Robert wants to have $$ \$ 2,300 $$ at the end of every three months for 8 years. The bank pays $$ 8 \% $$ interest, compounded quarterly. What would be the present value if payments were to be received at the BEGINNING of every period? (Use Table 12-2) Select one: a. $$ \$ 52,016.20 $$ b. $$ \$ 55,056.71 $$ c. $$ \$ 55,344.83 $$ d. $$ \$ 55,986.21 $$

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We are given quarterly payments of 
$$
\$2300
$$
for 8 years with an annual interest rate of $$8\%$$ compounded quarterly. Since the interest is compounded quarterly the effective rate per quarter is:
$$
i=\frac{0.08}{4}=0.02.
$$
The total number of periods is:
$$
n=8 \text{ years}\times 4=32.
$$

Because the payments are received at the BEGINNING of each period, the cash flow forms an annuity due. The present value of an annuity due is related to the present value of an ordinary annuity by multiplying by $$(1+i)$$.

Step 1. Write the formula for the present value of an annuity due:
$$
PV_{\text{due}}=R\left(\frac{1-(1+i)^{-n}}{i}\right)(1+i),
$$
where $$R=\$2300$$.

Step 2. Substitute the known values:
$$
PV_{\text{due}}=2300\left(\frac{1-(1.02)^{-32}}{0.02}\right)(1.02).
$$

Step 3. Compute $$(1.02)^{-32}$$. We first find $$(1.02)^{32}$$. Using logarithms or a calculator we get approximately:
$$
(1.02)^{32}\approx1.88417,
$$
so
$$
(1.02)^{-32}\approx\frac{1}{1.88417}\approx0.53034.
$$

Step 4. Evaluate the annuity factor:
$$
\frac{1-(1.02)^{-32}}{0.02}=\frac{1-0.53034}{0.02}=\frac{0.46966}{0.02}\approx23.483.
$$

Step 5. Multiply by $$(1.02)$$ to adjust for the annuity due:
$$
23.483\times1.02\approx23.95266.
$$

Step 6. Finally, compute the present value:
$$
PV_{\text{due}}\approx2300\times23.95266\approx55091.
$$

Looking at the answer choices, the value $$ \$55,091 $$ is closest to option

$$
\boxed{\$55,056.71.}
$$

Thus, the correct answer is b.
The present value is approximately \$55,056.71.

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