Level II Quant item sets

Leonor Castro is head of fixed-income research at Tavira Asset Management, a Lisbon-based manager of euro-denominated corporate bond funds. Castro wants a simple model that links monthly changes in credit spreads to changes in market conditions, so that the portfolio managers can see how much spread widening to expect when equity markets are under stress. She asks Rui Matos, a junior quantitative analyst, to estimate the model and summarise the results.

Matos collects 60 months of data. The dependent variable is the monthly change, in basis points, in the option-adjusted spread of an index of investment-grade euro corporate bonds. He uses three independent variables: the monthly change in the VIX volatility index, measured in index points; the monthly total return on a broad European equity index, measured in percent; and the monthly change in the ten-year German government bond yield, measured in basis points. He estimates the model by ordinary least squares. The ANOVA results are shown in Exhibit 1, and the estimated coefficients are shown in Exhibit 2.

Matos reviews the residual plots before he writes anything up. The residuals look roughly symmetric and show no obvious pattern against the fitted values or against time, and the correlations between the independent variables are all below 0.4 in absolute value. He decides that the standard assumptions of the multiple regression model are reasonable for this sample, and that no adjustments to the standard errors are needed.

In his first draft, Matos reports the coefficient of determination as the main measure of fit. Castro asks him to add the adjusted measure as well. She explains that the portfolio managers are considering adding further market variables to the model, and she wants a measure of fit that does not automatically rise every time a variable is added, whether or not that variable helps to explain spread changes.

Castro also wants to know whether the model as a whole has any explanatory power. She asks Matos to test the hypothesis that all three slope coefficients are equal to zero at the 5% significance level. Matos notes that the critical value of the F-distribution with 3 and 56 degrees of freedom at the 5% level is 2.77. For tests on individual coefficients, he uses a two-tailed test at the 5% significance level, for which the critical t-value with 56 degrees of freedom is 2.003.

To show the portfolio managers how the model could be used, Castro asks for a forecast under a stress scenario agreed with the risk team. In the scenario, the VIX rises by 4 index points in a month, the equity index returns −3%, and the ten-year German yield falls by 20 basis points as investors move into government bonds. Matos is asked to give the predicted change in the corporate spread for that month.

Finally, one portfolio manager tells Castro that the equity market and the government bond market both matter for spreads in their own right, and that the VIX adds little once those two variables are known. Castro asks Matos to check which of the three slope coefficients are statistically different from zero, so that she can respond with evidence rather than opinion. She also reminds him that a coefficient that is not significant in this sample does not prove that the variable has no effect; it means the data do not allow the effect to be distinguished from zero at the chosen level.

Matos plans to re-estimate the model every quarter as new data arrive, and to keep a record of how the coefficients change over time, so that the team can see whether the relationship between spreads and market conditions is stable.