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Portfolio optimalisation depends heavily on the estimation of the moments (and therefore has HUGE estimation uncertainty). Even though it's useful for comparing and analysing different existing strategies, I think practitioners are moving more towards the usage of factor portfolios for the strategies themselves (e.g. Fama-French). Also because the ...

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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 ...

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Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS regression beta: $$\beta = cov(r_i,r_i^M)/var(r_i^M).$$ Then if you write $r_i = \alpha + \beta r_i^M + \epsilon_i$ on the other hand $$\epsilon_i = r_i - ( \... 4 Sharpe's 1966 equation had R_b defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent. If we refer to the original definitions, then that is the primary difference - Sharpe's ratio looks at reward/risk of the excess return for an asset over the risk-free rate while the ... 4 Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it is a chance to discuss today's approaches. A nice approach that is very up-to-date where mementum investing seems very fashionable is the following: Momentum ... 4 One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica) You define w_{i,j} which is the weight of asset j in subportfolio i, furthermore you define w =(w_j)_{j=1}^{\text{no of assets}} the total weight of the portfolio in asset j. the objects for the ... 4 It depends on the distribution of the returns. If you assume that it's roughly normally distributed, then you have a ~68% chance for a return in the range of 1 standard deviation, ~95% chance for 2 standard deviations, and so on. 3 There is one minor mistake: If you compute sum(mean.var) you'll obtain -1 instead of 1. So it should be mean.var<-xt/sum(xt) in order to ensure that the weights sum up to one. The remainder is correct. Incorporating a risk aversion parameter into the framework requires the solution to the minVar problem (See for example here). Therefore, dividing ... 3 The problem as you formulate it above already allows for short-selling. You just have to add the constraint:$$ \theta_i \ge l $$where l is the lower bound. This is equivalent to$$ -\theta_i \le -l $$which if often the way linear constraints are formulated. Any solver that is able to work with box-constaints can solve this. 3 Theoretically speaking (as it's done in financial textbooks at b-school level), variance and covariance are calculated on historical performance of asset classes, forward looking returns are CAPM calculated returns. ARIMA. Practically speaking, ARIMA is useless for predicting long term returns (or portfolio management if you wish). Why? A short answer is ... 3 Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to none of the ... 2 This is an interesting problem. I don't think the problem is set up correctly quite yet. I rewrote it slightly to correspond to how it's generally written as a quadratic program. The optimization problem you write down fixes betas to be a certain value. That could make sense but instead I wondered if we could simply minimize beta across the portfolio while ... 2 How about letting the FX rates remain fixed, and recalculate the portfolio volatility. That seems very obvious - am i missing something? 2 You can do 2 things: incremental risk: Calculate the volatility with the asset and with the asset replaced by cash. The difference gives you the (non-linear) incremental risk contribution of the asset. They don't sum up to \sigma. contributions to volatility (Euler allocation) As \sigma = \sigma^2/\sigma you can define risk contributions by$$ \frac{...

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The general formula for the global minimum variance portfolio is $w=\frac{C^{-1} 1}{1^T C^{-1} 1}$ where C is the covariance matrix and 1 is a vector of 1's. In this case the covariance matrix is diagonal with $\sigma_i^2$ in the ith diagonal element. Its inverse is also diagonal and has $\frac{1}{\sigma_i^2}$ in the ith diagonal element. Evaluating the ...

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The last 2 calls must be changed from library(portfolio.r) library(portfolio_noshorts.r) to source(portfolio.r) source(portfolio_noshorts.r) The correct files must be availbale at http://faculty.washington.edu/ezivot/econ424/

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Your question is very important! In formal way to demonstrate it is very interesting ... but a bit complicated ... and boring for non mathematicians. We may move around this demonstration to explain most of portfolio theory. However, to give the idea, if we have N risky assets we obtain, as efficient frontier, a semi-parabola and the weights of the countless ...

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It doesn't make sense to use the (co)variance(s) of asset values; if you did, by cutting an investment's share of the allocation by half, you would also cut its variance by a factor of 4. In a meaningful portfolio design, the volatility (variance) of an asset, by itself, is the same no matter how much or how little of your portfolio you put in it. Why doesn'...

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Maybe. Certainly you shouldn't use their realized return ("past return") because that does not reflect expectations, it reflects events that became known after the client decided on their asset allocation. On the other hand: with a lot of (unrealistic?) assumptions, you CAN discern the client's risk aversion from their allocation. Suppose for example that ...

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$$u^Tx - m (x^T \Sigma x)^{1/2} \geq c$$ is the same as $$u^Tx-c \geq m (x^T \Sigma x)^{1/2}$$ which is the same as $$(u^Tx-c)^2 \geq m (x^T \Sigma x)$$ This has no square roots.

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Not many benefits actually. As you combine different strategies you abstract yourself from what you are trying to reach. E.g. What would you get if you combine a strategy that tries to maximize Sharpe Ratio, with a strategy that maximizes Certain Equivalent? You have no idea on what that would get you. Probably it might even get you a portfolio with high ...

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Many pension funds and mutual funds acquire small positions in many stocks, therefore just banking on the main results of the Markowitz framework: diversification. This could also just be seen as a plain 1/N rule: naive diversification. In the limit, this just equals the market portfolio. Alternatively, to overcome the sensitivity in changes in the return/ ...

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@Richard I assume your $V$ stands for variance so that your formula is correct. The question was about standard deviation, though, and there the square root needs to be taken.

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This will depend on the definition of "return on the long run". If we define the annualized return on the long run by $\frac{1}{T}\ln \frac{S_T}{S_0}$ for a certain time $T$ in the future, then \begin{align*} E\left( \frac{1}{T}\ln \frac{S_T}{S_0} \right) = \mu-\frac{1}{2}\sigma^2, \end{align*} as claimed. Note that $\mu$ is the instant, or instantaneous, ...

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Trying to shed some light here: What we also see using this here, is that if returns are log-normally distributed, ie. $$1 + r = \exp(\mu + \sigma Z),$$ with $Z$ standard-normal, then $$E[1+r] = \exp(\mu + \frac 12 \sigma^2)$$ holds. But the geometric mean $GM$ is given by $\exp(\mu)$ and we have $$\log(GM) = \mu = \log(E[1+r]) - \sigma^2 /2$$ and ...

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I agree on Richard. the simpler you choose, the better it is so as to get reliable estimates. What's your data frequency? purpose? For model construction as far as I am concerned, daily data from 2010 is enough. Otherwise, you could use a proxy asset for asset D depending on its nature. To clarify, if D is an ETF let's say CAC 40 ETF, concatenate its return ...

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You can consider old prices for Stocks B, C and D to be "missing data" and apply techniques used by Statisticians to deal with such missing data. One approach, the EM algorithm, suggests you estimate the covariance for the common period, use that covariance matrix and the available data to generate pseudo data for the back period for the third stock and re-...

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The short answer: Take all time series starting from 2010 (at most). The covarianc-matrix tells you something about the assets for a certain amount of time. E.g. if I estiamte the covaraince matrix of those 4 assets taking into account data from the last year (!) then I can expect that this matrix remains valid for the coming 1-3 months - if the markets don'...

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It is well known that the MV-optimal portfolio has some very bad properties in practice: Backtesting: The MV portfolio performs very bad in backtesting applications Diversification: The MV portfolio tends to invest all funds into the best asset (highest sharpe ratio) of the past, leading to very low diversification. Non-Normality: Return distributions are ...

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The total volatility of a portfolio is calculated as follows: Recall that Cov(a,b) is just (Correlation a,b)/(StD A * StD B). So in this case, no the portfolio could not have a total volatility of less than 15%. For this to happen, we would need negative correlation between the two assets. Think of volatility in this case as the amount of movement in ...

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