Using the changing interest rates


F=P (1+r ) (1+r ) ... (1+r )


You borrow $5 mln. at 10 % of interest on 5 years. Starting the second year the interest rate will increase on 2% every year. Calculate the future value of the sum.


F=5 (1.1 1.12 1.14 1.16 1.18)=9.612 млн. тенге


Discounting semi-annually, monthly and daily

Sometimes financial transactions take place on the basis that interest will be calculated more frequently than once a year. For instance, if a bank account paid 12 % nominal return per year, but credited 6% after half a year, in the second half of the year interest could be earned on the interest credited after the first six month. This will mean that the true annual rate of interest will be grater than 12 %.

The greater the frequency with which interest is earned, the higher the future value of the deposit.


If you put $10 in a bank account earning 12 % per annum then your return after one year is:


If the interest compounded semi-annually:

10(1+[0.12/2])(1+[0.12/2])=10(1+[0.12/2])2 =11.236


If the interest is compounded quarterly:

10(1+[0.12/4])4 =11.255


Daily compounding:

10(1+[0.12/365])365 =11.2747


Continuous compounding

If the compounding frequency is taken to the limit we say that there is continuous compounding. When the number compounding periods approaches infinity the future value is found by F = Pern where e is the value of the exponential function. This is set as 2.71828 9 (to five decimal places, as shown on a scientific calculator).

So, the future value of $10 deposited in a bank paying 12 % nominal compounded continuously after 8 years is:

10 x 2.718280.12 x 8 = 26.12


Converting monthly and daily rates to annual rates

Sometimes you are presented with a monthly or daily rate of interest and wish to know what that is equivalent to in terms of Annual Percentage Rate (APR) (or Effective Annual Rate (EAR)).

If m is the monthly interest or discount rate, then over 12 month:


(1+m)12 = 1+r where r is the annual compound rate.


r = (1+m)12 – 1


Thus, if a credit card company charges 1.5 % per month, the annual percentage rate (APR) is:

r = (1 + 0.0015)12 – 1 = 19.56 %


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