**Principles of Investments**

- The expected returns and standard deviation of returns for two securities are as follows:

Security Z | Security Y | |

Expected Return | 15% | 35% |

Standard Deviation | 20% | 40% |

The correlation between the returns is +0.25.

- Calculate the expected return and standard deviation for the following portfolios:

- All in Z
- 75 in Z and .25 in Y
- 5 in Z and .5 in Y
- 25 in Z and .75 in Y
- All in Y

- Draw the mean-standard deviation
- Which portfolios might be held by an investor who likes high mean and low standard deviation? (In other words, which portfolios are on the efficient frontier?)

- Here are some characteristics of two securities:

Security 1: 𝐸(𝑅_{1}) = 0.1 𝜎^{2}(𝑅_{1}) = 0.0025

Security 2: 𝐸(𝑅_{2}) = 0.16 𝜎^{2}(𝑅_{2}) = 0.0064

Answer the following questions:

- Which security should an investor choose if she wants to (i) maximize expected returns, (ii) minimize risk (assume the investor cannot form a portfolio)?
- Suppose the correlation of returns on the two securities is +1.0, what is the optimal combination of securities 1 and 2 that should be held by the investor whose objective is to minimize risk (assume short sales are not allowed)?
- Suppose the correlation of returns is -1.0, what fraction of the investor’s net worth should be held in security 1 and in security 2 in order to produce a zero-risk portfolio?
- What is the expected return on the portfolio in (c)? How does this compare with the riskless return on Treasury Bills of 10%? Would the investor want to invest in Treasury Bills?

**Principles of Investments**

- Assume that the expected return on the market portfolio is 10%. If a stock with a beta of 2 has an expected return of 15% in this economy, what is the expected return on a stock with a beta of 5?
- Using historical data you estimate the expected return and beta for Stock A as 9.5% and 1.2, and for Stock B as 14% and 1.8. The expected return of the market portfolio is 9% and the risk-free rate is 5%. If CAPM holds, which stock should you invest in?
- Suppose that a single factor APT holds. The risk-free rate is 4% and the expected return on a portfolio with unit sensitivity (sensitivity equal to 1) to the factor is 7%. Consider a portfolio of two securities with the following characteristics:

Security | Sensitivity to the factor |
Proportion in the Portfolio |

A | 0.8 | 0.3 |

B | 1.7 | 0.7 |

**Principles of Investments**

According to APT, what is the portfolio’s equilibrium expected return?

- Consider an economy where two factors are sufficient to describe the returns on common stocks. Stated formally, we have:

𝐸(𝑟_{i}) = 𝑟_{ƒ} + 𝑏_{i,1}ƒ_{1} + 𝑏_{i,2}ƒ_{2}

The following table gives the estimated sensitivities of stocks A and B to the two factors, as well as their expected returns.

Stock | 𝑏i,1 | 𝑏i,2 | 𝐸(𝑟_{i}) |

A | 0.7 | 1.4 | 12.7% |

B | 2.1 | 0.7 | 17.6% |

The risk-free rate of return is 5%. Suppose A and B are priced consistently with the two-factor APT above.

- Calculate the risk premiums of the two factors (ƒ
_{1}and ƒ_{2}). - Suppose there is another stock, C with sensitivities 𝑏
_{𝐶,1}= 3 and 𝑏_{𝐶,2}= 0.8. Your analysis shows that holding c your expected return will be 10%. Is c is overvalued, undervalued or correctly priced based on APT?

**Principles of Investments**

- For this exercise you would need to use the spreadsheet xls posted on the Blackboard.

The file contains information about monthly returns of twelve stocks, AMGN, AMZN, STX, MSFT, BA, WMT, PFE, ABB, YHOO, AAPL, VZ, GOOG, as well as the value-weighted market index excess return (MKT-RF), and the risk-free rate (RF). The sample period begins on January 2004 and ends December 2012.

The spreadsheet is set in a way that you would only need to change the values in cells H122 and T123 in order to control the portfolio weights and the stock/portfolio chosen for the regression calculation.

- Calculate the Beta and 𝑅
^{2}of each stock by running a regression of the monthly returns

of the stock (excess of the risk-free rate) on the monthly returns of the value-weighted market index (excess of the risk-free rate ). The latter are denoted MKT-RF and they are located on Column P (notice the risk-free rate has already been subtracted from the index). You can choose different stocks by changing the values in cell T123. This would immediately provide you with the excess returns of each stock in Column T. These returns should be used for the regression.

**Principles of Investments**

- Graph a scattered plot of the Security Market Line (SML), i.e. each point on this plot is a pair of the average returns of a stock and its Beta. Calculate the risk premium of the market index as the slope of the regression line in this graph. Is the premium statistically significant? Are these results consistent with the CAPM?
- Calculate the amount of idiosyncratic risk as a fraction of total risk (1-𝑅
^{2}) for five portfolios. The first portfolio includes only AMGN. The second portfolio includes the first three stocks, i.e. AMGN, AMZN, and Similarly, portfolios 3 through 5 include the first six, nine, and twelve stocks, respectively.

These portfolios can be chosen by changing the value of cell H122 (change also the value of T123 to 14 to run the regressions for this part). Use APA referencing style.

- Plot the fraction of idiosyncratic risk as a function of the number of stocks in the What is your conclusion?