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Suppose we have two assets, one safe and one risky
Suppose we have two assets, one safe and one risky

The Choice of Model Factors Under Multiple Definitions of Risk
The Choice of Model Factors Under Multiple Definitions of Risk

... • Estimated portfolio volatility is 14% – Using the “square root of time” rule, the expected value of the standard deviation of simulated five year cumulative returns is 31% – With 100 simulated paths, the standard deviation of the five year return is actually 65%, more than double what we would exp ...
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CHAPTER 6 Risk, Return, and the Capital Asset Pricing Model 1
CHAPTER 6 Risk, Return, and the Capital Asset Pricing Model 1

... No. Rational investors will minimize risk by holding portfolios. They bear only market risk, so prices and returns reflect this lower risk. The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk. ...
Daren Wei - Imperial College London
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... dampens excess volatility caused by speculation, while increasing social welfare under all reasonable probability measures specified by the belief-neutral welfare criteria. Difference in Beliefs and Equity Premium Term Structure, with Harjoat Bhamra We build an equilibrium model to capture downward ...
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... stimulus we saw the trend of flows reverse and this continued during the past quarter with inflows into global equities, specifically developed market equities, and outflows from global bonds. Global pension funds have increased their allocation towards equities, however it is still below the long t ...
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... • Avoid look-ahead bias: make sure sorting variables are observable to investors at formation (financial reporting lag) • Rebalance once a year (reduce transaction cost) ...
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Modern portfolio theory

Modern portfolio theory (MPT) is a theory of finance that attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. Although MPT is widely used in practice in the financial industry and several of its creators won a Nobel memorial prize for the theory, in recent years the basic assumptions of MPT have been widely challenged by fields such as behavioral economics.MPT is a mathematical formulation of the concept of diversification in investing, with the aim of selecting a collection of investment assets that has lower overall risk than any other combination of assets with the same expected return. This is possible, intuitively speaking, because different types of assets sometimes change in value in opposite directions. For example, to the extent prices in the stock market move differently from prices in the bond market, a combination of both types of assets can in theory generate lower overall risk than either individually. Diversification can lower risk even if assets' returns are positively correlated.More technically, MPT models an asset's return as a normally or elliptically distributed random variable, defines risk as the standard deviation of return, and models a portfolio as a weighted combination of assets, so that the return of a portfolio is the weighted combination of the assets' returns. By combining different assets whose returns are not perfectly positively correlated, MPT seeks to reduce the total variance of the portfolio return. MPT also assumes that investors are rational and markets are efficient.MPT was developed in the 1950s through the early 1970s and was considered an important advance in the mathematical modeling of finance. Since then, some theoretical and practical criticisms have been leveled against it. These include evidence that financial returns do not follow a normal distribution or indeed any symmetric distribution, and that correlations between asset classes are not fixed but can vary depending on external events (especially in crises). Further, there remains evidence that investors are not rational and markets may not be efficient. Finally, the low volatility anomaly conflicts with CAPM's trade-off assumption of higher risk for higher return. It states that a portfolio consisting of low volatility equities (like blue chip stocks) reaps higher risk-adjusted returns than a portfolio with high volatility equities (like illiquid penny stocks). A study conducted by Myron Scholes, Michael Jensen, and Fischer Black in 1972 suggests that the relationship between return and beta might be flat or even negatively correlated.
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