Optimal Portfolio Choice And The CAPM Chap 11

Question 1

Portfolio weights

Answer 1

How to describe a portfolio -> these weight add up to 1
Xi = value of investment i / total value of portfolio

Question 2

Covariance

Answer 2

The expected product of deviations from two returns from their means.
Cov (Ri,Rj) = E [(Ri- E[Ri])(Rj - E[Rj])]

Question 3

Correlation

Answer 3

The covariance of the returns divided by to standard deviation of each return, a measure of the common risk shared by stocks that does not depend on their volatility. The magnitude will be larger if the stocks are more volatile (and so have larger deviations from their expected return), and it will be larger the more closely to stocks move in relation to each other.

Corr(Ri,Rj) = Cov (Ri,Rj) / SD(Ri) x SD(Rj)

-> correlation is always between 1 and -1

Question 4

Correlation close 1

Answer 4

The closer the correlation is to 1 the more the returns tend to move together as a result of common risk

Question 5

Correlation 0

Answer 5

When the correlation equals zero, the returns are uncorrelated

Question 6

Correlation -1

Answer 6

The closer the correlation is to -1. The more the returns tend to move in opposite directions.

Question 7

Variance of a portfolio

Answer 7

The variance of a portfolio is equal to the weighted average covariance of each stock with the portfolio. This expression reveals that the risk of a portfolio depends on how each stocks return moves in correlation to it.

The overall variability of the portfolio depends on the total co-movement of the stock within it

Question 8

Equally weighted portfolio

Answer 8

A portfolio in which the same dollar amount is invested in each stock. An equally weighted portfolio consisting of n stocks has portfolio weights xi = 1/n

Var (Rp) = 1/n + ( 1-1/n)
Var(Rp) = average variance of the individual stock + average covariance between stocks

-> volatility declines as the number of stocks in the portfolio grows

Question 9

Inefficient portfolio

Answer 9

An inefficient portfolio describes a portfolio, for which it is possible to find another portfolio that has higher expected return and lower volatility.

Question 10

Effect of correlation

Answer 10

1. The lower the correlation, the lower the volatility we can obtain. As we lower the correlation, and therefore the volatility of the portfolios, the curve showing the portfolios will bend to the left to a greater degree
2. When stocks are perfectly positively correlated, we can identify the set of portfolios by the straight line between them. In this case, the volatility of the portfolio is equal to the weighted average volatility of the two stocks. There is no diversification.
3. When the correlation is less than one, however, the volatility of the portfolios is reduced due to the diversification, and the curve bends to the left. The reduction in risk (and the bending of the curve) becomes greater as the correlation decreases. At the other extreme of perfect negative correlation, the line again become straight, this time reflecting of the vertical axis. In particular, when the two socks are perfectly negatively correlated, it becomes possible to hold a portfolio that bears no risks.

Question 11

Efficient frontier

Answer 11

The set of portfolios that can be formed from a given set of investments with the property that each portfolio has the highest possible expected return that can be attained without increasing its volatility

Question 12

Sharpe ratio

Answer 12

Sharpe ratio is the excess return of an asset divided by the volatility of the return of the asset, a measure of the rewards per unit risk

E[Rp]-rf / SD(Rp)

Question 13

Tangent portfolio

Answer 13

A portfolio with the highest sharpe ratio; the point of tangency to efficient frontier of a line drawn from the risk free asset; the market portfolio if the CAPM holds
Tangent portfolio provides the biggest rewards per unit of volatility of any portfolio available

Question 14

CAPM Assumptions

Answer 14

1. Investors can buy and sell all securities at competitive market prices (without incurring taxes, or transaction cost) and can borrow and lend at risk free interest rate
2. Investors hold only efficient portfolios of traded securities - portfolios that yield the maximum expected return for a given level of volatility.
3. Investors have homogeneous expectations regarding the volatility, correlation and expected returns of securities

Question 15

2 Major conclusion of the CAPM

Answer 15

1. The market portfolio is the efficient portfolio. Therefore, the highest expected return for any given level of volatility is obtained by a portfolio on the capital market line, which combines the market portfolio with risk-free savings or borrowings.
2. The risk premium for any investment is proportional to its beta with the market. Therefore the relationship between risk and the required return is given by the security market line.