Single-factor and multi-factor models

Homework

 

Single-factor and multi-factor models

 

Unless stated otherwise, round your answers to two decimal points, and do not round intermediate calculations.

 

Problem 1. [20 points]

The following are estimates for two stocks.

 

Stock	Expected Return			Beta	Firm-Specific Standard Deviation
A		10	%	0.95		35	%
B		17		1.50		45

 

The market index has a standard deviation of 19% and the risk-free rate is 12%.

 

1. What are the standard deviations of stocks A and B?
2. Suppose we build a portfolio with the following proportion: 0.35 in stock A, 0.35 in stock B, and 0.3 in risk-free T-bills. Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio

 

 

Problem 2. [20 points]

 

Suppose that the index model for stocks A and B is estimated from excess returns with the following results:
 

and σ_M = 29%; R-squared_A = 0.29; R-squared_B = 0.14

 

Assume you create portfolio P with investment proportions of 0.60 in A and 0.40 in B.

 

1. What is the standard deviation of the portfolio? [Hint: R-squared is the variance explained by the market risk divided by the variance in the stock .]
2. What is the beta of the portfolio?
3. What is the firm-specific variance of the portfolio?(Round to 3 decimals.)
4. What is the covariance between the portfolio and the market index? (Round to 3 decimals.)

 

 

Problem 3. [10 points]

 

Consider a security of which we expect to pay a constant dividend of $18.49 in perpetuity. Furthermore, its expected rate of return is 20.1%. Using the equation for present value of a perpetuity, we know that the price of the security ought to be , where D is the constant dividend and k is the expected rate of return.
Assume that the risk-free rate is 3%, and the market risk premium is 6.4%.

What will happen to the market price of the security if its correlation with the market portfolio doubles, while all other variables, including the dividend, remain unchanged?

 

 

Problem 4: Fama-French Three-Factor Model. [50 points]

 

Download the Excel spreadsheet Portfolio_Returns.xlsx from Blackboard. The file contains the monthly excess returns in percent of a wide range of stocks and of QQQ (Nasdaq ETF), IWM (Russell ETF), and MCHI (China ETF).

Since some companies have not been publicly listed for the entire time period from January 2014 to December 2018, we cannot use them for our analysis. Please research the QQQ, IWM, and MCHI ETFs to see if you can use them as a replacement for missing stocks. For example, Alibaba (BABA) has not been listed for the entire time period, however, MCHI can serve as a decent albeit not perfect replacement in your analysis.

 

Additionally, the file contains the time series ExcMkt, SMB,and HML, representing the excess market return, the small-minus-big return, and the high-minus-low return.

 

For the following part, consider only your three largest holdings by market value.

 

1. Calculate using the single-factor model.
2. Estimate their risk (systematic and firm-specific) according to the single-factor model.

 

For the following part, consider your entire portfolio.

 

1. Calculate the weights of the stocks in your portfolio. Remember, stocks you’re selling short have negative weights, and the weights need to sum up to one.
2. Produce a time series for your portfolio of n stocks return using .
3. Estimate your portfolio’s sensitivity to Fama-French’s three factors, that is, run the regression

Remember:  is a proxy to how much market risk your portfolio carries;  is your exposure to small over big firms; is your exposure to firms with a high book value compared to their market capitalization. These three betas are assumed to be non-diversifiable risk factors, and in the long run investors should be rewarded for their risk of having greater betas.

 

1. Analyze your results. This includes, but is not limited to: classifying your portfolio in terms of aggressiveness, small vs large cap, value vs growth; identifying which of your holdings potentially contribute to the portfolio classification how; going back to your portfolio philosophy statement and discussing if your holdings reflect your investment goal; using the expected values of to compute the expected return of your portfolio in the coming month.