MPhil in Economics
MPhil in Economic Research
MPhil in Finance and Economics
Tuesday 23 March to Monday 5 April 2021
Candidates are required to answer two compulsory questions
Write your candidate number (not your name) on the cover of the Project.
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Questions 1 is weighted at 25%, Question 2 is weighted at 75%
The answer should be no longer than 3000 words inclusive of footnotes and
appendices but exclusive of bibliography
One A4 Page consisting largely of charts, statistics or symbols will be
regarded as the same as 250 words (pro rata for less than A4 page)
F540 : Topics in Applied Asset
Assessment / Project 2021
Answer both questions; 25% of the final mark will come from the theoretical
question 1 and 75% will come from the empirical exercise- question 2 .
1. Factor Risk Budgeting additively decomposes individual asset or portfolio re-
turn risk measures into factor contributions allowing a portfolio manager to
know the sources of factor risk for allocation and hedging purposes and allows
a risk manager to evaluate a portfolio from a factor risk perspective. Assume
an asset or portfolio return Rt can be explained by a factor model (FM), with
k factors, of the form
Rt = α + β
′ft + t
where ft ∼ iid(µf ,Ωf ), t ∼ iid(0, σ2 ), cov(fk,t, s) = 0 ∀ k, t, s . Rewrite the
factor model as
Rt = α + β
′ft + t = α + β′ft + σ × zt
=α + β˜′f˜t
β˜ = (β′, σ)′, f˜t = (ft, zt)′, zt =
∼ iid(0, 1)
σ2FM = β˜
′Ωf˜ β˜, whereΩf˜ =
Ωf 0
0 1
denoting the risk measure σFM (the factor model vol) by RM(β˜) show
(a) RM(β˜) is a linearly homogeneous function of β˜
(b) Apply Euler’s theorem to provide an expansion of RM(β˜) into k , β com-
ponents and an asset/portfolio specific risk factor
(c) find simple expressions for an individual factor j’s marginal contribution
to risk, then factor j’s contribution to risk and finally for each factor,
j = 1, ...., k, expressions for their percent contribution to risk and the
asset/portfolio specific factor contribution to risk, ie. j = k + 1
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(d) Show the same analysis can be applied for Portfolio Risk Budgeting when
you can additively decompose portfolio risk measures into asset contribu-
tions which allows a risk manager to evaluate a portfolio from asset risk
perspective; ie. with a portfolio return
Rp,t = w
′Rt =
and let RM(w) denote the portfolio vol, σp. The same analysis can in fact
be applied to any convex risk measure in place of vol, such as Value at
Risk and Expected Shortfall. You may wish to use this analysis in your
empirical exercise.
(e) Consider an investment universe of N assets with R = (R1, ..., RN)
′ ∼
N(µ,Σ) and a portfolio with weights x = (x1, ..., xN)
′ and
i=1 xi = 1
i. Derive the relationship between the β of the portfolio against a bench-
mark or market portfolio and the βi of the individual assets
ii. Given a quadratic utility function- verify that the optimal portfolio is
a linear function of the risk premium and derive an explicit expression
for the implied risk premium,pi = µ− r where r is the risk free rate.
iii. The investor assumes an ex-ante Sharpe Ratio for their portfolio,
SR(x|r) where r is the risk free rate. Show the risk aversion parameter
φ then satisfies the following relationship
φ =
iv. Deduce then that the implied risk premium of asset i is a linear func-
tion of its marginal volatility.
v. What is the economic interpretation of this previous relationship
vi. Find a new expression of the Sharpe Ratio in terms of marginal volatil-
The empirical exercise is based on developing strategies that seek to be robust
to different phases of the market or the global macro-economy i.e. “disaster proof”
or defensive strategies and hence factor timing, tilting or style rotation are the
issues to explore. You are free to follow your own path in the project as long as
you demonstrate knowledge of the material and techniques that have been covered
in the course.
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The steps outlined below should be covered and developing these beyond that
indicated below would increase the final grade awarded. While the more demand-
ing choices that you may make in your empirical analysis will gain greater credit
you must start simple and only attempt more advanced methods if you have al-
ready completed a basic analysis as far as step(e) for instance below, have the time
and are confident in the potential to deliver further results. Be careful to prevent
look ahead bias in your computations - do not let your strategies use data that
would not have been available at the time of strategy deployment- use suitable
lagged inputs and rolling windows for building and implementing your strategies.
1. A section on the course Moodle site has been set up to help you download
your data from CRSP but you may want to get data from other sources. The
development of your data base is your own responsibility but start by down-
loading stock price for the 100 random stocks (PERMNOs) allocated to you
from CRSP for the longest period you can with a common sample size with
no NAs. You can also if you wish download a subset of associated set of stock
characteristics (fundamentals) which will necessarily be for a shorter period-
see the data note on Moodle, a risk free rate, S&P500 index as well as data
on factors of your choice from a variety of sources that will probably include
Ken French’s and/or AQR’s web site, volatility factors can also be found on
Robeco’s web site - https://www.robeco.com/uk/themes/datasets/ . If you
want to use ETF’s they can be downloaded from CRSP- notice in particular
a number of potential hedge ETFs- Bonds, defensive sector indices etc.- again
feel free to expand on this set of data as you wish- for instance you could
also decide to construct technical indicators along the lines of the Neely et al
paper we considered in the lectures. For constructing technical indicators see
the bookdown book Technical Analysis in R by Chiu Yu Ko and R packages
Quantmod, Quantstrat or TTR. Equally you will most probably want to con-
sider ie. construct or download various “risk alarms” such as VIX or use macro
signals from the FRED web site. Your empirical work will be carried out on a
monthly basis. Locate periods of macroeconomic recession and market- “good”
and “bad” periods- crisis, expansion or downturn in your sample possibly using
the NBER indicators. Build your data set.
(a) Diversification: Examine the diversification of your universe of assets to be
considered for inclusion in your portfolio and from these select an investible
and well diversified set with consideration of the objective of building a
defensive strategy. Consider how this set may have changed through time
by exploring diversification over relevant sub periods using tools covered
in the course in a rolling window manner. Comment on what you find.
(b) Return Prediction:Next consider several different approaches to forming
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conditional forecasts of the expected returns for your assets- perhaps de-
velop the code for this step with just a few assets at the outset before
generalizing your analysis to a larger set whose size may be limited by
your own local computer power. Hopefully you will be able to consider
strategies for at least 30 diversified assets. At this point you need to build
the data set of your chosen features- predictor variables- factors, technical
indicators, characteristics..... If you include stock characteristics then your
sample period will be necessarily truncated as discussed on the data down-
load document as different consistent sets of characteristics are available
without NAs on a monthly basis over different periods. You may wish to
consider conflating variables as in the Brandt and Santa Clara paper on
Parametric Portfolio Policies we have studied, where they used the slope
of the yield curve to control for different regimes or alternative approaches
to factor timing. You could examine the use and the relative performance
of relaxation methods such as LASSO or ENET, Random Forests, factors
from PLS- Partial Least Squares, OLS, PCA,.... to compare their relative
forecasting ability along the lines covered in the paper by Kelly Pruitt and
Xiu, Empirical Asset Pricing via Machine Learning that we have consid-
ered in the course- you may wish to use this paper as a template, in some
respects, for your own analysis. Provide in your report an analysis of the
relative forecasting performance of the methods you have selected.
(c) Portfolio Construction: Drawing on your analysis in the sections above
build one or more defensive strategies designed to survive bad times- you
could start by simply building a conditional volatility targeting strategy
as we considered in the course with the paper - Conditional Volatility
Targeting, Bongaerts, Kang,van Dijk, Financial Analysts Journal, (2020 )
and then move on to a conditional strategy based on your findings in section
(b). You could consider min variance, mean variance or PPP and replicate
a similar analysis to Brandt and Santa Clara with Parametric Portfolio
Policies. Carry out an (comparative) analysis of the performance of the
defensive strategy or strategies you have developed including downside
risk measures and statistical comparisons where relevant.
(d) Performance Evaluation: Carry out a comparison with your preferred
defensive strategy from (c) with non defensive alternatives such as a stan-
dard Mean Variance portfolio based on FF3, 1
based on a set of your
original assets and the SP500. You might want to compare different ap-
proaches to covariance estimation. Graphical plots of backtests should be
on a common scale and use statistical performance comparisons to draw
conclusions regarding the value of the preferred defensive strategy com-
pared with the alternative non-defensive strategies you have considered.
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(e) A report should be drawn up, preferably in Markdown, containing your
results and conclusions specifically addressing the original objective- Can
we build a portfolio strategies that are robust to different cycles in the
markets and macro economy?
(f) All code and data used must be submitted with the report and your em-
pirical work must be able to be reproduced by a third party.
(g) As indicated above you are free to follow your own path at each stage but
this path should, in some form, cover the steps outlined above. Limit
your objectives initially to do just the simplest analysis and then
as time allows develop your ideas further.
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