# Calculation of interest rates

- There are many ways to calculate interest rates â€“ annual, semi annual, quarterly, continuously compounding and so on
- Each rate can be expressed in the form of another rate. For example an interest rate of 10% compounded semi-annually would fetch (1 + 10% / 2) * (1 + 10% / 2) = 1.1025 (remember 6months rate is 10% / 2) on $1 after one year. This is equivalent of 10.25% annual rate
- Amount compounded annually would be given by:
- A = P (1+ r)
^{t} - A- terminal amount
- P- principal amount
- r- annual rate of interest
- t- number of years for which the principal is invested

- A = P (1+ r)
- If amount compounded n times a year then:
- A = P ( 1+ r/n )
^{nt}

- A = P ( 1+ r/n )
- When n- âˆž then we call it continuous compounding:
- A = Pe
^{rt}(this formula is derived using limits and continuity)

- A = Pe

If the interest rate is 10% per annum compounded continuously, then what is the effective annual interest rate?

At continuous compounding, $1 after an year will become 1.e^{rt} = e^{0.1x1} = 1.10517.

Had it been just annual compounding, then the interest rate required for $1 to rise up to $1.10517 would have been 1.10517 â€“ 1 = 10.517% which is the effective annual rate.

If the interest rate is 10% per annum compounded semi-annually then what is the equivalent continuously compounded interest rate.

A = 1(1+10%/2)^2 = 1e^(rx1)

=> 1.1025 = e^r

=> r = 0.09758 = 9.758%

Alternatively, following formulae can be used to calculate the interest rates:

R_{c} = Continuous compounding interest rate

R_{m} = Periodic compounding interest rate with â€˜mâ€™ periods per year

R_{c} = m.ln(1+ R_{m}/m)

R_{m} = m[e^(R_{c}/m) â€“ 1]

Try solving the above problems using these formulae