STA457 Time Series AnalysisAssignment 2Date: March 18, 2019In this assignment, students will construct factor mimicking portfolios of economicvariables for portfolio management and hedging purpose. The structure of this assignment isas follows: Section 1 introduce the idea of economic variables in a multifactor asset pricingmodel, Section 2 discusses how to retrieve signals from given economic time series, Section3 discusses a method for constructing factor mimicking portfolios, and assignment questionsare given in Section 4.1. Economic variablesThe multifactor structure under ICAPM and APT provides a strong empiricalimprovement over CAPM. A multifactor model is usually given bywhere denotes the asset return at time , (, denotes the -th factor return at time , and is the error term. However, both theories are vague in defining specific factors to beincluded in the multifactor model1.A common way to find suitable factors is to look at the discounted cash flow (DCF)model. Under the DCF model, the present value of the asset may be calculated as 1 In fact, ICAPM of Merton (1973) did provide some rules for selecting factors—the market return and variablesthat proxy for the changes of investment opportunity set.Copyright 2019 Jen-Wen Lin, (2)where ,Ross (1986, hereafter) note that the common factors in returns must be variables which causepervasive shocks to expected cash flows [7,;popular choices of economic variables(but not limited to) are summarized in the table below.Table 1: Candidates for economic state variablesEconomic variables ReasonsMarket returnIn an efficient market, new information concerning future realactivities should be quickly reflected in the aggregate return ofmarket.InflationIf the effect of inflation is not perfectly neutralized in the cashflows and the valuation operator, it will influence the price of afinancial asset.Interest rate/termstructureRepresent opportunity costs and evaluate the impact ondiscounted cash flowsBusiness cycle risk1. Change in the expected real growth rate of the economy2. A positive realization signals an increase in the expectedeconomic growth (more future cash flows)2. Unanticipated shocks (signals)In theory, only unanticipated shocks to economic variables will contribute to assetpricing. In this section, we introduce how to create unanticipated shocks (or signals) ofeconomic factors in a multifactor model setting.Specifically, let denote the economic variable of interest in period, and thecorresponding signal can be defined as, where ?I- stands for anexpectation operator that uses information up to the end of period1.Several approaches are found in literature to generate signals (unanticipated shocks)to suitable economic state variables, including the vector autoregressive (VAR) approach, Copyright 2019 Jen-Wen Linsuch as Campbell (1996) and Petkova (2006) 2, and the Kalman filter approach3 of Priestley(1996). For simplicity, we only consider the VAR approach in this assignment.To facilitate our discussion, we briefly introduce the VAR approach below. Let�deonte the -th economic state variable in period , and�. The VAR approach assumes that the demeaned vector follows a firstorderVAR, as given by. (3)The residuals in the vector are the signals for our risk factors since they represent thesurprise components of the sate variables that proxy for changes of investment opportunityset.3. Factor mimicking portfoliosIf we would like to apply the aforesaid multifactor model for hedging or portfoliomanagement purposes, we need to convert the factor signals to factor mimicking portfolios,which are portfolios of investible assets. The method of Fama and MachBeth (1973) is one ofthe approaches commonly used for constructing factor mimicking portfolios.Let’s first define notation to facilitate our discussion of the Fama-MacBeth (FMhereafter) method. First, assume that asset returns are governed by a multifactor model:� 2 Petkova (2006), “Do the Fama–French Factors Proxy for Innovations in Predictive Variables?”, Journal ofFinance, Volume 61, Issue 2. 3 Priestley (1996) suggests using the residuals of a dynamic linear model on variables of interest as our estimateof innovations. Priestley claimed that this approach would avoid the concern about Lucas critique on the changeof optimal decisions of economic agents due to changes of polices. For example, we may consider� representing our expectation. Note that dynamic linear models can be easily estimated using Kalmanfilter.Copyright 2019 Jen-Wen Lin1) 7 = the return on asset in period (1 ≤ ≤ ),2) 7 = the realization4 of the th factor in period,3) 7 = the disturbance or random errors,and is the number of time series observations5.The FM method consists of a two-pass procedure. In the first stage of the two-passprocedure, we use OLS regression to estimate (, …, ?O) in eqn. (4) for each asset. Let f =(f-, … , fO) be the resulting × matrix of OLS (ordinary least squares) slope estimates6.In the second stage, we regress asset returns�Pon j = [1h, f]using OLS (for each period ). The corresponding regression coefficient can be given by�m represents the factor mimicking portfolio in period , where >jP�jP represents theweights allocating to each security at period .Remark 1: The j matrix is given byRemark 2: Some practitioners create factors without conducting the second stage of the FMmethod. Specifically, they first sort the values of betas (from the first stage) for each factor.They then construct the factor mimicking portfolios of a specific factor by longing the assetswith bigger betas (with respect to the factor) and shorting the assets with smaller betas (withrespect to the factor). 4 In our case, they are the signals or unanticipated shocks discussed in the last section. 5 For simplicity, in this assignment, we assume that that the disturbances are independent over time and jointlydistributed each period with mean zero and a nonsingular residual covariance matrix Σ, conditional on thefactors. The factors are assumed to be independent and identically distributed (iid) over time.Copyright 2019 Jen-Wen Lin4. QuestionsData retrieval:1. Retrieve data from the following resources:1) St. Louis Fed website (https://fred.stlouisfed.org)2) the Fama-French data library(http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html).2. Use the following macroeconomic variables for your assignment:1) : the change rate on the crude oil price (WTI);2) : the difference between the long-term government bond yield and the 1-Yearconstant maturity rate (term spread);3) : Moodys Seasoned Baa Corporate Bond Yield Relative to Yield on 10-YearTreasury Constant Maturity (default spread);4) y,: excess market return from the Fama-French (FF) data set;5) : Current General Activity (Diffusion Index for FRB - Philadelphia District).The description of the data is summarized in the above table.Table 2: Data description and sourcesEstimation of unanticipated shocksUse the VAR approach to construct unanticipated shocks (innovations). Specifically, consider, (6)Copyright 2019 Jen-Wen Linwhere represents a vector of innovations for each element in the state vector.1) Use the methods taught in class to select the optimal lags for Equation (4), includingmodel selection criteria and adequacy test.2) Orthogonalize the innovations to excess market returns as suggested by Petvoka(2006).7Construction of (economic) factor mimicking portfoliosUse the constructed signals from the above question and Fama-French industry portfolios toconstruct the factor mimicking portfolios. Use 60 months rolling-window to construct theportfolios and different re-calibration times, say ONE month, ONE quarter, or ONE year.1) Discuss the performance of constructed mimicking portfolios (using Sharpe ratio,mean and standard deviation, and maximum draw-down).2) Select the optimal re-calibration time based on Sharpe ratio.Construct the factor momentum portfolio as discussed Question B) in Assignment 1.Answer this question based on your analysis in the above question.1) Construct the equally weighted (EW) and risk-parity (RP) portfolio for the constructedfactor mimicking portfolios. Discuss the performance of both portfolios (using Sharperatio, mean and standard deviation, and maximum draw-down).2) Re-do Question B.3) in assignment 1 for the factor mimicking portfolios. Specifically,use = 12 and Equation (5) in assignment 2. (For simplicity, use the sample standarddeviation for this question.)3) Report the performance the time series momentum portfolio (using Sharpe ratio,mean and standard deviation, and maximum draw-down). 7 Doing so, the coefficient in front of the market factor in the multiple time series regression will be equal tothe simple market beta computed in a univariate time-series regression. This provides a convenient way toassess whether the innovations to the state variables add explanatory power to the simple CAPM model.本团队核心人员组成主要包括硅谷工程师、BAT一线工程师，精通德英语！我们主要业务范围是代做编程大作业、课程设计等等。我们的方向领域：window编程 数值算法 AI人工智能 金融统计 计量分析 大数据 网络编程 WEB编程 通讯编程 游戏编程多媒体linux 外挂编程 程序API图像处理 嵌入式/单片机 数据库编程 控制台 进程与线程 网络安全 汇编语言 硬件编程 软件设计 工程标准规等。其中代写编程、代写程序、代写留学生程序作业语言或工具包括但不限于以下范围:C/C++/C#代写Java代写IT代写Python代写辅导编程作业Matlab代写Haskell代写Processing代写Linux环境搭建Rust代写Data Structure Assginment 数据结构代写MIPS代写Machine Learning 作业 代写Oracle/SQL/PostgreSQL/Pig 数据库代写/代做/辅导Web开发、网站开发、网站作业ASP.NET网站开发Finance Insurace Statistics统计、回归、迭代Prolog代写Computer Computational method代做因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：99515681@qq.com 微信：codehelp
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