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• ### 1 Week 1: The Basics

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• ### 1.1 Portfolio Theory

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• #### Modern Portfolio Theory (MPT)

- Given expected returns, risks, and correlations:
• MPT maximizes expected return given a certain level of risk
• OR
• MPT minimizes risk given a certain level of expected return

- Focus on the optimal combination of assets
• Diversification
• Irrespective of the investor's utility function
• #### How to form expected returns out of historical returns?

Assume that historical average risk and returns are representative for the future.
• ### 1.2 Utility Functions

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• #### Von Neuman Morgenstern Expected Utility:

Decision-maker faced with risky outcomes of different choices will behave as if he is maximizing the expected value of some function defined over the potential outcomes --> so, probability-weighted average of utility over the possible outcomes
• #### Explain the formula: U[W] = aW - b/2V[W] = aW - b/2W^2

• U[W] describes my utility (happiness) over my wealth
• aW (with a > 0) describes that if I get more wealth, my happiness goes up.
• -b/2V[W] means that higher risk makes me less happy.
• b is the risk aversion factor. The higher b, the heavier the risk factor is taken into account
• First derivative: a - bW > 0, more wealth makes me more happy
• Second derivative: -b < 0: the extra wealth makes me less additionally happy
• #### What can we say about the utility function?

• If you invest in riskier assets (i.e., the w+ and w- are further apart, there is a bigger discount in utility)
• The difference in the utility function and actgual utility is the risk premium
• #### What can we conclude about utility wrt the demand for the market portfolio?

• Increases with expected return
• Decreases with risk-free rate
• Decreases with risk
• Decreases with risk aversion
• Decreases with wealth (?) -> no, other way around
• ### 1.3 CAPM

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• #### What is the empirical translation of the CAPM?

Alpha should equal the risk free rate!
• ### 1.4 Fama-MacBeth methodology

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• #### What are the three testable implications based on the CAPM equation?

1. The relation between expected return and risk is linear
2. Beta is a complete measure of risk of security i in portfolio m (beta is the only variable that predicts risk, any other measure should not have any predictive power)
3. Higher risk should be associated with higher expected returns
• #### What are the assumptions in the CAPM model?

• Markets are efficient, so prices full reflect available information
• Investors are risk averse
• #### Why is it difficult to estimate true beta?

• For out-of-sample testing (expected returns), need to estimate beta on a pre-testing period.
• Assume:
• Beta stays constant for next period
• Rational expectations: E(r) = r
• Trade off between stability and length of sample period -> remains arbitrary choice (to lengthy mihgt not be relevant anymore)
• Error in variables problem: estimated gamma is biased downwards