Corporate Finance (Chapter 11)

What are the two consequences of selling some of our risk-free assets (or borrowing money) and investing the proceeds in investment i?

1. Expected return: Because we are giving up the risk-free return and replacing it with i's return, our expected return will increase by i's excess return E[Ri]-rf 2. Volatility: We will add the risk that i has in common with our portfolio. P:SD(Ri)*Corr(Ri,Rp) Invest more in portfolio P: E[Rp]-rf/SD(Rp)

What are the three main assumptions underlie the CAPM?

1. Investors trade securities at competitive market prices (without incurring taxes or transaction costs) and can borrow and lend at the risk-free rate. 2. Investors choose efficient portfolios. 3. Investors have homogeneous expectations regarding the volatitilities, correlations and expected returns of securities

When is a portfolio efficient?

A portfolio is efficient if and only if the expected return of every available security equals its required return. E[Ri]=ri E[Ri]=ri≡rf+βi^eff*(E[Reff]-rf) Reff = return of the efficient portfolio.

What is a levered portfolio?

A portfolio that consists of a short position in the risk-free investment.

If investors are holding optimal portfolios, how will the portfolios of a conservative and an aggresive investor differ?

An investor can select a desired degree of risky by choosing the amount to invest in efficient portfolio relative to risk-free investments. Conservative: invest small amount Aggresive: investing more than conservative

How does the efficient frontier change when we use more stocks to construct portfolios?

As investors add stocks to portfolio, the efficient portfolio improves (greater diversification) - An investor seekig high expected returns and low volatility should invest only in efficient portfolios. - Investors will choose from the set of efficient portfolios based on their risk tolerance.

Explain why the market portfolio is effient according to the CAPM.

Because the supply of securities must equal the demand for securities

An investment's cost of capital is determined by its beta with what portfolio?

Beta indicates the sensitivity of the investment's return to fluctuations in the portfolio's return. The beta of an investment with a portfolio is: βi^p = (SD(Ri)*Corr(Ri,Rp))/SD(Rp)

Give the definition of buying stocks on margin.

Borrowing money to invest in stocks. This happens when we increase the fraction x beyond 100%. Then we are short selling the risk-free investment, so we must pay the risk-free return.

How does the correlation between two stocks affect the risk and return of portfolios that combine them?

Correlation has no effect on the expected return of a portfolio. However, the volatility of a portfolio wil differ depending on the correlation. In particular, the lower the correlation, the lower the volatility we can obtain. When the stocks are perfectly positively correlated (+1), we can identify the set of portfolios by the straight line between them (volatility portfolios = weighted average volatility).

How do we estimate the covariance from historical data?

Cov (Ri,Rj) = 1/(T-1)∑(Ri,t-Ri⁻)(Rj,t-Rj⁻)

How do we calculate the covariance between returns Ri and Rj?

Cov(Ri,Rj)=E[(Ri-E[Ri])(Rj-E[Rj])

When will a new investment improve the Sharpe ratio of a portfolio?

E[Ri]-rf>SD(Ri)*Corr(Ri,Rp)* E[Rp]-rf/SD(Rp) additional return from investment i>additional return from taking the same risk investing in P Increasing the ammount invested in i will increase the Sharpe ratio of portfolio P if its expected return E[Ri] exceeds its required return given portfolio P, defined as: E[Ri]>rf+βi^p*(E[Rp]-rf) ri≡rf+βi^p*(E[Rp]-rf)

According to the CAPM, how can we determine a stock's expected return?

E[Ri]=ri≡rf+βo*(E[Rmkt]-rf) βi=SD(Ri)*Corr(Ri,Rmkt)/SD(Rmkt) = Cov(Ri,Rmkt)/Var(Rmkt)

What is the expected return and volatility for portfolios that combining risk-free asset with a portfolio of risky assets?

E[Rxp]=(1-x)rf+xE[Rp] =rf+x(E[Rp]-rf) x invest in portfolio and (1-x) in risk-free assets SD(Rxp)=√(1-x)²Var(rf)+x²Var(Rp)+2(1-x)xCov(rf,Rp) = √x²Var(Rp)=xSD(Rp) rf is fixed and does not move with our portfolio!

How does the volatility of an equally weighted portfolio change as more stocks are added to it?

If the number of stocks, n, grows large, the variance of the portfolio is determined primarly by the average covariance among the stocks.

How does the volatility of a portfolio compare with the weighted average volatility of the stocks within it?

SD(Rp)=∑i xi*SD(Ri)*Corr(Ri,Rp) xi = amount of i held SD(Ri) = total risk of i Corr(Ri,Rp) = Fraction of i's risk that is common to P SD(Rp)<∑xiSD(Ri) risk of the portfolio will be lower than the weighted average volatility of the individual stocks.

What do we know about the Sharpe ratio of the efficient portfolio?

Sharp Ratio = Portfolio Excess Return / Portfolio Volatility = E[Rp]-rf/SD(Rp) The goal of an investor who is seeking to earn the highest possible expected return for any level of volatility is to find the portfolio that generates the steepest possible line when combined with the risk-free investment. The slope of this line is called the Sharpe ratio of the portfolio.

Why use investors short sales in their portfolio?

Short selling is profitable if you expect a stock's price to decline in the future, but it can greatly increase risk of the portfolio. A portfolio is short those stocks with negative portfolio weights. Short selling extend the set of possible portfolios.

What is the security market line (SML)?

The CAPM equation states that the risk premium of any security is equal to the market risk premium multiplied by the beta of the security. This relationship is called the security market line (SML), and it determines the required return for an investment: E[Ri]=ri≡rf+βo*(E[Rmkt]-rf)

What is a beta of a portfolio?

The beta of a portfolio is the weighted average beta of the securities in the portfolio. βp = Cov(Rp,Rmkt)/VAr(Rmkt)=Cov(∑xiRi,Rmkt)/Var(Rmkt) = ∑xi*(Cov(Ri,Rmkt)/Var(Rmkt))=∑xiβi

How does the correlation between the stocks in a portfolio affect the portfolio's volatility?

The correlation is defined as the covariance of the returns dividend by the standard deviation of each return. The correlation is always between -1 and +1. It represents the fraction of the volatility due to risk that is common to securities. Corr(Ri,Rj)=Cov(Ri,Rj)/SD(Ri)SD(Rj)

How do we calculate the return on a portfolio?

The expected return of a portfolio is the weighted average of the expected return of the investments within it, using the portfolio weights. E[Rp]=∑xiE[Ri] Rp = x1R1+x2R2+...+xnRn=∑xiRi xi = portfolio weights Ri = returns on investment

What is the required return?

The expected return that is necessary to compensate for the risk investment i will contribute to the portfolio.

Give the definition of tangent portfolio and efficient portfolio.

The portfolio that generates the tangent line. The risky portfolio with the highest Sharp ratio is calle dthe efficient portfolio. The efficient portfolio is the optimal combination of risky investments independant of the investor's appetite for risk.

What is a portfolio weight?

The portfolio weight is the initial fraction xi of an investor's money invested in each asset. Portfolio weights add up to 1. xi = Value of investment i / Total value of portfolio

What is the efficient frontier?

The set of efficient portfolios. Efficient portfolios offer investors the highest possible expected return for a given level of risk.

What does the covariance and correlation measure?

To find the risk of a portfolio, we need to know the degree to which stock returns move together. Covariance and correlation measure the co-movement of returns.

What is the capital market line (CML)?

Under the CAPM assumptions, the CML, which is the set of portfolios obtained by combining the risk-free security and the market portfolio, is the set of portfolios with the highest possible expected return for any level of volatility.

How do we calculate the variance of a portfolio?

Var (Rp) = ∑xiCov(Ri,Rp)=∑xi Cov(Ri,∑xjRj)=∑i∑j xixj Cov(Ri,"Rj) Var(Rp)=Cov(Rp,Rp)=Cov(∑xiRi,Rp)=∑xiCov(Ri,Rp)

How do we calculate the variance of an equally weighted portfolio of n stocks?

Var(Rp)=(1/n)*(Average Variance of the Individual Stocks)+(1-(1/n))*(Average Covariance between the Stocks)

How do you calculate the variance of a two-stock portfolio?

Var(Rp)=x₁²Var(R₁)+x₂²Var(R₂)+2x₁x₂Cov(R₁,R₂) Var(Rp)=x₁²SD(R₁)²+x₂²SD(R₂)²+2x₁x₂Corr(R₁,R₂)SD(R₁)SD(R₂)

When can we say that a portfolio is inefficient?

Whenever it is possible to find another portfolio that is better in terms of both expected return and volatility.

What is the difference between long position and short position?

long position = positive investment in a security short position = negative investment in a stock by engaging in a short sale.