Continuous Compound Interest



~If Po amount of money is compounded n times per year at an interest rate of r% for t years, the ending balance will be

Po(1+(r/n))nt dollars.

~See the following link for a derivation:

Compound Interest

~If the number of periods per year, n , approaches ∞, Po will be compounded continuously.

~The formula becomes, Poert

The following is the derivation:

Let P = Po(1+(r/n))nt

Take the ln of both sides to get, lnP = lnPo+nt[ln(1+(r/n))]

Now, let n approach ∞

Note: All limits in the derivation below has n approaching ∞

The equation becomes, lnP= lnPo + t{lim[(ln(1+(r/n))/(1/n)]}

L'Hospital's rule can be applied to the limit on the right side of the equation since it is of the form 0/0

Applying L'Hospital's rule, we get, lnP= lnPo + t{lim[{(1/[1+(r/n)]}(-rn-2/(-n-2]}=

lnPo + rt{lim[1/[1+(r/n)]}=lnPo+rt

Or, [lnP - lnPo]= rt, or ln(P/Po)= rt, so P/Po = ert

which gives, P = Poert