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Risk and probability are not mutually exclusive. Check out page 80 at the source: https://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf Many people ignore the fact that Harry Markowitz actually defines variance as a forward variance. Without knowing your probability distribution, you can't define your variance! They are interwoven together. ...


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There is no single right answer for this one, there are good reasons for and against using cash (either bills or LIBOR widely used) versus terming out the duration to match your equity/risk asset horizons. Whatever would be your default when not invested can never really be too far wrong. Strictly speaking, a treasury bond is not “riskless”. There may be ...


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For applying Fama/MacBeth (1973) regression, it is necessary to always run the cross-sectional regressions and then averaging the betas across years. In this case, as you run Fama/MacBeth regression, the first step is to get the cross-section regression, after which you get the betas for each characteristics. Then you do a rolling window of 5 years, every ...


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I would use 3-month bills as a measure of the risk free rate.


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There are a few different views out there in choosing your reference rate. Literature normally uses a T-Bill rate (1 month or 3 months). That's what Fama & French do in their online library. In practice, you're on the right track with matching the maturity of your investment horizon with the maturity of your risk free rate. In theory, you could go a step ...


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Let me begin with a disclosure; I am a staunch opponent of Modern Portfolio Theory. I believe there is a deep and profound error in its mathematics, which is why it does not work empirically. I also think that I have identified the error. That disclosure aside, let me defend its use of point estimators, confidence intervals, and risk. Let me provide an ...


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Use a rate that's representative of what you'd get on excess cash if you weren't fully invested. For most institutional investors this will be something like either Fed Funds or 1 month LIBOR less a spread (set by your PB / FCM). If you were a corporate treasurer it would likely be roughly equivalent to a money market fund rate. EDIT: LIBOR is unequivocally ...


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portfolio construction currently relies on the probability distribution of asset returns because the asset means $\boldsymbol{\mu}$ and asset volatilities $\boldsymbol{\sigma}$ are estimated from historical time series (data vectors) as $\hat{\boldsymbol{\mu}}$ and $\hat{\boldsymbol{\sigma}}$ and used as inputs in the Markowitz mean-variance model. The ...


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it might be the data still contains non-numerical elements such as left-over headers, titles or dates missing, nan, imaginary or incoherent values in the data file non-corresponding lengths of time series/columns that do have legible elements incomputable mishaps similar to division by zero that are causing the nans negative values, such as negative ...


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If the test (out-of-sample) variance for the new model is still higher than that of the classical model, then it does not offer outperformance, at least for the dataset used. Also, having a high training error and low test error from the new model by itself might mean it is underfitting the training data and somehow generalizing well to the test data, which ...


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If they were computed with the same criterion, the Sharpe ratio, you can simply compare the different portfolios' Sharpe ratios with one another: $\frac{\mu_{1}-r_f}{\sigma(r_{1})}$ vs $\frac{\mu_{2}-r_f}{\sigma(r_{2})} \dots$ vs $\frac{\mu_{P}-r_f}{\sigma(r_{P})}$, where $r_p\in\mathbb{R}^{T\times 1}$ is the weighted return time series (vector) for ...


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Identifying your variables: You will need a weight for each of the 26 assets in each of the 9 portfolios. Suppose you take each portfolio in turn and create a stacked vector: $\mathbf{w} = [w_{1,1} \; .. \;w_{1,26} \; w_{2,1} \; .. \; w_{2,26} \; .. \;w_{9,26}]$ Equality constraints: Each weight of an asset cross section has to sum to the holding, $W_j$: ...


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Mean-CVaR portfolio optimization is an alternative to the more widely known and simpler mean-variance model. Since there doesn't seem to be any median-variance model out there, the familiarity surrounding the traditional model stuck. The mean is an actual equation with convenient properties in statistics, whereas the median is obtained through a counting ...


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Hi: Footnote 15 of the paper at this link explains what the formulation is (in brief: it is based on the Sortino Ratio). It sounds like something that can be programmed as a quadratic optimization. R has a lot of facilities for doing that sort of thing. Addendum: I didn't read it but this paper provides a lot more detail than the one above.


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The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e. $\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(t) \; \text{and} \; \sigma(s)=\sigma(t) \}$, where $P$ is the set of all validly constructed portfolios. Therefore this also holds for the efficient frontier ...


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