You are in charge of the equity portfolio for the investment company Bronzeman Sachs Ltd and should develop a new investment strategy that provides a good risk-return tradeoff. In particular, you should try to find a portfolio that will outperform the market index.

The growing interest on smart beta funds has forced your firm to offer a smart beta product. This product will be formed as the combination of a particular investment style with the market portfolio. You can find your assigned investment style in a separate pdf named “Students_351b_Investment_Style_20_21.pdf”. On the other hand, you can find the returns of the market portfolio and your assigned investment style in the excel file named “Styles_data_20_21.xlsx”. Your objective is to combine the market with your assigned investment style such that the combined returns deliver a better risk-return tradeoff than the market.

In your report, respond, in order, to each of the following tasks:

1. Plot the cumulative returns of the market portfolio and your assigned investment style. Discuss the results from this graphical representation.

2. Compute and report the Sharpe ratio of your assigned investment style using the entire sample. Discuss this result relative to the Sharpe ratio of the market portfolio.

3. Using the entire sample, compute and report the VaR95 of your assigned investment style and discuss this result relative to that of the market portfolio.

4. a. Using the first 325 observations, construct an optimal mean-variance strategy with a risk aversion coefficient of 5 that combines the market portfolio and your assigned investment style. Then, evaluate this strategy in the following 325 observations. This is, with the optimal parameters that give the optimal combination in the first half of the sample, compute the returns in the second half of the sample of that same combination between the market returns and the investment style. Then, compute and discuss the Sharpe ratio obtained from this strategy, and those of your assigned investment style and the market portfolio in isolation.

b. Using the entire sample, construct the optimal mean-variance combination with a risk aversion coefficient of 5 between the market portfolio and your assigned investment style. Then, compute the Sharpe ratio of the resulting strategy (e.g. the combination between the market returns and your assigned investment style) and discuss this result relative to the Sharpe ratio of the market portfolio and your assigned investment style in isolation.

c. Briefly discuss the difference between the results in part a) and part b).

REMARK: When computing the optimal portfolio, you should make sure that the portfolio weights across all assets add up to one. You can read in the Appendix how an investment style is constructed.

5. Using the entire sample, regress the returns from the strategy that optimally combines the market portfolio and your assigned investment style on the three factors of Fama and French. These factors are: the market portfolio (Mkt-rf), the small minus big factor (SMB), and the high minus low factor (HML). Report the slope coefficients and the tstats and discuss your results. See the appendix for an explanation of the Fama and French factors.

6. Market impact costs: when putting X dollars in your style, suppose that you move prices against you (i.e. buying moves prices up and selling moves prices down). Imagine that prices move against you according with this formula:

where TCp represent the transaction costs from assigning X units to your style, and λ1 and λ2 are the transaction cost coefficients (i.e. the higher λ1 and λ are, the more expensive it is to trade your investment style). Now, suppose that these coefficients take a value of 0.007 and 0.006, respectively. Construct and optimal mean-variance strategy that takes into account market impact costs using the first half of the sample with a risk aversion coefficient of 5. Then evaluate the resulting optimal combination between the market and your investment style in the second half of the sample. Explain how your results change in comparison with your analysis in section/part 4.

7. Using the entire sample, look at the 20 worst market episodes (i.e. 20 periods with the lowest returns) and discuss how your assigned investment style helps you to cope with the poor market performance.

8. After the implementation: Imagine a situation where you have implemented your strategy for 6 months and you have experienced a loss of 50%. Give reasonable arguments for this bad performance that would convince your boss of not firing you. Be brief in your explanation and to the point.

• REMARK: Each time that you use solver, write down the mathematical problem that you are solving with solver. For instance, if you were to use solver to construct a VaR portfolio, you would need to write down in your report the mathematical problem that gives you the optimal VaR portfolio.

Appendix:

What is an investment style?

An investment style can be seen as a factor. This is a source of risk exposure in financial markets. When we talk about the market value weighted factor, we think of the returns on the market portfolio, which is usually represented by an index. The co-movement of a stock with the factor defines the risk profile of the asset. There are many different factors such as the value or the size factors. These factors are normally constructed with firm attributes that are related with the factor. For instance, the value factor is constructed by sorting stocks on the basis of the ratio book value to market capitalization (i.e. book value/market value). Once we sort the entire universe of stocks on the basis of this firm characteristic, we go long in a portfolio of companies in the top decile of book to market, and go short in a portfolio of companies in the bottom decile of book to market. The size of these portfolios is the same, and therefore the factor is constructed as the returns of a zero cost portfolio. The same idea applies for the size factor or other factors constructed on the basis of other specific firm characteristics.

Constructing an investment style

Consider at time t a given firm specific characteristic (e.g. value, size, momentum). In addition, consider a universe of N stocks. Therefore, you have at time t N different stocks. You sort those stocks using the specific firm characteristic that you are interested in. Once the sorting is done, you construct an equally (or value) weighted portfolio with the top N/10 companies, and another equally (or value) weighted portfolio with the bottom N/10 companies. Then you look at the spread of expected returns offered by these two portfolios, and if this spread is large enough, you go long in the portfolio with the largest expected return, and short in the portfolio with the smallest expected return. The return of this long-short (zero cost) portfolio corresponds with the factor.

What are the three Fama and French factors?

Eugene Fama and Kenneth French are the godfathers of style investing. Their three factor model is an asset pricing model that explains both the time series and the cross-section of stock returns. The first factor is the market portfolio. The second factor is named small-minusbig (SMB). This is an investment style that buys one dollar of an equally weighted portfolio of small companies, and shorts one dollar of an equally weighted portfolio of big companies. The third factor is named high-minus-low (HML). This is an investment style that buys one dollar of an equally weighted portfolio of companies with high book value relative to their market value, and shorts one dollar of an equally weighted portfolio of companies with low book value relative to their market value.