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Indiv. HW 1, FIN 3115/3711 AY 2020/2021 Semester 2 1 Individual Assignment #1 - Statistics and Calculus Review Ben Charoenwong International Financial Management AY 2020/2021 Semester 2 Instructions Please complete this homework assignment on you own. All homework assignments are open- book and open-internet. However, note that not all material on the internet is correct. Each student should turn in their own assignment. Only assignments with the honor code signed will be graded for credit. Complete answers in the space provided. Hand in assignments based on the class you are registered for. You can turn in assignments early but no late homework will be accepted. This is because in class I may go over some problems on the homework. This homework is a refresher for basic statistics and calculus that we will use in the course. There should be no new material for you in this homework. The ease of this homework is not representative of either future individual or group assignments. For derivations, show work for partial credit. You do not need to turn in a cover page for individual assignments. Just print out this completed assignment. Name:____________________________________________ Section #: _________________________________________ Student ID #:_______________________________________ Date:______________________________________________ I, _________________________________________, pledge by my honor that I worked on this assignment alone and have not misrepresented any work belonging to others. Indiv. HW 1, FIN 3115/3711 AY 2020/2021 Semester 2 2 Question 1. Consider a random variable ?~?(?, ?2). Part A. For each moment, explain the usefulness and meaning in words. What is its: Mean = ________ Usefulness: Standard deviation = ________ Usefulness: Skewness = ________ Usefulness: Kurtosis = ________ Usefulness: Part B. Suppose we observe some data {?1, … , ??} that we believe are realizations of ?. What is the sample estimates of: Mean = _______________ Standard deviation = _______________ Skewness = _______________ Kurtosis = _______________ Note: You may used either biased or unbiased estimators. Question 2. Consider a random variables ? and ?. Write down the general formula for ???(? + ?). http://onlinestatbook.com/2/estimation/characteristics.html Indiv. HW 1, FIN 3115/3711 AY 2020/2021 Semester 2 3 Question 3 Part A. Suppose X and Y are iid (?, ?2) . “iid” stands for Independent and Identically Distributed. Let ?? = ?? + (1 − ?)?. Calculate the standard deviation ???. Part B. Suppose X and Y are distributed ?(?, ?2) and ???(?, ?) = ?2. Let ?? = ?? + (1 − ?)?. Calculate the standard deviation ???. Question 4. Suppose you have N assets each with return ?? for ? = 1, … , ?. Consider the variance-covariance matrix ?. Part A. What is the maximum number of unique covariance terms between the N assets? Note: No need to derive. Just list the number. Part B. Suppose ??~?(?, ? 2) and the correlation between any 2 asset returns is ?(??, ??) = 1. Will you get any diversification benefit? Answer yes or no, and justify your answer by showing calculations using an equal-weighted portfolio. Part C. Now suppose ??~?(?, ? 2) and the correlation between any 2 assets is ?(??, ??) = ?. What is the limiting standard deviation as the number of assets gets arbitrarily large? Indiv. HW 1, FIN 3115/3711 AY 2020/2021 Semester 2 4 Question 5. Homework Grading. Recall that this class has 7 homework assignments total. Consider the following hypothetical homework weighting schemes and fill in the blank with >, =, or < for each blank: 1. if each homework is worth 10 points and the total homework grade is the sum of all homework points divided by 70, then the value of 1 point on hw 1 is ________ the value of 1 point on hw 3. the value of 1% on hw 1 is ________ the value of 1% on hw 3. 2. if the total homework score is equal-weighted across homework assignments, and hw 1 is out of 5 points and hw 3 is out of 20 points, then the value of 1 point on hw 1 is ________ the value of 1 point on hw 3. the value of 1% on hw 1 is ________ the value of 1% on hw 3. 3. if the total homework score is value-weighted across homework assignments, and hw 1 is out of 5 points and hw 3 is out of 20 points, then the value of 1 point on hw 1 is ________ the value of 1 point on hw 3. the value of 1% on hw 1 is ________ the value of 1% on hw 3. question 6. suppose a firm wants to maximize its profits ? by adjusting its price ?. the firm estimates its demand to be ?? = ? − ??, its average variable costs to be ? per unit produced, and its fixed cost is ?. part a. what is the optimal price ?∗ expressed in terms of the structural parameters ?, ?, ?? part b. fill in the blank: ??∗ ?? =____. interpret this result in words below. indiv. hw 1, fin 3115/3711 ay 2020/2021 semester 2 5 question 7. consider the following regression output from r. this is a regression of the form ???? = ? + ?????? + ? part a. use the estimated coefficient of ? in a sentence. refer explicitly to the numbers in the regression table. part b. use the ?-value for the speed coefficient in a sentence. refer explicitly to the numbers in the regression table. part c. use the ?2 in a sentence. refer explicitly to the numbers in the regression table. indiv. hw 1, fin 3115/3711 ay 2020/2021 semester 2 6 question 8. portfolio allocations suppose that you hold a market capitalization value-weighted portfolio across 2 stocks, ?1? and ?2?. assume that the market capitalization of the 2 stocks at time ? − 1 are ?1,?−1, ?2,?−1. part a. what is the mathematical formula for ???? part b. show that for any returns in time ? equal to ?1?, ?2? for each asset, the effective portfolio weight going from ? to ? + 1 is also value-weighted. hint: identify the relationship between market capitalization at time t and time t+1. indiv. hw 1, fin 3115/3711 ay 2020/2021 semester 2 7 bonus (hard). suppose that you allocate 1/n across n assets. show that as time goes to infinity, the weights to any asset ? will converge to the value-weight holding. show this by comparing the initially equal-weighted portfolio with the value weight and show that this difference converges to zero stock-wise. in other words, show this for a single arbitrary stock ?. this proof will apply to all stocks. for="" each="" blank:="" 1.="" if="" each="" homework="" is="" worth="" 10="" points="" and="" the="" total="" homework="" grade="" is="" the="" sum="" of="" all="" homework="" points="" divided="" by="" 70,="" then="" the="" value="" of="" 1="" point="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1="" point="" on="" hw="" 3.="" the="" value="" of="" 1%="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1%="" on="" hw="" 3.="" 2.="" if="" the="" total="" homework="" score="" is="" equal-weighted="" across="" homework="" assignments,="" and="" hw="" 1="" is="" out="" of="" 5="" points="" and="" hw="" 3="" is="" out="" of="" 20="" points,="" then="" the="" value="" of="" 1="" point="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1="" point="" on="" hw="" 3.="" the="" value="" of="" 1%="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1%="" on="" hw="" 3.="" 3.="" if="" the="" total="" homework="" score="" is="" value-weighted="" across="" homework="" assignments,="" and="" hw="" 1="" is="" out="" of="" 5="" points="" and="" hw="" 3="" is="" out="" of="" 20="" points,="" then="" the="" value="" of="" 1="" point="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1="" point="" on="" hw="" 3.="" the="" value="" of="" 1%="" on="" hw="" 1="" is="" ________="" the="" value="" of="" 1%="" on="" hw="" 3.="" question="" 6.="" suppose="" a="" firm="" wants="" to="" maximize="" its="" profits="" by="" adjusting="" its="" price="" .="" the="" firm="" estimates="" its="" demand="" to="" be="" =="" −="" ,="" its="" average="" variable="" costs="" to="" be="" per="" unit="" produced,="" and="" its="" fixed="" cost="" is="" .="" part="" a.="" what="" is="" the="" optimal="" price="" ∗="" expressed="" in="" terms="" of="" the="" structural="" parameters="" ,="" ,="" part="" b.="" fill="" in="" the="" blank:="" ∗="" =____.="" interpret="" this="" result="" in="" words="" below.="" indiv.="" hw="" 1,="" fin="" 3115/3711="" ay="" 2020/2021="" semester="" 2="" 5="" question="" 7.="" consider="" the="" following="" regression="" output="" from="" r.="" this="" is="" a="" regression="" of="" the="" form="" =="" +="" +="" part="" a.="" use="" the="" estimated="" coefficient="" of="" in="" a="" sentence.="" refer="" explicitly="" to="" the="" numbers="" in="" the="" regression="" table.="" part="" b.="" use="" the="" -value="" for="" the="" speed="" coefficient="" in="" a="" sentence.="" refer="" explicitly="" to="" the="" numbers="" in="" the="" regression="" table.="" part="" c.="" use="" the="" 2="" in="" a="" sentence.="" refer="" explicitly="" to="" the="" numbers="" in="" the="" regression="" table.="" indiv.="" hw="" 1,="" fin="" 3115/3711="" ay="" 2020/2021="" semester="" 2="" 6="" question="" 8.="" portfolio="" allocations="" suppose="" that="" you="" hold="" a="" market="" capitalization="" value-weighted="" portfolio="" across="" 2="" stocks,="" 1?="" and="" 2?.="" assume="" that="" the="" market="" capitalization="" of="" the="" 2="" stocks="" at="" time="" −="" 1="" are="" 1,?−1,="" 2,?−1.="" part="" a.="" what="" is="" the="" mathematical="" formula="" for="" part="" b.="" show="" that="" for="" any="" returns="" in="" time="" equal="" to="" 1?,="" 2?="" for="" each="" asset,="" the="" effective="" portfolio="" weight="" going="" from="" to="" +="" 1="" is="" also="" value-weighted.="" hint:="" identify="" the="" relationship="" between="" market="" capitalization="" at="" time="" t="" and="" time="" t+1.="" indiv.="" hw="" 1,="" fin="" 3115/3711="" ay="" 2020/2021="" semester="" 2="" 7="" bonus="" (hard).="" suppose="" that="" you="" allocate="" 1/n="" across="" n="" assets.="" show="" that="" as="" time="" goes="" to="" infinity,="" the="" weights="" to="" any="" asset="" will="" converge="" to="" the="" value-weight="" holding.="" show="" this="" by="" comparing="" the="" initially="" equal-weighted="" portfolio="" with="" the="" value="" weight="" and="" show="" that="" this="" difference="" converges="" to="" zero="" stock-wise.="" in="" other="" words,="" show="" this="" for="" a="" single="" arbitrary="" stock="" .="" this="" proof="" will="" apply="" to="" all="">