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
The note addresses whether an information ratio should use average monthly returns or cumulative monthly returns. Its answer recommends dividing the average monthly return by the standard deviation, pointing readers to an external response for further…
The document discusses how to identify common risk factors in private markets when asset prices are infrequently observed or unavailable. One suggested route is to describe companies using observable characteristics such as industry, business relationships,…
The document presents a derivation of the CAPM by treating the market portfolio as a portfolio on the capital allocation line that maximizes the Sharpe ratio. Under the assumption that investors choose mean–variance efficient portfolios, individual holdings…
The document explains how to generate multivariate Gaussian returns with a target covariance structure. Starting with a vector of independent standard normal draws, apply a matrix square root of the desired covariance matrix. A Cholesky factor is one such…
The document examines a spreadsheet that models portfolio wealth with a lognormal distribution derived from expected return and risk, then scales the log-return distribution across multiple years. The resulting wealth distributions spread out over time,…
This document asks whether a minimum-variance portfolio is more likely than a maximum-Sharpe, or tangency, portfolio to deliver out-of-sample performance close to its in-sample estimate. It defines disappointment as the gap between predicted in-sample…
The document explains how covariance-matrix inversion affects the standard mean-variance portfolio solution. In an eigenvector decomposition, each direction’s contribution is scaled by the reciprocal of its covariance eigenvalue. Consequently, directions…
The document shows how to calculate a money-weighted rate of return for a pension fund with dated inflows and outflows. The method sets the future value equation to zero at the fund’s closing date, compounds the starting balance over twelve months, adjusts…
The document explains why the S&P 500 price index and its total return counterpart produce different performance charts. The price index reflects constituent share prices, so it falls when a company goes ex-dividend; it does not credit the dividend to the…
The document resolves a confusion between variance as a function of a probability parameter and portfolio variance as a function of asset weights. A Bernoulli variable’s variance can be concave in its probability parameter, but portfolio variance, expressed…
The note derives the range of a common off-diagonal correlation that makes a matrix with unit diagonal entries a valid covariance or correlation matrix. It corrects a variance calculation: when summing pairwise covariances in the variance of a sum, each…
The document explains two possible budget constraints in mean-variance portfolio optimization when short sales are permitted. Requiring signed portfolio weights to sum to one treats short positions as negative weights whose proceeds can fund long positions.…
The document introduces mean-variance portfolio optimization, including a minimum-variance portfolio and a variance-minimizing portfolio constrained to a target expected return. It then describes practical variations: long-only holdings, caps on individual…
The document asks how to interpret the point where a line from the risk-free rate touches the efficient frontier for a portfolio built from three stocks. In mean-variance portfolio theory, that tangency point represents the risky portfolio with the highest…
The document asks whether investors who understand asset pricing should hold only portfolios on the mean-variance efficient frontier. It presents a relation between an asset’s expected return, its volatility, its correlation with the stochastic discount…
The document explains how to maximize expected return for a portfolio subject to a limit on expected shortfall (ES), also called conditional value-at-risk. It describes an auxiliary function of portfolio weights and a threshold variable: the function is…
The document asks how to combine multiple strategies into one portfolio and why volatility calculated from aggregate asset exposures may differ from volatility calculated using strategy return streams. It describes two views: treat each strategy’s returns as…
The document proposes revisiting the portfolio construction rule at every rebalance instead of committing to one allocation scheme throughout an investment horizon. It contrasts optimized approaches, such as minimum variance and maximum Sharpe portfolios,…
The document asks how to handle zero weights from a Markowitz allocation when every asset must be included, and how to determine the risk-free allocation when constructing a portfolio on the capital market line. The response focuses on selecting the risky…
The document proves convexity for a portfolio risk measure consisting of expected return plus a positive multiple of the expected absolute deviation from expected return. It assumes portfolio return is linear in portfolio weights, so expected return is also…
The document examines why the optimal risky portfolio can appear to differ when optimizing with or without access to lending and borrowing at a risk-free rate. Its central correction is that maximizing expected return divided by volatility is not equivalent…
The document presents an equity portfolio-selection problem based on monthly returns for a large stock universe and a market index. Its code estimates each stock's average return, beta, residual risk, and Treynor ratio, then applies a single-index cutoff…
This collection of answers challenges widely used ideas in quantitative finance, including correlation-based diversification, CAPM as an allocation rule, normally distributed independent returns, value at risk, efficient markets, copulas, and perfect delta…
The document explains how to turn a sequence of periodic returns into a continuous index or equity curve starting from a base value. It distinguishes adding return increments from compounding them and gives the accepted answer: multiply the index by one plus…