CourseNotes_P2

# CourseNotes_P2 - Course Notes for Statistics 365 Part II T...

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Course Notes for Statistics 365, Part II T. A. Severini Northwestern University April 8, 2016 Draft; please do not circulate. Copyright 2016, T. A. Severini All rights reserved.

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Contents Contents 1 1 Returns 3 1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Adjusted Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Statistical Properties of Returns . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Analyzing Return Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Random Walk Hypothesis 28 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Eﬃcient Markets and the Martingale Model . . . . . . . . . . . . . . . . . . 32 2.4 Random Walk Models for Asset Prices . . . . . . . . . . . . . . . . . . . . . 35 2.5 Tests of the Random Walk Model . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Do Stock Returns Follow the Random Walk Model? . . . . . . . . . . . . . . 48 2.7 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Portfolios 53 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Negative Portfolio Weights: Short Sales . . . . . . . . . . . . . . . . . . . . . 57 3.4 Optimal Portfolios of Two Assets . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Risk-Free Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.6 Portfolios of Two Risky Assets and a Risk-Free Asset . . . . . . . . . . . . . 67 3.7 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 75 1

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2 CONTENTS 3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Eﬃcient Portfolio Theory 76 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Portfolios of N Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Some Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.4 The Minimum-Variance Portfolio . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5 The Eﬃcient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Risk-Aversion Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.7 The Tangency Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.8 Portfolio Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.9 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 Estimation 114 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Simple Sample-Based Estimators . . . . . . . . . . . . . . . . . . . . . . . . 114 5.3 Estimation of the Mean Vector and Covariance Matrix . . . . . . . . . . . . 120 5.4 Model-Based Estimators and Shrinkage . . . . . . . . . . . . . . . . . . . . . 126 5.5 Using Monte Carlo Simulation to Study the Properties of Estimators . . . . 135 5.6 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Chapter 3 Portfolios 3.1 Introduction Suppose we have a given amount of capital to invest and a number of possible assets in which to invest. How should we place our investment in the various assets? This is known as the portfolio selection problem and the mathematical and statistical methods developed to solve it are known as portfolio theory . If it were possible to accurately forecast the future returns of the assets we could simply invest in the asset or assets with the largest predicted returns over the future time period of interest. However, empirical evidence, such as that presented in the previous chapter, suggests that such forecasting is diﬃcult at best. Hence, in portfolio theory, we attempt to choose the combination of assets that yields a portfolio return with desirable statistical properties; specifically, we seek a large expected return, while minimizing the “risk” of the portfolio, defined here as the standard deviation of the return. Thus, in an ideal case, the return on the portfolio will have a large expected value and a small standard deviation so that the portfolio realizes a large return with high probability. However, complicating the situation is the fact that the two goals are typically in conﬂict: riskier assets generally have a higher expected return, as a reward for assuming the risk.

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