Suppose you had three different offers for your used car. One person will give you $1,000 right now, another offers $1,200 six months from now and the other offers $1,600 two years from now. If interest is 16% compounded quarterly, which offer is worth the most.
To determine which offer is worth the most, we need to calculate the future value of each offer using compound interest.
First, let's calculate the future value of the first offer, which is $1,000 received right now. Since there is no compounding period, the future value (FV) will be equal to the present value (PV). Therefore, the future value of this offer is $1,000.
Next, let's calculate the future value of the second offer, which is $1,200 received six months from now. We have an interest rate of 16% compounded quarterly, which means a compounding period of 3 months. Since the compounding frequency is different from the interest rate period, we need to adjust the interest rate.
The adjusted interest rate per quarter (r) is calculated by dividing the annual interest rate by the number of compounding periods in a year. In this case, there are 4 quarters in a year, so the adjusted interest rate per quarter (r) is 16% / 4 = 4%.
Now we can calculate the future value using the compound interest formula:
FV = PV * (1 + r)^n
where PV is the present value, r is the interest rate per compounding period, and n is the number of compounding periods. In this case, PV is $1,200, r is 4%, and n is 2 compounding periods (6 months / 3 months).
FV = $1,200 * (1 + 0.04)^2 = $1,200 * (1.04)^2 = $1,200 * 1.0816 = $1,297.92
Therefore, the future value of the second offer is approximately $1,297.92.
Finally, let's calculate the future value of the third offer, which is $1,600 received two years from now. Again, we need to adjust the interest rate since it is compounded quarterly. The adjusted interest rate per quarter (r) is still 4%.
Using the compound interest formula:
FV = PV * (1 + r)^n
where PV is $1,600, r is 4%, and n is 8 quarters (2 years * 4 quarters).
FV = $1,600 * (1 + 0.04)^8 = $1,600 * (1.04)^8 = $1,600 * 1.3699 = $2,191.84
Therefore, the future value of the third offer is approximately $2,191.84.
Comparing the future values, we find that the third offer is worth the most, with a future value of approximately $2,191.84.