# Tag Info

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

Have a look at this classic paper: Honey, I Shrunk the Sample Covariance Matrix by O. Ledoit and M. Wolf The abstract answers your question already: The central message of this article is that no one should use the sample covariance matrix for portfolio optimization. It is subject to estimation error of the kind most likely to perturb a ...

5

The estimation of a covariance matrix is unstable unless the number of historical observations $T$ is greater than the number of securities $N$ (5000 in your example). Consider that 10 years of data represents only 120 monthly observations and about 2500 daily observations. Depending on the application, using data dating farther back than 10 years may be ...

5

Transaction costs - even for banks, funds etc, every trade has an associated cost, so if you would be buying a small number of shares, it's probably cheaper to carry the risk and not make those small trades. The source data is imperfect, and contains noise. A lot of the smaller components are simply artefacts of that noise so it would be both an unnecessary ...

5

This optimization is trivial $$w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases}$$ That is to say, when you optimize only one weight will be nonzero. That's because these ratios incorporate no notion of distributional width, and therefore do not reward diversification. With no concentration ...

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 ...

5

To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the ...

4

You can obtain the covariance between 2 portfolios by multiplying the row vector, containing the weights of portfolio A with the variance-covariance matrix of the assets and then multiplying with the column vector, containing the weights of assets in portfolio B. Equally you can set up a new portfolio A+B by creating a new column vector that contains the ...

4

Lots of wealth management firms still use MPT; in my experience regulators like it because they understand it. If asset returns are normally distributed, the standard deviation of the portfolio is a coherent risk measure (this can be seen by noting that the normal distribution's CVaR, which is a coherent risk measure, can be written as $$\mu+c \sigma$$ ...

3

One really nice book that comes to my mind is Little, Rubin, Statistical Analysis with Missing Data I read part of it but probably it is too much information in your case. For your application, i think you can categorize the problem into two possible subproblems: First, time series that have unequal starting points (when some stocks' history is ...

3

There are many papers on this subject (try googling portfolio optimization skewness kurtosis) that can describe the assumptions of including skewness and kurtosis in a utility function (if that's what you're interested in). I would highlight two main points. Mean-variance optimization does not make an assumption of normality. Assume returns are ...

3

Initial capital is not a real constraint in theoretical analysis, but might be a practical constraint in reality. The objective function you gave defines the efficient frontier corresponding to a given risk tolerance $q \in [0, \infty]$: $$\min\{w^T\Sigma w-qR^Tw\}$$ This criterion is among the other popular optimization criteria, such as minimum variance, ...

3

Of course you can choose the prior. As far as I understand the literature, the BL-model is characterized by using the equilibrium implied returns. Otherwise it would just be a Bayesian model. If you estimate the returns in a different way (not taking implied returns from the market portfolio), you could lose the stabilizing inverse optimization step ...

3

First the easy solution: Define the continuous weights of each asset: $w_i \in [0,1],i=1,\ldots,N$ and choose some meaningful lower bound for each weight. Then you have the objective $$w\mu - \lambda w^T \Sigma w \rightarrow Max,$$ all your constraints that you already apply and the additional (linear/box) constraint $$w_i \ge l, i=1,\ldots,N.$$ ...

3

Black Litterman might be a good solution to your problem, since it suffers less from corner solutions (concentrated portfolios). You already have active views in the form of return expectations, and you can control the confidence in your views explicitly; see for example Meucci's Risk and Asset Allocation chapter 9.2 for a description. Since you have a ...

2

These games are usually won by luck. If there is no fee for buying stocks I'd diversify, i.e. buy many different stocks, to get stable returns. After some weeks you'll see which profit you'll need to beat. Depending on the rules if options are allowed you could invest in highly leveraged derivatives and hope you win. As there is no point not to try to win I ...

2

Let there be n stocks, 2 portfolio a and b. c is a combined portfolio of portfolio a and portfolio b. $\Sigma$ is variance-covariance matrix of the n assets. Weight vectors for portfolios a and b are $$w_{pa},w_{pb}\in\mathbb{R}^{n} ,$$ $$\left\|w_{pa}\right\|_{1}=\left\|w_{pb}\right\|_{1}=1$$ then $$Var(a)= w_{pa}' \Sigma w_{pa}$$ Var(b)= w_{pb}' ...

2

It depends on the exact nature of the risk in question as well as the mandate of the options desk at the bank. Generally such products are "created" and hedged at exotic option sell-side desks. There are a myriad of different kinds of risk the bank and hence the insurance company may offer their clients insurance against. It could range from inflation risk, ...

2

By definition, an efficient portfolio is one that is "best in its (risk) class." That's the main rationale for holding it. There are some efficient portfolios for risk averse investors (low risk, accompanied by relatively low return), and others for risk loving investors (high risk, highest return). But in either case, they are (by definition) the highest ...

2

Go ahead and compute a sample covariance matrix with 5,000 stocks on a few years (or less) of daily or monthly returns data. This can be done almost instantly on a modern computer. There is a very good chance that this matrix will not be a covariance matrix. You can check by inspecting the eigenvalues. If any are negative then you don't have a covariance ...

2

In many cases, clients want to be fully invested and don't want their assets lying around in cash. Hence the budget constraint $\sum_i w_i = 1$ is fairly common in practice. By the way, there are also cases where the constraint $\sum_i w_i = 0$ is applied: the result is a dollar neutral portfolio with long and short positions, but no net investment (short ...

2

I see your argument with the math. "1" is an arbitrary choice of positive numbers, and you could choose anything. In the end, you're going to scale the whole thing to fit your capital anyway. If you are using a numerical optimizer, it will be happier with something noticeably away from 0 and away from infinity, so I recommend choosing a specific positive ...

2

@vanguard2k and @Theja provide useful information. In my experience, unequal starting points is most common, so I'll try to focus on that. The technique that @vanguard2k mentioned for unequal starting points can be thought of like a regression. You start with the longest available data and get the covariance matrix of that. For the next set of available ...

2

Given that by delta means that if the price goes up by 0.01% i.e. one basis point, you gain 15 and vice versa if the price goes down by one basis point. You know that the daily standard deviation is 2.2%, than again you know that $220*15 = 3300$ is the standard deviation of your portfolio. So, since we are using a normal distribution you can look at a table ...

2

Speaking from equity quant factor building experience, it is a common practice to build multi-factor models by regressing one component against other(s) and using the residual scores. This is done to avoid bias as you mentioned - these biases could be from the factor itself (in different regimes, Quality / Momentum influencing each other - or earnings, value ...

2

If you are investing an amount $M$, split over deals indexed by $i$ and with a weight $w_i$, then your dollar position in each share will be $w_i M$. The exposure to the index will be $\sum \beta_i w_i M$ You should realize that this will not hedge idiosyncratic risks. In general, the more deals you have, the better this type of hedge should work (assuming ...

2

The technique is sometimes referred to as full information maximum likelihood. It is more general than the technique you describe, but it is similar. Basically you start with the data with the longest horizon and get the covariance matrix, then for the data with the next longest horizon you regress them against the data with the longest horizon, finally you ...

2

It will bring diversification benefits to your portfolio. Mean and standard deviation alone only measures the first two moments of the individual asset returns, with no regards for their joint distribution and correlation structure. Assuming the mean and volatility measurements are the same for $2$ assets with correlation $corr<1$, then combining them ...

2

I am engineer studying Finance, therefore Im not an expert in Math/Stat, but not noob. I disagree with the previous answer. In fact, I know portfolio managers and hedge fund assesors that usses MPT. It must be said that you need to know what that represents, and also not only focus your investment in MPT, but consider other methods. Like in every other ...

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