a land company has a loan of $8,500 compounded quarterly for 5 years at 6% what is the effective interest rate for the loan?
P = Po(1+r)^n
r = (0.06/4) = 0.015 = Quarterly % rate
expressed as a decimal.
n = 4Comp./yr. * 5yrs. = 20 Compounding
periods.
P = 8500(1.015)^20 = 11,448.27
I = = (P-Po)/T = (11448.27-8500)/5 =
589.65 = Int. per yr.
APR = 589.65/8500 = 0.0694 = 6.94%
To calculate the effective interest rate for the loan, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the loan
P = the principal amount (loan amount)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
Let's plug in the given values and solve for the effective interest rate (r):
A = $8,500
P = $8,500
n = 4 (quarterly compounding)
t = 5 years
Substituting the values into the formula, we get:
$8,500 = $8,500(1 + r/4)^(4*5)
Simplifying the equation, we have:
1 + r/4 = (1 + r)^20
Now we need to solve for r.
Let's use an iterative approach to solve this equation. We start by assuming a value for r and calculate both sides of the equation:
Assume r = 0.06 (6% annual rate)
1 + 0.06/4 = 1.015
(1 + 0.06)^20 = 1.4185
The left side of the equation is smaller than the right side, indicating that our assumed interest rate is too low. We can adjust our assumption to a higher value and repeat the calculations:
Assume r = 0.10 (10% annual rate)
1 + 0.10/4 = 1.025
(1 + 0.10)^20 = 6.7275
In this case, the left side of the equation is larger than the right side, indicating that our assumed interest rate is too high. We can continue this iterative process until we find a value for r that satisfies the equation.
As an AI Bot, I can perform the iteration to find the approximate value of r that satisfies the equation using a programming code. Let me calculate it for you.
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To find the effective interest rate for a loan, we need to use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the future value of the loan
P = the principal (initial loan amount)
r = the nominal interest rate (annual rate)
n = the number of times interest is compounded per year
t = the time the loan is held or the number of years
In this case, the principal (P) is $8,500, the nominal interest rate (r) is 6%, the loan is compounded quarterly (n = 4), and the loan duration (t) is 5 years.
We can rearrange the formula to solve for the effective interest rate (r):
r = (A/P)^(1/(n*t)) - 1
Now let's plug in the given values and calculate:
A = P(1 + r/n)^(n*t)
A = $8,500(1 + 0.06/4)^(4*5)
A ≈ $11,098.39
r = (A/P)^(1/(n*t)) - 1
r = ($11,098.39/$8,500)^(1/(4*5)) - 1
r ≈ 0.0562 or 5.62%
Therefore, the effective interest rate for the loan is approximately 5.62%.