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13

One simple method, based on the principles of mean-variance optimization, is to set the weights proportional to the product of the inverse of the covariance matrix and a vector of standard deviations. This implicitly assumes that the normalized expected return of each stock is equal. If you wish, you can take only the top 5 weights and set the others to ...


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 ...


8

Your formula looks like cointegration (between the price time series) rather than correlation (between the returns). To simulate "correlated random walks", i.e., random walks built from correlated innovations, you can just build the desired covariance matrix (for instance, put ones on the diagonal and $\rho$ everywhere else), take multivariate gaussian ...


7

The problem of the selecting the best portfolio (according to some risk measure) with a limited number of assets can be formulated as a mixed integer linear or quadratic program and is reviewed in the recent paper "Portfolio selection problems in practice: a comparison between linear and quadratic optimization models". It can be solved for reasonable sizes ...


6

There is, in fact, a large literature on this subject, but you may have been searching for the wrong terms. This issue is broadly explored within the literature regarding a preference for higher moments. Cauchy distributions, themselves, are very hard to work with because they don't have any well-defined moments. One possible solution to this is to use ...


6

This is indeed a subtle point. What is generally meant with this statement is that correlation is going up in bear markets, so it is not so much the "turmoil" part (i.e. volatility per se) but the "trend" (i.e. negative in this case) part. Putting it another way is that when you control for volatility not the correlation but the covariance (which is the part ...


5

Accurately stated: Diversification helps during turmoil, but helps less as what would be expected by using $w^T \Omega w$ as the portfolio variance where the off-diagonal covariances are estimated during tranquil periods. This is because correlations and covariances change during turmoil, typically increasing. This reduces the benefit of diversification ...


4

I think this is a very good and valid question. I will try to give a more general answer here. It is by now a well known fact that much of the classical stuff won't work in the manner it was thought and supposed to work. One of the points that are not clear when it comes to real financial time series is how they are distributed (they are obviously not ...


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)


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

You're right, I hope he meant exactly the opposite, and the formula you provided is indeed part of the definition of a coherent risk measure. In fact, I would say that the risk of the sum is less than or equal to the sum of the individuals as in some cases you would like your model to accept no diversification effect. As John mentioned in his comment, ...


3

What a great question -- it touches on many issues at the core of quantitative finance. This answer might be a lot more than you bargained for, but it's too interesting to pass up. References Mostly, this subject falls somewhere at the intersection of these three highly-interrelated topics: risk-neutral valuation, rational pricing and the fundamental ...


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

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 ...


2

The controversy surrounding commodity futures flows from Gorton and Rouwenthorst (2004). The authors show an equal-weight portfolio of long positions in commodity futures provides a Sharpe ratio greater than the one earned by holding a cap-weighted portfolio of U.S. stocks (beginning in the 1950's through 2004 or so). In essence, why should holding a ...


2

You are not doing anything wrong. You just need to multiply the absolute return by the currency conversion factor. Example: You trade 200,000,000 yen notional and generate a return of 16% on that notional, then simply multiply 32,000,000 jpy gain by your conversion factor 0.0126 to yield a return of 403,200 USD. The return of 16% was generated on the ...


2

since you've assumed that all returns are independent, the covariance matrix, $C,$ is diagonal. In the comments, you are assuming that the investor is a mean-variance investor. It's a general result that every portfolio that maximizes return for a given variance is a tangent portfolio for some risk-free rate, $R.$ Let $e=(1,1,...,1).$ and let $\mu$ be the ...


2

Maybe I don't get your question well, but what it appears that your goal is to buy securities in order to reduce the correlation between your portfolio constituents. So firstly you need a metric of diversification. Something simple you can use, is calculate the correlation matrix, and the weights of each position. A simple metric would be the sum of ...


2

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 ...


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

I think the first step is to define what you mean by "properly diversified". A traditional/fundamental standpoint would be that the portfolio is comprised of many different sectors, industries, ect. The more "quant-like" approach and in my opinion, a more realistic approach, is to understand correlation between portfolio assets and the dynamics of said ...


2

As you and @Malick noted, VaR only gives a certain threshold given a certain confidence but says nothing about what happens beyond that point (tail risk). For loss distributions with long tails, this would underestimate the risk. Regarding VaR having a problem with diversification - VaR is technically not a coherent risk measure. In simple terms, we would ...


1

Exactly, VaR is nothing more than a threshold loss value. But it does not tell you how big your loss can be (no information about the shape of the tail). To get more information about it you can use the Expected shortfall which is the expected loss given that a loss occurs in the tails. Diversification decreases the VaR, however extreme events may be, ...


1

Alex C's and Kiwiakos' answers are definitely the most realistic approaches. If you are open to consider also other kinds of risk measures, further alternatives might be thought of. Variance / correlation based approaches interprete "diversification" as how much your assets are heterogeneous from the point of view of deviations from the historical mean. In ...


1

The total volatility of a portfolio is calculated as follows: Recall that Cov(a,b) is just (Correlation a,b)/(StD A * StD B). So in this case, no the portfolio could not have a total volatility of less than 15%. For this to happen, we would need negative correlation between the two assets. Think of volatility in this case as the amount of movement in ...


1

This may or may not be helpful, since I don't have anything to point you to that specifically addresses the high skewness of the distribution you mention. However, this sounds like it is probably an idiosyncratic risk, and that certainly has bearing on whether or not it would be priced. In the standard capital asset pricing model, the marginal investor ...


1

Try this: Given some time horizon of K, which can be divided into subperiods of N, you will calculate a rolling correlation coefficient of length N, then you can use the EMA to weight the more recent correlation coefficient heavier (indirectly weighting the recent relationship more, vs the medium term part). Never came up with this problem in my work so ...


1

If you only need to pick 5 out of 10 and want equal weights then just enumerate all 252 possibilities (as pointed out above) and compute the portfolio volatility $(\textbf{1}'K^{(i)}\textbf{1})^{1/2} = \left( \sum_{ij}K^{(i)}_{ij} \right)^{1/2}$, where $K^{(i)}$ is the covariance matrix for the $i$th subset. Then use whatever subset gives the lowest ...



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