To determine the value of an investment of $3000 at a 5% interest rate over 5 years, we will calculate it for six different compounding periods: annually, semiannually, monthly, weekly, daily, and continuously.
i. Compounded Annually
For annual compounding, the formula used is:
A = P(1 + r/n)^(nt)
Where:
- A = the amount of money accumulated after n years, including interest.
- P = the principal amount (the initial amount of money).
- r = the annual interest rate (decimal).
- n = the number of times that interest is compounded per year.
- t = the number of years the money is invested for.
Here, P = $3000, r = 0.05, n = 1, and t = 5.
Calculating:
A = 3000(1 + 0.05/1)^(1*5) = 3000(1.27628) ≈ $3828.84
ii. Compounded Semiannually
For semiannual compounding, n = 2.
Calculating:
A = 3000(1 + 0.05/2)^(2*5) = 3000(1.025)^(10) ≈ $3861.37
iii. Compounded Monthly
For monthly compounding, n = 12.
Calculating:
A = 3000(1 + 0.05/12)^(12*5) = 3000(1.00416667)^(60) ≈ $3906.40
iv. Compounded Weekly
For weekly compounding, n = 52.
Calculating:
A = 3000(1 + 0.05/52)^(52*5) = 3000(1.00096154)^(260) ≈ $3924.07
v. Compounded Daily
For daily compounding, n = 365.
Calculating:
A = 3000(1 + 0.05/365)^(365*5) = 3000(1.000136986)^(1825) ≈ $3947.92
vi. Compounded Continuously
For continuous compounding, the formula used is:
A = Pe^(rt)
Calculating:
A = 3000e^(0.05*5) ≈ 3000 * 1.28403 ≈ $3852.09
In summary, the values obtained for the investment at the end of 5 years are:
- Annually: $3828.84
- Semiannually: $3861.37
- Monthly: $3906.40
- Weekly: $3924.07
- Daily: $3947.92
- Continuously: $3852.09