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Posted: October 14th, 2022

The current prices of Stock A and Stock B are both $50. In a month, Stock A’s price may increase by $10 with probability 0.5 and decrease by $10 with probability 0.5. Meanwhile, Stock B’s price may increase by $5 with probability 0.5 and decrease by $5 with probability 0.5.

Suppose Stock A and Stock B are independent with each other:

(A) What are the expected return and volatility for Stock A next month? What are the expected return and volatility for Stock B next month?

(B) If you hold one share of Stock A and one share of Stock B, what are the expected return and volatility for your portfolio next month?

Now suppose Stock A and Stock B always move in the same direction:

(C) What are the expected return and volatility for Stock A next month? What are the expected return and volatility for Stock B next month?

(D) If you hold one share of Stock A and one share of Stock B, what are the expected return and volatility for your portfolio next month?

(E) If you hold one share of Stock A & short sell two shares of Stock B, what are the expected return and volatility for the portfolio next month?

Hint: It may save your time by thinking about the equivalence of assets and the scaling property of standard deviations.

12. (Each sub-question has 5 points) You attempt to study two populations that you know both are normal with identical variance (???? = ??). You know little about their population means, but you think that their means are different and attempt to test this conjecture. Based on only 18 observations for each

population, you have computed that the sample means are ??�
??� = ??.??, and the sample variances are ???? = ??.???? and ???? = ??.????.

(A) If the two populations are independent, formulate your hypothesis test, identify the appropriate test statistic and its distribution, and decide whether you can reject the null hypothesis at the level of significance 0.05.

Suppose now you find that the two populations are dependent:

(B) If the correlation coefficient between these two populations is known and equal to 0.5, can you reject the null hypothesis at the significance level 0.05? Show you calculations along with your reasoning.

(C) If the correlation coefficient between these two populations is known and equal to −0.5, can you reject the null hypothesis at the significance level 0.05? Show you calculations along with your reasoning.

Suppose now you are permitted to choose the sample size n which is the same

for both populations. Suppose that, regardless of the sample size you choose,

you always obtain the same sample means ??� = ??. ?? and ??� = ??. ?? and the ????

same sample variances ?????? = ??. ???? and ?????? = ??. ????.

(D) Is the sample size n = 18 large enough for you to reject the null hypothesis in Part B? If not, how large do you think the sample size need to be? Is the sample size n = 18 big enough for you to reject the null hypothesis in Part C? If not, how large does the sample size need to be? Explain briefly why your results are different.

Hint:itishelpfultothinkaboutwhatthedistributionof??� isinthisproblem. ??

??????

??

= ??. ?? and

13.

Gary Hansen is a securities analyst for a mutual fund specializing in small-capitalization growth stocks. The fund regularly invests in initial public offerings (IPOs). If the fund subscribes to an offer, it is allocated shares at the offer price. Hansen notes that IPOs are frequently underpriced, and the price rises when open market trading begins. The initial return for an IPO is calculated as the change in price on the first day of trading divided by the offer price. Hansen is developing a regression model to predict the initial return for IPOs. Based on past research, he selects the following independent variables to predict IPO initial returns:

Underwriter rank = 1–10, where 10 is highest rank
Pre-offer price adjustment = (Offer price – Initial filing price)/Initial filing price Offer size ($ millions) = Shares sold × Offer price

Fraction retained a = Fraction of total company shares retained by insiders a Expressed as a decimal.

Hansen collects a sample of 1,725 recent IPOs for his regression model. Regression results appear in Exhibit 1, and ANOVA results report in Exhibit 2. Hansen wants to use the regression results to predict the initial return for an upcoming IPO which has the following characteristics:

Underwriter rank = 6 Offer size = $40 (million) Pre-offer price adjustment = 0.04 Fraction retained = 0.70

EXHIBIT 1: Hansen’s Regression Results Dependent Variable: IPO Initial Return (Expressed in Decimal Form, i.e., 1% = 0.01)

Variable

Intercept

Underwriter rank Pre-offer price adjustment

Offer size Fraction retained

Coefficient(bj)StandardError t-Statistic

0.0477 0.0019 0.0150 0.0049 0.4350 0.0202 −0.0009 0.0011 0.0500 0.0260

25.11 3.06 21.53 −0.82 1.92

Question 11, 12, 13

EXHIBIT 2: Selected ANOVA Results for Hansen’s Regression

Degrees of Freedom (df) Sum of Squares (SS)

Explained 4 51.433

Residual 1,720 91.436

Total 1,724 142.869

Multiple R-squared = 0.49

Because he notes that the pre-offer price adjustment appears to have an important effect on initial return, Hansen wants to construct a 95 percent confidence interval for the coefficient on this variable. He also believes that for each 1% increase in pre-offer price adjustment, the initial return will increase by less than 0.5%, holding other variables constant. Hansen wishes to test this hypothesis at the 0.05 level of significance.

Selected values for the t-distribution and F-distribution are reported in Exhibits 3 and 4, respectively. “df” is the number of degrees of freedom.

EXHIBIT 3: Selected Values for the t-Distribution (df = ∞)

AreainRightTail t-value

0.050 1.645

0.025 1.960

0.010 2.326

0.005 2.576

EXHIBIT 4: Selected Values for the F-Distribution (α = 0.01) (df1/df2: Numerator/Denominator Degrees of Freedom)

df1 4∞

df2 4 16.00 13.50 ∞ 3.32 1.00

Given the above information, answer the following five questions:

(A) Based on Hansen’s linear regression model, what is the predicted initial return for the upcoming IPO? Suppose the Underwriter rank increases from 6 to 10 and the Fraction Retained decreases from 0.70 to 0.30, what would be the predicted initial return for this upcoming IPO?
(B) Given his regression results, what is the 95% confidence interval about the regression coefficient for the Pre-Offer Price? With you economic thinking, is it reasonable for the regression coefficient in front of the Offer Size to be negative? Is it reasonable for the regression coefficient in front of the Fraction Retained to be positive? Explain briefly.
(C) What should be the appropriate hypothesis test formulation regarding Hansen’s belief about the magnitude of the initial return relative to that of the Pre-Offer Price Adjustment (as reflected by the coefficient b2)? What is your conclusion about b2 at the level of significance 0.05?
(D) Given the reported ANOVA result, how much fraction of the variation in the dependent variable cannot be explained by Hansen’s regression model? What is the correlation coefficient between the predicted and actual values of the dependent variable in Hansen’s regression?
(E) Howwouldyoucarryoutatesttodeterminewhetherthelinearregression model designed by Hansen is overall significant or not? Formulate your hypothesis and specify the appropriate test statistic. Based on your method and calculation, what is your conclusion about the overall significance at the level of significance 0.01.

Hint: overall significance suggests a joint test of all regression coefficients.

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