use R studios for this assignment
1 Advanced Investments (FNCE40002) Assignment 1 - Semester 1 2021 ADMINISTRATIVE ARRANGEMENTS Due date: 3:00pm on Thursday 1 April 2021 Type: Individual assignment Late assignments: The penalty for late submission for each assessment task is 5% per day. Submissions that are late by 10 days or more will not be accepted. Extensions of time: Students with a genuine and acceptable reason for not completing an assignment (or other assessment task), such as illness, can apply for special consideration (see Subject Guide). Where to submit: Submit your assignment as a PDF document using the Assignments Tool on LMS. Cover sheet: Include the flowing details on the cover sheet: Subject Code and Name Student Number Full Name Assignment Number. Word limit: 1500 words Marks: This assignment counts 15% towards your final mark in this subject. Precision: Report all estimation results with four digits after the decimal point. Presentation: Use either Times New Roman 12pt or Arial 11pt. For paragraphs, use either double spacing or 1½ spacing. Software: Use a software of your choice for the data preparation and regression analysis. Please do not attach any programming code. 2 ASSIGNMENT The first assignment in Advanced Investments reviews your knowledge of the CAPM. In addition, you will become more familiar with regression analysis. The starting point for this assignment is the table from the Fu and Yang (2011) paper on page 26 of the Week 2 slides. Your analysis will complement their results. A) Collect the data Instead of individual stocks, use the “25 Portfolios Formed on Size and Book-to-Market (5x5)” from the online data library of Professor Kenneth R French (use a search machine to locate it). You will also need the “Fama/French 3 Factors” file from the same source. Download the data in a file format that is suitable for your software of choice. Prepare the data as follows: • Only keep the following information from the first file (scroll down to find these): o Average Equal Weighted Returns – Monthly from 1926.07 – 2021.01 o Average Market Cap – Monthly from 1926.07 – 2021.01 • Only keep the following information from the second file: o Monthly from 1926.07 – 2021.01 • Divide all returns and excess returns by 100. Change all net returns to gross returns (careful, do not add a one to the market excess return in the second file). • Combine all data in a single file (first column should be the date). B) CAPM regressions and annualization Now create a data matrix with 89 rows and 101 columns for the 25 portfolios in the following way (see the notes below for some help; use a loop if possible): o Use 1926.07 – 1931.06 data to estimate CAPM betas and idiosyncratic volatilities. o Use 1931.07 – 1932.06 data to obtain annual market and portfolio excess returns. o Use 1931.07 data to obtain the average market size for the 25 portfolios. o Save in row 1 of data: 26 returns, 25 betas, 25 idiosyncratic volas, 25 sizes. • Then continue with: o Use 1927.07 – 1932.06 data to estimate CAPM betas and idiosyncratic volatilities. o Use 1932.07 – 1933.06 data to obtain annual market and portfolio excess returns. o Use 1932.07 data to obtain the average market size for the 25 portfolios. o Save in row 2 of data: 26 returns, 25 betas, 25 idiosyncratic volas, 25 sizes. • Then continue like this until you reach: o Use 2014.07 – 2019.06 data to estimate CAPM betas and idiosyncratic volatilities. o Use 2019.07 – 2020.06 data to obtain annual market and portfolio excess returns. o Use 2019.07 data to obtain the average market size for the 25 portfolios. o Save in row 89 of data: 26 returns, 25 betas, 25 idiosyncratic volas, 25 sizes. 3 You now have a data matrix with 89 annual observations of excess returns, estimated betas, estimated idiosyncratic volatilities, and market sizes for the 25 portfolios. Note that you were using the 60 months preceding each annual return to estimate the CAPM betas and idiosyncratic volatilities. Thus, we can evaluate the out-of-sample performance of beta. Notes: Recall that you should use excess returns on the left-hand-side of the CAPM regression. Include an intercept. Use the square root of the estimated residual variance from the CAPM regression as an estimate of idiosyncratic volatility. To obtain an annualized excess return, first compound the risky and risk-free returns and then take the difference of these compounded returns. Prepare and present a graph like on page 22 of the Week 2 slides for the 25 portfolios based on your annual data set. Interpret the graph. (5 marks) C) Fama-MacBeth regressions Now run the following Fama-MacBeth cross-sectional regressions based on the annual data you created under B). Recall that you need to run T = 89 cross-sectional regressions for the N = 25 portfolios and then report the average of the estimated parameters. Always include an intercept in addition to the following (vola = idiosyncratic volatility): 1. Y = excess return, X = log(size) 2. Y = excess return, X = log(vola) 3. Y = excess return, X = CAPM beta 4. Y = excess return, X = log(size), log(vola) 5. Y = excess return, X = log(size), CAPM beta 6. Y = excess return, X = log(vola), CAPM beta 7. Y = excess return, X = log(size), log(vola), CAPM beta Report all Fama-MacBeth estimates, standard errors, and t-statistics in one table. Interpret your estimation results with respect to the following two questions: • Based on your results, is the size effect a size anomaly? (5 marks) • How do your results compare with Fu and Yang (2011)? (5 marks) End of Assignment Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 1 Advanced Investments Joachim Inkmann Week 2 Capital Asset Pricing Model Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 2 Content of this lecture 2.1 Derivation of the CAPM 2.2 Estimation and Testing Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 3 The Big Picture • In Week 1, we derived an investor’s demand for assets. • In this lecture we are looking at the equilibrium, where demand meets supply. • We use the mean-variance model to describe the demand side. • We use the standard MV model without extensions such as background risks and liabilities. • The leads to the Capital Asset Pricing Model (CAPM) in equilibrium. • A typical question this model can answer is “Which returns do investors demand for holding a certain asset?”. Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 4 2.1 Derivation of the CAPM Recall: Two-fund separation theorem All investors hold a combination (determined by their respective risk-aversion) of the risk-free asset and the tangency portfolio. Extreme cases: 100% risk-free, 100% risky (= tangency portfolio). With risk-free borrowing: > 100% risky All investors invest in the same risky portfolio. Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 5 Recall: MV Frontiers with and without a risk-free asset Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 6 Slope of the efficient frontier (with risk-free asset): Capital Market Line (CML) The location of all efficient portfolios in the �????, ????�-Graph. Mathematically, the line through the points �0,???? ??� and (???? , ????) is: ???? = ???? ?? + � ???? − ???? ?? ???? �???? Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 7 Properties of the tangency portfolio (Idea: solve for ??�????+1?? �) • Let ??????(????+1,????+1?? ) be the ?? × 1 vector of covariances of returns on all risky assets with the return on the tangency portfolio. • ????+1?? is a portfolio in ????+1. Thus, we can rewrite: ??????(????+1,????+1?? ) = ??????(????+1,????+1)?????? = ??[????+1]?????? • Recall that the tangency portfolio weights satisfy: ?????? = ??????[????+1]−1??[????+1?? ] Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 8 • If we replace the second equation in the first one, we obtain: ??????(????+1,????+1?? ) = ??????[????+1]??[????+1]−1??[????+1?? ] = ??????[????+1?? ] • The variance of the tangency portfolio return equals: ??[????+1?? ] = ?????? ′??[????+1]?????? = ?????? ′??[????+1](??????[????+1]−1??[????+1?? ]) = ?????????? ′??[????+1]??[????+1]−1??[????+1?? ] = ?????????? ′??[????+1?? ] = ????�??[????+1?? ] − ???? ??� Advanced Investments (FNCE 40002) 2021 © Dr Joachim Inkmann Department of Finance, The University of Melbourne 9 • Let’s write the covariance ??????(????+1,????+1?? ) for a single asset ??: ??????�????+1?? ,????+1?? � = ??????�????+1 ??,?? � ⇔ 1 ???? = 1 ??????�????+1?? ,????+1?? � ??�????+1 ??,?? � • Solving the variance ??[????+1?? ] for the same constant yields: ??[????+1?? ] = ????�??[????+1?? ] − ???? ??� ⇔ 1 ???? = ??[????+1?? ]