Transcript The Linear Model

Value at Risk
“What loss level is such that we are
X% confident it will not be exceeded
in N business days?”
Value at Risk is an attempt to provide a
single number summarizing the total
risk in a portfolio of financial assets.
THE VaR MEASURE
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I am X percent certain there will not be a loss of
more than V dollars in the next N days
The variable V is the VaR of the portfolio. It is a
function of two parameters: the time horizon (N
days) and the confidence level (X%).
For example, when N=5 and X=97, VaR is the
third percentile of the distribution of changes in
the value of the portfolio over the next 5 days.
THE VaR MEASURE
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All senior managers are very comfortable with
the idea of compressing all the Greek letter for
all the market variables underlying a portfolio
into a single number.
THE VaR MEASURE
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VaR is the loss level that will not be exceeded
with a specified probability
C-VaR (or expected shortfall) is the expected
loss given that the loss is greater than the VaR
level
Although C-VaR is theoretically more
appealing, it is not widely used
THE VaR MEASURE
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Regulators base the capital they require banks
to keep on VaR
The market-risk capital is k times the 10-day
99% VaR where k is at least 3.0
THE VaR MEASURE
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Time Horizon
Instead of calculating the 10-day, 99% VaR
directly analysts usually calculate a 1-day
99% VaR and assume
10 - day VaR  10 1- day VaR
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This is exactly true when portfolio changes on
successive days come from independent
identically distributed normal distributions
Historical Simulation
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Create a database of the daily movements in
all market variables.
Suppose that VaR is to be calculated for
portfolio using a 1-day time horizon, a 99%
confidence level, and 501 days of data.
Scenario 1 is where the percentage changes
in the values of all variable are the same as
they were between Day0 and Day1.
Historical Simulation
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This defines a probability distribution for daily
changes in the value of the portfolio.
Historical Simulation
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Today is Day 500, tomorrow is Day501
The ith trial assumes that the value of the market
variable tomorrow (i.e., on day m+1) is
vi
vm
vi 1
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In our example, m=500. For the first variable, the value
today, V500 , is 25.85. Also V0=20.33 and V1=20.78. It
follows that the value of the first market variable in the
first scenario is
25.85×20.78÷20.33=26.42
Historical Simulation
Historical Simulation
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We are interested in the 1-percentile point of
the distribution of changes in the portfolio
value.
Because there are total of 500 scenarios in
Table 20.2 we can estimate this as the
fifth-worst number in the final column of the
table.
Model-Building Approach
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Daily Volatilities
In option pricing we measure volatility “per
year”
In VaR calculations we measure volatility “per
day
 day 
 year
252
Model-Building Approach
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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
Model-Building Approach
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Single-Asset Case
Microsoft Example (page 448)
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
Model-Building Approach
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Microsoft Example (page 448)
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
Model-Building Approach
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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
Model-Building Approach
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AT&T Example (page 449)
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
Model-Building Approach
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AT&T Example (page 449)
The VaR is
158,114  2.33  $368,405
Model-Building Approach
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Two-Asset Case
Now consider a portfolio consisting of both
Microsoft and AT&T
Suppose that the correlation between the
returns is 0.3
A standard result in statistics states that
 X Y   2X  Y2  2 X  Y
Model-Building Approach
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In this case X = 200,000 and Y = 50,000
and  = 0.3. The standard deviation of the
change in the portfolio value in one day is
therefore 220,227
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
Model-Building Approach
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Less than perfect correlation leads to some of
the risk being “diversified away”
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
The Linear Model
n
P    i xi
i 1
n
n
 2P    i  j i  j ij
i 1 j 1
n
2
i
i 1
 2P    i2  2  i  j i  j ij
i j
where i is the volatility of variable i
and  P is the portfolio' s standard deviation
The Linear Model
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In the example considered in the previous
section, S1=0.02, S2=0.01, and the correlation
between the returns is 0.3.
The 10-day 99% VaR
is 0.22×2.33×√10=1.623 million
The Linear Model
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Handling Interest Rates:
One possibility is to assume that only parallel
shifts in the yield curve occur
△P = -DPΔy
This approach does not usually give enough
accuracy.
The Linear Model
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Handling Interest Rates: Cash Flow Mapping
We choose as market variables bond prices with
standard maturities (1mth, 3mth, 6mth, 1yr, 2yr,
5yr, 7yr, 10yr, 30yr)
a simple example of a portfolio consisting of a
long position in a single Treasury bond with a
principal of 1million maturing in 0.8 year
We suppose that the bond provides a coupon of
10% per annum payable semiannually
The Linear Model
The Linear Model
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When Linear Model Can be Used
Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
The Linear Model
The Linear Model and Options
Consider a portfolio of options dependent on a
single stock price, S. Define
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P

S
and
x 
S
S
The Linear Model
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As an approximation
P   S  S x
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Similarly when there are many underlying
market variables
P   Si i xi
i
where i is the delta of the portfolio with respect
to the ith asset
The Linear Model
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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
P  120  1,000 x1  30  20,000 x2
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As an approximation
where x1 and x2 are the percentage changes
in the two stock prices
Quadratic Model
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Gamma is defined as the rate of change of the delta with
respect to the market variable.
Gamma measures the curvature of the relationship
between the portfolio value and an underlying market
variable.
Quadratic Model
Quadratic Model
Quadratic Model
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For a portfolio dependent on a single stock
price it is approximately true that
1
2
P  S   (S )
2
this becomes
1 2
P  S x  S  (x) 2
2
Quadratic Model
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With many market variables we get an
expression of the form
n
n
1
 P   S i  i  xi   S i S j  ij  xi  x j
i 1
i 1 2
where
P
i 
Si
2P
 ij 
Si S j
This is not as easy to work with as the linear model
Monte Carlo Simulation
To calculate VaR using M.C. simulation we
 Value portfolio today
 Sample once from the multivariate
distributions of the xi
 Use the xi to determine market variables at
end of one day
 Revalue the portfolio at the end of day
Monte Carlo Simulation
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Calculate P
Repeat many times to build up a probability
distribution for P
VaR is the appropriate percentile of the
distribution times square root of N
For example, with 1,000 trial the 1 percentile
is the 10th worst case.
Monte Carlo Simulation
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Speeding Up Monte Carlo
Use the quadratic approximation to calculate
P
Comparison of Approaches
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Model building approach assumes normal
distributions for market variables. It tends to
give poor results for low delta portfolios
Historical simulation lets historical data
determine distributions, but is computationally
slower
Stress Testing And Back
Testing
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Stress Testing
This involves testing how well a portfolio performs
under some of the most extreme market moves
seen in the last 10 to 20 years
Back-Testing
Tests how well VaR estimates would have
performed in the past
We could ask the question: How often was the
actual 10-day loss greater than the 99%/10 day VaR?
Principal Components Analysis
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One approach to handing the risk arising from
groups of highly correlated market variable is
principal components analysis.
This takes historical data on movements in the
market variables and attempts to define a set of
components or factors that explain the movement
The approach is best illustrated with an example.
The market variable we will consider are 10 US
Treasury rates with maturities between 3 months
and 30 years.
Principal Components Analysis
Principal Components Analysis
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The first factor is a roughly parallel shift
(83.1% of variation explained)
The second factor is a twist (10% of variation
explained)
The third factor is a bowing (2.8% of variation
explained)
Principal Components Analysis
Example: Sensitivity of portfolio to rates ($m)
1 yr
+10
2 yr
+4
3 yr
-8
4 yr
-7
5 yr
+2
Sensitivity to first factor is from Table 18.3:
10×0.32 + 4×0.35 − 8×0.36 − 7 ×0.36 +2 ×0.36
= −0.08
Similarly sensitivity to second factor = − 4.40
Principal Components Analysis
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As an approximation
P  0.08 f1  4.40 f 2
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The f1 and f2 are independent
The standard deviation of P (from Table
20.4) is
0.082 17.49 2  4.40 2  6.052  26.66
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The 1 day 99% VaR is 26.66 × 2.33 = 62.12