compounded monthly, we first need to adjust it to fit the semiannual (every six months) payment structure. The monthly interest rate is:

\[
\text{Monthly interest rate} = \frac{6.9\%}{12} = 0.00575
\]

Now, to get the semiannual interest rate, we compound the monthly interest rate over 6 months:

\[
\text{Semiannual interest rate} = (1 + 0.00575)^6 - 1
\]

### Step 3: Determine the total number of periods.
Since the annuity is paid every **six months** for **seven years**, the total number of periods is:

\[
n = \frac{7 \text{ years}}{0.5 \text{ years per period}} = 14 \text{ periods}
\]

### Step 4: Calculate the present value.

Once we have $ r $ (semiannual interest rate) and $ n = 14 $, we plug these into the formula to get the present value.

Let me calculate the result.The semiannual interest rate is approximately **3.49998%**, and the present value of the annuity is **$982.85** (rounded to two decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the present value change if the payments were made monthly instead of semiannually?
2. What is the future value of this annuity after 7 years?
3. How does changing the interest rate to 5% affect the present value?
4. What if the annuity lasts for 10 years instead of 7?
5. How do different compounding frequencies (e.g., annual vs. quarterly) impact the annuity’s present value?

**Tip:** When solving annuity problems, always adjust the interest rate to match the payment period (monthly, quarterly, semiannual, etc.).

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 to less than six decimal places.)

Discover how to calculate the present value of an annuity with $90 payments made every six months for seven years at 6.9% APR with monthly compounding. Includes a detailed breakdown of the annuity formula, interest rate adjustments, and key calculations.

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