# 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 • If amount compounded n times a year then: • A = P ( 1+ r/n )nt • When n- âˆž then we call it continuous compounding: • A = Pert (this formula is derived using limits and continuity) Example If the interest rate is 10% per annum compounded continuously, then what is the effective annual interest rate? Solution At continuous compounding,$1 after an year will become 1.ert = e0.1x1 = 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.

Example

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

Solution

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:
Rc = Continuous compounding interest rate
Rm = Periodic compounding interest rate with â€˜mâ€™ periods per year
Rc = m.ln(1+ Rm/m)

Rm = m[e^(Rc/m) â€“ 1]

Try solving the above problems using these formulae