1. Quantitative Analysis for Business Lecture 7 August 23th, 2010

2. Exercise I With many US companies operating globally, the effect of the US dollar’s strength on a US company’s returns has become an important investment issue. You would like to determine whether changes in the US dollar’s value and overall US equity market returns affect an asset’s returns. You decide to use the S&P 500 Index to represent the US equity market Question I Write a multiple regression equation to test whether changes in the value of the dollar and equity market returns affect an asset’s returns. Use the notations below; Rit = return on the asset in period t RMt = return on the S&P 500 in period t ∆Xt = change in period t in the log of a trade-weighted index of the foreign exchange value of the US dollar against the currencies of a broad group of major US trading partners

3. Exercise I Question II You estimate the regression for Archer Daniels Midland Company. You regress its monthly returns for the period January 1990 to December 2002 on S&P 500 Index returns and changes in the log of the trade-weighted exchange value of the US dollar. The table below shows the coefficient estimates and their standard errors Determine whether S&P 500 returns affect ADM’s return. Then determine whether changes in the value of the US dollar affect ADM’s returns. Use a 0.05 significant level to make decision (t-critical is between 1.98 and 1.97)

4. Exercise i Question III Based on the estimated coefficient on RMt, is it correct to say that “For a 1 percent point increase in the return on the S&P 500 in period t, we expect 0.5373 percentage point increase in the return on ADM?”

5. Solution Exercise i Question I Rit = b0 + b1RMt + b2 ∆Xt + eit

6. Solution exercise i Question II – part I (S&P 500 returns) We have to test if the coefficient on the S&P 500 Index returns is statistically significant. Null Hypothesis H0: b1 = 0 Ha: b1 ≠ 0 Using t-test to test the hypothesis Where b1 = regression estimate of b1 b1 = the hypothesized value of the coefficient (0 in this case) Sb1 = the estimated standard error of b1 ^ ^ ^

7. Solution exercise i Question II (cont.) n = 156 3 regression coefficients t-test has 156 – 3 = 153 degrees of freedom 0.05 significance level (t-critical is between 1.98 and 1.97) |t-stat| = 4.0338 > 1.98 Reject the Null Hypothesis that b1 = 0

8. Solution exercise i Question II – part II (US dollar) Use t-test to test if H0: b2 = 0 Ha: b2 ≠ 0 |t-stat| is between 1.97 and 1.98 (t-critical) We cannot reject H0: b2 = 0 Based on the above t-tests, we conclude that S&P 500 Index returns do affect ADM’s returns but that changes in the value of the US dollar do not affect ADM’s return

9. Solution exercise i Question III The statement is not correct. The make it correct, we need to add the qualification “holding ∆Xt constant” to the end of the quoted statement.

10. Exercise ii One of the most important questions in financial economics is what factors determine the cross-sectional variation in an asset’s returns. Some have argued that book-to-market ratio and size (market value of equity) play an important role. Question I Write a multiple regression equation to test whether book-to-market ratio and size explain the cross-section of asset returns. Use the notation below (B/M)i = book-to-market ratio for asset i Ri = return on asset i in a particular month Sizei = natural log of the market value of equity for asset i

11. Exercise ii Question II The table below shows the results of the linear regression for a cross-section of 66 companies. The size and book-to-market data for each company are for December 2001. the return data for each company are for January 2002 Determine whether the book-to-market ratio and size are each useful for explaining the cross-section of asset returns. Use a 0.05 significance level to make your decision (t-critical is 2.0)

12. Solution exercise ii Question I Ri = b0 + b1 (B/M)i + b2Sizei + ei

13. Solution exercise ii Question II We have to test the coefficients on the book-to-market ratio and size are individually statistically significant using t-test Part I Test book-to-market ratio (B/M) H0: b1 = 0 Ha: b1 ≠ 0

14. Solution exercise ii Question II (cont.) Where b1 = regression estimate of b1 b1 = the hypothesized value of the coefficient (0 in this case) Sb1 = the estimated standard error of b1 |t-stat| = 0.9201 < 2 (t-critical) Cannot reject H0: b1 = 0 Book-to-market ratio is not useful ^ ^ ^

15. Solution exercise ii Question II (cont.) Part II Use t-test to test if size explains the cross-sectional variation in asset returns H0: b2 = 0 Ha: b2 ≠ 0 |t-stat| = 0.4686 < 2 (t-critical) Cannot reject H0: b2 = 0 Asset size is not useful in explaining the cross-sectional variation of asset returns

16. Exercise iii Researchers hypothesized that the percentage decline in Toronto Stock Exchange (TSE) spreads of cross-listed stocks was related to company size, the predecimalization ratio of spreads on NASDAQ to those on the TSE and the percentage decline in NASDAQ spreads. The following table gives the regression coefficient estimates from estimating relationship for a sample size of 74 companies, measured by company’s assets. The average company’s size in the sample is about C$900 million and a predecimalization ratio of spreads equal to 1.3. Based on the above model, what is the predicted decline in spread on the TSE for a company with these average characteristics, given a 1 percent decline in NASDAQ spreads?

17. Solution exercise iii The regression model Percentage decline in TSE = -0.45 +0.05Size – 0.06(Decimalization ratio) + 0.29(Decline in NASDAQ spreads) The prediction is = -0.45 + 0.05(900) – 0.06(1.3) + 0.29(1) = 0.45 The model predicts that for a company with given characteristics, the spread on the TSE decline by 0.45% for a 1% decline in NASDAQ spread.