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6

If you measure risk by the standard deviation of the portfolio return $$ \sigma = \sqrt{w^T \Sigma w}, $$ then it is usual to define risk contributions for each asset by $$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$ then diversified could mean that these $\sigma_i$ are evenly spread over the assets in the portfolio. You find this approach and more in this paper ...


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

most models in financial maths are linear so prices and Greeks just add. This is in particular true of Black--Scholes so Yes. However, once one starts taking into account value adjustments non-linearities appear and it is a lot more complicated.


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 $$\mu+c \sigma$$ ...


4

Both free and paid access to data sets conatianing company financial statement items is available from Quandl. The free data sets are sourced from the SEC based on compnay electronic filings and go back about five years. For example, you could obtain five years of MSFT's quarterly net income using the R call Quandl("RAYMOND/MSFT_NET_INCOME_Q") Lists of ...


4

If you could hedge continuously with zero transaction costs, the gamma would be irrelevant: you would perfectly replicate with delta hedging and be done. In practice, hedging is discrete and there is a certain amount of slippage giving a random outcome with mean zero. The larger the gamma, the bigger the variance of slippage. Trading more frequently ...


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

Actuarial science traditionally focuses on estimation of joint probabilities using real data where math finance is on valuation of contracts under an arbitrary distribution. It means the first one deals with methods of estimation of future distributions (the number of accidents of a given kind, the probability of someone with a given profile to have a ...


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

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 ...


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

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

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

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

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

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

@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

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 ...


2

Unfortunately I don't think it's possible to compute returns purely based on yields... There are a few options: If you're on the buy side, you can easily get access to Barclay, Citi, or BofA's bond indices. These are very high quality datasets for studying historical bond returns. If you have Bloomberg, they've started providing bond indices as well. They ...


2

Sure a lot of traditional (mutual) buy side funds use MPT. They also mostly subscribe to the efficient market hypotheses. And they also do not hide the fact that they have no interest to lobby many retirement investment and savings schemes to allow for long/short investments but hold on to long-only. And finally, most of them underperform simple benchmark ...


2

How to solve this, you can generate random portfolios based on constraints see method="random" in optimize.portfolio in PortfolioAnalytics in R See (1) as those would solve the above, however you do not have an objective function so ANY solution that meets your constraints would be accepted, see below for examples of objective functions as they would give ...


2

Well, you are asking something very subjective. In addition it should be mentioned that S&P500 are the companies with higher capitalization of S&P1500. Therefore a huge weight of S&P1500 is set by S&P500. In fact, as it can be seen in 2008 both went down a 37%, in the other hand S&P500 has 80% of the total of the US equity Market. After ...


2

You are asking two different questions: what would be the model result, and what would be the actual performance of an actual portfolio. The optimal model results with the S&P 1500 will be at least as good as the model results with the S&P 500. The S&P is a proper subset of the S&P 1500, so you can get the results of the S&P 500 model by ...


2

There are a couple of nice papers about the dot-com effect by Michael Cooper: full list, paper1, paper2


2

Here is a guide by morningstar: " A step by step guide to the black litterman model" https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf



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