7

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few more recent articles about portfolio theory, in no particular order (all accessible online): Jorion: Bayes-Stein Estimation for Portfolio Analysis, JFQA, 1986 ...


6

CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that $E[(R_m-R_f )]>0$). Every individual stock has some idiosyncratic risk in addition to ...


6

It kind of depends what your objective is. First, momentum 'bias' isn't well-defined. Are you looking to eliminate momentum exposure for some reason? Momentum itself isn't even well-defined really: momentum over the trailing 1 year? Trailing 6m? Looking over 3-5y periods where mean-reversion is more at play? Generally, in the absence of a clearer ...


6

As i understand your question you are confused as to why the expected parabola-shape of the frontier is not depicted clearly. If you want to see the shape more clearly you can do one of two things: Increase the number of random portfolios. As this numbers goes to infinity you will eventually plot all possible portfolio combinations, and your efficient ...


6

The Minimum Variance Portfolio (without constraints, other than the weights sum to one) is usually found as $$w=\frac{\Sigma^{-1} \iota}{\iota^T\Sigma^{-1} \iota}$$ where $\Sigma$ is the Covariance matrix and $\iota$ is a vector of all ones. However, there is another (equivalent) way to find it. Memmel and Kempf (2006) SSRN 940367 showed that you can find ...


4

I don't know that there is a "standard-solution crystalized in the community," but there are alternatives. The ones that I prefer are Omega, Sortino, and Kappa. All three of these ratios, unlike Sharpe, do not assume normally distributed returns. Omega Ratio: This is the probability-weighted ratio of gains versus losses for a given minimum acceptable ...


3

Have you checked the performance of the particular stocks? library("quantmod") library("PMwR") cmp <- "AAPL" aapl <- getSymbols(Symbols = cmp, auto.assign = FALSE)$AAPL.Adjusted cmp <- "FB" fb <- getSymbols(Symbols = cmp, auto.assign = FALSE)$FB.Adjusted returns(window(merge(aapl, fb), start = as.Date("2015-1-1")), period = "itd") ## ...


3

There are a lot of interesting articles... From a practitioner's point of view: Meb Faber's Global Tactical Asset Allocation Butler, Philbrick and Gordillo's Adaptive Asset Allocation Anything from Asness like: Fact, Fiction and Momentum Investing Value and Momentum Everywhere Or CTA/momentum related stuff: Time Series Momentum Momentum Strategies in ...


3

When you solve for a minimum variance portfolio you acquire some values, $\mathbf{\beta}$ corresponding to the weights of your assets, usually such that $\sum \mathbf{\beta} = 1$. Regularization means you try to limit these values such that your objective function also includes the norm of $\mathbf{\beta}$ (Ridge regression - L2-norm) or the sum of absolute ...


3

If you take $A^T∗COV∗B$ then the result will be 1 x1 ( a scalar). (1xN * NxN * Nx1 = 1x1). I believe you forgot to take the transpose of A. The vector which pre-multiplies COV needs to be a row vector, because in your example it isn't you may be getting this weird result.


3

Yes, it is easy to find the MDP of N risky assets if you have the covariance matrix V (assumed non-singular) if there are no constraints: Step 1. Compute the inverse of the covariance matrix: $CINV = V^{-1}$ Step 2. Find the standard deviations $\sigma$ by taking the square roots of the diagonal elemnts of $V$ Step 3. Find $X = CINV \times\sigma$ i.e. ...


3

You will have to add some constraints to get the weight vector of the eigen vector of the smallest eigen values, otherwise 0 is a trivial solution. Without going in the details of handling those extra constraints, the reason why the vector space associated with the smallest eigen value is relevant is because if you express variance of your portfolio in the ...


3

It would have been helpful had you provided links to those papers. But in general, you need to distinguish between the optimisation model, and the numerical technique used to solve the model. Suppose you wanted to estimate a linear regression, with the mean squared residual as the criterion of fit, and without further constraints. This is a model. Now you ...


3

If interest rates fall long-term bonds will benefit more (since they have higher duration). If investors believe that interest rates will fall they will demand lower yields on long-term bonds (equivalently, their buying will push down long-term bond yields) which can cause the curve to invert. See here for a more technical explanation, particularly point ...


3

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 ...


2

A basket of stocks which is long-only will always have a positive beta to the market. Shorting this basket will therefore mean you seek another portfolio with a negative beta. As per the first remark this impossible to achieve synthetically with a long only portfolio. Q.E.D.


2

Multiply the weight of the assets times the 1 + returns of the corresponding asset. This will give you the value of each asset at the end of your horizon. In your example: (0.2)(1+0.05) = 0.21; (0.3)(1+-0.05) = 0.285; (0.5)(1+0.10) = 0.55; Now add all of these values to get Total Assets: (0.2)(1.05) + (0.3)(0.95) + (0.5)(1.10) = 1.045 Finally ...


2

The tangent portfolio is the optimal portfolio of risk assets. So under the Modern portfolio theory world, all investors will buy and hold this portfolio if they want to make an investment in risky assets. The point where indifference curve and the CAL meets tells how much of your wealth would be invested in (1) the risk-free asset and (2) the optimal ...


2

This issue is dealt in detail in Atillio Meucci's paper titled Fully Flexible Views: Theory and Practice. See Appendix A.4.


2

In your other post's Python code, you already have the sample covariance matrix as: cov_matrix = returns.cov() To also get the covariance used by ridge regression, put on the next line: cov_ridge = cov_matrix + lamda*np.eye(N) The second term increases the values along the diagonal of cov_matrix, using a $\lambda$ (lamda) regularization factor for ...


2

Regularisation means that you impose structure on your problem; structure that could not be recognised from the data sample alone. In the context of mean-variance optimisation, regularisation is mostly discussed when people estimate quantities from historical data (e.g. variances) and plug them into their objective function. (However, the term "...


2

The correlation certainly has an impact on the price of your portfolio (of two options). If you simulate the prices at time $t < T$ then you get samples prices $X_t$ and $Y_t$ and the return between time $0$ and time $t$ reflects the correlations. This means that if $\rho$ is positive then the $X_t-X_0$ and $Y_t-Y_0$ are likely to have the same sign. ...


2

Whilst this doesn't answer your question you may be interested to know that you can vectorise your simulation and greatly improve the efficiency of your code like this: def portfolio_annualised_performance(weights, mean_returns, cov_matrix): """ Return annualised risk and return for a portfolio given asset weights and expected return and covriance ...


2

You can check my answer to this question for general details on how to solve this kind of problem. Let $C_X(S_T)$ and $P_Y(S_T)$ be a call and a put option with strikes $X$ and $Y$ respectively, then: $$\begin{align} (\text{i}) \quad V_T &= (S_T-A)1_{\{A\leq S_T\leq B\}}+(B-A)1_{\{S_T>B\}} \\ &=(S_T-A)1_{\{S_T\geq A\}}+(B-S_T)1_{\{S_T\geq B\}} \\...


2

Assuming we are talking about Pearson correlation, then we may apply the triangle inequality. Let $\rho(X,Y)$ denote the correlation between $X$ and $Y$. Then, $(1-\rho(X,Z))^{1/2}\le (1-\rho(X,Y))^{1/2} + (1-\rho(Y,Z))^{1/2}$


2

You can find a python implementation here https://github.com/tzhangwps/Turbulence-Suite The author refers to the absorption ratio as "systemic risk indicator" but the calculation is the same.


2

I know Information Ratio to be: $IR = {E[R_p - R_b] \over \sqrt{var[R_p - R_b]}}$ Meaning the ratio ($IR$) is equal to the average excess return (Portfolio return - Benchmark return) divided by the standard deviation of excess returns relative to a benchmark. You could also say that this is the active return divided by the tracking error. Note that this ...


2

In 1952, Markowitz published „Portfolio Selection“ introducing mean variance optimal portfolios („modern portfolio theory“) into finance and emphasising the effect of diversification. This work paved the way for Sharpe (1964) (and others) to develop the CAPM which marks the foundation of financial economics. Both obviously received the Nobel Prize. Beginning ...


2

If you are looking to construct an optimum portfolio, you would do neither. Assuming your portfolio of stocks, long duration treasuries, and short duration treasuries are the universe of investable assets, one would construct a portfolio of these assets utilizing a mean-variance optimization. The most efficient, highest sharp ratio portfolio would be a ...


2

Simple Directionality Spread Trade Hedge If the sum of the risks of the trade $t$ are zero (as in the case of the 2Y5Y10Y spread trade) that immediately gives a starting point from which to make a simple calculation for an adjustment. For example if one assumes that the first principal component is the outright market driver and that the factor loadings ...


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