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