Ordinary Differential Equations: An Introduction to by Herbert Amann

By Herbert Amann

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By Herbert Amann

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Additional info for Ordinary Differential Equations: An Introduction to Nonlinear Analysis (Degruyter Studies in Mathematics)

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Let rm,t+1 be the excess return on a market or benchmark index. For example, this could be the S&P 500, a “style” index such as “small cap growth,” or a vector of excess returns if a multi-factor model is used. 1) where rp,t+1 is the return of the fund in excess of a short term “cash” instrument, and Zt is the vector of lagged conditioning variables, in demeaned form. The symbol ⊗ denotes the kronecker product, or element-by-element multiplication when rm,t+1 is a single market index. 1) is the classical CAPM regression, where the terms involving Zt are omitted.

Jensen (1968) studies the performance of 115 open end mutual funds over the period 1945–1964. Data are obtained from Wiesenberger’s Investment Companies. 1% per year relative to the SP500 index. The distribution of the alphas across funds is skewed to the left, with 66% of funds having a negative alpha. Using data on expense ratios, Jensen (1968) calculates fund performance gross of expenses (but net of trading costs). 4% per year. 3 Professionally Managed Portfolios in Classical Measures 29 Jensen concludes that, in aggregate, the investment ability of mutual fund managers is not great enough to recover even brokerage commissions.

5%. The unconditional approach leads to the mistaken conclusion that the manager has negative abnormal performance. But the manager’s performance does not reflect poor investment choices or wasted resources, it merely reflects common variation over time in the fund’s conditional risk exposure and the market premium. The traditional model over adjusts for market risk and assigns the manager a negative alpha. However, investors who have access to information about the economic state would not use the inflated risk exposure and would therefore not ascribe negative performance to the manager.

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