Financial Risk Mgt - Lec 2 by Dr. Syed Muhammad Ali Tirmizi

FINANCIAL RISK MGT - FRM Lecture by;
Dr. Syed
Muhammad Ali
Tirmizi
DISCUSSION ON CONCEPTS
Part - 1
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INTRODUCTION
 In this lecture, we will cover the following basic
concepts relating FRM:
1. Valuation and Scenario Analysis
2. Interest Rate Risk
3. Volatility
4. Correlations and Copulas
5. Value at Risk (VaR)
6. Expected Shortfall (ES)

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VALUATION AND SCENARIO ANALYSIS
 Valuation and scenario analysis are two
important activities for financial institutions.
 Both are concerned with estimating future cash
flows, but they have different objectives.
 In valuation, a financial institution is interested
in estimating the present value of future cash
flows.

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 In scenario analysis, a financial institution is
interested in exploring the full range of situations
that might exist at a particular future time.
 Usually, it is the adverse outcomes that receive the
most attention because risk managers working for
the financial institution are interested in answering
the question: “How bad can things get?”

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• Suppose that a company sells one million one-
year European call options on a stock. The stock
price is $50 and the strike price is $55. The
company might calculate the theoretical value of
the options as +$4.5 million to the buyer and
−$4.5 million to itself. If it sells the options for,
say, $5 million, it can book $0.5 million of
profit.

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• However, a scenario analysis might reveal that
there is a 5% chance of the stock price rising to
above $80 in one year. This means that there is a
5% chance that the transaction will cost more
than $20 million, after the initial amount
received for the options has been taken into
account.

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 Scenario analysis focuses on extreme outcomes.
 In the example, $20 million is a possible net cost
of the transaction to the company.
 Investors do not live in a risk-neutral world.
 They do require higher expected returns when
the risks they are bearing increase, and this
applies to derivatives as well as to other
investments.

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 Risk-Neutral Valuation: The single most important result
in the valuation of derivatives is risk-neutral valuation.
 A risk-neutral world can be defined as an imaginary
world where investors require no compensation for
bearing risks.
 The single most important result in the valuation of
derivatives is risk-neutral valuation.
 A risk-neutral world can be defined as an imaginary
world where investors require no compensation for
bearing risks.

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 The world we live in is clearly not a risk-neutral
world.
 Investors do require compensation for bearing
risks.
 Investors do not live in a risk-neutral world.
 They do require higher expected returns when the
risks they are bearing increase, and this applies to
derivatives as well as to other investments.

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 In a risk-neutral world, all future expected cash flows
are discounted at the riskfree interest rate.
 Suppose that we are valuing a call option on a stock and
the risk-free interest rate is 3%.
 The steps in implementing risk-neutral valuation are:
1. Assume that the expected (average) future return on
the stock is 3%.
2. Calculate the expected payoff from the call option.
3. Discount the expected payoff at 3% to obtain the
option’s value.

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 We might be able to come up with a reasonable
estimate of the (real-world) expected future return on
the stock by estimating its beta and using the capital
asset pricing model.
 When we move from the risk-neutral world to the real
world, two things happen. The expected payoffs from
the derivative change, and the discount rate that must
be used for the payoffs changes.

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 The world we consider when carrying out a scenario
 analysis should be the real world, not the risk-neutral
world.
 To illustrate how scenario analysis is carried out,
suppose that the expected return on a stock in the real
world is 8%.
 The stock price is currently $30 per share and its
volatility is 25%. Assume you own 10,000 shares. How
much could you lose during the next year?

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VOLATILITY
 It is important for a financial institution to
monitor the volatilities of the market variables
(interest rates, exchange rates, equity prices,
commodity prices, etc.) on which the value of its
portfolio depends.
 A variable’s volatility, σ, is defined as the
standard deviation of the return provided by the
variable per unit of time when the return is
expressed using continuous compounding.

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VOLATILITY
 When volatility is used for risk management, the unit of
time is usually one day so that volatility is the standard
deviation of the continuously compounded return per day.
 Define Si as the value of a variable at the end of day i.
 The continuously compounded return per day for the
variable on day i is;

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VOLATILITY
 An alternative definition of daily volatility of a
variable is therefore the standard deviation of the
proportional change in the variable during a day.
 This is the definition that is usually used in risk
management.

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VOLATILITY
 Risk managers often focus on the variance rate rather than the
volatility.
 The variance rate is defined as the square of the volatility. The
variance rate per day is the variance of the return in one day. it
is correct to talk about the variance rate per day, but volatility
is per square root of day.

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VOLATILITY
 Risk managers usually calculate volatilities from
historical data, they also try and keep track of what
are known as implied volatilities.
 The one parameter in the Black–Scholes–Merton
option pricing model that cannot be observed directly
is the volatility of the underlying asset price.
 The CBOE publishes indices of implied volatility.
The most popular index, the VIX, is an index of the
implied volatility of 30-day options on the S&P 500.

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VALUE AT RISK (VaR)
 Value at risk (VaR) and expected shortfall (ES) are
attempts to provide a single number that summarizes the
total risk in a portfolio.
 VaR was pioneered by J P Morgan and is now widely
used by corporate treasurers and fund managers as well
as by financial institutions.
 When using the value at risk measure, we are interested
in making a statement of the following form:
 “We are X percent certain that we will not lose more than
V dollars in time T.”

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VALUE AT RISK (VaR)
 VaR can be calculated from either the probability
distribution of gains during time T or the probability
distribution of losses during time T.
 This section provides four simple examples to illustrate
the calculation of VaR.

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EXPECTED SHORTFALL (ES)
 A measure that can produce better incentives for traders than
VaR is expected shortfall (ES).
 This is also sometimes referred to as conditional value at risk,
conditional tail expectation, or expected tail loss. Whereas
VaR asks the question: “How bad can things get?” ES asks:
“If things do get bad, what is the expected loss?”
 ES, like VaR, is a function of two parameters: T (the time
horizon) and X (the confidence level).
 Indeed, in order to calculate ES it is necessary to calculate
VaR first. ES is the expected loss during time T conditional
on the loss being greater than the VaR.

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 For example, suppose that X = 99, T is 10 days,
and the VaR is $64 million.
 The ES is the average amount lost over a 10-day
period assuming that the loss is greater than $64
million.
 ES has better properties than VaR in that it
always recognizes the benefits of diversification.

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 Suppose that the VaR of a portfolio for a confidence
level of 99.9% and a time horizon of one year is $50
million.
 This means that in extreme circumstances
(theoretically, once every thousand years) the
financial institution will lose more than $50 million
in a year.
 It also means that if it keeps $50 million in capital it
will have a 99.9% probability of not running out of
capital in the course of one year.

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INTEREST RATE RISK
 Interest rate risk is more difficult to manage than
the risk arising from market variables such as
equity prices, exchange rates, and commodity
prices.
 One complication is that there are many
different interest rates in any given currency
(Treasury rates, interbank borrowing and
lending rates, swap rates, and so on).

30
INTEREST RATE RISK
 We need a function describing the variation of the
interest rate with maturity.
 This is known as the term structure of interest rates
or the yield curve.
 Considering the portfolio of many assets with
different maturities, there is always exposure to
movements in the one-year rate, the two-year rate,
the three-year rate, and so on.

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INTEREST RATE RISK
 There are a number of interest rates that are
important to financial institutions.
 Treasury Rates: Treasury rates are the rates an
investor earns on Treasury bills and Treasury
bonds.
 Treasury rates are regarded as risk-free rates in the
sense that an investor who buys a Treasury bill or
Treasury bond is certain that interest and principal
payments will be made as promised.

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INTEREST RATE RISK
 LIBOR: LIBOR is short for London interbank
offered rate.
 It is an unsecured short-term borrowing rate between
banks. LIBOR rates are quoted for a number of
different currencies and borrowing periods.
 It is now recognized that LIBOR is a less-than-ideal
reference rate for derivatives transactions because it
is determined from estimates made by banks, not
from actual market transactions.

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INTEREST RATE RISK
 The OIS Rate: An overnight indexed swap
(OIS) is a swap where a fixed interest rate for a
period (e.g., one month, three months, one year,
or two years) is exchanged for the geometric
average of overnight rates during the period.
 Overnight rates are the rates in the government-
organized interbank market where banks with
excess reserves lend to banks that need to
borrow to meet their reserve requirements.

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INTEREST RATE RISK
In the United States, the overnight
borrowing rate in this market is known as
the fed funds rate.
The effective fed funds rate on a particular
day is the weighted average of the
overnight rates paid by borrowing banks
to lending banks on that day.

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INTEREST RATE RISK
 Repo Rates: In a repo (or repurchase agreement), a
financial institution that owns securities agrees to sell
the securities for a certain price and to buy them back at
a later time for a slightly higher price.
 An important concept in interest rate markets is duration.
Duration measures the sensitivity of the value of a
portfolio to a small parallel shift in the zero-coupon
yield curve. The relationship is:

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CORRELATIONS AND COPULAS
 Suppose that a company has an exposure to two different
market variables.
 In the case of each variable, it gains $10 million if there
is a one-standard-deviation increase and loses $10
million if there is a one-standard-deviation decrease.
 If changes in the two variables have a high positive
correlation, the company’s total exposure is very high; if
they have a correlation of zero, the exposure is less but
still quite large; if they have a high negative correlation,
the exposure is quite low because a loss on one of the
variables is likely to be offset by a gain on the other.

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 It is important for a risk manager to estimate
correlations between the changes in market
variables as well as their volatilities when
assessing risk exposures.
 Copulas are the tools that provide a way of
defining a correlation structure between two or
more variables, regardless of the shapes of their
probability distributions.

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 Consider two correlated variables, V1 and V2.
 The marginal distribution of V1 (sometimes also
referred to as the unconditional distribution) is its
distribution assuming we know nothing about V2
and similar is the case with V1.
 If the marginal distributions of V1 and V2 are
normal, a convenient and easy-to work-with
assumption is that the joint distribution of the
variables is bivariate normal.

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 Suppose that variables V1 and V2 have the
triangular probability density functions.
 The density function for V1 peaks at 0.2. The
density function for V2 peaks at 0.5. For both
density functions, the maximum height is 2.0.
 To use what is known as a Gaussian copula, we
map V1 and V2 into new variables U1 and U2 that
have standard normal distributions.

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 The essence of the copula is therefore that,
instead of defining a correlation structure
between V1 and V2 directly, we do so indirectly.
 We map V1 and V2 into other variables that have
well-behaved distributions and for which it is
easy to define a correlation structure.
 The correlation between U1 and U2 is referred to
as the copula correlation.

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 The Gaussian copula is just one copula that can
be used to define a correlation structure.
 One of another copula is Student’s t-copula
which works in the same way as the Gaussian
copula except that the variables U1 and U2 are
assumed to have a bivariate Student’s
t-distribution instead of a bivariate normal
distribution.

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 Multivariate Copulas: Copulas can be used to
define a correlation structure between more than
two variables.
 The simplest example of this is the multivariate
Gaussian copula.
 Suppose that there are N variables, V1, V2,…,𝑉𝑁 and
that we know the marginal distribution of each
variable.
 For each i (1 ≤ i ≤ N),we transform Vi into Ui where
Ui has a standard normal distribution.

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 A Factor Copula Model: In multivariate copula
models, analysts often assume a factor model for
the correlation structure between the Ui. When
there is only one factor, which is:
Ui = 𝑎𝑖F + 1 − 𝑎𝑖
2
𝑍𝑖
 Where F and the 𝑍𝑖 have standard normal
distributions. The 𝑍𝑖 are uncorrelated with each
other and with F.

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 An important application of copulas for risk
managers is to the calculation of the distribution of
default rates for loan portfolios.
 Analysts often assume that a one-factor copula
model relates the probability distributions of the
times to default for different loans.
 The percentiles of the distribution of the number of
defaults on a large portfolio can then be calculated
from the percentiles of the probability distribution
of the factor.

Financial Risk Mgt - Lec 2 by Dr. Syed Muhammad Ali Tirmizi

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