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Zlzheng , Dept. of Finance,XMU
Chapter 20
Value at Risk
10:37 20.1
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?”
Zlzheng , Dept. of Finance,XMU10:37 20.2
VaR and Regulatory Capital
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
Zlzheng , Dept. of Finance,XMU10:37 20.3
VaR vs. C-VaR 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 VaR that is used by regulators in setting bank capital requirements
Zlzheng , Dept. of Finance,XMU10:37 20.4
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?”
Zlzheng , Dept. of Finance,XMU10:37 20.5
Historical Simulation to Calculate the One-Day VaR
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
Zlzheng , Dept. of Finance,XMU10:37 20.6
Historical Simulation continued
Suppose we use 501 days of historical data (Day 0 to Day 500)
Let vi be the value of a variable on day i There are 500 simulation trials The ith trial assumes that the value of the
market variable tomorrow is
5001
i
i
vv
v
Zlzheng , Dept. of Finance,XMU10:37 20.7
Example : Calculation of 1-day, 99% VaR for a Portfolio on Sept 25, 2008
8
Index Value ($000s)
DJIA 4,000
FTSE 100 3,000
CAC 40 1,000
Nikkei 225 2,000
Data After Adjusting for Exchange Rates
9
Day Date DJIA FTSE 100 CAC 40 Nikkei 225
0 Aug 7, 2006 11,219.38 6,026.33 4,345.08 14,023.44
1 Aug 8, 2006 11,173.59 6,007.08 4,347.99 14,300.91
2 Aug 9, 2006 11,076.18 6,055.30 4,413.35 14,467.09
3 Aug 10, 2006 11,124.37 5,964.90 4,333.90 14,413.32
… …… ….. ….. …… ……
499
Sep 24, 2008 10,825.17 5,109.67 4,113.33 12,159.59
500
Sep 25, 2008 11,022.06 5,197.00 4,226.81 12,006.53
Scenarios Generated
10
Scenario DJIA FTSE 100 CAC 40 Nikkei 225 Portfolio Value ($000s)
Loss ($000s)
1 10,977.08 5,180.40 4,229.64 12,244.10 10,014.334 −14.334
2 10,925.97 5,238.72 4,290.35 12,146.04 10,027.481 −27.481
3 11,070.01 5,118.64 4,150.71 11,961.91 9,946.736 53.264
… ……. ……. ……. …….. ……. ……..
499 10,831.43 5,079.84 4,125.61 12,115.90 9,857.465 142.535
500 11,222.53 5,285.82 4,343.42 11,855.40 10,126.439 −126.439
Example of Calculation: 08.977,1038.219,11
59.173,1106.022,11
Ranked Losses
11
Scenario Number Loss ($000s)
494 477.841
339 345.435
349 282.204
329 277.041
487 253.385
227 217.974
131 205.256
99% one-day VaR
The N-day VaR
The N-day VaR for market risk is usually assumed to be times the one-day VaR
In our example the 10-day VaR would be calculated as
This assumption is in theory only perfectly correct if daily changes are normally distributed and independent
Zlzheng , Dept. of Finance,XMU10:37 20.12
274,801385,25310
N
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
Zlzheng , Dept. of Finance,XMU10:37 20.13
Daily Volatilities
In option pricing we measure volatility “per year”
In VaR calculations we measure volatility “per day”
252y ear
day
Zlzheng , Dept. of Finance,XMU10:37 20.14
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
Zlzheng , Dept. of Finance,XMU10:37 20.15
Microsoft Example (page 440)
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
Zlzheng , Dept. of Finance,XMU10:37 20.16
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 10 456, $632,
Zlzheng , Dept. of Finance,XMU10:37 20.17
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 33 632 456 473 621. , $1, , Zlzheng , Dept. of Finance,XMU10:37 20.18
AT&T Example (page 441)
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
The VaR is50 000 10 144, $158,
158 114 2 33 405, . $368,
Zlzheng , Dept. of Finance,XMU10:37 20.19
Portfolio
Now consider a portfolio consisting of both Microsoft and AT&T
Assume that the returns of AT&T and Microsoft are bivariate normal
Suppose that the correlation between the returns is 0.3
Zlzheng , Dept. of Finance,XMU10:37 20.20
S.D. of Portfolio
A standard result in statistics states that
In this case X = 200,000 andY = 50,000 and = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227
YXYXYX 222
Zlzheng , Dept. of Finance,XMU10:37 20.21
VaR for Portfolio
The 10-day 99% VaR for the portfolio is
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?
657,622,1$33.210220,227
Zlzheng , Dept. of Finance,XMU10:37 20.22
The Linear Model
This assumes 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
Zlzheng , Dept. of Finance,XMU10:37 20.23
Markowitz Result for Variance of Return on Portfolio
24
sinstrument th and th of returns between ncorrelatio is
portfolio in instrument th on return of variance is
portfolio in instrument th of weightis
Return Portfolio of Variance
2
1 1
jiρ
iσ
iw
ww
ij
i
i
n
i
n
jjijiij
VaR Result for Variance of Portfolio Value (i= wiP)
day per value portfolio the in change the of SD the is return)daily of SD (i.e., instrument th of volatilitydaily the is
P
i
n
ijiji
jiijiiP
n
i
n
jjijiijP
n
iii
i
xP
1
222
1 1
2
1
2
25
Covariance Matrix (vari = covii)
26
nnnn
n
n
n
C
varcovcovcov
covvarcovcov
covcovvarcov
covcovcovvar
321
333231
223221
113121
covij = ijij where i and j are the SDs of the daily returns of variables i and j, and ij is the correlation between them
Alternative Expressions for P2
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 27
transpose its is and is element th whosevector column the is where
T
T
αα
αα
i
P
j
n
i
n
jiijP
αi
C
2
1 1
2 cov
Alternatives for Handling Interest Rates Duration approach: Linear relation between
P and y but assumes parallel shifts) Cash flow mapping: Cash flows are
mapped to standard maturities and variables are zero-coupon bond prices with the standard maturities
Principal components analysis: 2 or 3 independent shifts with their own volatilities
10:37 Zlzheng , Dept. of Finance,XMU 20.28
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)
Suppose that the 5yr rate is 6% and the 7yr rate is 7% and we will receive a cash flow of $10,000 in 6.5 years.
The volatilities per day of the 5yr and 7yr bonds are 0.50% and 0.58% respectively
Zlzheng , Dept. of Finance,XMU10:37 20.29
Example continued
We interpolate between the 5yr rate of 6% and the 7yr rate of 7% to get a 6.5yr rate of 6.75%
The PV of the $10,000 cash flow is
540,6
0675.1
000,105.6
Zlzheng , Dept. of Finance,XMU10:37 20.30
Example continued
We interpolate between the 0.5% volatility for the 5yr bond price and the 0.58% volatility for the 7yr bond price to get 0.56% as the volatility for the 6.5yr bond
We allocate of the PV to the 5yr bond and (1- ) of the PV to the 7yr bond
Zlzheng , Dept. of Finance,XMU10:37 20.31
Example continued
Suppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6
To match variances
This gives =0.074
)1(58.05.06.02)1(58.05.056.0 22222
Zlzheng , Dept. of Finance,XMU10:37 20.32
Example continued
The value of 6,540 received in 6.5 years
in 5 years and by
in 7 years.
This cash flow mapping preserves value and variance
484$074.0540,6
056,6$926.0540,6
Zlzheng , Dept. of Finance,XMU10:37 20.33
Zlzheng , Dept. of Finance,XMU
主成分分析法及其在利率变动分析中应用
10:37 20.34
Zlzheng , Dept. of Finance,XMU
主成分分析法主成分分析:通过构造一组变量的几个线性组合来解释这组变量的方差和协方差结构。
从数学的角度来讲,就是将一组相关变量转变为一组正交变量。这组正交变量能够反映存在于原始变量中的信息。
10:37 35
Zlzheng , Dept. of Finance,XMU
主成分分析法 比如说 , 我们现在要分析因变量 Y 与自变量 X(X1,X2……Xn) 的关系。建立计量模型:
在 X1,X2……Xn之间相互独立的情况下,可以用 OLS对上述模型进行估计
但是在 X1,X2……Xn 之间高度相关的情况下,利用 OLS 估计出来的系数不是有效的。
如何从 X 中抽取互不相关的信息,用于解释 Y ?
1 1 2 2t t t n nt tY X X X
10:37 36
Zlzheng , Dept. of Finance,XMU
主成分分析法 如果我们可以构造 X的线性组合:
使 F1,F2……Fn 之间彼此不相关
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
...........
n n
n n
n n n nn n
F X X X
F X X X
F X X X
10:37 37
Zlzheng , Dept. of Finance,XMU
主成分分析法 那么我们就可以对如下模型进行估计:
从而通过 Fi 对 Y 的影响形式间接获知 X 对 Y 的影响形式 问题是: 我们如何确定 Fi 含有多少 X 的信息?
1 1 2 2 n nY F F F
10:37 38
Zlzheng , Dept. of Finance,XMU
主成分分析法1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
...........
n n
n n
n n n nn n
F X X X
F X X X
F X X X
用 Fi 的方差来衡量其包含 X 信息的多少。方差越大,包含的信息越多
10:37 39
Zlzheng , Dept. of Finance,XMU
写成向量形式:
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
...........
n n
n n
n n n nn n
F X X X
F X X X
F X X X
1 1 2 2
1 2
1
1 2 1 2
( , ) '
' . ' ( )
( , ), ( , )
n n
i i i in
n n
F X X X
F A X X A F
A X X X X
或者矩阵形式:
原来的变量可以由主成份来解释
10:37 20.40
Zlzheng , Dept. of Finance,XMU
主成分分析法 因为我们要求 F 之间彼此不相关,所以
F ( F1 , F2…… )的协方差阵( )必须为对角阵。 用矩阵表示为:
现在的问题就转化为:如何将 对角化?
而与 中最大值对应的 A 的列向量即为第一主成分 F1 方程中的系数。依此列推可以获得 F2 , F3……
var( ) var( ' ) ' var( ) 'F A X A X A A A
10:37 41
Zlzheng , Dept. of Finance,XMU
主成分分析法 设 对角线元素为: 则第 i 个主成分的贡献率为:
前 m 个主成分的累积贡献率为:
ii
1
iin
jjj
1
1
mii
ni
jjj
10:37 42
Zlzheng , Dept. of Finance,XMU
用主成分分析利率变动将不同期限利率的变动分解为各个毫不相关的成分,进而解释利率变动的形式。
步骤 1、获取样本数据。采集不同期限利率变动的数据
步骤 2、计算其样本协方差阵,或者样本相关系数阵(可以避免指标和量纲的不同所引起的影响)
步骤 3、计算由 2得出的矩阵的特征向量及对应的(标准 )正交基。最大特征向量对应第一主成分,依此类推
步骤 4、计算各主成分的贡献率,和累计贡献率步骤 5、其他分析10:37 20.43
Zlzheng , Dept. of Finance,XMU
实证的结果 众多学者利用主成分分析法分析了利率变动,结果表明,利率变动的 95%
部分可以由前 3 个主成分所解释。 而这三个主成分的期限结构又强烈暗示着其经济含义 1. 第一个主成分的期限结构基本上是平坦的,因此可以解释为利率曲线的平
行移动 2. 第二个主成分的期限结构一般在 2 到 6 年之间开始变换符号,即从正转为
负值。说明了该因素对中短期利率和长期利率的影响不同。可以解释利率曲线的斜率
3. 第三个主成分的期限结构明显特点就是其中期部分的值明显大于或小于短期和长期时的值。它可以解释利率曲线的曲度。
10:37 44
Principal Components Analysis for Interest Rates
Zlzheng , Dept. of Finance,XMU10:37 20.45
Principal Components Analysis for Interest Rates
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)
Zlzheng , Dept. of Finance,XMU10:37 20.46
Using PCA to calculate VaR
Example: Sensitivity of portfolio to rates ($m)
Sensitivity to first factor is from Table 20.3:10×0.32 + 4×0.35 – 8×0.36 – 7 ×0.36 +2 ×0.36 = – 0.08
Similarly sensitivity to second factor = – 4.40
1 yr 2 yr 3 yr 4 yr 5 yr
+10 +4 -8 -7 +2
Zlzheng , Dept. of Finance,XMU10:37 20.47
Using PCA to calculate VaR continued
As an approximation
The f1 and f2 are independent The standard deviation of P (from Table
20.4) is
The 1 day 99% VaR is 26.66 × 2.33 = 62.12
21 40.408.0 ffP
66.2605.640.449.1708.0 2222
Zlzheng , Dept. of Finance,XMU10:37 20.48
When Linear Model Can be Used
Portfolio of stocks Portfolio of bonds Forward contract on foreign currency Interest-rate swap
Zlzheng , Dept. of Finance,XMU10:37 20.49
The Linear Model and Options
Consider a portfolio of options dependent on a single stock price, S. Define
and
S
P
S
Sx
Zlzheng , Dept. of Finance,XMU10:37 20.50
Linear Model and Options continued
As an approximation
Similarly when there are many underlying market variables
where i is the delta of the portfolio with respect to the ith asset
xSSP
i
iii xSP
Zlzheng , Dept. of Finance,XMU10:37 20.51
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
where x1 and x2 are the percentage changes in the two stock prices
21 000,2030000,1120 xxP
Zlzheng , Dept. of Finance,XMU10:37 20.52
But the distribution of the daily return on an option is not normal
The linear model fails to capture skewness in the probability distribution of the portfolio value.
Zlzheng , Dept. of Finance,XMU10:37 20.53
Impact of gamma
54
Positive Gamma Negative Gamma
Translation of Asset Price Change to Price Change for Long Call
55
Long Call
Asset Price
Translation of Asset Price Change to Price Change for Short Call
56
Short Call
Asset Price
Quadratic Model
For a portfolio dependent on a single stock price it is approximately true that
this becomes
2)(2
1SSP
22 )(2
1xSxSP
Zlzheng , Dept. of Finance,XMU10:37 20.57
Quadratic Model continued
With many market variables we get an expression of the form
where
This is not as easy to work with as the linear model
n
i
n
ijiijjiiii xxSSxSP
1 1 2
1
jiij
ii SS
P
S
P
2
Zlzheng , Dept. of Finance,XMU10:37 20.58
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
Zlzheng , Dept. of Finance,XMU10:37 20.59
Monte Carlo Simulation
Calculate P Repeat many times to build up a
probability distribution for P VaR is the appropriate fractile of the
distribution times square root of N For example, with 1,000 trial the 1
percentile is the 10th worst case.
Zlzheng , Dept. of Finance,XMU10:37 20.60
Speeding Up Monte Carlo
Use the quadratic approximation to calculate P
Zlzheng , Dept. of Finance,XMU10:37 20.61
Comparison of Approaches
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
Zlzheng , Dept. of Finance,XMU10:37 20.62
Stress Testing
This involves testing how well a portfolio performs under extreme but plausible market moves
Scenarios can be generated using Historical data Analyses carried out by economics group Senior management
Zlzheng , Dept. of Finance,XMU10:37 20.63
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?
Zlzheng , Dept. of Finance,XMU10:37 20.64