5:Continuous compounding formulas and Euler's number

Let see compounding interest formulas.

 If P is principal r is interest rate per annum (for 10% interest, r = 0.1) 

 interest for one year = principal *interest rate = P*r .

Total amount after one year =Principal + interest =P+Pr.

In annually compound interest rate regime P+Pr = P(1+r) become principal at the end of first year. 

Then the next year interest would be calculated on this principal

P(1+r) r would be interest for the second year.

At the end of second year, total amount would be = P(1+r)+P(1+r) * r = P(1+r)^2

After n years total amount = P(1+r)^n

For the same interest rate r, after 6 months how much interest would have been accumulated? Half of the annual interest , that is Pr/2  

 If interest is compounded to principal semi-annually, at the end of first 6 months

Total amont= P+Pr/2=P(1+r/2). For the next 6 months interest will be calculated on this. Interest =P(1+r/2)* r/2.

Total amount at the end of one year =P(1+r/2) +(P(1+r/2) * r/2) = P(1+r/2) ^ 2.

At the end of n Years ,total amount = P(1+r/2) ^ 2n

 If interest rates are semi-annually compounded, on annual basis you earn little more than the rate r.How much exactly? Solve for y to get effective annual rate  in the following equation

P(1+y) =P (1+r/2)^2 . It will give y = r+r^2/4. If r = 10 %, in semi-annual compounding is employed, you would get 10.25% p.a effectively.

 Many US Treasury notes are semi-annually compounded.

 If you compound interest quarterly, Total amount at the end of year = P(1+r/4) ^4

 Compounding monthly will give you P(1+r/12)^12 

 Many senior citizen deposits should pay interest on deposits monthly.

 Compounding weekly will give you P(1+r/52)^52 for a year

Weekly interest rates are not common in finance institutions . But local money lenders may collect interest weekly.

Jacob Bernoulli ,famous mathematician, studied compounding interest rates by increasing compounding frequency, with r =1 (100% interest rate). He noticed weekly compounding gives 2.69 times P, while daily compounding gives 2.71 times P. So even if you reduce interval and increase compounding frequency, increase in return is approaching some limit. Theoretically you can compound interest for every hour, second, nano second etc. When the time interval approaches zero, number of intervals (x)tends to infinity 

  Lt              (1+1/x) ^ x  can be evaluated in three steps.

x-> infinity

 1) Binomial expansion will you (1+ xC1 (1/x) + xc2 (1/x)^2    .....) 

 2) When x-> infinity x^p/x^p =1 and higher orders of x in denominator will make the term zero .

 3)Simplifying and evaluating for first few terms of 

 1+1+1/2!+1/3!...infinity will give you 2.71 approx... . The series is for famous Euler's number e.

When you continuously compound interest at 100% annual rate, instead of 2P in return, you get e*P  after one year.((e-1)*100 =171% approx is the effective annual interest rate here)

 From the above you can easily derive for any interest rate r, continuous compounding will have  (e^r-1)*100 effective rate.

Though continuous compounding is not normal in practice , it could be used in continuous time model for pricing instruments.

When Google went public in 2004, they offloaded shares to raise ‘e’ billion dollars .