FE630 Portfolio Theory and Applications
Final Project
Due date for the project: December 22nd, 2020
Prof. Papa Momar Ndiaye
December 1, 2020
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FE630 Portfolio Theory and Application Final Project
1 Overview
This project aims 1) to build a factor-based model allocation namely a Long/Short
Global Macro Strategy with a Target Beta and 2) to evaluate its sensitivity to
variations of Beta and its sensitivity to the length of the estimators for covariance
matrix and the expected returns under different market scenarios.
Students may work individually or in small teams (typically up to 3 peo-
ple). Each team is required to build an investment strategy that maxi-
mizes the return of the portfolio subject to a constraint of target Beta,
where Beta is the usual single factor Market risk measure. The port-
folio will be weekly reallocated (re-optimized every week to generate
new weights) from March 2007 to end of November 2020. For practical
considerations, we will assume that our universe of investment is a set of ETFs
large enough to represent the World global economy and that our factor model
is the French Fama 3-factor model. The data needed for the project are freely
available for download from Ken French’s website for the factors historical val-
ues (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french) and from Quandl
(www.quandl.com) or yahoo for the ETFs.
The performance and the risk profiles of such a strategy may be very sensitive
to the target Beta and the market environment. A low Beta meaning a strategy
that aims to be de-correlated to the global market represented by the S&P 500,
while a high Beta meaning that, having a big appetite for risk, we are aiming
to ride or scale up the market risk. In addition to that, such a strategy is likely
to to be quite sensitive to the estimators used for the Risk Model and the Alpha
Model (for example the length of the look-back period used for estimation risk and
expected returns), so it is important to understand the impact of those estimators
on the Portfolio’s characteristics: realized return, volatility, skewness, VaR/ CVaR
and risk to performance ratios such as the Sharpe ratio.
To simplify, we will assume in this project that once the factor model built, we
will use trend following estimators for the Expected returns, namely the sample
mean and the sample covariance. As the quality of those sample estimators depend
on the length of the look-back period, we will typically consider 3 cases: a long look-
back period ( ≥ 120 days), a short look-back period ( ≤ 40 days) and a medium
look-back period, and use the notational conventions Long-Term estimators (LT),
Short-Term estimators (ST) and Mid-Term estimators (MT), defining therefore a
Term-Structure for the Covariance and Expected Return. A similar remark on
the dependance on the look-back period can be made for the estimation of the
coefficients of the model as they are computed using a regression on the factors.
Consequently, a central question is to assess properly the impact of the length of
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FE630 Portfolio Theory and Application Final Project
those regression-based estimators on the realized performance and risk indicators
of the optimized portfolio.
In summary, the behavior of the optimal portfolio built from a specific com-
bination of estimators for Covariance and Expected Return may change with the
Market environment and the target Beta (a particular strategy being defined by a
specific combination, for example S9060(0.5) - say using 60 days for estimation of co-
variance, 90 days for estimation of Expected Returns and a target β = 0.5). The
goal of this project is to understand, analyze and compare the behavior
of strategies built using chosen combinations of return/risk estimators
and Target Beta during several historical periods : before the subprime
(2008) crisis, during that crisis and after the crisis. The factor model
we will use, known as the French Fama 3-factor model has 3 factors,
Momentum, Value and Size.
2 Investment Strategy
We will consider an portfolio optimization problem of the form:
max
ω∈Rn
ρTω − λ(ω − ωp)TΣ(ω − ωp)
n∑
i=1
βmi ωi = β
m
T
n∑
i=1
ωi = 1, −2 ≤ ωi ≤ 2,
(1)
where
• Σ is the the covariance matrix between the securities returns (computed from
the Factor Model), ωp is the composition of a reference Portfolio (the previous
Portfolio when rebalancing the portfolio and ωp has all its components equal
to 1/n for the first allocation) and λ is a small regularization parameter to
limit the turnover;
• βmi =
cov(ri, rM)
σ2(rM)
is the Beta of security Si as defined in the CAPM Model
so that βmP =
n∑
i=1
βmi ωi is the Beta of the Portfolio;
• βmT is the Portfolio’s Target Beta, for example βmT = −1, βmT = −0.5, βmT = 0,
βmT = 0.5, β
m
T = 1, β
m
T = 1.5.
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FE630 Portfolio Theory and Application Final Project
The French Fama factor models are well documented in the literature but
reminded here for reference. For instance, under the 3-factor model, the random
return of a security is given by the formula
ri = rf + β
3
i (rM − rf ) + bsirSMB + bvi rHML + αi + εi (2)
with E(εi) = 0 in such a way that we have in terms of Expected Returns
ρi = rf + β
3
i (ρM − rf ) + bsiρSMB + bvi ρHML + αi. (3)
In equation (2), the 3 coefficients β3i , b
s
i and b
v
i and estimated by making a linear
regression of the time series yi = ρi−rf against the time series ρM−rf (Momentum
Factor), rSMB (Size Factor) and ρHML (Value Factor)
1. Note that in general
βmi 6= β3i and needs to be estimated by a separated regression or computed directly.
3 Investment Universe and Analysis Setup
3.1 Investment Universe
We will consider the following set of ETFs that you can download from Yahoo,
Google or Quandl from March 1st, 2007 to June 30th, 2020.
1. CurrencyShares Euro Trust (FXE)
2. iShares MSCI Japan Index (EWJ)
3. SPDR GOLD Trust (GLD)
4. Powershares NASDAQ-100 Trust (QQQ)
5. SPDR S&P 500 (SPY)
6. iShares Lehman Short Treasury Bond (SHV)
7. PowerShares DB Agriculture Fund (DBA)
8. United States Oil Fund LP (USO)
9. SPDR S&P Biotech (XBI)
10. iShares S&P Latin America 40 Index (ILF)
11. iShares MSCI Pacific ex-Japan Index Fund (EPP)
12. SPDR DJ Euro Stoxx 50 (FEZ)
1ρM for Market hence Momentum, ρSMB for Small minus Big and ρHML for High minus Low
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FE630 Portfolio Theory and Application Final Project
3.2 Benchmark
The benchmark will be the Market Portfolio S&P 500 ( SPY ETF)
3.3 Analysis Periods and Bactkesting
• Divide the overall analysis period into 3 sub-periods: before, during and after
the subprime crisis.
• Run separate backtests for each sub-period when comparing strategies. Note
that there are different angles for such comparison: (1) Impact of Beta Target
on a given Term-Structure: compare S20040 (βT1) to S
200
40 (βT2) or (2) impact
of various term structure given Beta: for example, compare S20040 (0.5) to
S9040(0.5).
• Run also a comparison over the whole period from March 1st, 2007 to Novem-
ber 30th, 2019.
• For your backtesting, rebalance your portfolios once a week.
3.4 Performance and Risk Reporting for comparing Strategies
Use a Performance Analytics Module in R, Matlab or Python as much as possible
for the Risk and Performance Reporting. Below is the list of Key Indicators to
report your Optimal Strategies. All daily indicators will be annualized assuming
that each year has 250 business days. For example, the Reporting for a Strategy
over a given period (example: from 03/01/1997 to 12/30/2008) will be provided
by a summarizing Table with the following lines
• Cumulated PnL or Return
• Daily Mean Arithmetic / Geometric Return, Daily Min Return
• Max 10 days Drawdown
• Volatility
• Sharpe Ratio
• Skewness, Kurtosis
• Modified VaR, CVaR
In addition to that table:
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FE630 Portfolio Theory and Application Final Project
1. Plot the evolution the graph of cumulated daily Profit and Loss (PnL) as-
suming that you invest $100 at the first allocation date in Portfolio and $100
in the S&P 500 (when you benchmark your strategies against the Market,
the SPY is representing the S&P500 Index).
2. Plot and analyze the distribution of daily Returns.
3. A summarizing Table with the following lines for comparison with the un-
derlying
For comparison with between the strategies and the S&P, a summary table
may look like:
S9060(β
m
T = 0.5) S
30
120(β
m
T = 1) S
90
180(β
m
T = 0) SPY
Mean Return 12
...
Max DD 8
3.5 Tools
• Data can be downloaded from R, Matlab or Python using the APIs provided
by Quandl. Alternatively, you may use native functions when available,
for example in R, ”get.hist.quote”2, or ”get data yahoo” using Pandas in
Python.
• The strategies will be implemented using the Quadratic Solver in R, Matlab
or Python.
4 Submission of the Final Report
You have to submit the following.
1. A final report can be a Word, Latex File or PPT slides presenting your
findings and conclusions about the impact of the estimators on the behavior
of your strategy, and also what kind of estimators would recommend to use,
when and why (before, during and after the crisis). So to repeat again, a
global period of backtest from 2007 to November 2018, with 3 sub-periods
(before, during and after the crisis) and 2 axes of analysis: sensitivity to
the term-structure of estimators (short-term, mid-term and long-term) for
covariance and expected returns and sensitivity to β.
2 fxe < −get.hist.quote(instrument = ”fxe”, start = ”2007− 01− 01”, end = ”2018− 12−
01”, quote = ”Close”)
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FE630 Portfolio Theory and Application Final Project
2. The report should contain a clear description of the notations, models and
strategies you have analyzed, the graphs and summarizing tables supporting
your quantitative and qualitative analysis. You can include a brief descrip-
tion of the computational engine you have built but should not include any
code or Rmarkdown output.
3. It is mandatory to submit also the code developed for the project (R, Matlab,
Python or other) and all supporting graphs, tables and simulation results in
a Zip file. The code should ready to run when unzipped and with minimal
directions to the evaluators. The submitted code will be tested for compar-
ison and it is a requirement to build your code in a modular and clearly
documented manner..
Appendix
A Practical aspects
For estimation of the parameters of the factor model, you can use a cross sectional
regression model by gathering all the individual securities model in a single ”big”
factor model. If you assume, that you have 3 factors, then the model at time t for
each asset is given by
reit = αi + β
3
i (rMt − rft) + bsirSMBt + bvi rHMLt + αi + εit (4)
with reit = rit− rft, and moreover the εit are independent of the factors and satisfy
cov(εit, εjs) =
{
σ2i when i = j, and t = s
0 otherwise.
A.1 Time Series Model for a given Security
If we consider T observations of the excess return of Security Si stacked in a column
vector Ri =

rei1
rei2
...
reiT
, we have the time series regression model for Security i:
Ri = 1Tαi + Fβi + εi for i = 1, 2, . . . n (5)
where
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FE630 Portfolio Theory and Application Final Project
• βi =
β3ibsi
bvi
 is the (3 by 1) vector of Betas
• F =
f
′
1
...
f
′
T
 =
 rM1 − rf1 rSMB1 rHML1... . . . ...
rMT − rfT rSMBT rHMLT
 is the is the (T x 3) matrix of
observations of the factors.
• the residual term εi is a (T by 1) vector satisfying E(εiε
′
i) = σ
2
i IT
The previous model is well-suited for a regression to estimate the coefficients of
the model using data for the securities and the factors.
A.2 Cross Sectional Model
Alternatively, we can use a cross sectional formulation that can be useful for risk
analysis including the derivation of the covariance matrix of the returns. If we
define Rt =

re1t
re2t
...
rent
, the vector of all Securities excess returns at time t, then we
can write
Rt = α + Bft + εt for t = 1, 2, . . . T, (6)
where
• B

β
′
1
β
′
2
...
β
′
n
 =
β
3
1 b
s
1 b
v
1
...
. . .
...
β3n b
s
n b
v
n
 is a N by 3 matrix,
• ft =
rMt − rf1rSMBt
rHMLt
 is the vector of factor returns at time t.
• E(εtε
′
t|ft) = D = diag(σ21, . . . , σ2n)
The cross sectional model implies that if Ωf is the covariance of the factors, then
cov(Rt) = BΩfB
′
+D (7)
which implies that
cov(Rit) = βiΩfβi + σ
2
I (8)
and
cov(Rit, Rjt) = βiΩfβj (9)
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