QUESTIONS
Questions 1 through 5 pertain to the following information:
Assume that a portfolio managers constructs an equally-weighted portfolio of two stocks from different industries and wants to analyze the risk of this portfolio using a two-factor model in the following form:
where FIND is the industry factor and FSIZE is the size factor return, β and l are the security loadings to these two factors respectively and 
is the idiosyncratic return.
Using statistical techniques, the portfolio manager finds that the first stock has an industry beta of β1 = 1.4 and the second stock has an industry beta of β2 = 0.8. This means that the first stock moves more than average when its industry moves in a certain direction and the opposite is true for the second stock. The loading to the size factor (l) is a function of the market value of the company and is standardized such that it has a mean of 0 and a standard deviation of 1. Using this formulation, the portfolio manager finds that the first stock has a size loading of l1= −1. This means that the market value of this stock is 1 standard deviation smaller than the market average. The second stock has a size loading of l2 = 2, which tells us that it belongs to a large-cap company.
The industry factor corresponding to the first stock has a monthly volatility of σIND1= 5% while the industry factor of the second stock has a volatility of σIND2= 10%. The volatility of the size factor is σSIZE= 1%. The correlation between the two industry factors is ρIND1,IND2 = 0.5, while the size factor has a correlation of ρIND1,SIZE = −0.3 with the first industry and ρIND2,SIZE = −0.5 with the second industry. Generally speaking, correlations between industry factors tend to be significant and positive in such a model as industry factors incorporate the market effect. On the other hand, the correlation between the size factor and a given industry factor tends to be negative as large-cap stocks tend to be less volatile than small caps. Finally, the monthly idiosyncratic volatility of the first stock is σ1,IDIO= 12% and the one of the second stock is σ2,IDIO= 5%.
- What is the systematic risk of each stock?
- What is the total risk of each stock?
- What is the isolated risk of the portfolio coming from each systematic factor?
- What are the systematic risk, idiosyncratic risk, and the total risk of the portfolio?
- What is the correlation between the total return of the two stocks?
- What is the contribution of each stock to total portfolio risk?
* The authors would like to thank Andy Sparks, Anuj Kumar, and Chris Sturhahn of Barclays Capital for their help and comments.
1 The Barclays Capital Global Risk Model is available through POINT®, Barclays Capital portfolio management tool. It is a multicurrency cross-asset model that covers many different asset classes across the fixed income and equity markets, including derivatives in these markets. At the heart of the model is a covariance matrix of risk factors. The model has more than 500 factors, many specific to a particular asset class. The asset class models are periodically reviewed. Structure is imposed to increase the robustness of the estimation of such large covariance matrix. The model is estimated from historical data. It is calibrated using extensive security-level historical data and is updated on a monthly basis.
2 As an example, if the portfolio has 10 stocks, we need to estimate 45 parameters, with 100 stocks we would need to estimate 4,950 parameters.
3 This is especially the case over crisis periods where stock characteristics can change dramatically over very short periods of time.
4 Fixed income managers typically use cross-sectional type of models.
5 GICS is the Global Industry Classification Standard by Standard & Poor's, a widely used classification scheme by equity portfolio managers.
6 An application of macro variables in the context of risk factor models is as follows. First, we get the sensitivities of the portfolio to the model's risk factors. Then we project the risk factors into the macro variables. We then combine the results from these two steps to get the indirect loadings of the portfolio to the macro factors. Therefore, instead of calculating the portfolio sensitivities to macro factors by aggregating individual stock macro sensitivities—that are always hard to estimate—we work with the portfolio's macro loadings, estimated indirectly from the portfolio's risk factor loadings as described above. This indirect approach may lead to statistically more robust relationships between portfolio returns and macro variables.
7 The equity risk model suite in POINT consists of six separate models across the globe: the United States, United Kingdom, Continental Europe, Japan, Asia (excluding Japan), and global emerging markets equity risk models (for details see Antonio B. Silva, Arne D. Staal, and Cenk Ural, “The U.S. Equity Risk Model,” Barclays Capital Publication, July 2009). It incorporates many unique features related to factor choice, industry and fundamental exposures, and risk prediction.
8 See Anuj Kumar, “The POINT Optimizer,” Barclays Capital Publication, June 2010. All optimization problems were run as of July 30, 2010.
9 The setting of these exposures and its trade-offs are discussed later in this chapter.
10 As POINT® U.S. equity risk model incorporates industry level factors, a unit exposure to a sector is implemented by restricting exposures to different industries within that sector to sum up to 1. Also, note that as before, the objective in the optimization problem is the minimization of idiosyncratic TEV to ensure that the resulting portfolio represents systematic—not idiosyncratic—effects.
11 Note that we can sum the sector betas into the portfolio beta, using portfolio sector weights (not net weights) as weights in the summation.
12 For a detailed methodology on how to perform this customized analysis, see Antonio Silva, “Risk Attribution with Custom-Defined Risk Factors,” Barclays Capital Publication, August 2009.
13 Specifically, we can back out factor realizations from the portfolio or index returns by using their risk factor loadings.
14 For reference, as of July 30, 2010, scenario 1 would imply the VIX to move from 23.5 to 35.3 and scenario 2 would imply that the credit spread for the Barclays Capital European Credit Index to change from 174 bps to 261 bps.
15 The same scenario results in a −8.12% move in the Barclays Capital Euro Credit Index.