15.2 THE DISTRIBUTION OF THE RATE OF RETURN

The lognormal property of stock prices can be used to provide information on the probability distribution of the continuously compounded rate of return earned on a stock between times 0 and T. If we define the continuously compounded rate of return per annum realized between times 0 and T as x, then

ST=S0exT

so that

x=1TlnSTS0(15.6)

From equation (15.2), it follows that

xϕ(μσ22,σ2T)(15.7)

Thus, the continuously compounded rate of return per annum is normally distributed with mean μσ2/2 and standard deviation σ/T. As T increases, the standard deviation of x declines. To understand the reason for this, consider two cases: T=1andT=20. We are more certain about the average return per year ...

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