Value at Risk

Chapter 20

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull 2010

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

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

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

3

VaR vs. Expected Shortfall

(See Figures 20.1 and 20.2, page 431)

VaR is the loss level that will not be exceeded with a specified probability

Expected shortfall is the expected loss given that the loss is greater than the VaR level

Although expected shortfall is theoretically more appealing than VaR, it is not widely used

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

5

Historical Simulation

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

6

Historical Simulation continued

Suppose we use 501 days of historical data

Let vi be the value of a market variable on day i

There are 500 simulation trials

The ith trial assumes that the value of the market variable tomorrow is

v500

vi

vi −1

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

7

Historical Simulation continued

The portfolio’s value tomorrow is calculated for each simulation trial

The loss between today and tomorrow is then calculated for each trial (gains are negative losses)

The losses are ranked and the one-day 99% VaR is set equal to the 5 th worst loss

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

8

The Model-Building Approach

The main alternative to historical simulation is to make assumptions about the

probability distributions of return on the market variables

This is known as the model building approach or the variance-covariance approach

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

9

Daily Volatilities

In option pricing we express volatility as volatility per year

In VaR calculations we express volatility as volatility per day

σ day =

σ year

252

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

10

Daily Volatility continued

Strictly speaking we should define σday 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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

11

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

12

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

13

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

14

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

The VaR is

50,000 10 = $158,144

158,114 × 2.33 = $368,405

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

15

Portfolio (See Example 20.1)

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

16

S.D. of Portfolio

A standard result in statistics states that

σ X +Y = σ X + σY + 2ρσ X σ Y

2 and ρ =2 0.3. The standard deviation of the change in

In this case σX = 200,000 and σY = 50,000

the portfolio value in one day is therefore 220,227

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

17

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

18

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

19

Markowitz Result for Variance of

Return on Portfolio

n

n

Variance of Portfolio Return = ∑∑ ρ ij wi w j σ i σ j

i =1 j =1

wi is weight of ith instrument in portfolio

2

σ i is variance of return on ith instrument

in portfolio

ρ ij is correlation between returns of ith

and jth instrument s

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

20

VaR Result for Variance of

Portfolio Value (α i = wiP)

n

∆P = ∑ α i ∆xi

i =1

n

n

σ 2P = ∑∑ ρ ij α i α j σ i σ j

i =1 j =1

n

2

i

i =1

σ 2P = ∑ α σ i2 + 2∑ ρ ij α i α j σ i σ j

i< j

σ i is the daily volatility of ith instrument (i.e., SD of daily return)

σ P is the SD of the change in the portfolio value per day

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

21

Covariance Matrix (vari = covii)

(Table 20.6, page 441)

var1

cov 21

C = cov 31

cov

n1

cov12

cov13

var2

cov 23

cov 32

var3

cov n 2

cov n 3

cov1n

cov 2 n

cov 3n

varn

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

22

Alternative Expressions for σ P2

page 441

n

n

σ 2P = ∑∑ cov ij α i α j

i =1 j =1

σ 2P = α T Cα

where α is the column vector whose ith

element is αi and α T is its transpose

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

23

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: 1month, 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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

24

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

25

Chapter 20

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull 2010

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

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

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

3

VaR vs. Expected Shortfall

(See Figures 20.1 and 20.2, page 431)

VaR is the loss level that will not be exceeded with a specified probability

Expected shortfall is the expected loss given that the loss is greater than the VaR level

Although expected shortfall is theoretically more appealing than VaR, it is not widely used

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

5

Historical Simulation

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

6

Historical Simulation continued

Suppose we use 501 days of historical data

Let vi be the value of a market variable on day i

There are 500 simulation trials

The ith trial assumes that the value of the market variable tomorrow is

v500

vi

vi −1

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

7

Historical Simulation continued

The portfolio’s value tomorrow is calculated for each simulation trial

The loss between today and tomorrow is then calculated for each trial (gains are negative losses)

The losses are ranked and the one-day 99% VaR is set equal to the 5 th worst loss

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

8

The Model-Building Approach

The main alternative to historical simulation is to make assumptions about the

probability distributions of return on the market variables

This is known as the model building approach or the variance-covariance approach

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

9

Daily Volatilities

In option pricing we express volatility as volatility per year

In VaR calculations we express volatility as volatility per day

σ day =

σ year

252

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

10

Daily Volatility continued

Strictly speaking we should define σday 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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

11

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

12

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

13

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

14

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

The VaR is

50,000 10 = $158,144

158,114 × 2.33 = $368,405

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

15

Portfolio (See Example 20.1)

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

16

S.D. of Portfolio

A standard result in statistics states that

σ X +Y = σ X + σY + 2ρσ X σ Y

2 and ρ =2 0.3. The standard deviation of the change in

In this case σX = 200,000 and σY = 50,000

the portfolio value in one day is therefore 220,227

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

17

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

18

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

19

Markowitz Result for Variance of

Return on Portfolio

n

n

Variance of Portfolio Return = ∑∑ ρ ij wi w j σ i σ j

i =1 j =1

wi is weight of ith instrument in portfolio

2

σ i is variance of return on ith instrument

in portfolio

ρ ij is correlation between returns of ith

and jth instrument s

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

20

VaR Result for Variance of

Portfolio Value (α i = wiP)

n

∆P = ∑ α i ∆xi

i =1

n

n

σ 2P = ∑∑ ρ ij α i α j σ i σ j

i =1 j =1

n

2

i

i =1

σ 2P = ∑ α σ i2 + 2∑ ρ ij α i α j σ i σ j

i< j

σ i is the daily volatility of ith instrument (i.e., SD of daily return)

σ P is the SD of the change in the portfolio value per day

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

21

Covariance Matrix (vari = covii)

(Table 20.6, page 441)

var1

cov 21

C = cov 31

cov

n1

cov12

cov13

var2

cov 23

cov 32

var3

cov n 2

cov n 3

cov1n

cov 2 n

cov 3n

varn

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

22

Alternative Expressions for σ P2

page 441

n

n

σ 2P = ∑∑ cov ij α i α j

i =1 j =1

σ 2P = α T Cα

where α is the column vector whose ith

element is αi and α T is its transpose

Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C. Hull

2010

23

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: 1month, 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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

24

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, 7th Ed, Ch 20, Copyright © John C. Hull

2010

25

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