13 votes

Intuitive Explanation for Shannon's Demon?

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 ...
  • 3,480
8 votes
Accepted

Proving diversification return is always nonnegative

Assuming portfolio weights that sum to $1$, the geometric average portfolio return is given by $$(1+g_P)^T= \prod_{j=1}^T\left[1+ \sum_{i=1}^N w_i r_{ij} \right] =\prod_{j=1}^T \sum_{i=1}^N w_i(1+ r_{...
  • 3,480
7 votes
Accepted

Which portfolio is more "diversified": the $\frac{1}{N}$, the MDP or the max decorrelation?

First of all, I am not sure what you mean by the ratio in your second point. However, I will try to give you a partial answer at least. There is a very comprehensive overview of these by EDHEC, page 4....
  • 2,894
7 votes

Intuitive Explanation for Shannon's Demon?

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 ...
  • 27k
4 votes

What can I use to measure of diversification?

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 $x$ ...
  • 400
4 votes
Accepted

What can I use to measure of diversification?

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 ...
  • 9,167
4 votes
Accepted

Literature about optimal number of stocks in a diversified portfolio

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 ...
  • 9,656
3 votes

Non-linear correlation (co-dependence) and the efficient frontier

What you show here as an efficient frontier for a two-asset portfolio is presumably the usual return versus risk profile, where the vertical axis represents expected portfolio return $\mathbb{E}(r_P)...
  • 3,480
3 votes
Accepted

Maximum Diversification Strategy without risk free asset

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 = ...
  • 9,167
3 votes

What can I use to measure of diversification?

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 ...
  • 4,247
2 votes
Accepted

Risk minimization by investing in all assets with positive expected return

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 ...
  • 6,763
2 votes

How does one analyze diversification if stock prices follow a Cauchy distribution?

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 ...
  • 290
2 votes

How to measure if investors are diversified in a stock market?

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. ...
2 votes

Value-at-Risk "hiding risk in the tail" and diversification?

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 ...
2 votes
Accepted

Is it better to express a currency position through multiple pairs?

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 ...
  • 1,136
2 votes
Accepted

Total portfolio VaR greater than aggregated individual VaRs

Since this question does not seem to be a duplicate, I will make up a simple (but not entirely unrealistic) numeric example. Suppose some asset is now trading at some observable price, and suppose ...
2 votes

how to construct a diversified portfolio based on correlation

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-...
  • 2,825
2 votes
Accepted

Portfolio Allocation given Sharpe Ratio

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} \...
2 votes
Accepted

How to choose a stock?

This seems like a terrible idea. If you can have such an automated system for one stock, you can have it for many stocks. Then, since you're a serious investor, you want to take into account the ...
  • 7,731
2 votes

Do equity mutual funds typically have industry-level diversification constraints?

You would typically find such information in a prospectus. For example the prospectus of the Vanguard World Equity fund (VGHEX) doesn't have any wording putting hard limits on industry allocation or ...
  • 7,731
2 votes

How can beta be negative?

An index that represents all of the market is a CAPM assumption, but in reality $m$ is typically some stock index (like the S&P 500, which represent U.S. large cap stocks). It's not practical to ...
1 vote

Why is standard error used to show diversification effect for unsystematic risk?

OK, short answer, think of it this way. For a more formal explanation, google the distinction between what statisticians call a "confidence interval" versus a "prediction interval"....
  • 4,961
1 vote

How can beta be negative?

A negative beta just means there is a negative covariance (and thus correlation) between your asset in question and your reference “market” portfolio. Perhaps the most intuitive example of this is ...
  • 4,961
1 vote

Formula for underdiversification

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-...
  • 1,608
1 vote

Which Maximum Diversification Approach in MATLAB is correct?

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, ...
  • 2,825
1 vote

Decreasing dependence during the financial crisis?

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 ...
1 vote

Value-at-Risk "hiding risk in the tail" and diversification?

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 ...
  • 2,534
1 vote

What can I use to measure of diversification?

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 / ...
1 vote
Accepted

Can adding an uncorrelated high vol strategy to a low vol portfolio result in a portfolio with even lower volatility?

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 ...
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