Estimating covariances and returns using the Single Index Model

We saw in class that the SIM can provide us with a shortcut to estimating covariances and also returns. In this assignment, we are going to use some of those techniques to do a little portfolio work.

Using a pair of risky securities, the goal is to find the optimal risky portfolio. Along the way, we will need to compute the covariance of these two securities and also their expected returns. We will also estimate the efficient frontier. Remember that for two stocks, we don't need to use an optimizer. We can compute the optimal portfolio directly.

The deliverable: To produce a graph showing the optimal risky portfolios and efficient frontiers using: i) Covariances based on historical data, ii) Covariances using SIM. These graphs should be on the same chart. Answer questions.

Steps:

1. Pick two stocks from the Dow 30. Any two - Cisco and 3M.

2. Get holding period return data from CRSP (see video on website for access via WRDS). You'll need three data items: Monthly returns for each stock and the return for the market. The preferred choice for the market return is the Value Weighted Index (with Dividends). Do not use "Return on the S&P 500 Composite Index" as this is price only and doesn't include dividends. Get the data from 2001 -2010. This should be 120 months.

3. Combine the stock data and risk free data:

4. Now compute some numbers.

i) Estimate the annualized standard deviation and variance for each stock and the market.

ii) Estimate the parameters of the single index model - specifically the beta and alpha for the two stocks. You should refer to the "IBM Single Index Model" spreadsheet to do this. There are a variety of ways to compute these variables. I find that the 4th method listed is the quickest.

iii) Compute the covariance between the stocks in two ways. First compute it as the covariance that we looked at in Ch7. You can use the =COVAR() function. Second, use the single index model: Cov(I, j) = β_iβ_jσ_m². Remember that the covariances that you compute need to be annualized. Do this by multiplying them by 12.

iii) Estimate the expected returns. Here we have some choices. We could just use the historical returns, but as we noted in class, these are likely to be less informative about future returns. So instead we can use the SIM to create returns.

Basic SIM

R_i - R_f = α + β(R_m - R_f)

Rearrange to get expected returns

R^{^}_i = α + β(R_m - R_f) + R_f

We need some inputs: α alpha - from SIM make sure that this is annualized.

Β beta - from SIM.

(R_m - R_f) market risk premium - we can use the historical one, but you'll find that from the data this is pretty low (under 3%). It is computed as the average of the excess returns on the market, i.e. (R_m - R_f)^-. An alternative might be to use an implied risk premium. We haven't talked about this much yet, but use 6%.

R_f Risk free - use the average risk free rate estimated from data. It should be around 2% or so.

Plug these numbers into the expected return formula. Remember to make sure things are annualized - you don't annualize beta though.

5. You should now have all the pieces we need to estimate the efficient frontier:

E(R) for two stocks

SD for two stocks

Covariance for the stocks, (you'll have two of these, one computed using COVAR, one computed using the SIM).

Risk Free Rate.

6. You can now put (or link) these numbers to the first page of the spreadsheet template. You'll be creating two graphs together - one showing the optimal risky portfolio using COVAR and one showing the optimal risky portfolio using the SIM covariance.

Questions and Analysis -

1. How close are the two estimates of the efficient frontier and the optimal risky portfolio using the two different approaches to covariance estimation? Does the SIM represent a reasonable alternative to the prior method for estimating covariances?

2. Are the expected returns that you computed reasonable? For example, what if one or both of your stocks has a big alpha? Did that skew the expected return of the stock?

You have several choices in terms of the return. To use the historical RP or an implied RP (6%), to use alpha, or assume that the alpha will go away in the future.

R^{^}_i = α + β(R_m - R_f)^- + R_f

R^{^}_i = α + β(R_m - R_f)^- + R_f

R^{^}_i = α + β(Implied RP) + R_f

R^{^}_i = β(Implied RP) + R_f

Which of these do you think is the most appropriate for your data? In other words, which seems most reasonable and why?

Attachment:- Assignment Files.rar