Fundamentals of Futures and Options Markets, 7th Ed, Ch 20

Value at Risk
Chapter 20

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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