# MSc in Corporate Finance

## QUESTION 1                                                                                                                                                   [35 marks]

1. a) A researcher has formulated the simple linear regression model:

???? = ??1 + ??2????2 + ??3????3 + ????, ?? = 1, … , ??

to explain the cross-sectional variation in the abnormal performance of N=500 funds (???? is the Jensen’s alpha), using as drivers two characteristics of the fund manager: ????2 is the years of experience of the fund manager, ????3 is a dummy variable equal to 1 if the fund manager has a post-graduate degree from an top international university, 0 otherwise. The researcher deploys a heteroskedasiticy test and finds that the errors are not homoskedastic. Discuss:

1. i) What is heteroscedasticity?

## [3 marks]

1. ii) Discuss a test that the researcher may have used to detect this problem. Indicate clearly the null and alternative hypotheses and the probability distribution of the test statistic.

## [3 marks]

• In the presence of this heteroscedasticity problem, should she estimate the model parameters and corresponding standard errors using OLS? or should (s)he instead consider another estimation method? Justify your answers.

[4 marks]

1. Suppose that instead the researcher prefers to re-specify the model in order to eliminate the heteroscedasticity problem. Discuss potential model re-specifications that may result in homoscedastic errors.

## [4 marks]

1. In the context of the simple regression model ???? = ??1 + ??2???? + ????, ?? = 1, … , ?? explain the Ordinary Least Squares (OLS) estimation method and the Maximum Likelihood (ML) estimation method as regards: How are the corresponding OLS and ML estimators derived? In which circumstances is it appropriate to employ OLS instead of ML, and vice versa? (briefly explain your answers).

[4 marks]

1. Explain the main differences between an ordinary linear regression model and a logit regression model. Hint: Focusing on a single-regressor case, discuss for each model the interpretation of the fitted values, the estimation method, and the marginal effect.

## [9 marks]

1. Explain the concept of AutoCorrelation Function (ACF) in time-series regression analysis? Draw an example of ACF in a graph to explain what information it provides (label carefully the X axis and the Y axis).
• marks]

Cont.

1. Consider the       following             time-series             regression        based   on             daily    data

???? = ??1 + ??2???? + ????, ?? = 1, … , ??

where ???? are the returns of stock PIVECO plc and  ???? are the returns of the market portfolio. The researcher wants to test for the presence of autocorrelation up to 2 weeks. Discuss two different tests for this purpose. Outline in each case the null and alternative hypothesis of the test, the test statistic, and the corresponding probability distribution.

• marks]

## QUESTION 2                                                                                                                                                    [30 marks]

A researcher is interested in explaining the variation in abnormal performance across funds. For this purpose (s)he estimates the following regression model

???? = ??1 + ??2????(??????????????????) + ??3????(????????????????) + ??4????(????????) + ??5?????????? + ????

using data on ?? = 1, … , ?? funds (?? = 400) where ???? is the fund performance measure which is defined as the annualized CAPM one-factor alpha, ?????????????????? is total net assets under management in millions of dollars, ???????????????? is the number of years since the fund inception date, ???????? is the age of the fund manager,  and ?????????? is a gender dummy equal to 1 if the fund manager is a male, 0 otherwise. The researcher estimates a multiple linear regression model using OLS and obtains the estimation output shown in EXHIBIT 1.

EXHIBIT 1

 Dependent Variable: FUND PERFORMANCE Method: Least Squares; Sample: 1  400 observations Variable Coefficient Std. Error t-Statistic Prob. C 0.275104 0.031198 LN(FUNDSIZE) -0.032992 0.002924 LN(FUNDAGE) -0.045317 0.027201 LN(AGE) -0.169905 0.044995 MALE 0.104161 0.055652 R-squared Mean dependent var Adjusted R-squared S.D. dependent var 0.051137 S.E. of regression Akaike info criterion Sum squared resid 0.921937 Schwarz criterion Log likelihood 11.31757 F-statistic 13.52493 Prob(F-statistic) 0.000000

Cont.

1. a) Do funds managed by younger managers tend to perform better? Conduct a (one-sided) test at the conservative 1% significance level to answer this question. Begin by formalizing clearly the null and alternative hypothesis of the test, indicate the name of the test statistic, and its probability distribution. Draw a graph to discuss the outcome of the test, and visualize the following elements in the graph: test statistic value, critical value and p-value.

## (10 marks)

1. b) Which percentage of the variation in the abnormal performance across funds is this model able to explain? Indicate the name of the statistic that you are adopting to measure this explained variation.

## (10 marks)

1. c) Conduct a statistical test to evaluate whether the regression is significant overall. Indicate the name of the test statistic that you are using for this purpose and the probability distribution that it follows. Write down clearly the null and alternative hypotheses of the test in two different ways: i) with reference to the model’s coefficients, and ii) with reference to the percentage of the variation in the abnormal performance across funds that the model is able to explain. Explain verbally each of those hypotheses. Draw a graph to explain the test outcome; label clearly in the graph the sample value of the test statistic, the 5% critical value of the test and the p-value. What shall the researcher we conclude?

(10 marks)

## QUESTION 3                                                                                                                                                                                                                  [35 marks]

Consider the following linear regression model to investigate the determinants of CEOs compensations across firms within a specific industry:

?????????????????? = ??1 + ??2????(??????????) + ??3?????????????? + ??4???????? + ??

Where ?????????????????? measured in thousands of Euros, ?????????? is the stock market value of the company in millions of Euros, ?????????????? is the amount of profits in millions of Euros realized by each company, and male is a gender dummy equal to 1 if the CEO is a male, and 0 otherwise. Using a random sample of N=25 CEOs, we estimate equation (1) by OLS and obtain the following results:

Cont.

EXHIBIT 2

 Dependent Variable: CEOsalary Method: Least Squares; Sample: 1  25 observations Variable Coefficient Std. Error t-Statistic Prob. C 1.138569 0.031198 Ln(STOCK) 0.145317 0.027201 PROFITS 0.089905 0.044995 MALE 0.094161 0.055652 R-squared 0.178947 Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 0.103892 Schwarz criterion Log likelihood F-statistic Prob(F-statistic) 11.31757

1. Interpret the estimate of the intercept ??0. In which units is this measure expressed?
• marks]

1. According to the model estimates, what is the differential salary between male CEOs and female CEOs on average? Can you argue that male CEOs tend to earn more than female CEOs any other thing equal? Conduct a test to answer this question, indicate clearly the name of the test statistic, the null and alternative hypotheses, and the probability distribution of the test statistic.  Discuss the outcome of the test at the 1%, 5% and 10% significance levels.
• marks]

1. If the market value of the firm’s stock increases by 10 million Euros, how does a CEO salary changes on average ceteris paribus? [5 marks]

1. Using the information from the table compute the adjusted-??2 and interpret the meaning of this measure in relation to the ??2. Compute the AIC and discuss what can this measure be used for.
• marks]

Cont.

1. Suppose that years of experience, denoted exper, is a significant driver of CEO compensation and that adding this variable to the existing model results in an OLS estimate for the coefficient of exper equal to 0.21386 with standard error 0.10945. The estimated correlation between this variable and the first two dependent variables in the model (with significance p-value in square brackets) is corr(ln(stock), exper)=0.23[0.124), corr(profits, exper)=0.46[0.002], and the variance of each of those regressors is var(ln(stock))=0.8945, var(profits)=0.2394, and var(exper)=0.1345. Quantify the expected effect (if any) of omitting this variable in the coefficient estimates for the ln(stock) and profits

[10 marks]

1. Re-formulate the regression model in order to accommodate two effects: (i) the effect of Exper on CEOsalary, and (ii) a differential marginal effect of on CEOsalary for males and females. Write down the new model, and the null/alternative hypotheses associated to the two statistical tests that you would conduct to assess (i) and (ii), respectively, in the context of this model. Accordingly, what is the expected marginal effect of experience on CEO salary for malels? and for females?

[5 marks]

FORMULAE SHEET

## Jarque-Bera test statistic=(?? − ??)[??62 + (??−324)2] for regression residuals

AIC=???? ???????? + 2 ????

SBC=???? ???????? + ???????? ??

Z denotes the total number of observations used in the (cross-section or time-series) regression estimation.

## Functional forms                         Elasticity

??

?? = ??1 + ??2??                            ??2 ??

?? = ??1 + ??2????(??)                                     ??2 ??1

## Omitted-variable bias ( y = β1 +β2x2 +β3x3 +ε  versus  y= +β β ε1              2 2x + )

E(βˆ2) = β2 + β3 cov(var(x2x,2x)3 )

## Natural logarithm versus exponential transformation

A=log(B) is equivalent to eA=B where log(B) denotes the natural logarithm of B