Use the digits of your birthday as the amount of your initial investment (i.e., 6/25 is $625), calculate the value of this investment after 10 years at 3.5% APR for interest compounded yearly, quarterly, monthly, and daily. What do you notice?
well, we do know that
(1+.035)^10 = 1.4106
(1+.035/4)^(4*10) = 1.4169
...
e^.035 = 1.4191
To calculate the value of your investment after 10 years at different compounding frequencies, we'll need to use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the future value of the investment
P = the initial investment amount
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, the initial investment amount will be determined by the digits of your birthday. Let's assume your birthday is on March 15, so the initial investment will be $315. Now, let's calculate the future value of this investment after 10 years at different compounding frequencies:
1. Yearly Compounding:
For yearly compounding, the annual interest rate (r) is 3.5%, which is equivalent to 0.035 in decimal form. The compounding frequency (n) is 1 since the interest is compounded once a year.
A = 315(1 + 0.035/1)^(1*10)
A = 315(1 + 0.035)^10
A = 315(1.035)^10
A ≈ $430.57
2. Quarterly Compounding:
For quarterly compounding, the compounding frequency (n) is 4 since the interest is compounded four times a year.
A = 315(1 + 0.035/4)^(4*10)
A = 315(1 + 0.00875)^40
A = 315(1.00875)^40
A ≈ $431.95
3. Monthly Compounding:
For monthly compounding, the compounding frequency (n) is 12 since the interest is compounded twelve times a year.
A = 315(1 + 0.035/12)^(12*10)
A = 315(1 + 0.00292)^120
A ≈ $432.46
4. Daily Compounding:
For daily compounding, the compounding frequency (n) is 365 since the interest is compounded daily.
A = 315(1 + 0.035/365)^(365*10)
A = 315(1 + 0.00009589)^3650
A ≈ $432.62
What we notice is that as the compounding frequency increases (from yearly to daily), the investment's future value also increases. The more frequently interest is compounded, the more frequently the investment earns interest on the interest, leading to slightly higher overall returns. However, the difference between the values is relatively small, highlighting the diminishing impact of more frequent compounding for relatively low interest rates and shorter time periods.