Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
5,922 documents
Without a risk-free asset, there is no single portfolio on the efficient frontier that is automatically optimal: each point represents a different return and volatility trade-off. The document outlines three ways to choose among them: set a maximum…
The document explains the role of equality and inequality constraints in alternative portfolio-selection formulations. It compares maximizing return subject to a variance limit with minimizing variance subject to a minimum return, and relates these…
The document discusses two formulations of the information ratio: expected residual return relative to residual risk, and active portfolio return relative to its volatility, commonly called tracking error. It presents the latter as average portfolio excess…
The discussion considers how to allocate capital between two risky stocks when their returns are negatively correlated. It separates maximizing expected return from minimizing variance: under a fully invested, long-only constraint, expected return is a…
The document discusses reducing a bond portfolio’s many PV01 tenor buckets to a smaller set of risk factors. It compares three transformation approaches: a transparent linear allocation between selected maturities, PCA weights that project bucket exposures…
This short discussion addresses why indifference curves in mean-variance portfolio theory appear convex, and how their appearance depends on the graph’s axes. The answer suggests reversing the axes to make the relationship resemble a familiar utility curve:…
The document asks whether a James–Stein shrinkage coefficient for a sample mean remains suitable when returns or other inputs have heavy-tailed, non-normal distributions. It describes a formula that uses covariance eigenvalues and the squared distance…
The document compares three ways to estimate returns for portfolio weight decisions: averaging historical returns, using CAPM or other asset pricing models, and forecasting earnings through stock valuation. It presents each as a possible input, while…
The document compares Monte Carlo simulation, daily-return bootstrapping, and block bootstrapping as ways to estimate portfolio weights for monthly rebalancing. It distinguishes simulating correlated returns from fitted distributions, resampling observed…
The note describes how to formulate a robust mean-variance portfolio problem for a constrained optimizer such as MATLAB’s fmincon. Portfolio variance remains the objective, while the minimum expected return over an uncertainty region becomes a nonlinear…
The document examines how portfolio methods allocate between two assets with almost identical risk characteristics and slightly different estimated returns. It contrasts mean-variance optimization, Black-Litterman, and the Resampled Efficient Frontier,…
The document examines the role of diversification in deriving the Arbitrage Pricing Theory (APT) from a linear factor model. In the setup, each asset return consists of an intercept, factor exposures, and an idiosyncratic component. The target relation…
The document explains how the parameter gamma in constant relative risk aversion utility relates to both relative and absolute risk aversion. For a concave utility function, an investor is risk-averse; relative risk aversion measures the willingness to pay…
The document considers how to optimize a bond portfolio spanning government and corporate debt, credit ratings, and maturities. It outlines a mean-variance approach using asset returns, yields, durations, and risk tolerance, with constraints to control which…
The document explains how unequal lending and borrowing rates change the usual capital market line. Starting from the efficient frontier of risky assets, it describes a line from the lending rate that touches the frontier at one portfolio, followed by a…
The document derives the variance of the global minimum-variance portfolio from its closed-form weights. With a symmetric covariance matrix, substituting the weights into the quadratic portfolio-variance expression simplifies the result to the reciprocal of…
The document asks whether correlations among stocks increase when equity indices fall and which models could capture or test that pattern. The answer points readers to research on asymmetric correlations in equity portfolios and extreme correlations across…
The document explains how a portfolio’s Value at Risk can exceed the sum of its separate position VaRs, despite the intuition that diversification should reduce risk. Its example considers a short put and a short call with nearby out-of-the-money strikes on…
The document asks how to measure portfolio sensitivity to dependence in a multivariate t-copula model, in a setting involving electricity futures and rank correlations estimated with Kendall’s tau-b. The proposed finite-difference calculation perturbs the…
The document asks how to measure portfolio volatility when positions begin and end at different times, including when an investor pauses trading or closes some holdings before the rest. The key distinction in the answers is between changes in market value…
The document raises a portfolio-construction question about measuring exposure to property types across companies with multiple segments. Its proposed equal-weighted approach aggregates each company’s property-type share, calculated from segment net asset…
The document considers how to evaluate tax-loss harvesting by comparing an estimated tax benefit with the costs of realizing and reinvesting after a loss. The question proposes expressing the benefit relative to an asset class’s portfolio value and including…
The document addresses why the variance of a weighted return series may differ from the variance calculated from an asset covariance matrix. Its example compares a default variance calculation on portfolio returns with the quadratic form of the sample…
The document explains how to find a portfolio’s correlation with each of its component assets or subportfolios. It recommends starting with the assets’ covariance matrix and the portfolio weights. Multiplying the covariance matrix by the weight vector gives…