# Tag Info

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

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 ...

4

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 ...

4

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 ...

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

Duration is not linear. It is the weighted average of the duration of the underlyings with the weightings being their values. To get a linear system multiply the durations by the associated pvs and match that quantity instead.

4

After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected returns (Black-Litterman) as well as with the in-house views of Chief Investment Officer Burton Malkiel’s team. In addition, Wealthfront sets minimum and ...

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 ...

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

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

Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to none of the ...

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 ...

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

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

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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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 $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = ... 2 If you give a covariance matrix an inverse Wishart prior, then it simplifies a lot of math in the calculations. This is called a conjugate prior. If you don't understand conjugate priors, you might want to work through the math on the univariate normal case with an inverse gamma or chi square prior for the variance. The Wishart distribution is just a ... 2 The classical connection is the http://en.m.wikipedia.org/wiki/Esscher_transform developed for actuaries in 1932 which essentially transforms the objective probability measure into the risk neutral one used in quant finance. 2 I think you might be looking for the portfolio return variance:$$\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij},$$where \rho_{ij} is the Pearson product-moment correlation coefficient between the returns on assets i and j and \rho_{ij} = 1 for i=j. In your case you could either weigh the assets equally or according to the real ... 2 Assume the weights of the two assets are w,1-w respectively;the expected returns and standard deviations are denoted by \mu,\sigma with subscripts 1,2,p(for portfolio),i.e,we have \mu_1,\mu_2,\mu_p,\sigma_1,\sigma_2,\sigma_p.The correlation coefficent is \rho Then$$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ...

2

The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for financial modelling but it also goes through the actual estimation algorithm (Baum-Welch) step-by-step and even gives full Matlab-code. From the abstract: ...

2

This paper, Equity Portfolio Diversification by W. Goetzmann and A. Kumar, uses the following diversification measures to measure the diversification of retail investors: Normalized portfolio variance: $$NV = \frac{\sigma_p ^2}{\bar{\sigma} ^2}$$ Sum of Squared Portfolio Weights (SSPW). Since the weight in the market portfolio is very small ...

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

Your question accurately addresses of the practical problems of applying Modern Portfolio Theory (i.e. mean-variance optimization) in practice. Generally the correlations are considered much more difficult to accurately estimate in this context. I am not sure I understand the question. Isn't $E(r)$ is an unbiased estimator of $r$? You may want to look ...

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As regards the free sources, the best place where you can find material about credit risk management is defaultrisk.com; it is a website where are collected (almost) all academic (and not) articles and working paper, references and researchers. Moreover, as regards the forums, I think you should try visiting Credit Risk Group at Linkedin; it is a very ...

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In literature you'll find many approaches to compute the variance. As mentioned already, the standard ideas are to use MLE, Shrinkage on the Covariance Matrix (Ledoit, Wolf), Shrinkage on the inverse of the Covariance Matrix (Kourtis,Dotsis) which makes sense as in fact the inverse of the Covariance Matrix determines the shape of the efficient frontier. ...

2

I think generally there are two approaches: "calendar rebalancing" (such as monthly as you mention) and "optimal corridor width". For the first option, the danger is the portfolio could stray considerably from your benchmark between rebalancing dates. For the second option, track tactical deviation on a continuous basis. When you are outside the corridor, ...

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That's the way you apply. Usually you get the closest number of shares possible. However, if you use that strategy you are very likely to underperform the market. Check table 3 on this paper for the Out of sample performance of the Markowitz strategy. Over their sample the Sharpe Ratio is 0.07 whereas a simple naive strategy 1/N yielded 0.18.

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