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yesactuary
04-23-2009, 09:12 PM
I don't understand why we set up this way
dW(t)/W(t)=\phi*dS(t)/S(t)+(1-\phi)*r*dt ?

jraven
04-23-2009, 10:23 PM
The short answer is that the return on a portfolio is the value-weighted average of the returns on the individual assets. Over the infinitesimal time period dt the return on the portfolio will be dW/W, the return on the stock will be dS/S and the return on the risk-free asset will be r dt; on the other hand we have 100 \phi percent of our investment (by value) in the stock and 100 (1 - \phi) percent of our investment (again by value) in the risk-free asset. So the value-weighted average of the returns is

dW/W = (weight of stock) (return on stock) + (weight of rf asset) (return on rf asset)

\frac{dW}{W} = \phi \cdot \frac{dS}{S} + (1-\phi) r dt

yesactuary
04-24-2009, 08:54 PM
The short answer is that the return on a portfolio is the value-weighted average of the returns on the individual assets. Over the infinitesimal time period dt the return on the portfolio will be dW/W, the return on the stock will be dS/S and the return on the risk-free asset will be r dt; on the other hand we have 100 \phi percent of our investment (by value) in the stock and 100 (1 - \phi) percent of our investment (again by value) in the risk-free asset. So the value-weighted average of the returns is

dW/W = (weight of stock) (return on stock) + (weight of rf asset) (return on rf asset)

\frac{dW}{W} = \phi \cdot \frac{dS}{S} + (1-\phi) r dt

Thanks, Jraven.