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Financial Returns

We begin an investigation of financial returns with a very simple model, that of compound interest.  A starting principal is FinancialReturns_1.gif which grows at constant annual decimal rate, ar, with interest compounded each year.  The resulting formula, has time, t, in years.

FinancialReturns_2.gif

More typically for a savings account, the compounding interval might be daily with the formula becoming:

FinancialReturns_3.gif

The compounding interval can be made infinitely small leading to the formula for continuous interest.  r is now a continuous interest rate.

FinancialReturns_4.gif

To convert an annual decimal rate to a continuous rate, where log is the natural logarithm (the conversion will then give the same result as the first equation to the continuous equation):

r = log(1 + ar)

FinancialReturns_5.gif

Graphics:Annual Compound - Blue Daily Compound - Red

The effective annual rate of the daily compounding is actually:

0.072501 Annual Rate

0.0699933 Continuous Rate

FinancialReturns_7.gif



© Copyright 2008 mathestate    Fri 19 Dec 2008