9
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 discrete holdings is quite hard;
For small retail investors fixed trading costs will swamp the rounding in your example but also for larger amounts (at least ...
8
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
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 ...
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
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 ...
4
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.
4
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 ...
4
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 $x$ is a vector of $N$ portfolio asset weights.
One can easily see that $HHI(x) = 1$ if 100% is invested in a single asset, and $HHI(x) = 1/N$ if the portfolio is ...
4
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 article you quote "How many stocks make a diversified portfolio" is from 1987, it is still referenced and broadly agrees with the other.
A more recent ...
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
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
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
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
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
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 ...
3
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 distribution of $X$ may be found by $(N-1)$-fold convolution of the $N$-dimensional distribution $f$. Unfortunately, the integrals are not that easily solved anymore.
...
3
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, perturbation by rounding is inconsequential. If you would like to adopt a utility approach to personal investing, the following algorithm could work: (1) solve ...
2
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 propose a new scenario generation method, it preserves marginal (non-normal) distributions of asset returns and it precludes arbitrage opportunities.
References
...
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
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
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 ...
2
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-negative, where $\mathbf{C}$ is the correlation matrix of multivariate asset returns. If you also regularize the portfolio weights with an L2-norm by adding $\| \...
2
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 saw handling the risk (CVaR aka ES) as the most interesting part of the problem. Granted, doing so completely ignores that they should be looking at excess ...
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