I would like to know if there is a way (or theory) to manage a multi-strategy, multi-instruments portfolio that would calculate the optimal weight to allocate capital for each combination of strategy and instrument (sometimes we may find one strategy works for many instrument or vice versa). My first idea is that we can treat each combination of strategy and instrument as a imaginary instrument and introduce Markowitz's portfolio theory to find the optimal weight. However, I also learned that the estimated return and covariance is very noisy in practice and deduce very different results from CAPM. May not be a ideal way. Another problem is that my strategies could be various across types and timeframe (from intra-day to month holding times). Estimating average return on daily basis could be misleading and underestimate return of strategies that works infrequently (for instance, strategies that take advantage on annual earning announcement or monthly FOMC meeting). I checked Kelly formula and found the answer from it is exactly as Markowitz's theory. Thus, most issues on mean-variance theory (e.g. noise of estimation for mean and variance) applies here. I wonder if anyone can share some thoughts on this issue. Any idea/example? Many thanks.
My approach to this is to solve it numerically. Given N strategies S1, S2, ..., SN, solve for this: OG = MAX(w1 * LN(R1) + w2 * LN(R2) + ... + wN * LN(RN)) where OG is the optimal rate of growth R1, R2, ..., RN are the returns of strategies S1, S2, ..., SN w1, w2, ..., wN are the weights (fractions of capital) allocated to strategies S1, S2, ..., SN LN(.) is the natural logarithm Note that w1+w2+...+wN can be greater than 100% of the capital, to allow leverage. Markowitz did argue that using the mean-variance analysis is "almost the same" as maximizing the utility function. The advantage of maximizing the utility function is to avoid the problems that you mentioned, such as the return normalization for strategies which operate on different time scales.
Doing this without the asinine assumptions that make up the Markowitz theory is quite a challenge. You'll need three indices: one for the strategy number, one for the instrument number and one for the trade number. So something like R_i,j,k might represent the k'th trade return from applying the i'th strategy to the j'th instrument. You'll be looking for the weights w_i,j that maximize your overall gain. I'm no expert on optimization theory but this looks like a hairy one. For an example of multiple betting strategies applied to a single "instrument", check out the "What is your strategy?" thread.
Not so hairy afterall. What you want to maximize is this: (1 + (w_1,1)(R_1,1,1))*(1 + (w_1,1)(R_1,1,2))*...*(1 + (w_S,T)(R_S,T,V)) where S is the number of strategies (any positive integer) , T is the number of instruments (any positive integer) , V is the number of trades per strategy/instrument pair (at least ten, preferably more). At least one of the series sum[ R_i,j,k ]_k=1toV has to be positive. It is (theoretically) possible that a negative-sum series may have a positive value of w_i,j in the optimal solution due to some sort of arbitrage, as illustrated in the link above. Clearly a numerical approach is the most practical. No general analytic approach looks feasible.
What do you mean by MAX(w1 * LN(R1) + w2 * LN(R2) + ... + wN * LN(RN))? If it is exactly what it literally means, the weight will concentrate to the instrument with maximum historical return. This allocation completely ignores diversification, which is not ideal. Or if you mean MAX E(w1 * LN(R1) + w2 * LN(R2) + ... + wN * LN(RN)), then it is equivalent to Kelly's question: Max E(geometric return). I am afraid both ways are not what I am looking for. But still thanks for your suggestion.
After I read Markowitz's review (Markowitz 2010)on his theory, I would say Markowitz's assumption is not asinine. All he assumed is the expectation utility can be approximated by Taylor expansion to second order, i.e. the mean and variance. The functional form of utility is not very important if it is smooth. What you suggest is the same way as my first thought. But I am not quite satisfied for the problems I mentioned in the first post. However, I appreciate your suggestion.
If the Kelly formula you're talking about is the CK formula, it's been debunked. http://www.elitetrader.com/vb/showthread.php?t=283715
Thanks! It is a example to remind us of the limitation of mathematical approximation. As I mentioned before, Markowitz's theory is valid if return is not distributed too widely, e.g. interval (-40%, 50%). The example with 100% loss with 1% probability is out of reach of Markowitz's theory. In addition, as we are calculating geometric return, logarithm return is more appropriate in this case, which will show E(return)=- infinite, not positive.
Nope. E(return) == (.89)(.13) + (.10)(-.05) + (.01)(-1) == +0.1007 There is no geometric return. There is however a geometric growth rate. GGR[f] == ((1+f*.13)^.89)*((1-f*.05)^.10)*((1-f)^.01) where GGR is the geometric growth rate The goal is to find k, the value of f that maximizes GGR : GGR[k] > GGR[ f | f<>k ] This is where logarithms come into play: they facilitate finding the maximum. Log[GGR[f]] == 0.89*log[1+.13f] + 0.1*log[1-.05f] + 0.01*log[1-f] Setting the first derivative equal to zero leads directly to the Kelly equation I previously referenced.