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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 ...


9

Yes Strategic Asset Allocation: Determining the Optimal Portfolio with Ten Asset Classes Strategic Asset Allocation and Commodities The Case for Commodities An Asset Class for All Seasons: The Benefits of a Strategic Allocation to Commodities No Should Investors Include Commodities in Their Portfolios After All? New Evidence My Take Although there ...


9

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 ...


8

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 ...


8

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 ...


7

+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 ...


7

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 ...


5

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 ...


5

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.


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 ...


4

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.


4

Here example of practical application of Markov ideas to trading.


4

I think you might be interested by an article I mentioned in this post: Carlo Acerbi from MSCI presents in this presentation an innovative approach to liquidity risk. The idea is basically to model how liquid an asset is and how your portfolio allocation should take this risk into account. This way of seeing risk is in my opinion pretty interesting an ...


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

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

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 ...


4

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 ...


4

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 ...


3

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 ...


3

There are a couple components to this problem: Construct a portfolio incorporating relative views where weights shrink towards a default policy 1a. Some views are in currency-hedged terms 1b. Relative views are on a 5-point scale Maximize a mean-variance utility function incorporating a penalty for transactions costs and requiring that weights sum to ...


3

I would use something similar to Black Litterman where both confidence of manager views as well as dynamic correlations are used to re weight asset classes. For a good look at how transaction costs affect long term allocation decisions with changing parameters, you may be interested in Balduzzi and Lynch (1999). Another approach to consider is to look at ...


3

On the bond side, perhaps the yield on the 10-year constant maturity series available at the St. Louis Federal Reserve (FRED website). On the equities side, S&P 500 or a global index such as MSCI are good but you might not have history thru the early 80s on the latter.


3

Very informative and balanced is: The Strategic and Tactical Value of Commodity Futures by Claude B. Erb, CFA, and Campbell R. Harvey One well-known scientifically based passive investment fund in Germany (ARERO) draws a ratio of 15% for commodities (60% world stocks and 25% bonds, rebalanced on a yearly basis) as a conclusion out of this - see the live ...


3

1) Calculate exponential averages (EMA) for time series A & B. 2) Calculate exponential standard deviations for A & B. My little hack for this is to calculate an EMA of squared returns, then subtract the squared EMA of simple returns, then take the square root of this. sqrt( ema(return^2) - ema(return)^2 ) 3) Apply the same concept to calculating ...


3

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 ...


3

You can use a branch-and-cut algorithm (this is what mixed-integer solvers use). The idea is to solve the problem recursively, by considering subproblems in which the constraints are $w_i=0$ for some stocks and $0.05 \leq w_i \leq 1$ for others. This gives $2^n$ convex optimization problems, and you want the best solution among them. Even when $n$ is ...


3

If $\sum_{i=1}^k \alpha_i<1$, then you could just leave the remainder of the portfolio in cash. If $\sum_{i=1}^k \alpha_i>1$, that means you will have to take on some leverage in order to minimize tracking error. If you have a leverage constraint, then you can run this as a quadratic program with bounds on your coefficients. A regression should give ...


3

I encountered the same problem as I needed an index including equity, fixed income and alternative assets for my master thesis research. I needed to estimate the beta of my optimal portfolios with a representative index in order to compare the portfolios by treynor and Jensen measure. Combining for instance, the MSCI and Barcap is no option as the I already ...


3

Very simple answer: Duration, my friend. The notes present much higher interest sensitivity and if rates across the curve rise the investment in the longer duration notes will cause a mark to market loss larger than the outstanding treasury notes. Not always do banks have the luxury to hold all their assets until maturity, especially not when the regulatory ...


3

This however, goes against the conventional wisdom that variance becomes smaller as you hold the portfolio longer. Which conventional wisdom says this? If the variance decreases with time, then the likelyhood of getting a return close to the expected return increases (Cecbycev's inequality). So you are telling me, I know more about the long-time ...



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