Assume your business associate owes you $13650. Also assume they offer either $12020 now or $ 1950 per year for 7 years, starting now. Assume a 5.6% market interest rate, compounded continuously.
How much would you have at the end of 7 years if you choose to
take the $12020 offer now, and you use the market to earn interest
on the funds?
How much would you have at the end of 7 years if you choose to
take the installments each year, and you still used the market to earn
interest on the the funds?
Pt = Po * e^(r*t).
Pt = value after time t.
Po = Initial investment.
e = 2.7183 = base of natural log.
r = Annual percentage rate(APR).
t = lengtth of investment in years.
First Option
Pt = 12020 * 2.7183^(0.056*7) = 17788.90.
2nd Option
Year 1. Pt = 1950 * 2.7183^(0.056*1) = 2062.32 @ end of 1st yr.
2. Pt = (2062.32 + 1850) * 2.7183^(0.056*1) = 4243.42 @ end of 2nd yr
3. Pt = (4243.42 + 1950) * 2.7183^(0.056*1) = 6550.15 @ end of 3rd yr
4. Pt = (6550.15 + 1950) * 2.7183^(0.056*1) = 8989.74 @ end of 4th yr
5. Pt = (8989.74 + 1950) * 2.7183^(0.056*1) = 11569.85 @ end of 5th yr
6. Pt = (11569.85 + 1950) * 2.7183^(0.056*1) = 14298.57 @ end of 6th yr
7. Pt = (14298.57 + 1950) * 2.7183^(0.056*1) = 17184.46 @ end of 7th yr
Option 1(17788.90) is the best choice.
NOTE: Under option 2, interest was earned on the interest.
To calculate how much you would have at the end of 7 years if you choose to take the $12020 offer now, and use the market to earn interest on the funds, you can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the future value (how much you would have at the end of 7 years)
P = the principal amount (the initial offer of $12020)
e = Euler's number (approximately 2.71828)
r = interest rate per year (5.6% expressed as a decimal, so 0.056)
t = time in years (7 years)
Plugging in the values into the formula:
A = 12020 * e^(0.056 * 7)
Calculating this will give you the amount you would have at the end of 7 years.
To calculate how much you would have at the end of 7 years if you choose to take the installments each year, and still use the market to earn interest on the funds, you can calculate the future value of an annuity.
The formula for the future value of an annuity is:
A = P * (1 + r)^n * (e^(rt)-1)/(r*e^(rt))
Where:
A = the future value (how much you would have at the end of 7 years)
P = the annual payment (the installment amount of $1950 per year)
r = interest rate per year (5.6% expressed as a decimal, so 0.056)
n = number of periods (7 years)
Plugging in the values into the formula:
A = 1950 * (1 + 0.056)^7 * (e^(0.056 * 7)-1)/(0.056 * e^(0.056 * 7))
Calculating this will give you the amount you would have at the end of 7 years if you choose the installments each year option.