FRM - Financial Risk Management Lecture Summary

FINANCIAL RISK MGT – FRM
Lecture by;
Dr. Syed
Muhammad Ali
Tirmizi
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INTRODUCTION
 In this lecture, we will cover the following
topics:
4. Measuring Risk
i. Synopsis of risk measures
ii. Portfolio Risk Concepts
5. Modern Portfolio Theory
i. Markowitz Portfolios
ii. Empirical Mean Variance Portfolio

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MEASURING RISK
 The focus of investors has been shifted, not least
because of the financial crisis, to the need to
adequately measure risks.
 If an investor were too conservative with respect to
risk during such phases, s/he would jeopardize
potential investment opportunities.
 Furthermore, there are regulatory reasons and
economic arguments which necessitate a proper risk
assessment.

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MEASURING RISK
 The stricter requirements imposed by the Basel
Accords are worthy of mention, and for the
latter the normative presumption of an efficient
resource allocation between risky assets.
 If the riskiness of a financial instrument is not
captured correctly, its price would in general be
misleading and would therefore cause an
inefficient allocation of funds.

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MEASURING RISK
 Therefore, a proper assessment of market risks by
its participants helps to ensure smooth functioning
of the financial system.
 For example, if market participants have to revise
their risk estimates on a larger scale, as was the case
during the sub-prime mortgage crisis, and hence
want or have to rebalance their funds, a stampede-
like reaction will quickly dry up market liquidity
and aggravate the potential losses, due to the lack of
counter-parties.

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SYNOPSIS OF RISK
 The risk measures introduced in this section are based
upon a probability model for the potential losses an
investor would face.
 The investor’s wealth 𝑊(𝑡) is viewed as a random
variable, and the realization thereof at the time point t is
assumed to be known.
 The value of wealth at this specific point in time is
dependent on the vector of risk factors 𝑧𝑡 , which exert
an influence on the investor’s wealth position:

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SYNOPSIS OF RISK
 The prices of financial instruments, exchange rates,
and/or interest rates can be modelled as risk
factors, where it is assumed that these are known at
the time t.
 In contrast, the future value of wealth after a time
span is unknown.
 This could be the wealth position in 1 or 10 days’
time.

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SYNOPSIS OF RISK
 The loss that results after a time is denoted by 𝐿𝑡,𝑡+∆
and this is just the difference in wealth positions at
t +Δ and t.
 It is conventional that losses are expressed as
positive numbers:

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SYNOPSIS OF RISK
 Because wealth 𝑊(𝑡) is a random variable, the loss
function, as the difference between two wealth
positions in time, is also a random variable.
 As such it has a probability distribution, which will
henceforth be termed the loss distribution.
 This distribution can be made dependent upon the
information available at the time point t.

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SYNOPSIS OF RISK
 As a first step, only the modelling of market risk for
a single financial instrument is considered.
 Therefore, the price of the asset is the risk factor and
the loss depends on the time span and the price
change within this period.
 We will further confine the analysis to the
unconditional distribution of losses and the risk
measures that can be deduced from it.

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SYNOPSIS OF RISK
 In practice, the most commonly encountered risk
measure is the value at risk (VaR).
 For a given confidence level 𝛼 ∈ (0, 1), the VaR is
defined as the smallest number ℓ such that the
probability of a loss L is not higher than (1 − 𝛼) for
losses greater than ℓ.
 This value corresponds to a quantile of the loss
distribution and can be formally expressed as;

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SYNOPSIS OF RISK
 Where 𝐹𝐿is the distribution function of the
losses.
 For the sake of completeness the concept of the
mean VaR, 𝑉𝑎𝑅𝛼
𝑚𝑒𝑎𝑛, is also introduced.
 This risk measure is defined as the difference
between 𝑉𝑎𝑅𝛼 and the expected return 𝜇.
 If the chosen time period is 1 day, this measure
is also referred to as the daily earnings at risk.

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SYNOPSIS OF RISK
 One criticism against the use of VaR as a measure of
risk is that it is inconclusive about the size of the
loss if it is greater than that implied by the chosen
confidence level.
 Put differently, if there are 95 sunny days and 5
rainy days, one is typically not interested in the least
amount of rain to expect on these five rainy days,
but rather would like to have an estimate of the
average rainfall in these cases.

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SYNOPSIS OF RISK
 The expected shortfall (ES) risk measure,
introduced by Artzner et al. (1997) and Artzner
(1999), directly addresses this issue.
 This measure provides hindsight about the size of
the expected loss if the VaR has been violated for a
given level of confidence. It is defined for a Type
error I as 𝛼.

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SYNOPSIS OF RISK
 The stylized facts of skewed and fat-tailed return/loss
distributions can also be reflected directly in the calculation
of the VaR.
 Zangari (1996) proposed the modified VaR (mVaR), which
explicitly takes the higher moments into account.
 Here, the true but unknown quantile function is approximated
by a second-order Cornish Fisher expansion based upon the
quantile function of the normal distribution.
 Hence, the mVaR is also referred to as Cornish Fisher VaR
and is defined as;

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SYNOPSIS OF RISK
 Where S denotes the skewness, K the excess kurtosis,
and 𝑞𝛼 the quantile of a standard normal random
variable with level 𝛼.
 In the case of a normally distributed random variable,
m𝑉𝑎𝑅𝛼 = 𝑉𝑎𝑅𝛼 because the skewness and excess
kurtosis are zero.
 The Cornish Fisher VaR produces a more conservative
risk estimate if the return distribution is skewed to the
left (losses are skewed to the right) and/or the excess
kurtosis is greater than zero.

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PORTFOLIO RISK CONCEPTS
 The risk measures introduced in the previous section
can be computed for portfolio return data too.
 But in a portfolio context an investor is often
interested in the risk contributions of single
positions or a group thereof.
 Furthermore, s/he might be interested in how the
portfolio risk is affected if the weight of a position is
increased by one unit, that is, the marginal
contribution to risk.

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PORTFOLIO RISK CONCEPTS
 The concept of component VaR or component
ES addresses the former question.
 This notion of breaking down the overall risk
into single positions held in a portfolio can also
applied to the mVaR and mES.

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CHAP 5 - MODERN PORTFOLIO THEORY
 More than 60 years have passed since Harry Markowitz’s
groundbreaking article “Portfolio selection” was published
(see Markowitz 1952).
 Because the more recently advocated approaches to
portfolio optimization are still based on this approach.
 Given the turbulence in the financial markets witnessed
during the first decade of this century, the focus of
academia and practitioners alike has again shifted to the
Markowitz approach for selecting assets in a portfolio, in
particular minimum variance portfolios.

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MODERN PORTFOLIO THEORY
 Markowitz portfolios: The groundbreaking
insight of Markowitz was that the risk/return
profiles of single assets should not be viewed
separately but in their portfolio context.
 In this respect, portfolios are considered to be
efficient if they are either risk minimal for a
given return level or have the maximum return
for a given level of risk.

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 It is assumed that there are N assets and that they are
infinitely divisible and the returns of these assets are
jointly normally distributed.
 The portfolio return 𝑟 is defined by the scalar
product of the (N × 1) weight and return vectors 𝜔
and 𝜇.
 The portfolio risk is measured by the portfolio
variance 𝜎𝑤
2
= 𝜔′
𝛴𝜔, where 𝛴 denotes the positive
semi-definite variance-covariance matrix of the
assets’ returns.

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 The risk/return profile of an efficient portfolio can be
expressed in terms of a linear combination between
the global minimal variance (GMV) portfolio and any
other efficient portfolio.
 The covariance between these two portfolios equals
the variance of the minimum variance portfolio.
 Though it might not be evident at first glance, it
should be stressed that the only constraint with respect
to the portfolio weights is that their sum equals one.

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 Equation (5.3) describes a hyperbola for efficient
mean-variance portfolios.
 The risk/return points that are enclosed by the
hyperbola are referred to as the feasible portfolios,
although these are sub-optimal.
 In other words, portfolios exist that have either a
higher return for a given level of risk or are less risky
for certain portfolio return. Both instances would
yield a higher utility for the investor.

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 The optimal portfolio is located at the tangency point
of this line and the upper branch of the efficient
frontier.
 This is given when the slope is greatest and hence the
Sharpe ratio is at its maximum.
 The portfolio that is characterized at this tangency
point is therefore also referred to as the maximum
Sharpe ratio (MSR) portfolio.

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 Sharpe Ratio: The Sharpe ratio was developed by
Nobel laureate William F. Sharpe and is used to
help investors understand the return of an
investment compared to its risk. The ratio is the
average return earned in excess of the risk-free rate
per unit of volatility or total risk.

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 Sortino Ratio: The Sortino ratio is a variation of
the Sharpe ratio that differentiates harmful volatility
from total overall volatility by using the asset's
standard deviation of negative portfolio returns
termed as downside deviation or semi-deviation,
instead of the total standard deviation of portfolio
returns.

35
 Treynor Ratio: The Treynor ratio, also known as the
reward-to-volatility ratio, is a performance metric for
determining how much excess return was generated for
each unit of risk taken on by a portfolio. Risk in the
Treynor ratio refers to systematic risk as measured by a
portfolio's beta.

36
 The Calmar Ratio: The Calmar ratio is a gauge of
the performance of investment funds such as hedge
funds and commodity trading advisors (CTAs). It is a
function of the fund's average compounded
annual rate of return versus its maximum drawdown.
The higher the Calmar ratio, the better it performed
on a risk-adjusted basis during the given time frame,
which is mostly commonly set at 36 months.

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EMPIRICAL MEAN-VARIANCE PORTFOLIO
 The theoretical portfolio concepts outlined in the
previous section are unfortunately not directly
applicable in practice.
 In empirical applications these unknown parameters
must be replaced by estimates.
 In empirical simulations and studies it was found that
the weights of the former kind of portfolio
optimizations are characterized by wide spans and
unpredictable behavior over time.

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 The errors of the estimates for the expected returns
and the variance-covariance matrix could be
quantified heuristically beforehand by means of
Monte Carlo simulations.
 However, the case of constrained portfolio
optimizations discussed earlier can yield unintuitive
results.
 This implies that, in addition to estimation errors, a
model error often exists.

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 In general an investor is eager to achieve a portfolio
allocation that comes as close as possible to the efficient
frontier.
 But the implementation of long-only constraints is
undesirable for other reasons.
 The imposition of constraints is not a panacea for all kinds
of portfolio strategies and optimizations.
 Part III of this book will address these issues in more detail
and also offer examples of how these more recent advances
in portfolio construction can be explored with R.

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CHAP 3 - FINANCIAL MARKET DATA
 Stylized facts for univariate series: Before we turn to the
topic of modelling financial market risks, it is worthwhile
to consider and review typical characteristics of financial
market data.
 These are summarized in the literature as “stylized facts”
which are as follows:
1. Time series data of returns, in particular daily return
series, are in general not independent and identically
distributed (iid).

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2. The volatility of return processes is not constant with respect
to time.
3. The absolute or squared returns are highly autocorrelated.
4. The distribution of financial market returns is leptokurtic. The
occurrence of extreme events is more likely compared to the
normal distribution.
5. Extreme returns are observed closely in time (volatility
clustering).
6. The empirical distribution of returns is skewed to the left;
negative returns are more likely to occur than positive ones.

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 As an example, we will now check whether these
stylized facts are applicable to the returns of
Siemens stock.
 The data set of daily returns is contained in the
package evir. This series starts on 2 January 1973
and ends on 23 July 1996, and comprises 6146
observations.
 We will Practice R codes in R Studio.

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 Stylized facts for multivariate series: From the
portfolio point of view the characteristics of
multivariate return series are of interest.
 Here we will focus on these stylized facts:
1. The absolute value of cross-correlations between
return series is less pronounced and
contemporaneous correlations are in general the
strongest.
2. In contrast, the absolute or squared returns do
show high cross-correlations.

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3. This empirical finding is similar to the
univariate case.
4. Contemporaneous correlations are not
constant over time.
5. Extreme observations in one return series are
often accompanied by extremes in the other
return series.

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Implications for Risk Models: With respect to risk
models and the risk measures derived from them the
following normative requirements can be deduced so far:
1. Risk models which assume iid processes for the
losses are not adequate during all market episodes.
2. Risk models that are based on the normal
distribution will fall short in predicting the
frequency of extreme events (losses).

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3. Risk models should be able to encompass and
address the different volatility regimes. This
means that the derived risk measures should be
adaptive to changing environments of low and
high volatility.
4. In the portfolio context, the model employed
should be flexible enough to allow for changing
dependencies between the assets; in particular, the
co-movement of losses should be taken care of.

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 As an example, we will now check whether these
stylized facts are applicable to the European Equity
Market.
 The data set of daily returns is contained in the
package zoo, as well as package EuStockMarkets.
 We will Practice R codes in R Studio.

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