This document discusses various concepts related to market risk, including value at risk (VaR), stress testing, and back testing. It defines VaR as a threshold loss value such that the probability of portfolio loss exceeding this value over a time horizon is a given probability, like 5%. It describes how VaR is calculated using historical, variance-covariance, and Monte Carlo simulation methods. Stress testing and back testing are also introduced as tools to assess risk under unfavorable conditions and test trading strategies on historical data.

1. Chapter - IV
Introduction: Market Risk
Market Risk (includes asset liability management)
Yield Curve Risk Factor - Domestic and global
contexts-handling multiple risk factor-principal
component analysis - value at Risk (VAR) :
implementation of a VAR system - Additional Risk in
fixed income markets - Stress testing - Bank testing.

2. Introduction to Market Risk:
Market Risk may be defined as the possibility of loss to bank caused
by the changes in the market variables. It is the risk that the value of
on-/off-balance sheet positions will be adversely affected by
movements in equity and interest rate markets, currency exchange
rates and commodity prices.
Market risk is the risk to the bank’s earnings and capital due to
changes in the market level of interest rates or prices of securities,
foreign exchange and equities, as well as the volatilities, of those
prices.
Market Risk Management provides a comprehensive and dynamic
frame work for measuring, monitoring and managing liquidity,
interest rate, foreign exchange and equity as well as commodity price
risk of a bank that needs to be closely integrated with the bank’s
business strategy.

3. Meaning of Market Risk: Market risk is the risk that the value of an
investment will decrease due to moves in market factors. Volatility
frequently refers to the standard deviation of the change in value of a
financial instrument with a specific time horizon. It is often used to
quantify the risk of the instrument over that time period. Volatility is
typically expressed in annualized terms, and it may either be an
absolute number ($5) or a fraction of the initial value (5%).
1. Equity risk or the risk that stock prices will change.
2. Interest rate risk or the risk that interest rates will change.
3. Currency risk or the risk that foreign exchange rates will
change.
4. Commodity risk or the risk that commodity prices (i.e. grains,
metals, etc.) will change.
5. Equity index risk or the risk that stock or other index prices
will change adversely.

4. Definition: “The possibility for an investor to experience losses due
to factors that affect the overall performance of the financial
markets”, Market risk, also called "systematic risk," cannot be
eliminated through diversification, though it can be hedged against.
The risk that a major natural disaster will cause a decline in the
market as a whole is an example of market risk. Other sources of
market risk include recessions, political turmoil, changes in interest
rates and terrorist attacks.
Scenario analysis and stress testing is yet another tool used to assess
areas of potential problems in a given portfolio. Identification of
future changes in economic conditions like – economic/industry
overturns, market risk events, liquidity conditions etc. that could have
unfavorable effect on bank’s portfolio is a condition precedent for
carrying out stress testing. Market risk is typically measured using a
Value at Risk methodology.

5. Types of Market Risk: There are four major types of market risk:
a. Interest Rate Risk –
a. Basis Risk: LIBOR & MIBOR
b. Reprising Risk: Loan and Receivables
c. Yield Curve Risk: Differences between long and
short run interest rates.
d. Options Risk: Interest rate changes – Prepayment
of interest on loan amount.
b. Equity Price Risk
c. Foreign Exchange Risk
d. Commodity Price Risk
The Yield Curve: The yield curve is a graph which plots time (from
shortest to longest maturity date) on the horizontal access, and yield
on the vertical access. It is used to show the relationship between
yield and maturity. Different types of Yield Curves are as follows:-

6. The Normal Yield Curve: Normally the yield curve is upward sloping
showing that, all else being equal, a bond with a longer maturity pays a
higher yield than the same bond with a shorter maturity. Another way of
saying this is that the longer the term of the loan or bond, the greater the
chance something unexpected will happen.
Generally speaking, individuals and institutions prefer to lend money for
shorter periods of time, rather than longer periods of time.

7. The Steep Yield Curve: When investors are expecting interest rates to rise
in the future, it makes sense that they are going to demand a higher rate of
return when buying longer term bonds. If longer term bonds did not pay a
higher rate of interest in this situation, investors would simply buy shorter
term bonds; with the expectation that when the bonds mature, they would be
able obtain a higher return on the next purchase. Often times, when the
economy is coming out of a recession, future interest rate expectations will
increase. This is because economic recoveries are normally accompanied by
corporations wanting to borrow more (for investment) which increases the
demand for money, putting upward pressure on interest rates.

8. The Flat Yield Curve: The yield curve is flat when yields of all
maturities are close to one another. This happens when inflation
expectations have decreased to the point where investors are
demanding no premium for tying their money up for longer periods of
time. Like with the inverted yield curve, when the yield curve moves
from normal to flat, this is generally a sign of a pending, or ongoing
economic slowdown.

9. The Humped Yield Curve: The yield curve is humped when short
and long term rates are closer to each other than with medium term
rates. This generally happens when there is either an increase in
demand, or decrease in supply of longer term bonds. In recent years
there has been a larger increase in demand for 30 year treasury bonds
for example, than for 20 year treasury bonds, causing the yield curve
for treasuries to often form a humped shape.

10. The Inverted Yield Curve: The yield curve inverts when longer term
rates are actually lower than short term interest rates. This happens
rarely, but when it does, it is one of the surest sings of an oncoming
economic slowdown, as investors anticipate less future demand for
money and therefore lower interest rates.

11. The level and structure of bond yields and the yield curve based upon
three principles.
1) Across different maturities along the yield curve, bond yields change
with the proportional change in the risk or potential volatility of the
bonds.
2) The incremental yield required as a bond's volatility increases by an
infinitesimal amount is determined by the riskless interest rate.
3) The relationship between the yield of a bond and the riskless interest
rate is governed by expectations of future riskless interest rates over
the term of the bond.
Determinants of Yield Curve Shape:
1. Pure Expectation of the Investors.
2. Liquidity Preference given by
Investors.
3. Preference and will of an Investors.

12. INTRODUCTION: VAR MODEL
In financial mathematics and financial risk management, value at
risk (VaR) is a widely used risk measure of the risk of loss on a
specific portfolio of financial assets. For a given portfolio, time
horizon, and probability p, the 100% VaR is defined as a threshold
loss value, such that the probability that the loss on the portfolio
over the given time horizon exceeds this value is p. This assumes
mark-to-market pricing, normal markets, and no trading in the
portfolio.
For example, if a portfolio of stocks has a one-day 5% VaR of $1
million, there is a 0.05 probability that the portfolio will fall in
value by more than $1 million over a one day period if there is no
trading. Informally, a loss of $1 million or more on this portfolio is
expected on 1 day out of 20 days (because of 5% probability). A
loss which exceeds the VaR threshold is termed a "VaR break.“

13. Value at Risk Advantages: Why Use VAR in Risk Management:
VaR has four main uses in finance: risk management, financial
control, financial reporting and computing regulatory capital.
VaR is sometimes used in non-financial applications as well.
Important related ideas are economic capital, back testing,
stress testing, expected shortfall, and tail conditional
expectation.
1. Value at Risk is easy to understand.
2. VAR is often available in financial software.
3. Comparing VAR of different assets and portfolios - Value at Risk is
applicable to stocks, bonds, currencies, derivatives, or any other
assets with price.
4. Everybody else uses VAR

14. Value at Risk (VAR) Limitations:
1. Value at Risk can be misleading: false sense of security
2. Value at Risk gets difficult to calculate with large portfolios
3. Value at Risk is not preservative
4. The resulting VAR is only as good as the inputs and assumptions
5. Different Value At Risk methods lead to different results
6. VAR does not measure worst case loss
How to calculate VaR
The generality of value-at-risk poses a computational challenge. In
order to measure market risk in a portfolio using value-at-risk, some
means must be found for determining the probability distribution of
that portfolio’s market value. Obviously, the more complex a portfolio
is the more asset categories and sources of market risk it is exposed to
the more challenging that task becomes.

15. It is worth distinguishing two concepts:
1. A value-at-risk measure is an algorithm with which we
calculate a portfolio’s value-at-risk.
2. A value-at-risk metric is our interpretation of the output
of the value-at-risk measure.
A value-at-risk metric, such as one-day 90% USD VaR, is specified
with three items:
a. A time horizon;
b. A probability;
c. A currency.
We know a portfolio’s current market value 0p. Its market value
1P at the end of the horizon is unknown. Define portfolio loss 1L as
If 0p exceeds 1P, the loss will be
positive. If 0p is less than 1P, the
loss will be negative, which is
another way of saying the
portfolio makes a profit.

16. Portfolio value 1P;
Asset values 1Si;
Key factors 1Ri.

17. APPROACHES / METHODS OF VAR:
You can see how the "VAR question" has three elements: a relatively
high level of confidence (typically either 95% or 99%), a time period
(a day, a month or a year) and an estimate of investment loss
(expressed either in dollar or percentage terms). There are three
normal methods of calculating VAR: the historical method, the
variance-covariance method and the Monte Corlo Simulation.
Historical Method: The historical method simply re-
organizes actual historical returns, putting them in order from
worst to best. It then assumes that history will repeat itself,
from a risk perspective.

19. Notice the red bars that compose the "left tail" of the histogram.
These are the lowest 5% of daily returns (since the returns are ordered
from left to right, the worst are always the "left tail"). The red bars
run from daily losses of 4% to 8%. Because these are the worst 5% of
all daily returns, we can say with 95% confidence that the worst daily
loss will not exceed 4%. Put another way, we expect with 95%
confidence that our gain will exceed -4%. That is VAR in a nutshell.
Let's re-phrase the statistic into both percentage and dollar terms:
• With 95% confidence, we expect that our worst daily loss will
not exceed 4%.
• If we invest $100, we are 95% confident that our worst daily
loss will not exceed $4 ($100 x -4%).
You can see that VAR indeed allows for an outcome that is worse than
a return of -4%. It does not express absolute certainty but instead
makes a probabilistic estimate.

20. The Variance-Covariance Method: The variance-covariance, or
delta-normal, model was popularized by J.P Morgan (now J.P.
Morgan Chase) in the early 1990s when they published the Risk
Metrics Technical Document. In the following, we will take the
simple case, where the only risk factor for the portfolio is the value of
the assets themselves. This method assumes that stock returns are
normally distributed. In other words, it requires that we estimate only
two factors - an expected (or average) return and a standard
deviation - which allow us to plot a normal distribution curve. Here
we plot the normal curve against the same actual return data
The idea behind the variance-covariance is similar to the ideas behind
the historical method - except that we use the familiar curve instead of
actual data. The advantage of the normal curve is that we
automatically know where the worst 5% and 1% lie on the curve.
They are a function of our desired confidence and the standard
deviation (
):

22. Confidence # of Standard Deviations (σ)
95% (high) - 1.96 x σ
99% (really high) - 2.58 x σ
The blue curve above is based on the actual daily standard deviation
of the QQQ, which is 2.64%. The average daily return happened to be
fairly close to zero, so we will assume an average return of zero for
illustrative purposes. Here are the results of plugging the actual
standard deviation into the formulas above:
Confidence # of σ Calculation Equals
95% (high) - 1.96 x σ - 1.96 x (2.64%) = -5.17%
99% (really high) - 2.58 x σ - 2.58 x (2.64%) = -6.81%

23. Monte Carlo Simulation: The third method involves developing a
model for future stock price returns and running multiple hypothetical
trials through the model. A Monte Carlo simulation refers to any
method that randomly generates trials, but by itself does not tell us
anything about the underlying methodology.
For most users, a Monte Carlo simulation amounts to a "black box"
generator of random outcomes. Without going into further details, we
ran a Monte Carlo simulation on the QQQ based on its historical
trading pattern. In our simulation, 100 trials were conducted. If we ran
it again, we would get a different result--although it is highly likely
that the differences would be narrow. Here is the result arranged into a
histogram (please note that while the previous graphs have shown
daily returns, this graph displays monthly returns):

25. To summarize, we ran 100 hypothetical trials of monthly returns for
the QQQ. Among them, two outcomes were between -15% and -20%;
and three were between -20% and 25%. That means the worst five
outcomes (that is, the worst 5%) were less than -15%. The Monte
Carlo simulation therefore leads to the following VAR-type
conclusion: with 95% confidence, we do not expect to lose more than
15% during any given month.
If the VAR is systematically “too low”, the model is underestimating
the risk and you tend to have too many occasions where the loss in the
portfolio exceeds the VAR. This can lead to an increase in the
“multiplier” for the capital calculation.
If the VAR is systematically “too high”, the model is over estimating
the risk and your regulatory capital charge will be too high

26. Stress Testing: Stress testing is the process of determining the ability
of a computer, network, program or device to maintain a certain level
of effectiveness under unfavorable conditions... Reasons can include:
- To determine breaking points or safe usage limits.
- To confirm intended specifications are being met.
- To determine modes of failure (how exactly a system fails)
- To test stable operation of a part or system outside standard usage
Techniques of Stress Testing:
• Simple Sensitivity Test: Short term impact on portfolio. Ex: +2 or
-2 changes in prices.
• Scenario Analysis: Based on historical event or a hypothetical
event.
• Maximum Loss: Most potentially damaging combination of
moves of market risk.
• Extreme Value Theory: Extreme possible circumstances & works
on “tails”.(High/Low)

27. Essentials of Good Stress Testing:
1. Be relevant to the Current Position
2. Consider changes in all relevant market rates.
3. Examine potential regime shifts (whether the current
risk parameters will hold or breakdown).
4. Stress tests should spur discussion.
5. Consider market illiquidity.
6. Consider the interplay of market and credit risk.
Steps of Stress Testing:
A. Generate Scenarios:
B. Revalue Portfolio:
C. Summaries Results:

28. Introduction to Back Testing:
The process of testing a trading strategy on prior time periods. Instead
of applying a strategy for the time period forward, which could take
years, a trader can do a simulation of his or her trading strategy on
relevant past data in order to gauge its effectiveness.
When you back test a theory, the results achieved are highly
dependent on the movements of the tested period. Back testing a
theory assumes that what happens in the past will happen in the
future, and this assumption can cause potential risks for the strategy.
Back testing is a valuable tool available in most trading platforms.
Dividing historical data into multiple sets to provide for in-sample
and out-of-sample testing can provide traders a practical and efficient
means for evaluating a trading idea and system. Since most traders
employ optimization techniques in back testing, it is important to then
evaluate the system on clean data to determine its viability.