Financial Returns

We begin an investigation of financial returns with a very simple model, that of compound interest.  A starting principal is (Subscript[p, 0]) which grows at constant annual decimal rate, ar, with interest compounded each year.  The resulting formula, has time, t, in years.

P =(Subscript[p, 0]) (((1 + ar))^t)

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

P=(Subscript[p, 0]) (((1 +(ar/365)))^(365 t))

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

(lim_(n->∞)) (Subscript[p, 0]) (((1+ (r/n)))^(n t))=(Subscript[p, 0]) (e^(r t))

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)

ar=(e^r)-1

Plot[{100 (1.07)^t, 100 (1 + .07/365)^(365 t)}, {t, 0, 10}, PlotStyle -> {Blue, Red}, PlotLabel -> "Annual Compound - Blue\nDaily Compound - Red", FrameLabel -> {"Years", "Amount"}, Frame -> True]

The effective annual rate of the daily compounding is actually:

dr = (1 + 0.07/365)^365;
CellPrint[TextCell[ToString[dr - 1] <> " Annual Rate", "Text"]]
CellPrint[TextCell[ToString[Log[dr]] <> " Continuous Rate", "Text"]]

0.072501 Annual Rate

0.0699933 Continuous Rate

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© Copyright 2008 mathestate    Fri 19 Dec 2008