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

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

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

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

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

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

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 ...


3

Hedging strategies, even on the same underlying asset, can vary. It’s nearly impossible to say what exactly your company is doing to hedge something as general as an F.I. portfolio. For instance, they may sell futures on a bond index with a similar duration, or they may hedge using derivatives tied to rates themselves. They can duration match or key-rate ...


3

There are some subtleties / difficulties. Implied volatiliy, which depends on the strike of a vanilla option, is only a forward looking measure in the sense that it can be regarded as the risk-neutral expectation of break-even delta-hedging profit and loss of a vanilla option of strike $K$ which is delta hedged to expiration using a constant volatility. If ...


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

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

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

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 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

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 ...


2

For a linear F.I bond/swaps portfolio, if you execute a new trade you have the following considerations for hedging: Existing/residual portfolio risks (i.e. those remaining unhedged from previous trades) Liquidity of current hedging instruments (i.e. the bid-offer prices available to hedge the required notional size) The covariance of hedging with different ...


1

For question (i), you simply buy one EU call with strike A and sell one EU call with strike B - this is called a bull call spread. Try using the put-call-parity to construct the corresponding bull put spread yourself.


1

The correlation will impact the random numbers generated for the simulation. Use Cholesky Decomposition on the original correlation matrix to recalculate what the correlated random normal numbers will be and use those in the simulated path(s). If you're using Matlab or another canned scripting language, they usually have the function pre-coded. In matlab: R ...


1

One common way to construct portfolio is a high - low factor portfolio. First you sort the asset classes based on a particular factor. For example if the regression co-ef is positive implying positive risk premia, you sort them in ascending order of the factor, and opposite for negative co-ef. After that you percentile this sorted series and decide a ...


1

The Markowitz mean-variance model takes in some target expected portfolio return $\mu_T$ as an input and returns optimal portfolio weights $\boldsymbol\omega$ that minimize risk for that return. Repeating this for a series of target returns, $\boldsymbol\mu_T$, manifests two different efficient frontier curves (series of efficient portfolios) depending on ...


1

The maximum attainable return is unbounded as in the model you can borrow without limit. However, what matters in that model is what is the maximum sharpe ratio you can attain. That has bounds, which are given by the Hansen-Jagannathan distance. Let me show you what the JH distance looks like. From the law of one price: \begin{equation} 1 = E [R_{i,t+1} ...


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