Download 2 hundred million +2 hundred million

CHAPTER 2
1
MARKET-RISK
MEASUREMENT
Introduction
2
 Market risk for a bank
 Trading group in a bank can trade the financial
instruments (for example, bonds and stocks) in the
market
 When the values of traced instruments changes, the
bank might get a loss

The common approaches to measuring market risk
Standard deviation, variance: total risk
 Capital Asset Pricing Model (CAPM): systematic risk
 Value at Risk (VaR)


In this class, we focus on VaR
 Risk vs. Uncertainty
3
Average (1/2)
4
 Equation: sum of probability * average dollars
20%
80%
-2 hundred
million
+2 hundred million
 Compute: 20%*-2+80%*+2=1.2 (hundred million)
Average(2/2)
5
 Determine : Low average dollars has higher risk
 Advantage: easy to understand
 Disadvantage:
FX:
Company A
Company B
20%
-2 hundred
million
100% firm gets 1.2
hundred million
80%
+2 hundred million
The average dollars is the same (1.2 hundred million), but the
risk is different
Variance (Standard Deviation)(1/2)
6
 Equation: sum of probability * (dollars-average dollars)^2
20%
80%
-2 hundred
million
+2 hundred million
 Compute:
 Variance: 20%*(-2-1.2)^2+80%*(+2-1.2)^2=2.56
(hundred million)
 Standard deviation: 2.56^0.5=1.6 (hundred million)
Variance(Standard Deviation)(2/2)
7
 Determine : high variance or standard deviation has
higher risk
 Disadvantage:
FX:
Company C
20%
80%
-2 hundred
million
+2 hundred million
Company D
80%
20%
-2 hundred
million
+2 hundred million
The variance or standard deviation is the same (1.6 hundred
million), but the risk is different
VALUE AT RISK
8
 Value at Risk (VaR) is a measure of market risk that tries
objectively to combine the sensitivity of the portfolio to
market changes and the probability of a given market
change.
 In overall, VaR is the best single risk-measurement
technique available
 As such, VaR has been adopted by the Basel Committee to
set the standard for the minimum amount of capital to be
held against market risks
VALUE AT RISK
9
 VaR summarizes the predicted maximum loss (or
worst loss) over a target horizon within a given
confidence interval. (Jorion（2000))
 How to compute VaR



Know the distribution of the asset
Know the confidence interval or significant level
Know the time horizon (time period)
 Typically, a severe loss is defined as a loss that has
a 1% (significant level) chance of occurring on any
given day
VALUE AT RISK
10
 Feature
 Unit: dollars. Not standard deviation or ratio (Sharpe ratio)
 Estimate value. is computed by statistics, given the
confidence interval to compute the estimate value, not
certain value.
 VaR is estimated in normal market. VaR fails to tell you the
maximum loss in stress market, such as 2008 financial crisis.
VALUE AT RISK
11
 A common assumption is that movements in the
market have a Normal probability distribution,
meaning there is a 1% (significant level) chance that
losses will be greater than 2.32 standard deviations.
 Assuming a Normal distribution, 99% (confidence
interval) VaR can be defined as follows:
standard deviation
of the portfolio's
value
The subscript T in the VaR expression refers to the time period over which
the standard deviation of returns is calculated. VaR can be calculated for any
time horizon. For trading operations, a one-day horizon is typically used.
12
VALUE AT RISK
13
For an example of a VaR
statement, consider an equity
portfolio with a daily standard
deviation of $10 million. Using
the assumption of a Normal
distribution, the 99%
confidence interval VaR is $23
million. We would expect that
the losses would be greater
than $23 million on 1% of
trading days.
VALUE AT RISK
14
 Senior management should clearly understand that
VaR is not the worst possible loss.
 Losses equal to the size of VaR are expected to
happen several times per year
 VaR is therefore not equal to capital
 We will discuss the relationship between VaR and
capital in great depth in later chapters
 but a very rough rule of thumb is that the capital
should be 10 times VaR
VaR for Bonds
15
 For a bond, VaR can be approximated by multiplying
the dollar duration by the "worst-case" daily interest
move. This gives the value change in the "worst
case."
VaR for Bonds
16
the "worst-case" daily interest
move for 1% chance in one day
If we assume that interest-rate
movements have a Normal
probability distribution, then the 1%
worst case will correspond to2.32
standard deviations of the daily
rate movements
VaR for Bonds
17
 As an example. If the duration is 7 years (time duration in
term of year), the current price is $100 (the dollar duration
is 7*100), and the daily standard deviation in the absolute
level of interest rates is 0.2%, then the VaR is
approximately $3.24:
VaR = $100 *7 *2.32 * 0.2 = $3.24
 The approximations that we made here were as follows:
the changes in the rate is Normally distributed
 The change in the price can be well-approximated by the
linear measure of duration (no consider the convexity).

VaR for Equities
18
 The VaR for an equity is easy to calculate
 If we assume that equity prices have a Normal distribution.
The VaR is then the number of shares held (N), multiplied
by 2.32 and the standard deviation of the equity price (σE):
VaR = 2.32* σE * N
So, for example, if we held 100 shares of IBM, and the daily
standard deviation of the price was 10 cents, the VaR would
be
$23.2: VaR = 2.32 *$0.1 * 100 = $23.2
VaR for Options
19
 A simple approximation of the VaR to an option can
be obtained using the linear sensitivities
 The standard deviation of the option price caused by
changes in the stock price is simply the standard
deviation of the stock price multiplied by delta
VaR for Options
20
Delta: the derivative of value
of option with respect to the
price of underlying stock
The standard deviation of the
underlying stock’s price
How to calculate volatility of each asset?
 JP Morgan's RiskMetrics system
 Equally-Weighted Moving Average (that is Simple Moving
Average (SMA); variance)
2
(
R

R
)
ˆ t2   t
, volatility  ˆ t
T 1
i 1
 Exponentially-Weighted Moving Average (EWMA)
T
ˆ t2  ˆ t21  (1   ) Rt2 , volatility  ˆ t
 λ: decay rate, 0<λ<1. The more the λ value, the less last
observation affects the current dispersion estimation.
 The formula of the EWMA model can be rearranged to the

following form:
2
t 1 2
ˆ t  (1   )  Rt i
i 0
21
How to calculate volatility of each asset?
22
 The EWMA model has an advantage in comparison with
SMA, because the EWMA has a memory.
 Using the EWMA allows one to capture the dynamic
features of volatility. This model uses the latest
observations with the highest weights in the volatility
estimate. However, SMA has the same weights for any
observation.
 JP Morgan suggests: the optimal value for current daily
dispersion (volatility) is =0.94; the optimal value for
current monthly dispersion (volatility) is =0.97
 Using moving window method to calculate the daily
volatility and then calculate the daily VaR
General Considerations in Using VaR
23
 In the discussion above, we gave approximations
for calculating the one-day 99% VaR.
 However, there are several conventions in use for
the VaR probability, which implies a different
multiplication factor for the standard deviation.
 The most common alternative is to set the tail
probability at 2.5%. If a Normal distribution is
assumed, this implies a multiplier of 1.96 rather
than 2.32
General Considerations in Using VaR
24
 In some cases, we may wish to know the VaR for
the potential losses over multiple days.
 A reasonable approximation to the multi day
VaR is that it is equal to the one-day VaR
multiplied by the square root of the number of
days:
General Considerations in Using VaR
25
 This relationship requires the following assumptions:
 Changes
in market factors are Normally distributed.
 The one-day VaR is constant over the time period.
 There is no serial correlation. Serial correlation is
present if the results on one day are not independent
of the results on a previous day.
General Considerations in Using VaR
26
 In general, for trading operations it is safe to
assume that if the term VaR is used without a
specified time, it means one-day VaR.
 the term VaR is also used to refer to the potential
loss from asset liability management, in which case
a monthly or yearly horizon is used.
 Also, the term "credit VaR" is sometimes used to
describe the loss distribution from a credit
portfolio. This is quite different from the VaR used
for trading portfolios.
General Considerations in Using VaR
27
 The major limitation of VaR is that it describes what
happens on bad days (e.g., twice a year) rather than
terrible days (e.g., once every 10 years).
 VaR is therefore good for avoiding bad days, but to
avoid terrible days you still need stress and scenario
tests.