Questions tagged [portfolio]

A portfolio is a collection of financial instruments. We often collect instruments together to represent the complete holdings of an investor and to analyze the overall risk (which may be lower due to diversification, i.e the portfolio holding multiple instruments).

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Reverse optimization: How to generate the expected portfolio returns given the weights and a series of constraints on those weights?

I have the below function in Python. My objective is to back out the expected returns associated with certain portfolio weights given a series of assumptions. From this I want to generate the expected ...
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GARCH for Mean Variance Optimization

I am currently trying to carry out a mean variance optimisation, with the implementation of GARCH. I'm not sure if this is going to make complete sense as my understanding of GARCH is limited. In the ...
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Help me understand super replicating portfolio

Lets consider a hypothetical stock with current price of $S_t$ at time t and it can take any positive value with a strictly positive probability. There exists a derivative that pays $ e^{S_T}$ at ...
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Generalization of pairs trading to portfolios?

Standard pairs trading only trades one security against another. In principle, nothing is stopping one from trading a portfolio of N stocks. There must be work on this? If so, can you point me to any ...
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Constant maturity bond fund returns in downward sloping curve

Assume the following: we are running a constant maturity bond fund (10yrs) all zero coupon bonds to maintain the maturity, we buy bonds of 10y and sell the 9y bonds 1y later price of 10y bond is 100, ...
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Optimal weights in portfolio after rebalancing

I have a quite simple question but while looking for answers in research papers I couldn't find anything. The question can be summarized as : if you expect a shock on an asset, why don't you rebalance ...
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Value-at-Risk of a portfolio with a stock after recent IPO

I have a task to calculate VaR for a portfolio of stocks and bonds. The main problem is that there is 1 stock which IPO was in November 2023 so there is few data points. To cope with that I came up ...
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How to calculate historical returns and variance for a non-BAH trading strategy?

Suppose i have a strategy that is not buy-and-hold type of strategy. It can have unique entry timing and unique exit timing for a single asset and both long and short positions will be allowed, and ...
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Claim-augmented market is arbitrage-free

Suppose we have a no-arbitrage market with a risk-free rate $r$ and $d$ risky assets with prices $(S_t)_{t \in \{0,1\}}$. Let's introduce a claim with payout $Y$, and assume that there exists a price $...
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Standard deviation of large equal-weighted portfolios

Say I've got a portfolio of shares with the following parameters: Let $n$ be the number of shares in the portfolio, let $\bar\sigma$ be the average standard deviation (volatility/risk) for each share, ...
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What is Ei in paper "How to Combine a Billion Alphas" by Zura Kakushadze? [closed]

I am reading paper "How to Combine a Billion Alphas" by Zura Kakushadze. In the paper, it has Ei which are the expected returns for alphas. It also has Ri hat as follows. I wonder what the ...
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Why not inequality constraint in mean-variance portfolio optimization?

Question 1: In Modern Portfolio Theory, the case where we minimize variance given a set return and that the weights sum to 1, why is the return set as an equality constraint, not an inequality? ...
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How can equilibrium weights be found for momentum factor in Black-Litterman model?

I have a momentum factor which consists of going long in three rising ETFs and going short in three falling ETFs. I want to use this factor as part of my portfolio for Black-Litterman model, however I ...
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What is the meaning of Beta of an individual asset in relation to a portfolio, not the market?

Assume I've got a portfolio "A" with an expected return of 14% and a volatility of 20% and my broker suggests to add a new share "H" to my portfolio which has an expected return of ...
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Intuition behind portfolio weights with lower RMSE but higher variance

I have recently encountered a phenomena in portfolio optimization that has baffled me for days. I was experimenting with different ways of transforming a covariance matrix to get a stable minimum ...
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Discuss how you would allocate your budget between the two assets if their correlation is 1, 0, or -1

An asset A is expected to yield a $2\%$ return with a standard deviation of $1\%$, and another asset B is expected to yield a $1\%$ return with a standard deviation of $1\%$. Discuss how you would ...
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Constraints in a Mean-Variance Optimization Case

Might be a repeat question, feel free to close if it is. I am trying to perform a mean-variance optimization (maximizing the Sharpe ratio) for lets say 5 assets. Besides the weights of the assets ...
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Reinvesting the dividends of a dividend paying stock

Suppose we have a dividend paying stock that has the following dynamics: $$dS_t=S_t((\mu-q)dt+\sigma dW_t)$$ With a continuous dividend yield $q$. What is the portfolio $Y_t$ that results out of ...
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When you have negative weights in the context of portfolio construction, what is the correct way normalize them?

For context, I am building an eigenportfolio following the conventions of Avellaneda and Lee Statistical Arbitrage in the U.S. Equities Market (2008), and I get negative weights for eigenportfolios 2,...
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CML equation - from where does the square come from?

In his textbook Asset management Andrew Ang uses the following CML formula (chapter 6) E(rm) - rf = y * σ^2 Where y is risk aversion factor What is the source of square? When I look at CML graph there ...
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Interpretation of optimal weights in portfolio for risk-adjusted return maximization

To start, I'm not an expert in portfolio management. My research involves examining the effects that one financial asset has on another, specifically looking at the spillovers between cryptocurrency ...
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Creating a Bloomberg bquant portfolio

As an ex-investment banker (salesperson to be specific) and amateur programmer, I’d like to move into quant work. I’d like to create a coding portfolio using Bloomberg’s bquant, but as I’ve left the ...
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How do I find daily expected portfolio return that has both short and long assets?

For example, let's say we have 3 assets (A, B, C) with weights (0.3, 0.3, -0.4). Hence the portfolio is 20% net long. Assets have daily price returns of [[0.01, -0.02], [0.03, 0.04], [-0.01, 0.03]]. i....
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How can this problem be defined formally?

Let's consider a straightforward example in which I possess a portfolio consisting of two stocks: $ R(t) = S_{1}(t) \cdot x_1 + S_{2}(t) \cdot x_2, $ Here, $t$ represents the time index, $R(t)$ ...
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How can one quantify the incremental value of better covariance matrix modeling in portfolio optimization?

Let's say we have two estimators of the covariance matrix, $\hat{C}_1$ and $\hat{C}_2$, and the latter is an improvement on the former. Is there any measure of the improvement that can be sensibly ...
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Why do we need the covariance when calculating portfolio VaR?

I was recently learning about value at risk and how to calculate it, and one of the steps was to calculate the covariance of the returns of the securities making up the portofolio. This makes sense ...
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Solving Equation for estimation risk averse parameter

Let the portfolio value follow the SDE: $$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t $$ where $\mu$ = drift of the portfolio, $\sigma$=standard deviation of the portfolio, ...
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Difference between the two definitions of Ulcer Index

The Ulcer Index (UI) is defined as follows on page 89 of the book "Practical Portfolio Performance Measurement and Attribution, 2E" by Carl Bacon: $$ UI= \sqrt{\sum_{i=1}^{i=n} \frac{D_{i}^...
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Difference between Maximum Drawdown and Largest Individual Drawdown

Bacon in Practical Portfolio Performance Measurement and Attribution distinguishes between the two, specifying that "Maximum drawdown represents the maximum loss an investor can suffer in the ...
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Is the self-financing condition necessary/"useful" in practice outside of replication/valuation?

I know that the need for a portfolio/strategy to be self-financing (the purchase of a new asset needs to be funded by selling of an older one/ones) is very helpful when attempting to price derivatives ...
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Why reduce number of short contracts to reduce beta, and take long positions to increase it?

My question is about chapter 3 in the ninth edition of "Options, futures and other derivatives" by John C. Hull, subchapter 5 under the heading "Changing the Beta of a Portfolio". ...
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How to express the process of number of stock (nt) in a portfolio using ito's lemma

We have a regular self-financing portfolio $W_t$: $$dW_t = n_t dS_t + (W_t − n_t S_t) r dt$$ Where $W_t$ is total wealth, $n_t$ is amount of stock, $S_t$ is stock price, $r$ is the risk-free rate. And ...
nearhome's user avatar
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Confusion about the formula for gain process in a financial market

In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio $\pi$ is defined such that From what I understand, ...
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calculating portfolio weight for long short

Sorry for this dumb question but what are the mathematical-finance /academic conventions for calculating portfolio weights in a long/short portfolio where the longs are fully funded by the shorts? ...
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Derivation of optimal portfolio weights using Risk Budgeting approach

In Thierry Roncalli's book Introduction to Risk Parity and Budgeting (2013), he gives an example of particular solutions to the Risk Budgeting portfolio such as for the $n=2$ asset case. The risk ...
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How to calculate the ex-ante beta of a portfolio between several rebalancing?

I have a portfolio composed of $ N $ assets. I know the one-year beta of these assets, I also know the past (ex-post) beta ($\beta$) of my portfolio. My portfolio changes allocation every month. So I ...
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Calculating tangency portfolio weights with the given information? (2risky +riskfree asset)

We have 2 risky and 1 risk-free asset. E1 = 4%, STD1=10% E2 = 5.5%, STD2 = 20% rf=1.5% The covariance matrix and it's inverse are given: ...
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Calculating variance of long/short portfolio

Say I have a portfolio of stocks, stock A, stock B and stock C, with the below positions: stock A: long 100 USD stock B: long 50 USD stock C: short 200 USD How do I calculate the portfolio variance ...
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Equity and Credit Portfolio Return

This might sound like a trivial question but would appreciate the answer. How would you calculate the return of the portfolio consisting of only equity and credit instruments? For example, consider ...
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Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?

Suppose the consumer Solves $\max -e^{-\gamma W}$ where $W=X^T D -X^Tp R_f$ where $X$ is the vector invested in a risky asset and $D\sim N(E[D],\Sigma^2_D)$ and $R=\sim N(E[R],\Sigma^2_R)$. Then ${ X=(...
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How to create Industry Portfolios per country? [duplicate]

Regarding the industry portfolio as created by Kennth R. French like here: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_17_ind_port.html I was wondering how I could do this ...
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Basket variance with correlation 1

I'm reading the famous nuclear phynance primer on dispersion trading and finding difficulty in understanding the author's simplification for the variance of a basket with correlation 1. See below: I ...
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How to predict a portfolio's reversion?

Sorry if this has been asked before. I've been baffled by a question I'm facing. Assuming I know there are some certain demands for some stocks in near future, and I put them in a basket as a ...
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How to calculate the portfolio risk and return if daily share prices, volume held on that day is given for all assets?

The problem is with the changing volume of assets which changes the weights. I want to use the formula for portfolio risk but cannot figure out the weights. Should taking average weight of an asset ...
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Portfolio with Put Options - VaR, Std. Dev

I did a Monte Carlo simulation to evaluate my portfolio. I used different Strikes and Weights for the Put options. Now to my problem: All statistical measures (like expected return, volatility) ...
Financeguy's user avatar
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Derivations of the pricing PDE for the Heston-Hull-White or Heston-CIR models

Consider the hybrid model given by $$dS=(r-q) S dt + \sqrt{v} S dZ_1$$ $$dv = \kappa_v (\theta_v - v) dt + \sigma_v \sqrt{v} dZ_2$$ $$dr = \kappa_r (\theta_r - r) dt + \sigma_r r^p dZ_3$$ with ...
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How to use coherent risk measure for evaluating price?

Coherent risk measures are defined by number of axioms (see e.g. Coherent Risk Measure) but a question that does not seem well studied is how to use them. Let's take a coherent risk measure $\rho$ and ...
Mathieu Dutour's user avatar
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How do you cross-sectionally standardize a variable?

i am doing research on a so-called carbon beta where I made a portfolio that takes a short position in 'green' stocks and a long position in 'brown' stocks, from a period of 2005 until 2020. So from ...
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How to calculate the log return of portfolio?

Suppose that we have five trades each day with these returns ($R_{day,trade}$) and we have 300 days in total: $R_{1,1}$, $R_{1,2}$, $R_{1,3}$, $R_{1,4}$, $R_{1,5}$ $R_{2,1}$, $R_{2,2}$, $R_{2,3}$, $R_{...
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Marginal Risk Contribution Implementation Questions

Sorry if this is too obvious to you. The marginal risk contribution mentioned here is the same as in this post Marginal Risk Contribution Formula . I understand the concepts and derivation on the ...
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