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5

if you take the variance of a single asset it scales as a quadratic, $$ var(\lambda X) = \lambda^2 var(X) $$ so it's not surprising that the general case gives a quadratic form.


4

If you can add linear constriants (as you can do in quadprog) then you can formulate $w \mu = c_1$ as linear constraint, no matter what $\mu$ is (and first delete it from the objective by setting the parameter to zero. The only problem is the one norm. Let my clarify, this is: $$ \sum_{i=1}^n |w_i| < c_2 $$ Thus you allow for short sales but you want to ...


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

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 ...


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 ...


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

This problem is not interesting enough, because putting your money in the bank guarantees you zero volatility (and a zero return on investment). In practice, whatever set of assets you chose you would get a very extreme solution (e.g. 100% weight on one asset with very low volatility.) With a minor tweak, you can get a very interesting problem. You can ...


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

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

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

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 ...


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.


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^* = ...


2

Let $s_1 = r_1 -r_f$ and $s_2 =r_2-r_f$. Then, this is the maximization problem: \begin{align*} & \ \max_{w_1, w_2} SR = \frac{\mu_p}{\sigma_p}, \, \mbox{ subject to}\\ \mu_p = & \ w_1 s_1 + w_2 s_2,\\ \sigma_p^2 = & \ \sigma^2\big(w_1^2 + w_2^2 + 2 w_1 w_2 \rho\big),\\ 1 = & \ w_1+w_2. \end{align*} By certain substitution, we convert the ...


2

PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf


2

I wonder if it's possible to use solve.QP from quadprog by using dummy variables. One dummy variable $y_i$ would be used for each $w_i$, each $y_i$ would be constrained to be greater than zero, and the leverage constraint would be applied to the sum of the $y_i$. Problem formulation would look like $$ \text{min } w^tΣw $$ subject to the ...


2

If you take the a sample of historical asset returns as model for the risk then you can do two things: You calculate $r_j = \sum_{i=1}^n w_i r_i^j$ thus for each scenario $j$ you aggregate the individual asset returns to get a scenario for the portfolio. Then you can calculate $Var(r_j)$ the variance of the sample of portfolio returns. This is the same as ...


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

Assuming that $\Sigma$ is invertible, then \begin{align} 2\omega' = \lambda_1\overrightarrow{1}'\Sigma^{-1}+\lambda_2\beta'\Sigma^{-1}. \end{align} We can then solve $\lambda_1$ and $\lambda_2$ from the system of equations \begin{align*} 2 &= \lambda_1\overrightarrow{1}'\Sigma^{-1}\overrightarrow{1}+\lambda_2\beta'\Sigma^{-1}\overrightarrow{1}\\ 2c ...


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

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.


2

You are correct in your basic approach. Given the correlation matrix $\textbf{C}$ and standard deviation matrix $\textbf{S}$ where standard deviations occupy the diagonal and zeros the rest (i.e. $s_{i,j} = \sigma_i | i = j$ and $s_{i,j} = 0 | i \neq j$), the covariance matrix can be found as $\textbf{R} = \textbf{SCS}$. Then your portfolio standard ...


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, ...


2

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 ...


1

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 ...


1

Maybe you could find pretty interesting the following papers: Laureti, P., Medo, M., and Zhang, Y.-C. (2010). Analysis of Kelly-optimal portfolios. Quantitative Finance, 10(7): 689–697. and Nekrasov, Vasily, Kelly Criterion for Multivariate Portfolios: A Model-Free Approach (September 30, 2014). The last is available at SSRN. Particularly, ...


1

It is supposed to be multiplied by 5/100 (5%). You should then be able to get $57,870.37 if you multiply it by the fund value.


1

Suppose we have no dividends like in Black-Scholes-Merton and in your example. Expected return between time $t$ and $t+\Delta t$ is defined as $$ \mathbb{E}_t\left[R_{t+\Delta t}\right]\equiv\mathbb{E}_t\left[\frac{S_{t+\Delta t} - S_t}{S_t}\right] = \mathbb{E}_t\left[\frac{\Delta S_t}{S_t}\right] $$ You can see that, as $\Delta t \to dt$, ...


1

vega captures the two most common solutions to this problem. There are some valid criticisms of corridors as well. Because assets are correlated within a portfolio the decision to trade a particular asset should actually depend on the movements of other assets rather than having a corridor per asset. Also, finding the right corridor is often done using ...


1

This is a strange question. Usually questions about expected utility involve some uncertainty about the future wealth of the investor. If there is no uncertainty in the outcome and the investor is not doing anything which might change his or her future wealth then the expectation of utility is a constant, that is $E[U(w)] = U(w)$.



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