What is the future worth of $1,000 in month 1, $1,040 in month 2, and amounts increasing by $40 per month
through month 12, at the end of year 2 if the interest rate is 23.7631528% per year, compounded continuously?
To calculate the future worth of the amounts increasing by $40 per month over a two-year period with a continuously compounded interest rate of 23.7631528% per year, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the future worth
P = the initial amount
e = Euler's number (approximately 2.71828)
r = the interest rate
t = the time in years
Let's calculate the future worth step-by-step for each month:
Month 1:
- P = $1,000
- r = 23.7631528% per year, which is approximately 0.237631528 in decimal form
- t = 1/12 years (since it's one month)
- A = $1,000 * e^(0.237631528 * (1/12))
Month 2:
- P = $1,040 (previous amount + $40 increase)
- r = 23.7631528% per year, which is approximately 0.237631528 in decimal form
- t = 2/12 years (since it's two months)
- A = $1,040 * e^(0.237631528 * (2/12))
Repeat this process for each month up to month 12, increasing the time by 1/12 each time.
At the end of year 2, we need to calculate the future worth of the amount at the end of month 12 + the future worth of the amount at the end of month 24.
Finally, add the two future worth calculations together to find the total future worth at the end of year 2.
To calculate the future worth of the given amounts at the end of year 2 with continuous compounding, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the future worth
P = the initial amount
e = the base of natural logarithms (approximately 2.71828)
r = the interest rate (as a decimal)
t = the time period (in years)
In this case, we have monthly amounts, so we need to convert the interest rate to a monthly rate by dividing it by 12. We also need to multiply the interest rate by the time period in years (2) to get the total interest rate.
Let's calculate the future worth of $1,000 in month 1 first:
P = $1,000
r = 23.7631528% per year / 12 = 0.0198 (monthly rate)
t = 2 years
Using the formula, we have:
A = $1,000 * e^(0.0198 * 2)
Calculating the exponential term:
e^(0.0198 * 2) ≈ 1.0396199
Now we can calculate the future worth:
A = $1,000 * 1.0396199 ≈ $1,039.62
Next, we'll calculate the future worth of $1,040 in month 2:
P = $1,040
r = 0.0198 (monthly rate)
t = 2 years
Using the formula:
A = $1,040 * e^(0.0198 * 2)
Calculating the exponential term:
e^(0.0198 * 2) ≈ 1.0396199
Calculating the future worth:
A = $1,040 * 1.0396199 ≈ $1,081.63
Since the amounts increase by $40 per month, we can calculate the future worth for each month in year 2 using the same method:
Month 3:
A = ($1,080 + $40) * e^(0.0198 * 2) ≈ $1,123.65
Month 4:
A = ($1,120 + $40) * e^(0.0198 * 2) ≈ $1,165.98
Month 5:
A = ($1,160 + $40) * e^(0.0198 * 2) ≈ $1,209.62
Month 6:
A = ($1,200 + $40) * e^(0.0198 * 2) ≈ $1,254.60
Month 7:
A = ($1,240 + $40) * e^(0.0198 * 2) ≈ $1,300.93
Month 8:
A = ($1,280 + $40) * e^(0.0198 * 2) ≈ $1,348.61
Month 9:
A = ($1,320 + $40) * e^(0.0198 * 2) ≈ $1,397.65
Month 10:
A = ($1,360 + $40) * e^(0.0198 * 2) ≈ $1,448.06
Month 11:
A = ($1,400 + $40) * e^(0.0198 * 2) ≈ $1,499.85
Month 12:
A = ($1,440 + $40) * e^(0.0198 * 2) ≈ $1,553.03
Therefore, at the end of year 2, the future worth of the given sequence of amounts is approximately $1,553.03.