Question

at what annual interest rate compound semi annually will a certain amount triple itself in 20 years​

Answer 1

Step-by-step explanation:

The formula for the future value (FV) of some some present value (PV) compounded at an interest rate of i per annum over a period (n) of 20 years is:

FV = PV (1 + i ) ^ n

Where the symbol ^ means to the power of.

In your question we are given:

n = 20

FV = 3 x PV (if we take PV = $1, then FV = $3)

Your question asks for i, so we have to do some mathematical rearranging to solve for i:

FV = PV (1 + i ) ^ n

FV / PV = (1 + i ) ^ n

(FV / PV) ^ 1/n = ((1 + i ) ^ n) ^ 1/n

(FV / PV) ^ 1/n = (1 + i )

i = (FV / PV) ^ 1/n - 1

So for your question we have:

i = (FV / PV) ^ 1/n - 1

i = (3/1) ^ 1/20 - 1 = 3^0.05 - 1 = 5.65

So the annual compounded interest rate to triple $1 today to $3 in 20 years is 5.65%.

Now there are variations of the above formula where you can account for the timing of the compounding.. the above formula assumes interest is calculated once per annum. In reality it is compounded continuously with Euler’s e or daily or weekly/fortnightly/monthly.

Fun fact: an interest calculated annually will be less than when it is calculated say weekly which is less than daily, which is less than continuous compounding.

Does that answer your question?