Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
and to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2
where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdz
and the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdz
which is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdz
so now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.
Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
E[R(T)]=E[V(T)]V(0)=T−1∏i=0E[V(i+1)V(i)|Fi]=T−1∏i=0E[kS(i+1)S(i)−k|Fi]=kT(exp(μ)−1)Tand to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdzand the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdzwhich is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdzso now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.