Obligor R 2 in CreditMetrics
Jerry Yi Xiao
November 12, 2002 Summary: This note discusses the study on R 2 of obligors based on equity and asset information. Keywords: CreditManager, R 2 , total asset
The CreditMetrics methodology adopts an asset value model to capture the correlation of credit quality change betweentwo obligors. This will require an estimate of asset value correlation between any two obligors. To get around the logistical impossibility of providing a correlation matrix for all obligors which a user might request, CreditMetrics resort to estimating the correlations between obligors and a set of market indices. This is equivalent to decomposing an obligor’s volatility into market volatility andidiosyncratic volatility. Afterwards, the obligor-by-obligor correlations can be built from this decomposition and correlations between indices. The decomposition used to be expressed in terms of “Obligor Speciﬁc Volatility" (OSV). This was the parameter used in CreditManager before Version 3.0. Since then, later versions have switched to using R 2 as the parameter, which is more intuitive forcommunication. Nothing has changed in the model, and in fact there is a very simple relationship between R 2 and OSV; R 2 + O SV 2 = 1. (1)
The interpretation of R 2 in CreditManager is just the goodness-of-ﬁt between the obligor return and the index returns.
This study of obligor R 2 is a follow-up of studies done in 1997 and 2000 on the obligor-speciﬁc volatility (OSV), in which a rule ofthumb relating obligor-speciﬁc volatility to company size was proposed and derived.1 This rule can serve as a guide when assigning R 2 values for obligors when no other method is available. The goal of this study is to reexamine this rule in terms of R 2 , and move a step further to check if the same parameters of the rule remain unchanged across time or industries.
The data used is the741 NYSE-traded stocks contained in the CreditManager Obligor Database. As in the previous studies, the stocks are from a full range of industries and total asset values. The data used in this study contain daily closing prices from January 1, 1999 to August 29, 2002. The set of indices used is the same set of MSCI indices used in the Obligor Database, which includes the MSCI USA country indexand the following 23 sector indices:
Automobiles & Components Commercial Services & Supplies Energy Health care Equip & Services Insurance Pharmaceuticals & Biotech Software & Services Transportation Banks Consumer Durables & Apparel Food & Drug Retailing Hotels Restaurants & Leisure Materials Real Estate Technology Hardware & Equip Utilities Capital Goods Diversiﬁed Financials Food Beverage &Tobacco Household & Personal Products Media Retailing Telecommunication Services
1.2 Calculation of R 2
The daily returns of each stock are normalized to mean zero and standard deviation one, and then ﬁt to the similarly normalized returns of the MSCI index set. We use the same assumptions for the ﬁtting as before: the relationship is linear, and the coefﬁcients for each index were estimated tominimize squared errors subject to the constraint that each coefﬁcient be nonnegative.
1 See Finger, C., Obligor-Speciﬁc Volatility in CreditMetrics, RiskMetrics Product Technical Note, and the summary
2 Asset rule
In this study, we focus on R 2 instead of the obligor-speciﬁc volatility. As an indicator of goodness of ﬁt, R 2 can also be viewed as the ratio of varianceexplained by a regression over variance of the dependent variable. To establish a relationship between asset size and R 2 , we chose to ﬁt the data to the same logistic function as used in previous studies. The logistic function is a common nonlinear regression used when the dependent variable varies between zero and one. In our case, we end up estimating parameter γ and λ of the following logistic...