Refresher slides:
Expected return, Variance, and Standard Deviation:
Historical returns of stocks and bonds:
Computing Historical Returns:
- Realized Return:
→ The return that actually occurs over a particular time period.
Average annual return:
Variance and Volatility of returns:
Variance Estimate using realized returns:
The estimate of the standard deviation (SD) is the square root of the variance.
Common vs. Independent risk:
- Common risk:
Risk that is perfectly correlated
→ risk that affects all securities
- Independent risk:
Risk that is uncorrelated
→ risk that affects a particular security
- Diversification:
The averaging out of independent risks in a large portfolio.
,No Arbitrage and the Risk Premium:
- The risk premium for diversifiable risk is zero, so investors are not compensated for
holding firm-specific risk.
- The risk premium of a security is determined by its systematic risk and does not
depend on its diversifiable risk.
Historical volatility and return for 500 individual stocks, ranked annually by size:
Measuring systematic risk:
- Efficient Portfolio:
A portfolio that contains only systematic risk. There is no way to reduce the volatility
of the portfolio without lowering its expected return.
- Market portfolio:
An efficient portfolio that contains all shares and securities in the market.
The expected return of a portfolio:
Then the return on the portfolio, Rp, is the weighted average of the returns of the
investments in the portfolio, where the weights correspond to portfolio weights.
The volatility of a two-stock portfolio:
- Combining risks:
By combining stocks into a portfolio, we reduce risk through diversification.
The amount of risk that is eliminated in a portfolio depends on the degree to which
the stocks face common risks and their prices move together.
,Determining Covariance and Correlation:
Covariance: the expected product of the deviations of two returns from their means.
Covariance between returns Ri and Rj:
Estimated covariance from historical data:
- If the covariance is positive, the two returns tend to move together.
- If the covariance is negative, the two returns tend to move in opposite direction.
Correlation: a measure of the common risk shared by stocks that does not depend on their
volatility:
Computing a portfolio’s variance and volatility:
- For a two-security portfolio:
- For a large portfolio:
The variance of a portfolio is equal to the weighted average covariance of each stock
with the portfolio:
, Risk versus Return: Choosing an efficient portfolio.
- Efficient Portfolios with two stocks;
In an inefficient portfolio, it is possible to find another portfolio that is better in terms
of both expected return and volatility.
Volatility versus expected return for portfolios of two stocks – an example:
Effect on volatility and expected return of changing the correlation between two stocks:
Risk free saving and borrowing:
- Risk can also be reduced by investing a portion of a portfolio in a risk-free
investment, like T-bills. However, doing so will likely reduce the expected return.
- On the other hand, an aggressive investor who is seeking high expected returns
might decide to borrow money to invest even more in the stock market.
Expected return, Variance, and Standard Deviation:
Historical returns of stocks and bonds:
Computing Historical Returns:
- Realized Return:
→ The return that actually occurs over a particular time period.
Average annual return:
Variance and Volatility of returns:
Variance Estimate using realized returns:
The estimate of the standard deviation (SD) is the square root of the variance.
Common vs. Independent risk:
- Common risk:
Risk that is perfectly correlated
→ risk that affects all securities
- Independent risk:
Risk that is uncorrelated
→ risk that affects a particular security
- Diversification:
The averaging out of independent risks in a large portfolio.
,No Arbitrage and the Risk Premium:
- The risk premium for diversifiable risk is zero, so investors are not compensated for
holding firm-specific risk.
- The risk premium of a security is determined by its systematic risk and does not
depend on its diversifiable risk.
Historical volatility and return for 500 individual stocks, ranked annually by size:
Measuring systematic risk:
- Efficient Portfolio:
A portfolio that contains only systematic risk. There is no way to reduce the volatility
of the portfolio without lowering its expected return.
- Market portfolio:
An efficient portfolio that contains all shares and securities in the market.
The expected return of a portfolio:
Then the return on the portfolio, Rp, is the weighted average of the returns of the
investments in the portfolio, where the weights correspond to portfolio weights.
The volatility of a two-stock portfolio:
- Combining risks:
By combining stocks into a portfolio, we reduce risk through diversification.
The amount of risk that is eliminated in a portfolio depends on the degree to which
the stocks face common risks and their prices move together.
,Determining Covariance and Correlation:
Covariance: the expected product of the deviations of two returns from their means.
Covariance between returns Ri and Rj:
Estimated covariance from historical data:
- If the covariance is positive, the two returns tend to move together.
- If the covariance is negative, the two returns tend to move in opposite direction.
Correlation: a measure of the common risk shared by stocks that does not depend on their
volatility:
Computing a portfolio’s variance and volatility:
- For a two-security portfolio:
- For a large portfolio:
The variance of a portfolio is equal to the weighted average covariance of each stock
with the portfolio:
, Risk versus Return: Choosing an efficient portfolio.
- Efficient Portfolios with two stocks;
In an inefficient portfolio, it is possible to find another portfolio that is better in terms
of both expected return and volatility.
Volatility versus expected return for portfolios of two stocks – an example:
Effect on volatility and expected return of changing the correlation between two stocks:
Risk free saving and borrowing:
- Risk can also be reduced by investing a portion of a portfolio in a risk-free
investment, like T-bills. However, doing so will likely reduce the expected return.
- On the other hand, an aggressive investor who is seeking high expected returns
might decide to borrow money to invest even more in the stock market.
