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Minimum variance can be solved simply and efficiently via a quadratic optimizer as the only key input is a covariance matrix. Drawdown or Sortino cannot be optimized via a covariance matrix unless you assume some functional relationship between co-variances/variances and your risk metric of interest. Likely you'll wind up with a similar portfolio to the ...

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Using solve.QP in R, a straightforward approach is to add a binary exposure vector as an inequality constraint to your Amat matrix for each group that you want to constrain. The only catch is that values in the exposure and b_0 vectors should be negative, since the function is really satisfying the constraints: A^T b >= b_0. For a simple mean-variance ...

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CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that $E[(R_m-R_f )]>0$). Every individual stock has some idiosyncratic risk in addition to ...

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I would look to run a pre-optimization routine over the whole universe of 200+ ETFs. I would use this pre-optimization to reduce the universe to a cardinality that provides optimal diversification effects. You can do that by first looking at pair-wise correlations and then also run optimizations to reduce portfolio variance by utilizing the covariance ...

5

If you're using Python, you may want to take a look at this question, to which the cvxopt library was the most popular answer. If not, or if you don't want to use cvxopt, then the basic setup is no different than using mean-variance optimization. You will almost always characterize your problem as a function taking a single vector argument (the portfolio ...

5

This optimization is trivial $$w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases}$$ That is to say, when you optimize only one weight will be nonzero. That's because these ratios incorporate no notion of distributional width, and therefore do not reward diversification. With no concentration ...

5

The Kelly Criterion aims to maximise the expected value of the logarithm of terminal wealth. The derivation starts off by assuming that there is a risky asset that is following a Geometric Brownian Motion: $$\frac{\,dS}{S} = \mu \,dt + \sigma \,dZ_t$$ This is combined with a riskless asset that is continuously compounding: $$\frac{dB}{B} = r \,dt$$ ...

4

Just includling my thoughts and the link in a proper answer. The goal function I suggest for this optimization is the following. $$\underset{w}{\arg \min} \sum_{i=1}^N [\frac{\sqrt{w^T \Sigma w}}{N} - w_i\partial_i\sigma(w)]^2$$ I added the square root compared to the comment as you are actually using the euler decomposition on $\sigma$ (not on $\sigma^2$)...

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There is a formula for calculating ES from a normal distribution. There is also a formula for ES of arbitrary distributions using a Cornish-Fisher expansions (easy for univariate processes but frustrating for multivariate). However, the most common approach is a scenario representation of the distribution. This could include using the historical distribution ...

4

If I understand you correctly, then you have a filter defined for your portfolio that is defined by "1.". A) So you either filter out these bonds before you start anything that has to do with the optimization. This should be the way to go if you are interested in speeding up your program. B) If you want to do everything in the optimization, then you need ...

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The Lyxor white paper Regularization of Portfolio Allocation contains a lot on this topic. The head of quant research there, Thierry Roncalli, also held a talk about this recently.

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Consider these two simple portfolios: Portfolio 1 returns -10% in month 1 and 10% in month 2. Average arithmetic return is zero, and cumulative return is $(1-10\%)(1+10\%)=0.99$. Portfolio 2 returns -50% in month 2 and 50% in month 2. Average arithmetic return is still zero, but cumulative return is $(1-50\%)(1+50\%)=0.75$, a much lower terminal value! In ...

3

I would use a Metropolis Monte Carlo / simulated annealing approach to solve your problem. Start with an arbitrary fully invested portfolio which satisfies constraints (2), (3) and the cardinality constraint $N \le K$. Then choose one of the following trial moves: Select two bonds $i,j$ at random and perform a random weight shift $w_i \rightarrow w_i + \... 3 In many cases, clients want to be fully invested and don't want their assets lying around in cash. Hence the budget constraint$\sum_i w_i = 1$is fairly common in practice. By the way, there are also cases where the constraint$\sum_i w_i = 0$is applied: the result is a dollar neutral portfolio with long and short positions, but no net investment (short ... 3 There is a simple reason to use prefer$CE$to pure utility:$CE$is independent of utility units. Thus it allows direct comparison. The cash equivalent of a risky portfolio is the certain amount of cash that provides the same utility that portfolio. So for portfolio$w$we can define$CE$via$U(CE)=E[U(w)]$or$CE=U^{-1}(E[U(w)])$. Note that for risk-... 3 When I select assets for a portfolio given an universe, I tend to pick ones that span the beta spectrum, given your selected benchmark. I find that if your portfolio of assets have varying volatility or correlation, you can achieve better diversification. I didn't come up with the idea but it comes from a rotational system's framework from the link below: ... 3 In case anyone is interested, I solved it using Nelder-Mead's algorithm instead. The performance could be better, but I didn't want to waste any more time in it. Here's the final solution: using System; using System.Collections.Generic; using System.Linq; using System.Text; using Extreme.Statistics; using Extreme.Mathematics; using Extreme.Mathematics.... 3 Even if some buy side funds are not allowed to short sell it does not mean they must buy. They could long sell, they can do nothing and stand on the sidelines and they can hedge, selling index futures or buy put protection on broad indexes or on the underlying of core holdings. Why this is an important point becomes apparent when you start to think about ... 3 First you need to define what you need a risk measure for. It is usually to take a decision, so you have an operational criterion that defines your risk. You should go back at this point and see what is the impact of a change of distribution on it. Just say for instance that you need a risk measure to take decisions according to a Sharpe ratio and define it ... 3 One really nice book that comes to my mind is Little, Rubin, Statistical Analysis with Missing Data I read part of it but probably it is too much information in your case. For your application, i think you can categorize the problem into two possible subproblems: First, time series that have unequal starting points (when some stocks' history is shorter)... 3 I use the 'implied correlation' defined as $$\rho = \frac{V^2_P-\sum V^2_j}{(\sum V_j)^2-\sum V^2_j}$$ for$V_p$the VaR (or volatility) of the portfolio, and$V_j$the VaRs (or volatilities) of the individual components. Essentially it shows what would be the common correlation that I would need to use in order to aggregate the stand-alone risks to the ... 3 It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the standard$\sum w_i = 1\$ constraint). Once he devised a computer algorithm [the Critical Line Method] for solving this problem (he was a math programming whiz) he seems ...

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It could be useful to set ex-ante what is your investable universe. It is pretty typical that at any period of time you would track only names wich are part of a Benchmark (SP500, R1000, FTSE, Nikkei 225 etc). Or set your own rule for what would consider an investable company. Usually the main criteria are : Market Cap, Market Cap accessible to global ...

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There is only one MVP and only one MDP portfolio so, unless these are the same, this will not be possible.

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You are looking for a Risk Parity based utility function. Risk Parity assigns weights to assets in the portfolio such that the marginal contribution to risk of all assets is equal. As a result, Risk Parity penalizes assets with high volatility and high correlation. AQR has a multi-decade research on the performance of Risk Parity portfolios and it is quite ...

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I played around with this a little using Portfolio Probe. The way to get risk parity portfolios (in the sense you are using) with that is to constrain the fractions of variance for each asset to be slightly more than one over the number of assets. Slightly more because trading is done in integer amounts. I took 20 assets and tried forming dollar neutral ...

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I've used the graphical lasso for exactly this kind of thing in the past. You can control the degree of shrinkage, which determines the how tight the clusters become.

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I do not see any advantage in this approach whatsoever, nor would I believe, as you suggested, that "many" use this kind of approach. In fact I find it horribly wrong. Using a single variable (CE in this case) to represent a non-trivial risk-return construct implies the ability to map such relationship to one variable representations. Everybody values risk ...

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I see your argument with the math. "1" is an arbitrary choice of positive numbers, and you could choose anything. In the end, you're going to scale the whole thing to fit your capital anyway. If you are using a numerical optimizer, it will be happier with something noticeably away from 0 and away from infinity, so I recommend choosing a specific positive ...

2

These games are usually won by luck. If there is no fee for buying stocks I'd diversify, i.e. buy many different stocks, to get stable returns. After some weeks you'll see which profit you'll need to beat. Depending on the rules if options are allowed you could invest in highly leveraged derivatives and hope you win. As there is no point not to try to win I ...

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