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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

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

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

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

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

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

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

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 ...


3

An introductory presentation by Michael Brandt from a seminar of Inquire Europe is Bayesian Portfolio Construction. His review Portfolio Choice Problems has a section on decision theory which could also be useful to you. Another good choice is Attilio Meucci's Risk and Asset Allocation book which contains a whole chapter (ch 9) on Bayesian techniques in ...


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

@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

7 years ago I had to solve the problem of a efficiency frontier under linear constraints on the asset weights and also stumbled upon Markowitz Critial Line Algorithm. I still have a directory with some resources in it. Since Bryce already gave a practical implementation with R code by Eric Zivot, I will concentrate on some papers which might help. I ...


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

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

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

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


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

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

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

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

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


2

I'll assume the rest of the world doesn't have access to a similar oracle. Indeed if it did future returns would converge to the risk free rate instantly. In this case, I would prefer holding the AAA bond instead of the stock because the rest of the world would consider it to be much less risky. As a financial institution, reducing the risk of your ...


2

If you give a covariance matrix an inverse Wishart prior, then it simplifies a lot of math in the calculations. This is called a conjugate prior. If you don't understand conjugate priors, you might want to work through the math on the univariate normal case with an inverse gamma or chi square prior for the variance. The Wishart distribution is just a ...


2

You do note require a sum up constraint that gives you that the weights sum up to 1? Then the problem is equivalent to a maximization without constraints: $$Z(\omega)=w'\mu - \frac{\gamma}{2}w'Vw$$ then it holds that $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = ...



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