OPTime Applet

This applet displays (in red)a user-editable projected value function for an asset, along with an exponential growth function (in gold) corresponding to a user-specified interest rate. The exponential is scaled to match the value of the asset at the point which the user chooses by dragging a slider along the graph. Following back along the exponential gives the present value of the asset (ie the money which invested now would match the value of the asset at the time 'x' years from now). The asset should be purchased at any time when its value is rising faster than the exponential and sold when the exponential is rising faster. The optimal purchase and selling times occur when the asset value function is tangent to an exponential corresponding to the current interest rate. These also correspond to the minimum and maximum respectively of the present value function (displayed in green in the applet)
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The Math

The condition that the graphs of y=f(x) and y=Aexp(rx)be tangent at time x is that f(x)=Aexp(rx) and f '(x)=rAexp(rx)[=rf(x)], So f '(x)/f(x)=r.

The condition that exp(-rx)f(x) have a critical point at x is that 0=d/dx[exp(-rx)f(x)]=(-r)exp(-rx)f(x)+exp(-rx)f '(x)=exp(-rx)[-rf(x)+f '(x)] So dividing out exp(-rx) we get 0=f '(x)-rf(x), or again f '(x)/f(x)=r.