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There are some cases where you can blend your portfolios using weights directly. One case involves corner portfolios. In this case a linear combination of weights is also efficient. Another case is where you can treat the two separate weights you have produced each as distinct portfolio under the assumption that the correlation between these portfolios is ...
19
Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
16
Risk Parity is not about "having the same volatility", it is about having each asset contributing in the same way to the portfolio overall volatility.
The volatility of the portfolio is defined as:
$$\sigma(w)=\sqrt{w' \Sigma w}$$
The risk contribution of asset $i$ is computed as follows:
$$\sigma_i(w)= w_i \times \partial_{w_i} \sigma(w)$$
You can then ...
15
I can think of three reasons.
First, and simplest, is that people care about variance.
Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing ...
14
+1 for asking an excellent question. I agree with the answers of @Owen and @chrisaycock - I'm late to the party but perhaps this will shed some light.
How practitioners or academics answer this question will tell you a lot about their view on the nature and sources of returns and risk. For example, the Fama-French "equilibrium" school of thought would argue ...
14
I took a quick look at Matlab's Financial Toolbox and attempted to map the features to corresponding Python packages –
For asset allocation, portfolio optimization, and risk analytics:
Standard packages such as scipy provide a large number of optimizers that should suit your needs. There are also pre-canned packages that do portfolio optimizations more ...
12
There is a vast, growing body of literature on risk parity, much of which is predicated on this idea (i.e. of optimizing a portfolio allocation without including expected return). As an example, The Journal of Investing put out an entire issue dedicated to the subject last year: see "Latest Approaches to Risk Parity and Diversification".
From "Risk ...
12
I'll answer by way of example. Suppose I want to buy a stock that is relatively cheap. Firstly, I need to define what is meant by cheap, so I might choose to look at the price-to-earnings ratio. Then I need to define what is meant by relative, so I might compare stocks only within a given sector.
This may work well at first, but then I notice that as I try ...
9
There has been a split in the community ever since Mandelbrot published his paper "On the Variation of Certain Speculative Prices."
See:
Mandelbrot, B. (1963). The variation of certain speculative prices.
The Journal of Business, 36(4):394–419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
8
Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero.
Discussion
A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ...
8
It is more complicated than that: It is not the optimization per se that leads to inferior results but the data you use.
Kritzman et al. makes a strong case in defense of optimization vs. 1/N in this popular paper:
In Defense of Optimization: The Fallacy of 1/N, Financial Analysts Journal, Vol. 66, No. 2, 2010 by Mark Kritzman, Sebastien Page and David ...
6
My two cents - The risk models are used to explain the variations (volatility) while the alpha models try to forecast drifts (mean). This explanation also works outside the framework of relative valuation.
6
The blog post http://www.portfolioprobe.com/2011/10/03/predictability-of-kurtosis-and-skewness-in-sp-constituents/ suggests that there is some predictability in kurtosis, but it isn't clear (to me at least) that there is enough predictabiilty to be useful.
If there is a place for higher moments, my guess is that it is in asset allocation problems where ...
6
I am very happy with the following equivalent formulation for the risk budgeting problem (as presented in Bruder, Roncalli, 2012, Managing Risk Exposures using the Risk Budgeting Apporach):
Let $b_i$, $\Sigma_{i=1}^n b_i =1$ be the risk budgets, $y_i$ the unscaled portfolio weights and $S$ the variance covariance matrix and $c$ arbitrary.
$$ y^* = \text{...
6
Meucci covers this example precisely in his paper "Fully Flexible Views: Theory & Practice". You can find his code here for three examples related to the paper. The Butterfly Trading example covers the CVAR scenario.
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Actually, Ralph Vince's Leverage Space Trading Model does utilise draw down. A short introductory pdf is available here, and the R-forge package is here.
Briefly, a genetic algorithm is used to model the maximum expected portfolio return based on a joint probability distribution of the portfolio component returns, subject to an overall maximum draw down ...
6
Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 there is 100% certainty that he will pay you back in a year. Before the pay date, he can do whatever he wants with your $100, while you have no access to it. Even ...
6
In recent years there has been much attention given to defining indexes other than market-cap based indices. While market-cap based indices approximate the theoretical Market Portfolio enshrined in textbooks, some people believe we could do better than that. One popular idea is that "market indexes overweight the most overvalued stocks", though this is ...
6
The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years:
The standard model (Markowitz, 1952, 1959) originally only considered:
Constrained model where short sales are forbidden
Only risky assets considered for investment (no risk-free asset)
Scenarios that the mean-variance ...
5
Adding a bit to the references mentioned by Quant Guy - apart from the aforementioned paper by Keating and Shadwick, Kazemi et al. introduce an alternative formulation of the Omega ratio (Sharpe-Omega) similar to the Sharpe ratio.
As noted by Patrick Burns, higher moments could have some use when instruments other than equity are involved (hedge fund ...
5
There is a paper by Goldstein and Taleb (2007) which tries to address this question of what number captures investors intuitive feelings of the volatility of series of returns and whether this coincides with the standard deviation of returns.
What they found was that Median Absolute Deviation does a much better job of capturing this intuition in a small ...
5
With minimum variance, the covariance matrix does not change when you change the holdings. So all the optimizer needs to do is change the weights. This makes it easy to calculate the gradients.
To construct the drawdown statistic, you would need the distribution of returns in each period to your horizon. You would then need to calculate the path of profits ...
5
I would create categories, and work on risk parity among the categories.
Otherwise, variance is not really a good measure of downside risk:
Change your risk measure, use a rolling window historical VaR or Expected Shortfall at some horizon that matches your investment style. downside semi-variance could do the trick too if don't want to change your algo ...
5
There are very powerful software solutions out there, so you should not reinvent the wheel.
One notable R package is PortfolioAnalytics.
You can find a very good introduction here, where your concrete constraints requirement is addressed in section 3.3, p. 6:
Benett, R.: Introduction to PortfolioAnalytics (2015)
5
Many long term investors use historical events and the market moves associated with such events to stress test their portfolios. For example, they use the dot-com bust, the latest "great recession", LTCM, Asian Crises, Black Monday, etc and any other dramatic events in history and see how their portfolios would have performed under those conditions. Of ...
4
Do you know the Black-Litterman Model?
In principle Modern Portfolio Theory (the mean-variance approach of Markowitz) offers a solution to this problem once the expected returns and covariances of the assets are known. While Modern Portfolio Theory is an important theoretical advance, its application has universally encountered a problem: although the ...
4
Yes, this is what the idea behind Omega as a portfolio optimization objective is all about.
Keating and Shadwick (2002a, 2002b) first introduced this notion. An introduction by Keating is here.
In fact, the Performance Analytics package in R includes a function to calculate Omega.
For your second question, one can compute the moments of higher orders ...
4
Very simple answer (in line with the second statement of Quant Guy): Bayesian modelling of the weights.
The data and the model provide the likelihood function of the data $y$ as a function of the model parameters
$p(y|\theta)$.
The researcher then specifies the prior distribution of the parameter or weight
$p(\theta)$).
The combined view of the ...
4
You can use the Exponentially Weighted Average directly aswell, finding the covariances and then normalizing back to the correlations:
$ \sigma_{t+1,jk} = (1-\lambda) \sum_{n=0}^\infty \lambda^{n} r_{j,t-n} r_{k,t-n} $
(this assumes average returns 0 etc etc. More general versions can be derived)
4
Another approach to construct a risk parity portfolio would be to use the formulation proposed by Spinu [1]: $$\begin{array}{ll}
\underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}\Sigma\mathbf{w} - \sum_{i=1}^{N}b_i\log(w_i)\\
\textsf{subject to} & \mathbf{1}^T\mathbf{w}=1.
\end{array}$$
where $\mathbf{w}$ is the vector of portfolio ...
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