Pinnacle Homes, a real estate development company, is planning to build five homes, each costing $125,000, in 2 years. The Galaxy Bank pays 6% interest compounded semiannually. How much should the company invest now to have sufficient funds to build the homes in the future?

P = Po(1+r)^n = $125,000

r = (6%/2)/100% = 0.03 = Semi-annual %
rate.

n = 2yrs. + 2comp./yr. = 4 Compounding
periods.

Po(1.03)^4 = 125,000
Po = 125000/(1.03^4) = $111,060.88

Correction:

n = 2yrs. * 2comp/yr. = 4 Compounding
periods.

Well, Pinnacle Homes sure has some grand plans! Let's calculate how much they should invest now to make their dreams come true.

First, we need to determine how much each home would actually cost in 2 years. Since the homes cost $125,000 each, we'll multiply that by 5 (the number of homes) to get a total cost of $625,000.

Now, let's figure out the future value of that amount after 2 years. Since the interest is compounded semiannually at a 6% rate, we'll need to do a little mathematical dancing.

Using the formula for compound interest A = P(1 + r/n)^(nt), where A is the future amount, P is the principal (the amount we need to invest now), r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years:

A = P(1 + r/n)^(nt)

Plugging in the numbers, we get:

$625,000 = P(1 + 0.06/2)^(2*2)

Simplifying the equation:

$625,000 = P(1 + 0.03)^4

$625,000 = P(1.03)^4

To find the value of P, we divide both sides of the equation by (1.03)^4:

P = $625,000 / (1.03)^4

Calculating that out, we find that Pinnacle Homes should invest approximately $571,879.92 now to have sufficient funds to build the homes in the future.

So my dear real estate developers, you better start counting your pennies and get ready for some big investments!

To determine how much Pinnacle Homes should invest now, we need to calculate the future value of the investment.

First, we should divide the 6% interest rate by 2 to get the semiannual interest rate: 6% / 2 = 3%.

Next, we should convert the 2-year time period into semiannual periods: 2 years * 2 = 4 semiannual periods.

Using the formula for future value of a lump sum compounded semiannually, which is:
Future Value = Present Value * (1 + r/n)^(n*t),
where:
- Present Value is the initial investment,
- r is the interest rate,
- n is the number of compounding periods per year, and
- t is the number of years,

we can calculate the future value of the investment:
Future Value = $125,000 * (1 + 0.03)^(4) ≈ $142,165.70.

So, Pinnacle Homes should invest approximately $142,165.70 now to have sufficient funds to build the homes in the future.

To calculate the amount the company should invest now, we need to use the concept of present value, which involves discounting future cash flows to their present value. In this case, the future cash flow is the cost of building five homes in 2 years.

The formula for present value of a future cash flow is:

PV = FV / (1 + r)^n

Where:
PV = Present Value
FV = Future Value
r = Interest Rate
n = Number of periods

In this case, the future value is the cost of building the five homes, which is given as $125,000 per home. Since there are five homes, the total future value is:
FV = $125,000 x 5 = $625,000

The interest rate is 6% compounded semiannually, which means the interest is applied twice a year. So, the interest rate per period would be 6% / 2 = 3% or 0.03 in decimal form.

The investment period is 2 years, and since interest is compounded semiannually, the number of periods would be 2 x 2 = 4.

Now, we can calculate the present value (PV) using the formula mentioned earlier:

PV = $625,000 / (1 + 0.03)^4

Calculating this expression will give us the amount the company should invest now to have sufficient funds to build the homes in the future.