Find the value of $15,000 at the end of one year if it is invested in an account that has an interest rate of 4.50% and is compounded:
a) monthly
b) daily (assuming a 365-day year)
c) quarterly
What is the formula that should be used for these 3? I just need help getting started.
If compounded n time per year, the annual rate has to be divided by n, so after t years there have been n*t periods of compounding at the smaller rate.
15000(1+.045/n)^(n*t)
You have t=1
To find the future value of an investment using compound interest, you can use the formula:
A = P * (1 + r/n)^(n*t)
Where:
A = the future value
P = the principal amount (initial investment)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
Now, let's calculate the future value for each case:
a) Monthly compounding:
Using the formula, we have:
A = 15000 * (1 + 0.045/12)^(12*1)
In this case, interest is compounded monthly (n = 12), and the investment is held for one year (t = 1).
Calculate the exponent first:
(1 + 0.045/12)^(12*1) ≈ 1.04577
Now substitute the values back into the formula:
A ≈ 15000 * 1.04577 ≈ $15,686.55
Therefore, the value of $15,000 at the end of one year, compounded monthly, is approximately $15,686.55.
b) Daily compounding (assuming a 365-day year):
Using the formula, we have:
A = 15000 * (1 + 0.045/365)^(365*1)
In this case, interest is compounded daily (n = 365), and the investment is held for one year (t = 1).
Calculate the exponent first:
(1 + 0.045/365)^(365*1) ≈ 1.04595
Now substitute the values back into the formula:
A ≈ 15000 * 1.04595 ≈ $15,688.13
Therefore, the value of $15,000 at the end of one year, compounded daily, is approximately $15,688.13.
c) Quarterly compounding:
Using the formula, we have:
A = 15000 * (1 + 0.045/4)^(4*1)
In this case, interest is compounded quarterly (n = 4), and the investment is held for one year (t = 1).
Calculate the exponent first:
(1 + 0.045/4)^(4*1) ≈ 1.04594
Now substitute the values back into the formula:
A ≈ 15000 * 1.04594 ≈ $15,688.07
Therefore, the value of $15,000 at the end of one year, compounded quarterly, is approximately $15,688.07.
To summarize, the formulas used for the three cases are:
a) Monthly: A = P * (1 + r/n)^(n*t)
b) Daily (365-day year): A = P * (1 + r/365)^(365*t)
c) Quarterly: A = P * (1 + r/4)^(4*t)