Portfolio Theory and Asset Pricing

If we Rebalance our Portfolio

If we dont rebalance our Portfolio, we choose the “Buy & Hold” Strategy. It takes the compounding into account

It is the logaretmic price change and takes the compunding interest affect in to account.

It is Additive to the monthly return.

For a given Risk (standard deviation), there is a possibility for a better return. The combination of both assets is better than a single asset.

The Avergage Covariance is independet from the Number of Assets while the term (1/N)*(Average Variance-Average Covariance) depends on the number of assets. We can diversify our Risk by increasing the number of assets

• Utility functions are concave

• Utility functions increase with wealtch

• If Investor prefer certain outcome over uncertain outcome -> Risk avers

• If Investor is indifferent -> Risk neutral

• If Investor prefer uncertain outcome oder certain outcom -> Risk Loving

The highest return of a given risk.

The Tangent of the utility function and the Efficient frontier line

• Risk free asset has zero volatilty

• Correlation between the risk free asset and risky portfolio is zero

• Risk averse and utility maximizing investors

• Perfect capital markets (no taxes or transactions costs)

• Risk free rate exists

• Investors have homogennous expectation of volatilites, correlations an expected return of securites

The set of possible combinations of the risk free asset and the efficient portfolios.

It can only be used to price efficient portfolios whre the total risk equal systematic risk

The lower the beta, the lower the correlation the stock has with the market portfolio. Thus we have higher diversification. Low Beta stocks are less risky

• Dont hold individual assets, hold the market

• Each investor has his own optimal market risk

• Average investors holds the market

• Systematic risk is beta

They have high risk premiums:

• High betas

• High betas are more risky, dont offer diversification and require high expected values

It is a portfolio whith one factor sensitivity of 1 against all other factor

• Macroeconomic factor (inflation, war, interest rate)

• Fundemental factor (Attributes of firms or stocks like investemnt styl or book-to-market size

If there is no variation in return.

It is the efficient Portfolio with the lowest variance

The Portfolio Variance converges to the average covariance. So the the diversifiable risk equals the systemtic risk.

1. No, due to the fact that every Investor has his own risk tolerance.

2. The individually optimal portfolio is the one with the highest mean-variance utility

• The composition of the optimal risky portfolio does not depend on the risk preferences of the investor

• the mix of the tangent portfolio and the risk free assets does depend on the risk preferences

In equilibrium the investor holds a combination of the market portfolio and the risk free asset.

All investors hav homogenous expectation. -> Investors will hold an identical risky portfolio. If all investors hold the same portfolio, in equilibrium it will contain all assets.

Tangecy Portfolio = Market Portfolio

Risk avers people need a higher expected return to be compesated otherwise their utility would not match with lower risk avers people

It is based on the “law of one price” and assumes that invesotrs have homogenous expectations without relying on the utility maximation or mean variance model

We weight our assets to delete the factors impact. As a result the factor is immune to our portfolio. There is a arbitrage against the risk free rate possible

CAPM has only one driving factor, where APT has more factors. The Risk Premium of each facotr reflects the exposure of the asset and the factor. Each factor has its own definition on bad times.

The efficient frontier has the highest sharpe ratio

• Time series regression

  • It violates the CAPM, High Beta -> Low Returns and Low Beta -> High returns

• Cross Sectional

  • Confirm the linear relationship between excess return and beta, but in some period it was not consistent with the capm

• World Market Portfolio is the most important factor

• Every Country has its own factor sensitivity

• Y0 = 0

• Y1 = 1 = Market-Risk

• Relationship between Expected Return and Beta is positive

• But Y0 is > Rf and Beta < Market-Risk