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8

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 ...


6

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 ...


5

Since you are asking for low correlation of the assets, I'm guessing that you are really trying to get a low (or minimum) volatility portfolio. If that is the case, then the steps for one approach are: estimate the variance matrix of the universe of assets use a portfolio optimizer to select the minimum variance portfolio given your constraints This ...


5

Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution. Here's another approach by Kuhn. These are all one-period ...


4

One of the major assumptions is that you have zero transaction costs. Another one is that your returns are tax-free. Otherwise it looks to me to be a windowed version of CBAL (constant rebalanced). A more technical analysis can be found at: Castonguay, Portfolio Management: An empirical study of the Anticor algorithm (An MS thesis) Covan and Gluss, ...


4

The question is somewhat vague (lacking a well-defined objective), so this advice may not apply. Be mindful that you may be simultaneously considering multiple futures contracts that contain overlapping underlying constituents (e.g. futures that track the EuroStoxx 600 and DAX). If you are using a risk model, the idiosyncratic risk may not, in fact, be ...


4

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 ...


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 ...


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 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 ...


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

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 ...


3

Mean-Variance optimization is the standard finance answer to this question. However, solutions can be costly since the weights will likely be dispersed across many instruments raising fixed transactions costs. I would consider Sparse PCA as another solution where you can specify cardinality constraints on the number of securities in your basket to better ...


3

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 ...


2

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 ...


2

MSCI has country indices for developed markets going back to 1970 in many cases and a decent history for emerging markets (starting 1988). iShares has pretty liquid ETFs for many of the most popular countries and regions, such as EAFE (EFA), Emerging Markets (EEM), Japan (EWJ), Germany (EWC), Canada (EWC), etc. Other major indices with very long histories ...


2

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 ...


2

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.


2

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 ...


2

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 ...


2

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 ...


2

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: ...


1

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 ...


1

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 ...


1

Portfolio management is about solving problems in the real world. In the real world, it is highly unlikely that EVERY asset has a negative expected return. If all the assets in your universe have negative returns, expand your universe to include a short-term fixed income security that is bound to produce a return greater than (or at a minimum equal to) ...


1

Assume that instead of a possible portfolio of 1000 bonds, the portfolio may only contain M bonds. The input vector then needs to contain both the weightings and the choice of bonds, but how can you present the choice of bonds as an input vector? Consider sorting the candidate bonds by Macaulay duration. Given a single Macaulay duration value, then, you ...


1

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 ...


1

Yahoo! Finance allows you to list all ETFs and sort by inception date or size. Considering your follow-up question, though, I think you are asking the wrong question here. ETFs are a relatively recent innovation. Only 32 of these ETFs were around in 2000. For example, the Russell 2000 ETF (IWM), today's third most actively traded ETF overall, was only ...



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