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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1answer
211 views

Optimal investment strategy problem with competing bet-sizing options and limited budget

Apologies for a potentially naive question and unusual wording. I am from another field and would be very grateful for help! I am looking for the optimal investment strategy that maximizes an overall ...
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41 views

Relationship between risk and return for GBM and riskless bond

Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$. In ...
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Why is a Delta-hedged option always profitable even in case of a sharp drop of value of the underlying?

I am trying to understand the following concept on a practical level. Given a Delta-hedged long call position, so holding a portfolio $$ Port_0 = C(S_0, \sigma) - \Delta_C(S_0) \, S_0 $$ If there is a ...
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198 views

Monte Carlo simulations of correlated stocks by Geometric Brownian motion

I am trying to simulate using a Geometric Brownian Motion process three autocorrelated stocks. In particular, I need to simulate three different matrices with 1000 scenarios each using a Monte Carlo ...
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42 views

Creating daily rebalancing stock portfolios based on analyst recommendations

I am doing research on analyst recommendations on Finnish stock market, using Can Investors Profit from the Prophets? Security Analyst Recommendations and Stock Returns by Barber etc. as reference ...
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Is there no fix to improving portfolio risk estimation under small sample size?

When asked if copula are needed to calculate portfolio Value-at-Risk, it is said that "You can use historical method if you have sufficiently enough data". But actually copula are also ...
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Assumptions of the CAPM

As to my understanding, the CAPM assumes that all investors behave as described in the portfolio theory. Consequently, all investors hold a combination of the risk-free investment and the efficient ...
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50 views

Relation between CAPM and Portfolio Theory

can any of you explain to me in simple terms how CAPM and portfolio theory are related to each other? To my understanding: Portfolio theory helps to select the "right" stocks under risk/...
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Minimum variance portfolio's analytical solution, but assuming $t$-distribution

$$ \boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}} $$ is the well known closed-form analytical solution to the minimum ...
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51 views

Why is standard error used to show diversification effect for unsystematic risk?

quite long text incoming, sorry for that: While reading a corporate finance textbook, i came across a section describing the effect of diversification as well as the systematic and unsystematic risk. ...
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60 views

Covariance/correlation matrix from data with missing data points

I have a data set with index fund quotes, and I'm trying to compute the efficient portfolio frontier for it. But some data points are missing. In some cases there are few funds that trading even on ...
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44 views

internal rate of return ((M/X)IRR ?) of a fund

I have the following data for a fund. The contributions come from the LPs (i.e., the investors invest more in the fund, or withdraw money from the fund), MV stands for market value. The timing is not ...
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348 views

Portfolio Performance Attribution Using Carino Smoothing

I'm trying to conduct portfolio performance attribution using Carino smoothing, but it seems that the active returns do not match and I don't know why. Here is the example I use: \begin{array} {|r|r|r|...
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Notation of long/short positions in defining a portfolio [closed]

I am a bit confused with the sign-related abbreviations used when we refer to long or short position on assets in a portfolio. For example denote the stock price, $S_t$ and the bond price, $B_t$ and ...
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Portfolio vs individual security Sharpe and Sortino ratios

For an individual security calculating it's Sharpe and Sortino ratios is straightforward. What I'm curious about is the following: Let's say I have a portfolio of several securities, which is a ...
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How to deal with missing stock returns?

If I want to calculate the Covariance between two stocks but there are missing days in both, how can I deal with missing data? I want to use Pairwise deletion and only use the days of which both ...
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88 views

Ito's differential in portfolio dynamics

I try to be as concise as possible. Basically I'm following the text "Arbitrage Theory in Continuous Time", by Tomas Bjork. I put here the point where I'm stuck: Chapter 6 - Portfolio ...
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1answer
58 views

How to adjust a portfolio's rate of return for contributions and withdrawals?

Suppose we have a portfolio with many assets. Since this portfolio receives monthly contributions and withdrawals, what is the best method to evaluate its global rate of return and avoid computing ...
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What is the formula for the global minimum variance portfolio with positive weights?

I know how to algebraically solve for the weights when short selling is allowed but I can’t seem to find the formula for when it’s strictly positive an the weights sum to 1 anywhere online.
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Why do only portfolios of indices show elliptical dependence?

Elliptical distributions imply an asymmetric relationship between variables such as financial returns of different assets. I'm guessing this is mainly due to skewness, although I might be wrong and ...
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58 views

Can I build an efficient frontier using matrix algebra?

If i have a vector of expected returns $A$, a covariance matrix $C$ and a vector of the corresponding weights $W$ for each investment, is it possible to generate the efficient frontier with vector ...
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43 views

Optimizing Portfolio Return by Targeting Variance

I understand Markowitz and targeting returns to minimize our variance. I know this optimization problem well and its constraints. However when the reverse scenario is to be considered I get very ...
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85 views

Option Based Portfolio Insurance OPBI Simulation Excel

I want to simulate an example of OBPI in Excel. I can't find any example online with random figures that show how to simulate it. I tried to understand the appendix of Perold (1995): Dynamic ...
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Density of a portfolio's returns is the weighted average of asset distributions?

The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns: $$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$ Does it also follow that the ...
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189 views

How to compute portfolio returns when constructing a dollar-neutral portfolio

I am trying to wrap my head around this statement: dollar-neutral portfolios are built: dollar amounts of both long and short positions are equal. Furthermore, it is also true at the stock level: ...
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Finding a PDE for an option $V(t,r(t),S(t))$

I have 2 approaches in my mind for finding a pde of an option that depends both on the short rate as well as the stock price- $V(t,r(t),S(t)$. Are these equivalent? Find a hedging portfolio by ...
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Is there a clear mathematical statement of what problem Hierarchical Risk Parity is solving?

Prado's paper is really just an algorithm for solving some inverse problem. Has anyone seen a clear statement of that inverse problem? Or do you know how to write it simply? The first step is just a ...
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111 views

Annualised returns and volatility for 3 month data

I have a portofolio with 30 indexes and I want to calculate the annulised returns and volatility because I want to compare it with another portofolio with different number of indexes (but same time ...
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67 views

Global portfolio alpha

How does one compute the alpha of a global portfolio. Let's say we are using the Fama French 3 factor model and we have a portfolio of 50% US stocks and 50% German stocks. Should the regression be ...
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1answer
41 views

How to build a portfolio following the Smart beta process by dividend? [closed]

I'm having trouble finding the method to track smart beta dividend management. I have an Excel file which contains the prices and the dividends of certain companies, and I want to build a portfolio ...
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111 views

What’s the best Backtest Software/method? [closed]

I have a CSV which looks like this. Ticker | Buy Date | Sell date AAPL | 2018-01-03 | 2019-03-30 TSLA | 2019-03-01| 2019-04-05 What’s the best way to backtest this CSV performance given that ...
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How does negative performance of a portfolio constituent affect its weight?

This is an easy question, I hope. Suppose we have a swap A with a long position, which, originally, has a weight of 30%. Over time, it has a positive performance of 3%, meaning we have a multiplier ...
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80 views

Equity risk factors with daily rebalancing

I am building some well known equity factors on the S&P for research purposes. It means those are going to be used for general evaluation purposes but do not need to be replicable. Would it be a ...
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1answer
50 views

Beta of sum or sum of betas

When interested in the beta of a portfolio, I see people make a weighted sum of the portfolio components' betas. Intuitively, I would have calculated the beta of the portfolio based on its aggregate ...
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80 views

CVaR portfolio optimization with risk aversion parameter

I'm trying to implement the Rockafellar's function described in this paper http://past.rinfinance.com/agenda/2009/yollin_slides.pdf with a risk aversion parameter for my thesis. The function to ...
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1answer
55 views

Why does Merton's fraction give unintuitive quantities using real world data?

The solution to Merton's portfolio problem suggests that an investor invest $\frac{\mu - r}{\sigma^2 \gamma}$ percent of their wealth in the stock market, where $\mu$ is the rate of return of the ...
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32 views

Time and asset weighted rate of return of a portfolio

If I have a portfolio with 3 initial assets on day 1 (say, stock 1 with beginning market value of \$100, stock 2 \$150 and stock 3 \$175) and after 10 days the stock 2 is sold for \$200, how can I ...
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163 views

R - Portfolio construction based on own calculations, with rebalancing of components

I have used random forest in R to get probabilities for stocks being in a certain class. With those probabilities i would like to construct portfolios containing the 5 stocks with the highest ...
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2answers
130 views

Understanding what is 'special' about the security market line

I am trying to get my head around the CAPM model and all the intricacies of portfolio management. I have written some code to help me visualise what happens to the risk-return characteristics of my ...
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1answer
137 views

Is it possible to construct a hedge that matches value Delta Gamma and Vega?

Given a strike price, current price, risk free rate, dividend yield and volatility, I have been asked to calculate: - a hedge which matches the value Delta and Gamma - a hedge which matches the value ...
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44 views

Variance/VaR calculation for a Portfolio

I'm considering a portfolio of multiple stocks (>2), and calculating their Standard Deviation/Variance and VaR for the portfolio. My question is about the below two ways to calculating them Consider ...
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2answers
70 views

How to compute a portfolio value?

I am learning fundamentals of option market and ran into an example I do not understand : Let's assume I have a portfolio of 3 shares priced \$22, and a European call option to buy a share for \$21 ...
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1answer
71 views

portfolio volatility over time

When estimating portfolio vol. with: $\sigma = \sqrt{w^T \cdot cov \cdot w}$ How does the sample length of returns affect $\sigma$? Is it possible to exponentially weight something to give more ...
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1answer
183 views

Value at Risk for portfolio with different maturities

I am new to StackExchange and relatively new to quantitative finance. I work at a commodity trading company and we have an extensive portfolio of futures and options on commodities (traded on the CME, ...
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71 views

Visualise returns of n-asset portfolio as 2D plot, like heatmap?

It's possible to visualise both asset allocations and returns of 2-assets portfolio as 2D heatmap. Like in image below, the visualisation of $[Gold, Silver]$ portfolio with restriction $Gold + ...
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211 views

What if all the weights are negative in mean-variance optimization during a crisis?

Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization. But it contradicts ...
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109 views

Is optimising for the Final Wealth is the same as optimising log of growth rate in Kelly Criterion?

A direct, brute force approach could be used to find the Optimal Portfolio. Consider simple play. There's a biased coin with 55% probability of win. The simulator play as a single person with 100$ ...
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160 views

What is the appropriate breakpoints for portfolio sorting (using CRSP stocks) based on size?

I want to sort stocks based on size. I use CRSP database from 1962 to 2018. My question is that in order to create quintiles, what is the appropriate breakpoints? and how can I calculate them? I saw ...
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184 views

Constructing an arbitrage opportunity for a company involving Forwards

Let's say an investor enters a long forward contract on 100 units of underlying assets $S$ and maturity $T$ = 4 years. The asset $S$ pays no dividends and the spot price of one asset is $S_0$ = £5. ...
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333 views

Calculating Delta of option portfolio using average of inputs

Trying to think through two options portfolio scenarios, which are highly similar. I'm wondering if you can take a portfolio of options, all written against the same underlying product, and use ...

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