Combining Individual Securities Into Portfolios (Chapter 4) Individual Security Return and Risk Portfolio Expected Rate of Return Portfolio Variance Combination Lines Combination Line Between a Risky Asset and a Risk-Free Asset

Individual Security Return and Risk Expected Rate of Return where: E(rA) = Expected rate of return on security (A) rA,i = i(th) possible return on security (A) hi = probability of getting the i(th) return Variance and Standard Deviation

Portfolio Expected Rate of Return A weighted average of the expected returns on the portfolio’s component securities. where: E(rp) = Expected rate of return on portfolio (p) m = Number of securities in portfolio (p) xj = Weight of security (j) Note: The contribution of each security to portfolio expected return depends on: 1. the security’s expected return 2. the security’s weight

Portfolio Variance To compute the variance of a portfolio, you need: (1) the covariances of every pair of securities in the portfolio, and (2) the weight of each security. Example: (Three Security Portfolio)

Take each of the covariances in the matrix and multiply it by the weight of the security identified on the row (security j) and then again by the weight of the security identified on the column (security k). Then, add up all of the products.

The Covariance Between a Security and Itself is Simply Its Own Variance

Each Element Above the Diagonal is Paired With an Identical Element Below the Diagonal

Using the Correlation Coefficient Instead of Covariance Recall: As a Result: Therefore:

COMBINATION LINES A curve that shows what happens to the risk and expected return of a portfolio of two stocks as the portfolio weights are varied. Example 1 (Perfect Negative Correlation)

Now, for various weights, portfolio risk and return can be calculated:

Students are encouraged to prove that the following portfolio standard deviations and expected rates of return are indeed correct for the weights given. Note: With perfect negative correlation, we can create a riskless portfolio by taking positive positions in both stocks.

COMBINATION LINE (Perfect Negative Correlation) Expected Rate of Return (%) xA = .5, xB = .5 xA = -.3, xB = 1.3 All (B) xA = .75, xB = .25 xA = 1.5, xB = -.5 All (A) Standard Deviation of Returns (%)

Example 2 (Perfect Positive Correlation) E(rA) = 5% E(rB) = 11% (rA) = 2.236% (rB) = 6.708% Cov(rA,rB) = +15 A,B = +1.00

Note: When the stocks’ standard deviations are not equal and the stocks are perfectly positively correlated, we can always create a riskless portfolio by selling one of the two stocks short.

COMBINATION LINE (Perfect Positive Correlation) Expected Rate of Return (%) xA = -.3, xB = 1.3 xA = .5, xB = .5 All (B) All (A) xA = 1.5, xB = -.5 xA = 1.75, xB = -.75 Standard Deviation of Returns (%)

E(rA) = 5% E(rB) = 11% (rA) = 2.236% (rB) = 6.708% Example 3 (Zero Correlation) E(rA) = 5% E(rB) = 11% (rA) = 2.236% (rB) = 6.708% Cov(rA,rB) = 0 A,B = 0

COMBINATION LINE (Zero Correlation) Expected Rate of Return (%) xA = -.3, xB = 1.3 xA = .5, xB = .5 All (B) xA = .9, xB = .1 All (A) xA = 1.5, xB = -.5 Standard Deviation of Returns (%)

PATHS OF COMBINATION LINES E(rp) is influenced by E(rj) and xj (rp) is influenced by (rj), j,k, and xj j,k determines the path between two securities Moving along the path occurs by varying the weights.

COMBINATION LINES Expected Rate of Return (%)  = -1.00 Stock (B)  = +1.00 Stock (A)  = 0 Standard Deviation of Returns (%)

Combination Line Between a Risky Stock (or Portfolio) and a Risk-Free Bond Example: Risky Stock (A): E(rA) = 10%, (rA) = 20% Risk-Free Bond (B): E(rB) = 6%, (rB) = 0% Note on Risk:

Combination Line Between a Risky Stock (or Portfolio) and a Risk-Free Bond (Continued)

Combination Line When one of the Assets is Risk-Free Expected Rate of Return (%) Borrowing Lending xA = 1.5, xB = -.5 xA = 1.0, xB = 0 xA = .5, xB = .5 xA = 0, xB = 1.0 Standard Deviation of Returns (%)

Combination Line Between a Risky Stock (or Portfolio) and a Risk-Free Bond (Continued) Note: When one of the two investments is risk-free, the combination line is always a straight line. Lending: When you buy a bond, you are lending money to the issuer. Borrowing: Here, we assume that investors can borrow money at the risk-free rate, and add to their investment in the risky asset.