This question explores the effects of estimation error on apparent arbitrage opportunities in a controlled simulation setting. We simulate returns for N = 20 assets over T = 15 years. We will use the first 10 years for estimation and the selection of the hedge portfolio ("in-sample"), and the remaining 5 years as "out-of-sample" performance evaluation period. We will assume throughout that the risk-free rate is zero so there is no need to distinguish returns and excess returns. All assets follow the single index model Rn,t = αn + βnRM,t + εn,t with a normally distributed market return RM.

The parameters to generate returns are as follows:

• µM = 0.05 and σM = 0.15.
• All assets have βn = 1 and σ(εn,t) = 0.25
• α1 = . . . = α10 = -0.04, α11 = . . . = α20 = 0.04

Perform 100 simulations (data table) for returns on market-neutral zero-investment portfolios and report the mean return and how often you investment loses money over the performance evaluation period (years 11 to 15) assuming that i) you know the true alphas and betas and invest equal amounts in every stock with positive alpha and short equal amounts of every stock with negative alpha

ii) you estimate alphas (but know betas) using the first 10 years of data and invest equal amounts in every stock with positive alphas and short equal amounts of every stock with negative alphas

Compare and explain your findings.

iii) Think of a way to improve the strategy in (ii) and implement it.

Notes

• Your investment should hedge out all market risk (using known betas)
• The portfolio weights in (ii) are a bit tricky. Use "= if(α > 0, 1, 0)" to identify assets to buy. Summing this across assets tells you how many assets have positive alphas, and therefore the proportion to assign to each.