Tri-Star Airlines intends to pay off a $20,000,000 bond issue that comes due in 4 years. How much must the company set aside now, at 6 % interest compounded quarterly, to accumulate the required amount of money?
20000000 = x [1 + (.06 / 4)]^(4 * 4)
P = Po(1+r)^n,
P = $20M,
Po = Initial investment,
r = 0.06/4 = 0.015 = Quarterly % rate expressed as a decimal,
n = 4 comp./yr. * 4yrs. = 16 Compounding periods.
To find out how much Tri-Star Airlines must set aside now to accumulate $20,000,000 in 4 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the accumulated amount (in this case, $20,000,000)
P = the principal amount (the amount Tri-Star Airlines needs to set aside now)
r = the annual interest rate (6%, or 0.06 as a decimal)
n = the number of times the interest is compounded per year (quarterly, so 4 times)
t = the number of years (4 years)
Now, let's substitute the values into the formula:
$20,000,000 = P(1 + 0.06/4)^(4*4)
Simplifying the equation:
$20,000,000 = P(1 + 0.015)^16
Next, we need to isolate P by dividing both sides of the equation by (1 + 0.015)^16:
P = $20,000,000 / (1 + 0.015)^16
Calculating the value within the parentheses:
(1 + 0.015)^16 = 1.26322524108
Finally, divide $20,000,000 by 1.26322524108 to find the amount Tri-Star Airlines needs to set aside now:
P ≈ $15,823,768.40
Therefore, Tri-Star Airlines must set aside approximately $15,823,768.40 now, at 6% interest compounded quarterly, to accumulate the required amount of $20,000,000 in 4 years.