This document describes one of the available topics for the MSc-project in
Financial Mathematics. The focus is on an investor who holds a portfolio of
assets and who wants to compute and interpret certain risk measures in order
to guide future actions in an environment with negative rates.
The project has three parts: first part is a literature review that would de-
scribe and explain the contracts in the portfolio, the models used for equity/
interest rates, the methods available for the modelling of the default, the meth-
ods used to estimate the parameters, and the most common risk measures. The
second part consists of a numerical analysis of a specified portfolio, where the
models would be implemented on a practical level. The third part treats more
advanced issues related to this topic, with a view of enhancing the understanding
of the problem, the model and the results.
Implementing such a project in real life would require at a minimum identi-
fying the risk factors and appropriate models for them, checking if counterparty
credit risk is present, and how to model it if necessary, identifying which real
market data to be used for parameter estimation and how long the historical
time series should be.
To facilitate the analysis we provide guidance for some of the steps mentioned
above. The risk factors are modelled with stochastic models that have been
introduced in previous modules, and the parameters of the models are estimated
using real data from Bloomberg over the specified time horizon. Future paths
are generated according to these models, and the possible future values are
incorporated in a risk analysis through the computation of risk measures for the
portfolio.
The investor is subject to credit risk, where the counterparty of a certain
contract in the portfolio can default before the maturity of the contract, thereby
affecting the payoff of the contract. For our portfolio we consider a reduced-form
model with constant intensity of default that will model the occurrence of the
default. The parameters for this model of default can be calibrated to market
data (e.g. CDS market prices), but we will assume specific values for them.
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Part 1: Literature review
In the first part the student is asked to write a literature review that should
include a description of the contracts in the portfolio, particularly the EONIA-
based interest rate swap, a brief outline of the models used for equity/ interest
rates, the methods available for the modelling of the default (i.e. structural
vs reduced form models, advantages and disadvantages of each class), a brief
outline of the methods used to estimate the parameters, and a review of the
most common risk measures.
The student is invited to consult a number of publications on EONIA/ECB
rates, Credit Risk modeling, and on Value at Risk and risk measures in general.
We present some suggestions below as a starting point:
• For EONIA and EONIA based contracts see the European Money Markets
Institute (EMMI) website https://www.emmi-benchmarks.eu
• For ECB deposit rate see the European Central Bank (ECB) website
http://www.ecb.europa.eu
• ”Getting started guide”, Bloomberg, 2012 (available at:
https://www.kcl.ac.uk/nms/depts/mathematics/research/finmath/
bloomberg/docs/education-userguide-a4.pdf)
The references at the end of this document are classical books on risk man-
agement, interest rate models, least squares parameter estimation and related
topics, and give good starting points to the literature. The student should be
proactive in researching the literature, which involves published journal papers
and books. Working papers should be used mostly for orientation, given that
their content has not been peer reviewed.
It is particularly important that the student synthesizes the information
gathered from these sources and presents it as a flowing story that is consistent
both in terms of notation and mathematical and financial content.
Part 2: Numerical analysis
This part applies the theoretical notions from Part 1 on an analysis of a specific
portfolio. The assets in the portfolio are:
1. Equity: 1 share of the Deutsche Boerse AG German Stock Index (Bloomberg
ticker DAX INDEX)
2. EONIA based interest rate swap with a counterparty that is default free.
The maturity is one month, the notional amount is 10 million, and the
swap rate can be obtained from Bloomberg market data on the day t = 0.
The investor is the swap payer, i.e. pays the fixed rate and receives the
floating rate pegged to daily EONIA values. The formula for the EONIA
swap variable rate can be found in EMMI publications.
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https://www.emmi-benchmarks.eu
http://www.ecb.europa.eu
3. European call option (long position) with a counterparty that may default.
The underlying is the equity above (DAX), the strike is 12,450 and the
maturity is 50 days from t = 0. Initial price of 235.1 at t = 0 comes from
Bloomberg. For later prices we use a pricing measure where the equity
follows a Black-Scholes model with drift equal to the simulated EONIA
rate at 30 days from t = 0, and the volatility is the square root of the
element of Σ corresponding to equity. The counterparty of the option
can default with zero recovery rate (in case of default the entire option
becomes worthless). The default is modelled by a reduced form model
with constant annual intensity of default 0.12.
The goal of the project is to analyse the risk and return characteristics of the
portfolio using a stochastic model for the underlying risk factors.
Stochastic model
Consider the risk factors to be the equity (DAX) and the EONIA spread over
the ECB deposit rate:
Xt = (log Yt logSt)
′,
and assume they follow under the subjective measure P a discretized version of
a stochastic differential equation (SDE) of the type:
∆Xt = (AXt−∆t + b)∆t + ε, ε ∼ N(0,Σ),
where ∆t = 1 day (for tractability make the simplifying assumption that week-
ends or holidays are equivalent to 1 day periods).
• Estimate model parameters: A, b, Σ using two years of historical daily
data from Bloomberg (see, for instance, the packages lm, dynlm in R). Fix
the date of the analysis (t0 = 0) as 27/04/2017. Assume the data spikes at
the end of most months (colloquially called the beat/the pulse) are caused
by expired regulatory requirements, so we exclude all end of the month
observations from the data. Plot the data with and without the beats.
• If a parameter has a significance level above 5%, then temporarily set it
to zero (we will use alternative information to historical data to estimate
them). Write the resulting discrete dynamics of log Yt and logSt.
• Identify b so that the log equity has an annual drift of 5%, and log EONIA
spread over ECB satisfies E(logSt0+1 − logSt0) = 0. Assume the central
bank deposit rate remains constant at its last observed level at t = 0.
• The covariance matrix Σ is derived as the cross-product of the residuals,
normalized appropriately with respect to the number of observations.
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Computational results
• Analyse the distribution of the potential losses incurred by the portfolio
over a 30 days holding period, and use VaR and ES (CVaR) to quantify
them with confidence level 99%. The relative losses are defined wrt to the
value V1 of the portfolio in 30 days from t = 0 and the value V0 at time
t = 0 (you can consider the percentage change in the value of the portfolio
for instance). Analyse the numerical accuracy of the results.
• To compute the value V0 find the equity price, the option price and the
swap rate on the date t = 0 in the Bloomberg terminal, and explain how
you obtained them.
• For V1 simulate 30 days for the process Y , and simulate also the default
of the counterparty of the call. Plot one future sample path together with
the historical path for each component of the process X.
• Include in your analysis the histogram of the relative losses, a plot of the
return distribution of the portfolio, computation of VaR and ES, and the
expected and median returns of the portfolio.
• Analyse the impact of credit risk by repeating the calculations without
credit risk (zero intensity) and comparing the results. Discuss.
Summarize your answers by completing a table in the format shown in Table 1.
Portfolio Expected Median V@R CV@R
returns returns
credit risk
no credit risk
Table 1: Format to use for the display of the results
Part 3: Advanced issues (to be done after part 2)
This part should include any pertinent analysis that would contribute to enhanc-
ing the understanding of the topic. Ideally this would be focused on negative
rates and portfolio risk management. Among the possible extensions that could
be studied in relation to the proposed topic we mention (but these are just
suggestions, and the list is not comprehensive):
• Analyse the impact that negative rates have on the portfolio. You can
try to repeat the calculations, but with a constant zero ECB deposit rate
instead of the real market value, and then compare the results. Or you
can propose alternative ways to analyse this, but explain your reasoning.
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• Give histograms of the residuals and perform statistical tests to check for
their normality. Test also for serial autocorrelation, and comment on any
GARCH implications.
• Use variance reduction techniques to simulate future scenarios and im-
prove numerical accuracy.
• Analyze the impact of # of scenarios on the accuracy of the results.
• Change the weights of the assets, starting with same initial portfolio value,
to find a better portfolio (i.e. higher expected return and smaller CVaR).
References
• D. Montgomery, E. Peck, G. Vining, ”Introduction to linear regression
analysis”, 5th Ed, Wiley, 2012.
• D. Filipovic, ”Term structure models: a graduate course”, Springer, 2009.
• M. Crouhy, D. Galai, and R. Mark, ”A Comparative Analysis of Current
Credit Risk Models”, available at http://www.defaultrisk.com/pp model 12.htm
• CreditMetrics Technical Document, available at defaultrisk.com
• D. Brigo, F. Mercurio, ”Interest Rate Models: Theory and Practice”, 2nd
Edition, 2006, Springer Verlag.
• T. Bielecki and M. Rutkowski, ”Credit risk: Modeling, Valuation and
Hedging”, Springer Verlag, 2002. This book has a rigorous treatment of
credit risk.
• P. Jorion, ”Value at Risk”, 3rd ed, McGraw Hill. This covers VaR at
elementary level.
• For models of default see the numerous papers by M. Jeanblanc; or the
paper by Jarrow and Protter “Structural versus reduced form models: a
new information based perspective”, J of Inv. Manag., 2(2), 1-10, 2004.
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