10

Whether it's called volatility pumping, rebalancing premium, or Shannon's Demon it would just be a form of replicating a short gamma option strategy (eg. selling straddles). Intuitively, you are systematically selling at higher levels and buying at lower levels. The payoff for continuously rebalancing an equity/cash portfolio without friction when the ...


7

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

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 ...


4

You may find the following paper worthwhile. It addresses most of the above points (and many more) in a systematic way: Dubikovsky, Vladislav and Susinno, Gabriele, Demystifying Rebalancing Premium and Extending Portfolio Theory in the Process (May 20, 2015). Available at SSRN: https://ssrn.com/abstract=2927791 or http://dx.doi.org/10.2139/ssrn.2927791 ...


3

Yes, it is easy to find the MDP of N risky assets if you have the covariance matrix V (assumed non-singular) if there are no constraints: Step 1. Compute the inverse of the covariance matrix: $CINV = V^{-1}$ Step 2. Find the standard deviations $\sigma$ by taking the square roots of the diagonal elemnts of $V$ Step 3. Find $X = CINV \times\sigma$ i.e. ...


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

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 ...


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

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

As a start. it is easy to prove that diversification always helps whenever the variances are all finite. To see this, consider two stocks A and B with variances var(A) and var(B). Then the variance of the portfolio where you mix them in equal parts is: $$var((A+B)/2) = var(A)/4+var(B)/4+cov(A,B)/2 =\\ =var(A)/2 + cov(A,B)/2.$$ The largest that $cov(A,B)$ can ...


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

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 ...


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

My 10 cents is to think about how, if the market is buying GBP, for example, then it will buy GBP against any or all other currencies. It's not a matter of buying GBP only against USD or EUR although those are, say, the major terms currencies. Alternatively, if the market is selling GBP then they will sell against all other currencies. Thus if you use a ...


2

The optimal Sharpe you can achieve, by the Markowitz portfolio, is $$ \sqrt{\frac{1}{1-\rho^2} \left( 1.2^2 - 2 \rho (1.2) (0.5) + 0.5^2 \right)}. $$ The optimal portfolio is $$ \frac{1}{1-\rho^2} \begin{bmatrix} 1 & -\rho \\ -\rho & 1 \end{bmatrix} \begin{bmatrix} 1.2\\ 0.5 \end{bmatrix}, $$ where $\rho$ is the correlation of the assets. You can ...


1

A variant of the Herfindahl-Hirschman index, specifically its inverse, is probably the most widely used for this sort of thing. It's a measure of portfolio or market concentration, where an equally-weighted portfolio (most diversified) has an effective sample size equal to the number of assets. It's a cousin of the Gini coefficient used a lot in income ...


1

Is the formula for code #1 $\max D(S)=\frac{S^{\top}\Sigma_S}{\sqrt{S^{\top}V_S S}}$? or is it $\max D(S)=\frac{1}{\sqrt{S^{\top}V_S S}}$ s.t. constraints $\Gamma$? Both appear on the same page, 41, in paper #1. and is formula for code #2 $\min \frac{1}{2}\mathbf{w}^{\top}\Sigma\mathbf{w}$ s.t. $w_i\geq 0$, $\mathbf{w^{\top}}\boldsymbol{\sigma}=1$ from ...


1

From my experience, I think your results are plausible. Due to the globalization, economies and stock markets from different countries are much more connected than in the past. Furthermore, since the financial crisis of 2008, central bank policies have also converged more and more (e.g. in the US, Euro-Zone, Japan and UK you have zero-interest-rate policy). ...


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 ...


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