To find the present value (PV) of a five-period annuity with payments of $3,200 received at the end of each year under a 12% annual interest rate compounded quarterly, follow these steps:

Step 1: Identify the Parameters

• Annual Interest Rate ($r$): 12% or 0.12
• Payment Amount ($P$): $3,200
• Number of Payments ($n$): 5
• Compounding Periods per Year ($m$): Quarterly, so $m = 4$

Step 2: Adjust the Interest Rate for Compounding Periods

Since interest is compounded quarterly, the interest rate per period ($i$) is: $i = \frac{r}{m} = \frac{0.12}{4} = 0.03 \text{ (per quarter)}$

Step 3: Calculate the Number of Compounding Periods

Given that the annuity spans 5 years and interest compounds quarterly: $\text{Total Compounding Periods} = m \times n = 4 \times 5 = 20 \text{ periods}$

Step 4: Find the Present Value of the Annuity

The formula for the present value ($PV$) of an annuity when payments are made annually but interest is compounded more frequently is: $PV = P \times \frac{1 - (1 + i)^{-mt}}{i}$

Where:

• $P = 3200$
• $i = 0.03$
• $m = 4$
• $t = 5$

$PV = 3200 \times \frac{1 - (1 + 0.03)^{-20}}{0.03}$

Step 5: Calculate the Values

1. $(1 + i)^{-mt} = (1.03)^{-20}$

   Using a calculator: $(1.03)^{-20} \approx 0.5537$

2. Substituting back into the formula: $PV = 3200 \times \frac{1 - 0.5537}{0.03}$ $PV = 3200 \times \frac{0.4463}{0.03}$ $PV \approx 3200 \times 14.8767$ $PV \approx 47,605.44$

Conclusion

The present value of the annuity is approximately $47,605.44.

Would you like more details or have any questions?

Related Questions:

1. How would the present value change if the payments were received at the beginning of each year instead?
2. What would be the effect of monthly compounding instead of quarterly compounding?
3. How do you find the future value of the same annuity under the same conditions?
4. What if the interest rate changed to 10%? How would that affect the present value?
5. How is the present value affected if the annuity duration extends to 10 years?

Tip:

When dealing with annuities, always match the payment frequency and compounding frequency carefully for accurate calculations.