Transcript 1 Value at Risk

Value at Risk
Chapter 18
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.1
The Question Being Asked in VaR
“What loss level is such that we are X%
confident it will not be exceeded in N
business days?”
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.2
VaR and Regulatory Capital


Regulators base the capital they require
banks to keep on VaR
The market-risk capital is k times the 10day 99% VaR where k is at least 3.0
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.3
VaR vs. C-VaR
(See Figures 18.1 and 18.2)



VaR is the loss level that will not be
exceeded with a specified probability
C-VaR is the expected loss given that the
loss is greater than the VaR level
Although C-VaR is theoretically more
appealing than VaR, it is not widely used
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.4
Advantages of VaR



It captures an important aspect of risk
in a single number
It is easy to understand
It asks the simple question: “How bad can
things get?”
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.5
Historical Simulation
(See Table 18.1 and 18.2)




Create a database of the daily movements in all
market variables.
The first simulation trial assumes that the
percentage changes in all market variables are
as on the first day
The second simulation trial assumes that the
percentage changes in all market variables are
as on the second day
and so on
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.6
Historical Simulation continued




Suppose we use m days of historical data
Let vi be the value of a market variable on day i
There are m-1 simulation trials
The ith trial assumes that the value of the
market variable tomorrow (i.e., on day m+1) is
vi
vm
vi 1
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.7
The Model-Building Approach


The main alternative to historical simulation is to
make assumptions about the probability
distributions of return on the market variables
and calculate the probability distribution of the
change in the value of the portfolio analytically
This is known as the model building approach or
the variance-covariance approach
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.8
Daily Volatilities


In option pricing we express volatility as
volatility per year
In VaR calculations we express volatility
as volatility per day
 day 
 year
252
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.9
Daily Volatility continued


Strictly speaking we should define day as
the standard deviation of the continuously
compounded return in one day
In practice we assume that it is the
standard deviation of the percentage
change in one day
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.10
Microsoft Example



We have a position worth $10 million in
Microsoft shares
The volatility of Microsoft is 2% per day
(about 32% per year)
We use N = 10 and X = 99
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.11
Microsoft Example continued


The standard deviation of the change in
the portfolio in 1 day is $200,000
The standard deviation of the change in 10
days is
200,000 10  $632,456
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.12
Microsoft Example continued



We assume that the expected change in
the value of the portfolio is zero (This is
OK for short time periods)
We assume that the change in the value of
the portfolio is normally distributed
Since N(–2.33)=0.01, the VaR is
2.33  632,456  $1,473,621
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.13
AT&T Example




Consider a position of $5 million in
AT&T
The daily volatility of AT&T is 1%
(approx 16% per year)
The S.D per 10 days is
50,000 10  $158,144
The VaR is
158,114  2.33  $368,405
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.14
Portfolio (See Trading Note 18.1)


Now consider a portfolio consisting of both
Microsoft and AT&T
Suppose that the correlation between the
returns is 0.3
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.15
S.D. of Portfolio

A standard result in statistics states that
 X Y  2X  Y2  2r X  Y

In this case X = 200,000 and Y = 50,000
and r = 0.3. The standard deviation of the
change in the portfolio value in one day is
therefore 220,227
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.16
VaR for Portfolio

The 10-day 99% VaR for the portfolio is
220,227  10  2.33  $1,622,657
The benefits of diversification are
(1,473,621+368,405)–1,622,657=$219,369
 What is the incremental effect of the AT&T
holding on VaR?

Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.17
The Linear Model
We assume
 The daily change in the value of a portfolio
is linearly related to the daily returns from
market variables
 The returns from the market variables are
normally distributed
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.18
The General Linear Model
continued (equations 18.1 and 18.2)
n
P   wi xi
i 1
n
n
   wi w j  i  j r ij
2
P
i 1 j 1
n
 2P   wi2  i2  2 wi w j  i  j r ij
i 1
i j
where wi is the amount invested in asset i,
 i is the volatility of asset i, r ij is the
correlatio n between returns on assets i and j ,
and  P is the portfolio' s standard deviation
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.19
Handling Interest Rates



We do not want to define every bond as a
different market variable
We therefore choose as assets zero-coupon
bonds with standard maturities: 1-month, 3
months, 1 year, 2 years, 5 years, 7 years, 10
years, and 30 years
Cash flows from instruments in the portfolio are
mapped to bonds with the standard maturities
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.20
When Linear Model Can be Used




Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.21
The Linear Model and Options

Consider a portfolio of options dependent
on a single stock price, S. Define
P

S

and
S
x 
S
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.22
Linear Model and Options
continued (equations 18.3 and 18.4)

As an approximation
P   S  S x

Similar when there are many underlying
market variables
P   S i  i xi
i
where i is the delta of the portfolio with
respect to the ith asset
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.23
Example


Consider an investment in options on
Microsoft and AT&T. Suppose the stock
prices are 120 and 30 respectively and the
deltas of the portfolio with respect to the two
stock prices are 1,000 and 20,000
respectively
As an approximation
P  120 1,000x1  30  20,000x2
where x1 and x2 are the percentage
changes in the two stock prices
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.24
Skewness
(See Figures 18.3, 18.4 , and 18.5)
The linear model fails to capture skewness
in the probability distribution of the
portfolio value.
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.25
Quadratic Model
For a portfolio dependent on a single stock
price
1
2
P  S  g (S )
2
where g is the gamma of the portfolio. This
becomes
1 2
P  S x  S g (x) 2
2
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.26
Usual Approach to Estimating
Volatility (equation 18.6)




Define n as the volatility per day between day n-1
and day n, as estimated at end of day n-1
Define Si as the value of market variable at end of
day i
Define ui= ln(Si/Si-1)
The usual estimate of volatility from m observations
is:
1 m
2
 
(
u

u
)
 n i
m  1 i 1
2
n
1 m
u   u n i
m i 1
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.27
Simplifications
(equations 18.7 and 18.8)



Define ui as (Si–Si-1)/Si-1
Assume that the mean value of ui is zero
Replace m–1 by m
This gives
1 m 2
  i 1 un i
m
2
n
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.28
Weighting Scheme
Instead of assigning equal weights to the
observations we can set
  i 1  i u
m
2
n
2
n i
where
m

i 1
i
1
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.29
EWMA Model


(equation 18.10)
In an exponentially weighted moving
average model, the weights assigned to
the u2 decline exponentially as we move
back through time
This leads to
  
2
n
2
n 1
 (1  )u
2
n 1
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.30
Attractions of EWMA




Relatively little data needs to be stored
We need only remember the current
estimate of the variance rate and the most
recent observation on the market variable
Tracks volatility changes
RiskMetrics uses  = 0.94 for daily
volatility forecasting
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.31
Correlations
Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1
 Also
u,n: daily vol of U calculated on day n-1
v,n: daily vol of V calculated on day n-1
covn: covariance calculated on day n-1
covn = rn u,n v,n
where rn on day n-1

Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.32
Correlations continued
(equation 18.12)
Using the EWMA
covn = covn-1+(1-)un-1vn-1
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.33
Comparison of Approaches


Model building approach has the
disadvantage that it assumes that market
variables have a multivariate normal
distribution
Historical simulation is computationally
slower and cannot easily incorporate
volatility updating schemes
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.34
Back-Testing


Tests how well VaR estimates would have
performed in the past
We could ask the question: How often was
the loss greater than the 99%/10 day
VaR?
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.35
Stress Testing

This involves testing how well a portfolio
would perform under some of the most
extreme market moves seen in the last 10
to 20 years
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
18.36