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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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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 ...

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 ...

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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 ...

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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 ...

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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

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

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 ...

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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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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

In 2006 Choueifaty proposed a measure of portfolio diversification, called the Diversification Ratio (DR), which he defined as the ratio of the weighted average of the volatilities of the assets in the portfolio, to the portfolios overall volatility. The DR of a long only portfolio is greater than or equal to one, and equals unity for a ...

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)...

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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)])$. Note that for risk-... 3 It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the standard$\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 ... 3 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 ... 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 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 ... 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: ... 2 Yes, it is normal for a L/S fund to have a lot of cash. When you short securities your account is credited with the proceeds from the sales. So if you short 1 million of stock you end up with 1 million cash and -1 million short stock position. Another way to look at it is: as you mentioned, the weights as a fraction of NAV have to add up to 1.0 by definition ... 2 @vanguard2k and @Theja provide useful information. In my experience, unequal starting points is most common, so I'll try to focus on that. The technique that @vanguard2k mentioned for unequal starting points can be thought of like a regression. You start with the longest available data and get the covariance matrix of that. For the next set of available ... 2 You can also use the Herfindahl-Hirschman-Index (HCI) as a measure for concentration. In portfolio analysis, you can calculate it as$\frac{1}{N} \leq HCI(x) = \sum_{i=1}^N x_i^2 \leq 1$where$x$is a vector of$N$portfolio asset weights. One can easily see that$HCI(x) = 1$if 100% is invested in a single asset, and$HCI(x) = 1/N$if the portfolio is ... 2 Both approach gives same results for stocks weights but different results for lamda values Correct approach is multiplying the covariance matrix by 2 but you only need stocks weights so its the same 2 You could try a heuristic approach. The problem can be split into two nested optimisations: i) in the inner optimisation, given a set of selected assets, compute mean--variance efficient weights; ii) in the outer optimisation, you iterate through combinations of assets. The inner optimisation can be solved via a quadratic programme (QP). For the outer ... 2 Your question is very confusing. But let's take it by parts: You say you have power utility so your utility is:$\frac{W_{t+1}^{1-\gamma}}{1-\gamma}$You have a risk-free rate number You have an option implied distribution for stock returns so that should give you a two vectors one with returns$r_{t+1}$and another with probabilities$dF(r_{t+1}$). Given ... 2 From b. we get$Vh = \lambda a$, so$h=\lambda V^{-1}a$(assuming V is invertible). Using this to evaluate a. we get$h^Ta = \lambda a^T V^{-1}a=1$(assuming$V^{-1}$is symmetric). We can solve this for lambda:$\lambda=\frac{1}{a^T V^{-1}a}$Now we can use this lambda in the previous expression for h to find the final explicit expression for h: $$h=\... 2 Consider the equation of two variables (basically your obj func):$$f(x,y) = x^2 + y^2$$The unconstrained minimisation is$x=y=0\$. If you now constrain this sum to be equal to one, post optimisation, well it doesn't quite work since you multiply by infinity. But, even if the obj func was slightly different and it was finite it wouldn't return the minimum ...

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