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 if the annuity is set up for this purpose, recognizing that this is 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 if the annuity is set up for this purpose, recognizing that this is an annuity due.

2. Solution Steps

Since the company wants to have $20,000 at the *beginning* of each period, this is an annuity due.

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 $6.61\%$, and it's compounded semiannually, so the interest rate per period is:

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

Next, we need to find the number of periods. The annuity lasts for $4\frac{1}{2}$ years, and payments are made semiannually, so the number of periods is:

$n = 4.5 \times 2 = 9$

Now we can plug these values into the formula for the present value of an annuity due:

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

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

$PV = 20000 \times \frac{1 - 0.745633}{0.03305} \times 1.03305$

$PV = 20000 \times \frac{0.254367}{0.03305} \times 1.03305$

$PV = 20000 \times 7.708517 \times 1.03305$

$PV = 20000 \times 7.963252$

$PV = 159265.04$

Therefore, the amount that must be invested now is $159265.

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3. Final Answer

$159265.04