Inclusion of a Risk Free Asset

Suppose that there is a risk-free asset with constant interest rate ry. This asset can be considered to be a bond whose price po(t) satisfies the equation dpoiO .

Assuming that there is no other combination of assets that produces zero variance, the risk-free asset is on the efficient frontier. Indeed, it is the minimum-variance point. To find the entire efficient frontier it is therefore only necessary to find the log-optimal point, and we shall do that now

The log-optimal portfolio is defined by a set of weights u^, iu2,. . , wn for the risky assets and a weight wq = 1 — i wj *or risk-free asset , The weights for the risky assets are chosen to maximize the overall growth rate; that is, to solve the problem max

Setting the derivative with respect to Wk equal to zero, we obtain the equation for the log-optimal portfolio ¡jl-, ~ ry — Ylj= i <JUwj ~ which we highlight:

The log-optimal portfolio When there is a risk-free asset, the log-optimal portfolio ~~y has weights for the ri-sky assets that satisfy n

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