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Calculation of interest rates

 

  • There are many ways to calculate interest rates – 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-   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 – 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 ‘m’ periods per year
Rc = m.ln(1+ Rm/m)
 
Rm = m[e^(Rc/m) – 1]
 
Try solving the above problems using these formulae
 

 





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