Mr. Nielson wants to borrow $1,000 for 2 years. He is given the choice of
i) simple interest at 12%, or
ii) a loan at 10% compounded monthly.
Which loan results in less interest due ?
b) What interest rate compounded quarterly will give an effective interest rate of 7% ?
1. I = Prt = 1000 * 0.12 * 2 = $240.
2. Pt = Po*r*t / (1 - (1+r)^-t,
r=(10% / 12) / 100% = 0.00833 = Monthly
% rate expressed as a decimal.
t = 2yrs * 12mo/yr = 24 Months.
Pt=1000*0.00833*24/(1 - (1.00833))^-24
= 199.992 / 0.180583955 = $1107.47.
I = 1107.47 - 1000 = $107.47
Option 2gives lowest Interest.
To determine which loan results in less interest due, we need to calculate the interest amount for each option.
Option i) Simple Interest at 12%:
The formula to calculate simple interest is: Interest = Principal * Rate * Time
In this case, the principal (P) is $1,000, the rate (R) is 12% (0.12), and the time (T) is 2 years.
Using the formula, the interest for option i) is: I = 1,000 * 0.12 * 2 = $240
Option ii) Loan at 10% compounded monthly:
To calculate the interest for this option, we need to use the compound interest formula:
A = P * (1 + r/n)^(n*t)
Where:
A = the final amount
P = principal
r = interest rate (in decimal form, so 10% = 0.1)
n = number of times interest is compounded per year
t = time in years
We need to solve for the interest (I), so rearranging the formula, we get:
I = A - P
First, let's calculate the future value (A) of the loan:
A = 1,000 * (1 + 0.1/12)^(12 * 2) ≈ $1,210.68
Then, we can calculate the interest:
I = 1,210.68 - 1,000 ≈ $210.68
So, the interest for option ii) is approximately $210.68
Comparing the two options, we can see that option ii) (loan at 10% compounded monthly) results in less interest compared to option i) (simple interest at 12%).
Moving on to the second question:
To find the interest rate compounded quarterly that will give an effective interest rate of 7%, we can use the formula for effective interest rate:
Effective interest rate = (1 + r/n)^n - 1
In this case, the effective interest rate is given as 7% (0.07), and we need to solve for the quarterly interest rate (r), with quarterly compounding (n = 4).
Rearranging the formula, we get:
r = (1 + 0.07)^(1/4) - 1 ≈ 0.0166 or approximately 1.66%
Hence, an interest rate compounded quarterly of approximately 1.66% will result in an effective interest rate of 7%.