Question

find the present value of an ordinary annuity with deposits of \\$7,853 quarterly for 3 years at 6.0\\% compounded quarterly.

what is the present value?
\\$
(round to the nearest cent.)

Explanation:

🆕 New Concept Discovered: Present Value of an Ordinary Annuity
Finding the current value of a series of equal future payments

Step 1: Identify the given values

An ordinary annuity involves regular payments made at the end of each period. To find its present value, we first extract the parameters from the problem:

• Periodic payment (\(R\)): \(\$7,853\)
• Annual interest rate (\(r\)): \(6.0\% = 0.06\)
• Compounding frequency per year (\(m\)): \(4\) (since it is compounded quarterly)
• Time in years (\(t\)): \(3\)

From these, we calculate:

• Interest rate per period (\(i\)):

$$ i = \frac{r}{m} = \frac{0.06}{4} = 0.015 $$

• Total number of payment periods (\(n\)):

$$ n = m \times t = 4 \times 3 = 12 $$

Step 2: Apply the Present Value formula

The formula for the present value of an ordinary annuity (\(PV\)) is:

$$ PV = R \times \frac{1 - (1 + i)^{-n}}{i} $$

Substitute the values into the formula:

$$ PV = 7853 \times \frac{1 - (1 + 0.015)^{-12}}{0.015} $$

Step 3: Calculate the final value

First, calculate the term inside the numerator:

$$ (1.015)^{-12} \approx 0.836387 $$

$$ 1 - 0.836387 = 0.163613 $$

Next, divide by \(i\):

$$ \frac{0.163613}{0.015} \approx 10.90751 $$

Finally, multiply by the payment amount \(R\):

$$ PV \approx 7853 \times 10.90751 \approx 85656.68 $$

Rounding to the nearest cent gives \(\$85,656.68\).

Answer:

85,656.68