Which is worth more after 5 years, an investment of $1000 at 5% interest compounded semi - annually(twice a year). or an investment of $1000 at 5% interest compounded continuously?
a)
amount = 1000(1.025)^10
= 1280.08
b)
amount = 1000 e^(5(.05))
= 1000 e^.25
= 1284.03
To determine which investment is worth more after 5 years, we can calculate the value of each investment using the compound interest formula.
1. Investment compounded semi-annually:
The compound interest formula for an investment compounded semi-annually is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, P = $1000, r = 5% (or 0.05), n = 2 (compounded semi-annually), and t = 5 years.
Let's calculate the future value (A) of the investment compounded semi-annually:
A = 1000(1 + 0.05/2)^(2*5)
A = 1000(1 + 0.025)^(10)
A ≈ 1000(1.025)^(10)
A ≈ 1000(1.280084)
A ≈ $1,280.08
Therefore, the investment of $1000 at 5% interest compounded semi-annually will be worth approximately $1,280.08 after 5 years.
2. Investment compounded continuously:
The compound interest formula for continuous compounding is:
A = P*e^(rt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
t = number of years
e = Euler's number (approximately 2.71828)
In this case, P = $1000, r = 5% (or 0.05), and t = 5 years.
Let's calculate the future value (A) of the investment compounded continuously:
A = 1000*e^(0.05*5)
A = 1000*e^(0.25)
A ≈ 1000*1.284025
A ≈ $1,284.03
Therefore, the investment of $1000 at 5% interest compounded continuously will be worth approximately $1,284.03 after 5 years.
Comparing the two investments, we see that the investment compounded continuously is worth slightly more ($1,284.03) than the investment compounded semi-annually ($1,280.08) after 5 years.