9 votes
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Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?

There are a few related reasons: The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with ...
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  • 7,652
8 votes
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How are modern portfolio theory (MPT) and CAPM related?

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 ...
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  • 496
5 votes
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average return Vs cumulative return interpretation

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 -...
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  • 10.9k
5 votes

Create optimal portfolio by Treynor and Jensens Alpha

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 ...
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5 votes
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Question about quadratic form of f* in the Continuous Kelly Criterion

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 ...
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  • 1,356
4 votes

Sampling problem in portfolio optimization

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 ...
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4 votes
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What can I use to measure of diversification?

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 ...
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  • 9,077
4 votes

What can I use to measure of diversification?

You can also use the Herfindahl-Hirschman-Index (HHI) as a measure for concentration. In portfolio analysis, you can calculate it as $$\frac{1}{N} \leq HHI(x) = \sum_{i=1}^N x_i^2 \leq 1$$ where $x$ ...
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  • 400
4 votes
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Why did Markowitz not derive an equation for the efficient frontier?

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 ...
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  • 9,077
4 votes
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Literature about optimal number of stocks in a diversified portfolio

The first article on this was Fisher and Lorie "Some studies of variability of returns on investments in common stocks" JB April 1970. https://www.jstor.org/stable/2352105 The Statman ...
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  • 9,462
3 votes
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Analyzing stock performance - keep companies after bankruptcy?

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 ...
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3 votes

What can I use to measure of diversification?

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 ...
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  • 4,217
3 votes

Sampling problem in portfolio optimization

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 ...
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  • 866
3 votes

Handling Missing values in stocks returns when estimating the co variance matrix

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 ...
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  • 2,874
3 votes

Budget Constraint in Sharpe Ratio Optimization

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 ...
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  • 866
3 votes

Is the portfolio return distribution a weighted combination of individual asset return distributions?

As @Martin has pointed out in his answer, of course it is. Let $X=\sum_{i=1}^N w_ix_i$ denote the return of a portfolio of $N$ assets with multivariate distribution $f(x_1,x_2,\ldots,x_N)$. The ...
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  • 5,763
3 votes

Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?

As correctly mentioned in an earlier answer, portfolio optimization is something used for books in hedge funds and other institutions. As a smooth utility function changes slowly near its maximum, ...
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2 votes
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Budget Constraint in Sharpe Ratio Optimization

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 ...
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2 votes

Making portfolios better than others for a 16 week portfolio game?

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 ...
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2 votes

Determining the portfolio return distribution to calculate CVaR/ES

In the paper (Klaassen, 2002) the author propose a scenario generation method as well as a rule in order to precludes arbitrage opportunities. In the paper (Davari-Ardakani, et al. 2016) authors ...
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  • 255
2 votes
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market neutral weights and cash values

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 ...
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  • 9,462
2 votes

Handling Missing values in stocks returns when estimating the co variance matrix

@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 ...
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  • 5,311
2 votes

Dmat argument in solve.QP R function: Cov or 2*Cov?

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
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2 votes

Portfolio optimisation with conditional weight restrictions among asset

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) ...
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2 votes
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What is the return of risky asset in direct utility optimization probem?

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 ...
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  • 6,820
2 votes
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Characteristic Portfolio for an Attribute

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 ...
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  • 9,077
2 votes
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Difference between constraining pre and post optimization

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 ...
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  • 8,037
2 votes

how to construct a diversified portfolio based on correlation

The maximum decorrelation portfolio can ensure your portfolio is not so correlated in one general asset class: min $\mathbf{w^{T} C w} $ subject to constraints that weights sum to 1 and are non-...
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  • 2,835
2 votes

Portfolio selection with no risk-free asset

There are a few reasons the authors may have only looked at risky assets. First, they are trying to find a faster way to solve a mean-CVaR optimization through relaxations. Therefore, they probably ...
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  • 2,750
2 votes

Quantatively identifying stocks to short when overall market starts to roll-over

my take would be -- short the index. I think when markets turn most people will sell the index and that will have been the best trade to be in. Not sure you can rely on historical/statistical beta ...
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