THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MMAT5250 Financial Mathematics Homework 2 Due Date: March 24, 2018 Name: Student ID.: I declare that the assignment here submitted is original except for source material explicitly acknowledged. I also acknowledge that I am aware of University policy and regulations on honesty in academic work, and of the disciplinary guidelines and procedures applicable to breaches of such policy and regulations, as contained on the website http://www.cuhk.edu.cn/departsite/ar/en/academic.html/ Signature Date For lecturer/ta s use only 1 3a 5 2 3b 6 4 7 Total
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[Objective:] The objective of this homework assignment is to learn how to understand: the meanings of risk and return; the meaning of attainable set; the meanings of the expected return and the variance of the portfolio; the concept of positive definite matrix; solve portfolios consisting of two assets; solve portfolios of multiple assets;
1. Given the six years of percentage returns for Stocks 1 and 2 in the following table, calculate the mean return, sample variance, sample covariance, and correlation for the two returns: Year Stock 1 Stock 2 R t R 1 R t R 2 (R t R 1 )(R t R 2 ) Return Return 20X4 +0.10 +0.20 +0.05 +0.10 +0.005 20X5 0.15 0.20 0.20 0.30 +0.060 20X6 +0.20 0.10 +0.15 0.20 0.030 20X7 +0.25 +0.30 +0.20 +0.20 +0.040 20X8 0.30 0.20 0.35 0.30 +0.105 20X9 +0.20 +0.60 +0.15 +0.50 +0.075 Rt = 0.30 R 2 = 0.60 ( ) = 0.255 R 1 = 0.05 R2 = 0.10 2. Consider two risky assets that have returns variances of 0.0625 and 0.0324, respectively. The assets standard deviations of returns are then 25% and 18%, respectively. Calculate the variances and standard deviations of the portfolio returns for an equalweighted portfolio of the two assets when their correlation of returns is 1,0.5,0, and 0.5. 3. Suppose that the returns K 1 and K 2 of the two assets S 1 and S 2 are as in the following scheme: Scenario Probability K 1 K 2 ω 1 0.2 10% 10% ω 2 0.3 5% 2% ω 3 0.5 20% 15% (a) Compare the risk (assume the variance as a risk measure) of a portfolio V composed by the two assets above with weights w 1 = 0.3, and w 2 = 0.7, with the risk of the two single assets considered separately. (b) Using the same table, determine the composition of a portfolio V still consisting of the two previous assets, but with expected return equal to 9%. Calculate the variance of the portfolio V.
4. Let three risky assets be given. Their returns have the expectations and covariance matrix described in the following scheme: µ 1 = 0.20 σ 1 = 0.25 ρ 12 = 0.20 µ 2 = 0.12 σ 2 = 0.30 ρ 23 = 0.50 µ 3 = 0.15 σ 3 = 0.22 ρ 13 = 0.30 Find the minimum variance portfolio and compute its expected return and variance. 5. Consider a portfolio V with return K V and the market portfolio M with return K M, where K V and K M take the following values Scenario Probability K V K M ω 1 0.3 3% 8% ω 2 0.2 2% 7% ω 3 0.3 4% 10% ω 4 0.2 1% 9% Find the coefficients β V and α V of the regression line. 6. Consider the market consisting of three assets S 1,S 2, and S 3 with returns K 1,K 2, and K 3, uncorrelated and all with unit variance, with expectations E(K 1 ) = 1/4,E(K 2 ) = 1/2, and E(K 3 ) = 3/4, respectively. Determine the efficient portfolio with expected return ˆK, under the constraint of no short selling (i.e. w i 0 for i = 1,2,3).
7. Consider a portfolio composed of five assets with expected returns S 1,S 2,S 3,S 4, and S 5 E(K 1 ) = 0.183,E(K 2 ) = 0.085,E(K 3 ) = 0.121, E(K 4 ) = 0.112, and E(K 5 ) = 0.096, respectively, and with return covariance matrix 0.0690 0.0279 0.0186 0.0222 0.0069 0.0279 0.042 0.0066 0.0168 0.0078 C = 0.0186 0.0066 0.054 0.0234 0.0081 0.0222 0.0168 0.0234 0.102 0.168. 0.0069 0.0078 0.0081 0.168 0.078 Determine two funds whose linear combinations generate any efficient portfolio. 8. (Optional) Let x and y be the coordinates so that we have the following description of the attainable set: y = wµ 1 +(1 w)µ 2 and x 2 = w 2 σ 2 1 +(1 w2 )σ 2 2 +2w(1 w)σ 12. Assume that µ 1,µ 2 and ρ 12 ( 1,1). Show that the attainable set is a hyperbola with its centre on the vertical axis. 9. (Optional) Let R be the risk-free return. Show that the weights of the market portfolio are µ = (w,1 w), with where and w = c c+d, 1 w = d c+d, c = σ 2 2 (µ 1 R) σ 12 (µ 2 R), c = σ 2 2(µ 1 R) σ 12 (µ 2 R).
Print out your scripts and submit them with your answers to the homework papers. 10. Consider the following year of data history for three assets: Asset A Asset B Asset C Jan 48.5% 22.4% 14.8% Feb 10.3% 29.0% 26.0% Mar 23.5% 21.5% 41.9% Apr 5.5% 27.2% 6.9% May 8.5% 14.5% 17.0% Jun 5.6% 10.7% 3.5% Jul 3.8% 32.1% 13.3% Aug 8.9% 30.5% 73.2% Sep 9.0% 19.5% 2.1% Oct 8.2% 39.0% 3.0% Nov 3.5% 7.2% 0.6% Dec 17.6% 71.5% 90.8% Please refer to the Excel file three assets.xlsx. This problem splits into two parts: Part I: Data Preparation and Data Management Answer the following questions: (a) Use the MATLAB build-in functions, mean and std (select w = 1), to compute the mean (meana, meanb, meanc ) and the standard deviation row vectors (stda, stdb, stdc) of three assets. (b) Use the MATLAB build-in function, corrcoef, to compute the correlation matrix of the three assets, called corrcoefa. (c) Use the results to (a) and (b) and diag to compute the covariance matrix of the returns of the three assets, called cova. (d) Use the MATLAB build-in function, cov(select w = 1), to compute the covariance matrix of the returns of the three assets as a check, call covm.
Part II: Portfolio optimization A minimum variance portfolio with 16% expected rate of return and fully invested in the three assets (i.e., with no cash position) can be set up by investing a percentage w i of the total value of the portfolio in assets i, with i = 1,2,3, where w 1,w 2 and w 3 can be found by solving the following linear system: where and 2Σ µ 1 µ T 0 0 1 T 0 0 µ p,0 = 0.16, µ = w = µ 1 µ 2 µ 3 Answer the following questions: w λ 1 λ 2 w 1 w 2 w 3 = = 0.1041 0.2136 0.2269 0 µ p,0 1, 1 = (a) Using the result of Part I (e), apply the leading principal minor approach and use the MATLAB build-in function, det, to verity whether Σ = covm is a positive definite matrix or not. 1 1 1.
(b) Show that using the MATLAB build-in function, lu, the matrices from the LU decomposition with row pivoting of the matrix on the left-hand side are P = 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 and L = 1.0000 0 0 0 0 0.1041 1.0000 0 0 0 0.0226 0.6875 1.0000 0 0 0.0238 0.7596 0.2755 1.0000 0 0.0401 0.1488 0.0104 0.3685 1.0000 and U = 1.0000 1.0000 1.0000 0 0 0 0.1095 0.1228 0 0 0 0 0.0699 0.2269 1.0000 0 0 0 0.2761 1.2755 0 0 0 0 0.5196. (c) Find the weightsof each asset in the minimumvariance portfolio with 16% expected return using three different approaches of MATLAB build-in functions as follows: i. using the \ command ii. using the inv command iii. using the lu command Find the standard deviation of the return of this portfolio such that σ p,0 \ inv lu
11. (Optional) You are given ten securities as follows: (a) S 1 : Apple Inc. (AAPL) (b) S 2 : Amazon.com, Inc. (AMZN) (c) S 3 : The Boeing Company (BA) (d) S 4 : The Walt Disney Company (DIS) (e) S 5 : General Motors Company (GM) (f) S 6 : Alphabet Inc. (GOOG) (g) S 7 : S&P 500 index ( GSPC) (h) S 8 : Intel Corporation (INTC) (i) S 9 : Microsoft Corporation (MSFT) (j) S 10 : Netflix (NFLX) Please download the historical data of the above securities from 1-1-2013 to 31-12-2017 from Yahoo Finance Define the vector random variable S 1 S 2 S =. S 10 Please use R with packages installed as below install.packages("xts") install.packages("quantmod") install.packages("quadprog") to answer the following questions: (a) Find the mean vector and covariance matrix for S. (b) If one owns a portfolio with the following weighting vector [10%,7%,14%,3%,12%,6%,19%,5%,11%,13%], what is the expected return and standard deviation of this portfolio?
(c) What is the weighting of the portfolio that yields the minimum expected return? What is the corresponding risk (as measured by the standard deviation of this portfolio)? (d) What is the weighting of the portfolio that yields the maximum expected return? What is the corresponding risk (as measured by the standard deviation of this portfolio)? (e) Graph an efficient frontier for S. Suppose the 3-Month Constant Maturity Treasury yield of 31-12-2017 is 1.39%(i.e. the annual risk free rate is r f = 1.39%), and plot the tangency portfolio on the same graph. (f) Suppose one has a target of 0.2% expected return with, of course, minimum risk. Your advice?