A company wants to have $20,000 at the beginning of each 6-month period for the next $4\frac{1}{2}$ years. The annuity earns 6.61%, compounded semiannually. We need to find how much must be invested now. The problem is identified as an annuity due.

1. Problem Description

A company wants to have $20,000 at the beginning of each 6-month period for the next$4\frac{1}{2}$ years. The annuity earns 6.61%, compounded semiannually. We need to find how much must be invested now. The problem is identified as an annuity due.

2. Solution Steps

The formula for the present value of an annuity due is:

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

Where:

$PV$ = Present Value (the amount to be invested now)

$PMT$ = Payment per period ($20,000)

$i$ = Interest rate per period

$n$ = Number of periods

First, we need to find the interest rate per period. The annual interest rate is 6.61%, compounded semiannually.

$i = \frac{0.0661}{2} = 0.03305$

Next, we need to find the number of periods. The annuity lasts for $4\frac{1}{2}$ years, and payments are made every 6 months (semiannually).

$n = 4.5 \times 2 = 9$

Now, plug these values into the present value formula:

$PV = 20000 \times \frac{1 - (1+0.03305)^{-9}}{0.03305} \times (1+0.03305)$

$PV = 20000 \times \frac{1 - (1.03305)^{-9}}{0.03305} \times (1.03305)$

$PV = 20000 \times \frac{1 - 0.743628}{0.03305} \times 1.03305$

$PV = 20000 \times \frac{0.256372}{0.03305} \times 1.03305$

$PV = 20000 \times 7.7571 \times 1.03305$

$PV = 20000 \times 8.01302$

$PV = 160260.40$

3. Final Answer

$160260.40