Copyright 2019 Jen-Wen Lin
STA457 Time Series Analysis
Assignment 2
Date: March 18, 2019
In this assignment, students will construct factor mimicking portfolios of economic
variables for portfolio management and hedging purpose. The structure of this assignment is
as follows: Section 1 introduce the idea of economic variables in a multifactor asset pricing
model, Section 2 discusses how to retrieve signals from given economic time series, Section
3 discusses a method for constructing factor mimicking portfolios, and assignment questions
are given in Section 4.
- Economic variables
The multifactor structure under ICAPM and APT provides a strong empirical
improvement over CAPM. A multifactor model is usually given by, (1)
where ” 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 be
included 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 variables
that proxy for the changes of investment opportunity set.
Copyright 2019 Jen-Wen Lin, (2)
where “,< is the discount rate at time t for expected cash flows at time. Chen, Roll and
Ross (1986, hereafter) note that the common factors in returns must be variables which cause
pervasive shocks to expected cash flows [7,”;<] or risk-adjusted discount rate ?”,<. Some
popular choices of economic variables(but not limited to) are summarized in the table below.
Table 1: Candidates for economic state variables
Economic variables Reasons
Market return
In an efficient market, new information concerning future real
activities should be quickly reflected in the aggregate return of
market.
Inflation
If the effect of inflation is not perfectly neutralized in the cash
flows and the valuation operator, it will influence the price of a
financial asset.
Interest rate/term
structure
Represent opportunity costs and evaluate the impact on
discounted cash flows
Business cycle risk - Change in the expected real growth rate of the economy
- A positive realization signals an increase in the expected
economic growth (more future cash flows) - Unanticipated shocks (signals)
In theory, only unanticipated shocks to economic variables will contribute to asset
pricing. In this section, we introduce how to create unanticipated shocks (or signals) of
economic factors in a multifactor model setting.
Specifically, let ” denote the economic variable of interest in period, and the
corresponding signal can be defined as””I-“, where “I- stands for an
expectation operator that uses information up to the end of period 1.
Several approaches are found in literature to generate signals (unanticipated shocks)
to suitable economic state variables, including the vector autoregressive (VAR) approach,
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such 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
+,”, = 1, … , and = 1, … , deonte the -th economic state variable in period , and” = 9-,”, … , O,”=P. The VAR approach assumes that the demeaned vector ” follows a firstorder
VAR, as given by. (3)
The residuals in the vector are the signals for our risk factors since they represent the
surprise components of the sate variables that proxy for changes of investment opportunity
set. - Factor mimicking portfolios
If we would like to apply the aforesaid multifactor model for hedging or portfolio
management 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 of
the approaches commonly used for constructing factor mimicking portfolios.
Let’s first define notation to facilitate our discussion of the Fama-MacBeth (FM
hereafter) method. First, assume that asset returns are governed by a multifactor model, (4)
where
2 Petkova (2006),“Do the Fama–French Factors Proxy for Innovations in Predictive Variables?”, Journal of
Finance, Volume 61, Issue 2. 3 Priestley (1996) suggests using the residuals of a dynamic linear model on variables of interest as our estimate
of innovations. Priestley claimed that this approach would avoid the concern about Lucas critique on the change
of 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 Kalman
filter.
Copyright 2019 Jen-Wen Lin
1) 7″ = the return on asset in period,
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-pass
procedure, we use OLS regression to estimate (-, …, O) in eqn. (4) for each asset. Let
O) be the resulting matrix of OLS (ordinary least squares) slope estimates6
.
In the second stage, we regress asset returns ” = (-“, … , h”)P
on 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 >jPjP represents the
weights allocating to each security at period .
Remark 1: The j matrix is given by
Remark 2: Some practitioners create factors without conducting the second stage of the FM
method. 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 assets
with bigger betas (with respect to the factor) and shorting the assets with smaller betas (with
respect 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 jointly
distributed each period with mean zero and a nonsingular residual covariance matrix Σ, conditional on the
factors. The factors are assumed to be independent and identically distributed (iid) over time.
Copyright 2019 Jen-Wen Lin - Questions
Data retrieval: - 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… - 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-Year
constant maturity rate (term spread);
3) : Moody’s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year
Treasury 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 sources
Estimation of unanticipated shocks
Use the VAR approach to construct unanticipated shocks (innovations). Specifically, consider, (6)
Copyright 2019 Jen-Wen Lin
where ” 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), including
model selection criteria and adequacy test.
2) Orthogonalize the innovations to excess market returns as suggested by Petvoka
(2006).7
Construction of (economic) factor mimicking portfolios
Use the constructed signals from the above question and Fama-French industry portfolios to
construct the factor mimicking portfolios. Use 60 months rolling-window to construct the
portfolios 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 constructed
factor mimicking portfolios. Discuss the performance of both portfolios (using Sharpe
ratio, 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 standard
deviation 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 to
the simple market beta computed in a univariate time-series regression. This provides a convenient way to
assess whether the innovations to the state variables add explanatory power to the simple CAPM model.
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