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Modern Portfolio Management:
from Markowitz to Probabilistic
Scenario Optimisation
Goal-Based and Long-Term Portfolio Choice
Paolo Sironi

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Contents
About the Author ix
Foreword xi
Introduction xiii
1 Beyond Modern Portfolio Theory 1
PART I RISK MANAGEMENT FRAMEWORK 13
2 A Modern Risk Management Perspective 15
3 The Probability Measure 43
4 Real Securities and Reinvestment Strategies: Fixed-Income and
Inflation-Linked Securities and Structured Products 55
5 Derivation and Modelling of Risk–Return Time Profiles 85
PART II PORTFOLIO OPTIMISATION METHODS 103
6 À la Markowitz: A Tale of Simple Worlds 105
7 The Black–Litterman Approach: A Tale of Subjective Views 123
8 Probabilistic Scenario Optimisation 139
PART III PORTFOLIO OPTIMISATION CASE STUDIES 161
9 Case Studies: Mean–Variance and Black–Litterman 163
10 Case Studies: Probabilistic Scenario Optimisation 175
Symbols and Notation 195
Index 197
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About the Author
Paolo Sironi is practice leader of wealth management solutions and
risk content services at IBM Risk Analytics, where he is responsible
for quantitative methods and asset allocation advisory for financial
institutions (retail banking, private banking, ultra-high-net-worth
and institutional advisory clients). Combining risk analytics and
technology, Paolo’s expertise spans wealth management, asset man-
agement, investment banking, market and credit risk management,
regulatory reporting, cognitive computing, on-cloud and banking
digitalisation. Before joining IBM, Paolo worked as managing direc-
tor of Capitects, the company (a provider of risk management solu-
tions) that he founded in 2008 as a joint venture between Sal. Oppen-
heim Private Bank and Algorithmics and that became part of IBM
following the Algorithmics acquisition. Prior to Capitects, Paolo
worked as head of market and counterparty risk modelling at Banca
Commerciale Italiana and Banca Intesa Sanpaolo.
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Foreword
At the heart of every investment decision lies the question “what
will be the value of a given portfolio at some future time horizon?”.
By definition, the future value is uncertain. There are a range of
future possible outcomes depending on the market scenarios that
are possible. The decision to invest in a given portfolio will depend
on the trade-off between the possible downside and upside, or risk
and reward. This is subjective for each investor and is a function of
their preferences: tolerance for risk and desire for performance.
This book features an excellent description of Modern Portfolio
Theory, which still forms the basis for many investment decisions. It
also does an excellent job of describing the Black–Litterman method-
ology, a more modern enhancement. Paolo Sironi’s key contribution,
however, is in making scenario analysis and the very general Mark-
to-Future approach accessible to goal-based investing. He describes
in great detail how to simulate investment strategies over time while
accounting for an investor’s risk–return profile.
This is not only a theoretical treatise but one based on many years
of experience of real-world investment decision-making.
I believe it will be an excellent addition to any portfolio manager’s
library.
Ron Dembo
January 2015
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Introduction
Investment banking, asset management and wealth management
are sophisticated industries that correspond to the investment
needs of a large population of investors (institutional and private
individuals) who require quantitative but intuitive solutions for
investment decision-making. Products with mathematically com-
plex payoffs (eg, structured notes) are nowadays broadly traded on
financial markets and distributed to final investors. Yet, institutional
portfolio management is often based upon rules of thumb and
simplifications, such as the usage of benchmarks to proxy real
investments. This can affect the coherence of optimal portfolio
analysis and lead to inefficient capital allocations across risk factors
and asset classes. This book addresses a renewed interpretation of
portfolio choice based upon a modern risk management perspective
and a clearer definition of the investors’ risk–return profile. The
probability of achieving a desired target return (ie, a return target
for an investment fund, a return ambition for a private investor) or
minimising risk (ie, a value-at-risk (VaR) limit for a trading desk,
a potential capital loss for a private investment) is chosen as the
statistical measure that enforces optimal portfolio allocations by
explicitly stating investment goals and downside boundaries. Port-
folio managers, asset managers and wealth managers, who engage
in long-term and goal-based portfolio construction, are concerned
not only about today’s perception of risk and opportunity, but also
about the way risk and return evolve over time. Such investors
might ponder over statistical analysis and institutional research
discussing optimal allocation and diversification among global asset
classes, and might adopt benchmark-based frameworks to represent
actual portfolios. However, they ultimately trade actual products
whose payoffs can no longer be disregarded when discussing port-
folio choice. Modern optimisation techniques therefore must fully
embrace the risk–return characteristics of fixed-income securities
and derivatives to overcome the limitations of classical approaches.
Modern Portfolio Theory relies on the Markowitz (1952) formula-
tion, which combines the basic objectives of investing: maximising
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
expected return or minimising risk. This leads to an efficient fron-
tier that indicates the set of portfolios with the best combination of
risk–return characteristics. Portfolio managers have certainly ben-
efited from this insightful formulation, but they did not grant it
the expected practical success due to known limitations: profes-
sional investors may be believed to possess asymmetrical infor-
mation; mean and variance are very restrictive indicators of risk–
return characteristics of fixed-income payoffs and derivatives; the
mean–variance efficient frontier often indicates extreme portfolio
weights;theuncertaintyoftheinputvariablesisnotembeddedinthe
approach. Hence, the framework does not include the results of cal-
culating these dynamics, leading to insufficient risk–return manage-
ment for life-cycle portfolio insurance and goal-based investments.
Black and Litterman (1992) proposed an elegant approach to alle-
viate some of these limitations and indicated the positive weights
stemming from the market equilibrium as the initial reference port-
folio, thus combining return expectations with investors’ subjective
views of the market. Although the Black–Litterman approach helps
individuals to identify a more reasonable, less extreme and less sen-
sitive portfolio weighting scheme, it still cannot address some of
the relevant risk management challenges posed by modern finance:
the approach still relies on the dynamics of benchmarks, which
are an incomplete representation of the full universe of risk factors
and opportunities; real investments and the way they can change
over time are neglected; embedding investors’ views in consistent
formats is not an easy exercise; investors’ characteristics, denoted
by profiles of return ambition and risk appetite, do not enter the
optimisation method explicitly.
This book discusses ways to mitigate such limitations and covers
portfolio choice from the perspective of goal-based investing and
probabilistic scenario optimisation (PSO). In particular, we address
the challenges of long-term investments in order that a myopic
approach to portfolio choice need not dominate the asset alloca-
tion exercise. Investors might not know enough about future states
of the world, which is why they may focus on short-term money-
management although they express goals for longer investment
horizons (eg, a yearly budget or a multi-year portfolio insurance
strategy). The investment horizon, as well as the frequency of the
intermediate steps of portfolio rebalancing, is important in portfolio
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INTRODUCTION
choice and cannot be disregarded. The introduction of multi-period
stochastic simulations, common practice in counterparty risk mea-
surement and credit value adjustment (CVA), may make behaviour
less short-sighted by including professional knowledge about future
potential returns, so that investment decisions can be tested ex ante
and verified throughout the life of the investment.
ORGANISATION OF THE BOOK
The book is organised into an introductory chapter and three parts:
(I) risk management framework;
(II) portfolio optimisation methods;
(III) portfolio optimisation case studies.
Chapter 1: Beyond Modern Portfolio Theory
The three parts are preceded by an introductory chapter that exam-
ines modern investment environments and outlines the reasons
for portfolio choice to evolve beyond the Markowitz and Black–
Litterman approaches. Probabilistic scenario optimisation is briefly
reviewed as a valuable alternative and its main traits are discussed;
these traits are linked across all subsequent chapters, guiding the
reader in their studies throughout the book.
Part I: Risk Management Framework
The first section is a precursor to our review of goal-based optimi-
sation principles and covers aspects of financial risk management.
Chapter 2 describes the main characteristics of a modern risk man-
agement perspective based on scenario simulation. While invest-
ment banks have implemented enterprise-wide risk management
architectures in order to comply with best practices and banking
regulation, asset managers and wealth managers often rely upon
simplified approaches for the risk management of portfolio expo-
sures and the optimisation of the risk–return profile. Therefore, we
start our discussion by presenting the most common risk measure-
ment methods of computing VaR and expected shortfall (paramet-
ric, historical, bootstrapping and Monte Carlo) and comparing them
with numerical examples. This discussion outlines why Monte Carlo
scenarios are chosen to simulate the assets used in the multi-period
optimisation.
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
Chapter 3 introduces the a posteriori probability measure, which
can be estimated by overlapping the investor’s risk–return profile
with the density function of the portfolio potential returns gener-
ated by a multi-period Monte Carlo simulation. This emphasises
the appealing advantages of choosing the probability measure as
the objective function of goal-based optimisations, as it allows us to
compare ex post and ex ante performance in a synthetic and graphical
representation.
Chapter 4 discusses the advantages of building portfolio choice
on a risk management framework that directly models real secu-
rities as opposed to benchmarks and market indexes. Modelling
the reinvestment rules of fixed-income and derivative products
allows portfolio managers to supplement long-term simulations of
maturity-bearing securities. This is particularly relevant for port-
folio managers wanting to optimise long-term portfolio allocations
with fixed-income holdings, derivatives, structured products and
inflation-linked exposures.
In Chapter 5, we present aspects of modelling the investors’ risk–
return profiles, so that we can map the vectors of the actors’ pref-
erences onto the full space of potential total returns of portfolios,
which is a building block of probabilistic scenario optimisation.
Part II: Portfolio Optimisation Methods
This section examines portfolio choice from the point of view of
the main approaches available to market practitioners: Markowitz,
Black–Litterman and probabilistic scenario optimisation.
Chapter 6 presents Modern Portfolio Theory, a classical diver-
sification framework. The key traits of Markowitz-type optimisa-
tions are outlined, taking mean–variance as a starting point, track-
ing error minimisation as an alternative for asset managers, using
semi-variance to overcome the statistical limitations of the volatility
measure and expected shortfall as a more advanced formulation of
the objective function.
Chapter 7 relaxes the classical assumptions of information sym-
metry embedded in the Markowitz approach and reviews the Black–
Litterman alternative. We start by formalising the market equilib-
rium portfolio (CAPM) and then introduce the investors’ views in
order to estimate the posterior distribution of the expected returns
of assets and optimal portfolios.
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INTRODUCTION
Chapter 8 discusses probabilistic scenario optimisation; this is
seen as a turning point in goal-based investing, because it com-
bines the mathematical properties of investment products with the
preferences of actual investors. PSO is an exhaustive enumeration
technique that aims at maximising the probability of achieving or
beating an investment target, thus complying with a risk profile.
Our description of the methodological steps is enriched by examin-
ing low-discrepancy sequences and lexicographical representations,
which allow computational performance to be properly addressed.
Part III: Case Studies of Portfolio Optimisation
This section presents a set of case studies using numerical examples
that allow us to compare the three optimisation methods presented
in the previous section with respect to the model inputs and the
outputs of the optimisation routines.
Chapter9examinesbothamean–variancecaseandaBlack–Litter-
man optimisation, and Chapter 10 examines probabilistic scenario
optimisationandcomparesthefindingsofthemulti-periodexercises
forasetofalternativerisk–returnprofiles:riskaverse,riskmitigating
and risk tolerant.
SUMMARY OF THE BOOK
Portfolio choice and goal-based investing are attractive cutting-edge
topics for a large and international audience. We discuss the related
aspects of quantitative finance with the intention to make them as
digestible as possible. This book does not aim to provide direct
advice to portfolio managers, private investors or their intermedi-
aries.Instead,itprovidesanempiricalframeworkbasedonprobabil-
ity measurement for those practitioners willing to apply their intu-
ition together with an understanding of the dynamics of the trade-off
between the portfolio risks and returns, as part of a decision-making
process designed for long-term investments.
Some limitations should be acknowledged. First, the risk man-
agement methods are only outlined. Second, trading costs are not
formally discussed because the focus of the book is an argumenta-
tion of life-cycle optimal investments, as opposed to myopic trading,
so that the cost implications due to short-term trading become less
relevant. Third, taxation is generally ignored, although it can signif-
icantly influence decisions of wealth allocation. Last, inflation can
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
play a key role in long-term investing but this topic is only partly
discussed, leaving space for further applications.
ACKNOWLEDGMENTS
I thank the numerous colleagues and friends that have inspired
my professional activity. In particular, I am grateful to Gabor Topa,
Dominik Flierl and Andres Hernandez for their thoroughness and
dedication as they contributed to this work with open discus-
sions, formalisation and constructive criticism. I am indebted to
Ron Dembo, Michael Zerbs and colleagues at Algorithmics: their
visionary work in risk management has inspired my career. A sin-
cere thank you to Sarah Hastings, Commissioning Editor, and to
Lewis O’Sullivan, Managing Editor, for believing in this project.
Most importantly, I am grateful to my family, who helped me to
dedicate time to this work.
This book contains the formulations, evidence and opinions of
the author alone; these do not necessarily represent the practice or
the views of his current or previous employer, or the beliefs of his
present and past colleagues.
January 2015
REFERENCES
Black, F., and Litterman, R., 1992, “Global Portfolio Optimization”, Financial Analysts
Journal, pp. 28–43.
Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91.
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1
Beyond Modern Portfolio Theory
An investment in knowledge pays the best interest.
Benjamin Franklin (1706–90)
This chapter sketches the main arguments of this book, which are
related to portfolio choice for long-term and goal-based invest-
ing, and provides a summary of Modern Portfolio Theory, the
Black–Litterman approach, probabilistic scenario optimisation and
knowledge-based principles of optimal investing.
INTRODUCTION
Financial markets underwent a profound transformation during the
last decade of the 20th century. The integration of international mar-
kets, fostered by broader deregulation of cross-border capital flows,
was accompanied by strong financial innovation: the landscape of
investmentopportunitiesbecamemoreaccessibleyetheterogeneous
(ie, derivatives, structured products, securitisation) and also more
interdependent, as revealed by the contagion risk that characterised
the global financial crisis in 2007–12. This affected the dynamics
of the correlations among global asset classes, as it appeared not
only that risks become over-concentrated more often than expected,
instead of being diversified away across a larger number of players,
but also that asset classes co-move faster than forecasted, as capital
flows in and out of international markets.
A direct consequence is that there is a growing demand from
investors to shift their priorities in the direction of more customised
asset–liability management and to be more ambitious in modelling
risk appetite; this ambition cannot be addressed by existing mar-
ket equilibrium approaches, elegant in nature as they are. Regula-
tors also demand more risk transparency in financial intermediation,
stimulating the financial services industry to revise existing method-
ologies of portfolio choice towards risk-based approaches. These
elements reinforce the call for optimal portfolio modelling to be
based on actual products, actual investors’ preferences and actual
investment goals over the life cycle.
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
Yet, financial markets get more and more sophisticated and
volatile, so investors are invited to make increasingly complex deci-
sions about their wealth allocation and require greater knowledge of
quantitative finance. Can the optimal portfolio be the same for long-
term investors and short-term players? Is cash a risk-free heaven
when looking at longer investment horizons, in which reinvestment
occurs at today’s unknown real interest rates? Can money man-
agers provide long-term capital protection but yield returns stem-
ming from tactical opportunities, in such a way that investments are
always optimal during all periods?
Behavioural finance has documented patterns of individual be-
haviour that do not reconcile with rational models, so that actual
portfolios tend to be a function of short-term market opportunities
only, making it unfeasible to optimise portfolios over the life cycle. It
is now acknowledged that conditions for the market to be efficient,
in the sense that investors have accurate information and use it cor-
rectly to their advantage, and the statement that the market portfolio
is an efficient portfolio should be discussed differently. In fact, mar-
ket efficiency implies portfolio efficiency only under some specific
assumptions which are proved to be inappropriate: transaction costs
and liquidity constraints must not be ignored, most investors do not
hold efficient portfolios or the same (correct) beliefs about the risk–
return profiles of securities and cannot lend or borrow without limits
at the risk-free rate.
Investment practices at established investment banks and asset-
management firms often rely upon overly simplified rules of thumb
to assess the trade-off between investment risks and potential
returns, which leads to the indication of strategic asset allocations
that do not always reconcile with real investment opportunities. The
divergence between an enlarged set of investment requirements and
the need for consistent responses has widened the information gap
between the so-called optimal market portfolio and an investor’s
attainable portfolio. The strategic market portfolio, which arises
from a theoretical asset allocation, emerges from optimisation exer-
cises based on the statistical properties of the market variables (ie,
expected return and standard deviation) and the asset classes that
map to them. The investor’s attainable portfolio instead emerges
from the operational asset allocation, which is the result of a self-
directed or an intermediated process that implicitly bounds the
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BEYOND MODERN PORTFOLIO THEORY
investors’ choices to a defined set of real products (out of the larger
set available) bearing non-linear risk–return profiles.
The provision of more intuitive and consistent information about
potential future states of the world and the simulation of investment
returns (net of commissions, transaction costs and possibly tax) can
contribute to improved market efficiency and reconcile operational
and theoretical portfolio allocations. As a matter of fact, a new inter-
pretation of Modern Portfolio Theory based on scenario optimisa-
tion seems to be emerging: probabilistic scenarios, which are part
of established risk management practices, grant investment man-
agers the chance to employ time-varying characteristics of investors’
preferences and achieve a more consistent risk–return description.
Explicit modelling of the investor’s profile can change the traditional
landscape of optimisation models, whose main inputs are market
variables or their subjective reinterpretation (equity tilt) at a single
point in time. The inclusion of the investor’s profile makes it easier
to realign the actors’ preferences not only with the prevailing market
outlook, but also with the most appropriate mix of long-term costs/
benefits that originate from the simulation of the potential returns
of real investments.
Empowering individual investors to take transparent care of their
own assets, directly or indirectly via the professional work of finan-
cial advisors, is also a developing idea. Keynes (1931) had already
imagined central bankers as orthodontists, intervening with humble
fiscal and monetary policy to optimise the dynamics of the economy
at large: “If economists could manage to get themselves thought of
as a humble, competent people, on a level with dentists, that would
be splendid”. As Campbell and Viceira (2002) brightly indicated, it
is now common wisdom that dentists shall also pursue the goal of
advising on oral hygiene, rather than simply intervening once the
pain becomes unbearable. Similarly, investors should be given the
tools and the means to reallocate investments with an ex ante view
of the potential drawbacks and opportunities, which is the essence
of proactive risk management.
Probabilistic scenarios are the cornerstone of this new interpre-
tation. By simulating total returns of actual investments and lia-
bilities over time, we are granted direct access to the information
hidden in the potential dynamics of the probability densities of
actual products. Thus, we can verify whether a given set of an
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
individual’s constraints complies with the simulated total return
space of portfolios, by measuring the probability of achieving or
underperforming a defined investment goal so that the probabil-
ity measure becomes the key variable of the min/max objective
function. Furthermore, the evolution of the potential total returns
of optimal portfolios can be stress tested with subjective views or
alternative hypotheses of market behaviour to strengthen the risk
management aspects.
This chapter introduces the main traits of probabilistic scenario
optimisation (PSO), a risk-based optimisation framework for long-
term and goal-based investing. First, the main traits of portfolio
theory are outlined by reviewing the essential elements of the
Markowitz and Black–Litterman approaches to portfolio choice.
Then, scenario optimisation is introduced as an exhaustive enumera-
tion technique requiring Monte Carlo simulation of actual products,
modelling of actual investors’ risk–return profiles, low-discrepancy
sequences and lexicographical representations to achieve compu-
tational efficiency. Finally, five knowledge-based principles are
outlined to address goal-based portfolio investing in the long term.
THE MAIN TRAITS OF MODERN PORTFOLIO THEORY
Modern Portfolio Theory relies on Markowitz’s (1952) formula-
tion, which combines the basic objective of investing: maximising
expected return while minimising risk. This leads to an efficient fron-
tier that indicates the set of portfolios with the best combination of
risk–return characteristics. The theory suggests that investors, who
care only about the mean and the variance of portfolio returns over
a single period, can choose an optimal portfolio that is the unique
combination of risky assets combined with an appropriate amount
of risk-free cash, so that personal propensity to risk can be dealt with.
Portfolio managers have certainly benefited from this insight,
but they did not grant it the expected practical success, due to
known limitations. First, professional investors might be believed
to possess superior information about financial markets, or require
more customised decision-making to better reflect personal ele-
ments. Second, portfolio managers might not have a complete set
of return expectations for the entire universe of asset classes that
is required to generate optimal portfolio weights of global asset
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BEYOND MODERN PORTFOLIO THEORY
allocations. Third, the mean–variance efficient frontier often indi-
cates extreme portfolio weights, either long or short, which are
excessively sensitive to changes in the estimate of expected returns.
Moreover, the uncertainty in the input variables is not embedded in
the approach (estimation error). Last, a realistic and practical asset
allocation that encompasses global investment opportunities, espe-
cially fixed-income securities and derivative payoffs, cannot easily
be identified.
Although market practice has improved the original mean–var-
iance proposition with the use of better risk measurements such as
regret, expected shortfall, semi-variance and tracking error, these
approaches tend to be restricted to an oversimplified representation
of real securities by means of benchmarks and market indexes.
Therefore, Markowitz-type optimisations are not fully suited to
addressing risk–return management for life-cycle portfolio insur-
ance and goal-based investments, since the implications of total
return dynamics of actual securities stand outside the framework.
THE MAIN TRAITS OF THE BLACK–LITTERMAN APPROACH
Black and Litterman have further extended the original mean–
variance formulation (1992) and have indicated the positive weights
stemming from the market equilibrium as the initial reference port-
folio, thus combining return expectations with investors’ subjective
views of the market. Portfolio managers have been given the chance
to indicate a confidence level for each view and re-optimise the equi-
librium portfolio by shifting the asset weights towards the preferred
strategies.
Although the Black–Litterman approach helps individuals to
identify a more reasonable, less extreme and less sensitive portfolio
weighting scheme, it still cannot address some of the relevant risk
management challenges posed by modern finance. First, the mar-
ket equilibrium is a theoretical formulation of how financial mar-
kets function, and it relies on the dynamics of benchmarks, which
are an incomplete representation of the full universe of risk fac-
tors and opportunities. Second, the distinctive risk–return proper-
ties of real investments and the way they can change over time are
neglected. Third, embedding investors’ views in consistent formats
is not an easy exercise, so institutionalised processes of portfolio
choice cannot be enforced. Last, investors’ characteristics, denoted
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
by profiles of return ambition and risk appetite, do not enter the
optimisation method explicitly.
THE MAIN TRAITS OF PROBABILISTIC SCENARIO
OPTIMISATION
PSO is an exhaustive enumeration technique that allows us to miti-
gate some of the limitations of Markowitz-type and Black–Litterman
approaches. Owing to advances in computing power it has become
increasingly accessible, allowing institutional investors and wealth
managers to find solutions to the problems of multi-period port-
folio choice based on discrete-state approximations. This technique
requires the simulation of the potential returns of real securities
over time, which permits fixed-income products, derivatives and
structured products to be conveniently represented in making opti-
mal allocations. The introduction of dynamic reinvestment strate-
gies allows us to make long-term simulations of optimal portfolios
beyond the contractual expiry of maturity-bearing securities, so that
portfolio choice can be made conveniently across asset classes and
payoffs. Investors’ ambitions and fears can also be elicited, so that
their risk–return profile over time can be drawn and overlapped
with the potential total return space of strategic and tactical asset
allocations. Stress tests and investment views can be modelled freely,
and the potential dynamics of actual payoffs can be reviewed with-
out loss of information. This allows portfolio managers to comple-
ment strategic portfolio optimisation with asymmetrical opinions
and make decisions regarding “suboptimal” portfolios (with respect
to the theory) through a clearer understanding of the confidence lev-
els associated with stressed market changes. The key element of PSO
is a reinterpretation of the objective function, which becomes the
maximal probability of achieving (or beating) an investment target
while complying with a given risk limit, so that goal-based invest-
ing is supported. The explicit statement of the probability measure
helps to combine past and future performance and track the devi-
ation from optimality, as time passes and investment goals become
more likely or less likely to be attained.
PSO is a step-by-step process of portfolio filtering and order-
ing according to probability measurement criteria, as constructed
in Table 1.1.
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Table 1.1 Probabilistic Scenario Optimisation process
ΦU, ϕU
(generate all potential portfolios)
↓
ΨU
(identify only the admissible portfolios)
↓
ΘU
(filter the risk-adequate portfolios)
↓
Θ∗
U
(indicate the optimal goal-based portfolio)
The computing power challenges posed by such an exhaustive
enumeration technique are still relevant for the treatment of large
portfolio allocations. However, this is no longer a limitation in the
context of wealth management and portfolio insurance, as the opti-
mal portfolio is generated out of a reduced universe of investment
opportunities. Quasi-random methods can be applied in such a way
that the resulting space of the admissible portfolio compositions is
made of equidistant outcomes that represent well all possible port-
folio combinations, thus avoiding large gaps and clustering. Halton
(1960), Sobol (1967), Faure (1982) and Niederreiter (1987) are all well-
known alternatives, among the variety of low-discrepancy methods
proposed by this growing field of mathematical research. We argue
for the non-binding adoption of Halton sequences in the making
of the examples and case studies presented in the following chap-
ters.Haltonsequencesaredeterministicsequencesofnumbersbased
on increasingly fine prime-based division (eg, 2, 3, 7, 11, 13, . . . ) of
subunit intervals, which produce well-spaced draws from the unit
interval so that the quasi-random variables sampled from a larger
population are ex post evenly spread (equidistant). Quasi-random
methods still require that the optimisation routine generates the full
explicit list of the ordered portfolios from which to sample. One way
to further improve the calculation efficiency is offered by computa-
tional science, as we can model a lexicographical representation with
a more parsimonious tree of the relationships among the ordered
numbers, so that the explicit list of all possible allocations can be
sampled by a smaller number of iterations, without having to gen-
erate the full space. More importantly, knowing that the objective
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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO
Figure 1.1 Example of PSO portfolio simulation
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1.0
1Y 2Y 3Y 4Y 5Y
0
0.5
1.0
Probability
Time
Monte Carlo
Ambition profile
Risk profile
Optimisation node
CIO view H1
CIO view H2
Risk/return
Positive return Beating target
function is not convex, we can make use of genetic algorithms to
surf the multi-dimensional space generated by the verification of
the objective function with even greater speed and accuracy. In such
a case, the step-by-step approach indicated in Table 1.1 would be
different.
Exhaustive enumeration techniques are very unrestrictive meth-
ods and can be applied to any type of investment problem. However,
for the convenience of the applications, in the remainder of this book
we often refer to simpler cases of model portfolio optimisation that
optimise private wealth.
Figure 1.1 shows an example of PSO portfolio analysis (as in
Chapter 10).
FIVE KNOWLEDGE-BASED PRINCIPLES
Decision-making for goal-based investments is in itself a thorough
exercise that certainly requires dedication, knowledge and time, as
if planning a journey. In the course of this work, the reader will
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be taken through this journey, composed of five knowledge-based
steps.
1. Know the products: we identify the universe of potential
investments and simulate them over time with stochastic
scenarios in order to investigate their risk–return properties.
2. Know the investor: we identify the optimisation constraints
according to the investor’s preferences, such as minimum allo-
cation, investment step size and maximum exposure to a cer-
tain market, and generate the set of all potential portfolios
that comply with the given constraints. Also, we indicate the
investor’s risk–return profile and the time discretisation along
which optimisation should be performed (investment horizon,
liquidity term and reallocation steps).
3. Know the portfolio risks: we discard all potential portfolios
that do not comply with the investor’s risk appetite and focus
on the potential allocations that are risk-adequate.
4. Know the portfolio returns: we measure the probability of
each potential risk-adequate portfolio to beat/achieve the
investor’s ambitions and order portfolio results in terms of
probability levels to indicate the optimality. We can also stress
test the chosen optimal allocation and challenge its robustness
and meaningfulness.
5. Knowtheperformance:wetracktheperformanceofthenewly
invested portfolio by drawing the ex post (historical) and ex
ante (prospective) dynamics of total returns. The probability of
reaching a chosen target is a function of the ex post performance
(net capital loss or gain plus/minus cashflows) and the density
of potential future risks/returns. We can therefore identify the
most appropriate time steps for revising the asset allocation
and rolling forward the financial bets.
An appealing feature of PSO is that we can operate multiple
problems without having to recalibrate the full set of simulation
inputs: we can redefine time horizons, time steps, allocation con-
straints, client ambitions or risk appetite levels and operate on the
same stochastic distribution of the total returns of individual prod-
ucts. This should facilitate the institutionalisation of the methodol-
ogy across advisory networks, well outside the specialised desks of
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quantitative professionals, and hence allow sell-side institutions to
deliver better and transparent support to buy-side players. More-
over, we can identify an intuitive metric that allows comparison of
strategic and tactical asset allocations with a certain level of intuition.
The probability, such as that of beating a financial goal, of yielding a
minimum total return or avoiding a capital loss, is such a measure.
CONCLUSIONS
Modern financial environments require mitigation of the limitations
of Modern Portfolio Theory to make portfolio choice easier in the
contextoflong-termandgoal-basedinvesting.PSOisemergingasan
alternative to the classical methods, such as the Markowitz-type and
Black–Litterman approaches. The adoption of probabilistic scenar-
ios requires a thorough understanding of modern risk management
techniques, based upon full revaluation methods of actual secu-
rities by means of multi-period stochastic simulations. Therefore,
the first section of this book will introduce the reader to the princi-
ples of portfolio simulation, the generation of scenarios and scenario
paths, the calculation of product and portfolio total returns, the dif-
ference between the most common risk management approaches
(parametric, historical, bootstrapping and Monte Carlo), the time
properties of fixed-income securities, the asymmetry of the poten-
tial returns of derivatives and structured products as well as the
added-value of dynamic strategies for the simulation of maturity-
bearing securities. The elicitation of the risk–return profile of actual
investors (being professional traders or private individuals) is also
presented in such a way, that it constitutes fundamental input to the
Probabilistic Scenario Optimisation exercise.
REFERENCES
Barberis, N., 2000, “Investing in the Long Run when Returns Are Predictable”, Journal of
Finance 55(1), pp. 225–64.
Campbell, J. Y., and L. M. Viceira, 2001, Strategic Asset Allocation: Portfolio Choice for Long-
Term Investors (Oxford University Press).
Dembo, R., and A. Freeman, 1998, Seeing Tomorrow: Rewriting the Rules of Risk (Chichester:
John Wiley & Sons).
Faure, H., 1982, “Discrépance de Suites Associées à un Système de Numération (en
Dimension s)”, Acta Arithmetica 41, pp. 337–51.
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Halton, J. H., 1960, “On the Efficiency of Certain Quasi-Random Sequences of Points in
Evaluating Multi-Dimensional Integrals”, Numerische Mathematik 2(1), pp. 84–90.
Keynes, J. M., 1931, Essays in Persausion (New York: Norton).
Kahneman, D., P. Slovic and A. Tversky, 1982, Judgment under Uncertainty: Heuristics and
Biases (Cambridge University Press).
Litterman, R., and G. He, 1999. “The Intuition behind Black–Litterman Model Portfolios”,
Report, Goldman Sachs Investment Management Series.
Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91.
Markowitz, H. M., 2000, “Foundations of Portfolio Theory”, Nobel Lecture (New York:
Baruch College, The City University of New York).
Niederreiter, H., 1987, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-
NSF Regional Conference Series in Applied Mathematics (Philadelphia, PA: SIAM).
Sobol, I., 1967, “On the Distribution of Points in a Cube and the Approximate Evaluation
of Integrals”, USSR Computational Mathematics and Mathematical Physics 7(4), pp. 86–112.
Zimmermann, H., W. Drobetz and P. Oertmann, 2003, Global Asset Allocation: New Methods
and Applications (Wiley Finance).
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