Chapter 11 1

Need for Risk Management

Example: (from Jorion (2001), Value at Risk)

• David Askin managed $600 million fund invested in

collateralized mortgage obligations (CMOs)

– somewhat like derivatives

– difficult to price

• Askin claimed his funds were market neutral “with

no default risk, high triple-A bonds, and zero

correlation with other assets”

• Askin used his own model to identify, purchase, and

hedge underpriced securities

Chapter 11 2

• objective was 15% return

• leveraged so a total of $2 billion was invested

• Askin was betting on interest rates remaining low

• Feb – Apr 1994:

– rates go up

– collateral call

Chapter 11 3

• $600 million hedge fund reduced to $30 million

• irate investors

– claimed they were mislead

– it is true that they had little idea of the risks

• Askin used valuation models to price his position

– first reported a 2% loss

– revised to 28%

• investors were subject to

– market risk

– liquidity risk

– model risk

Chapter 11 4

Barings

• Feb 26, 1995: 233-year old Barings PLC goes

bankrupt

• rogue trader – 28-year old Nicholas Leeson lost $1.3

billion (more than the firm’s equity capital)

• Leeson bought stock index futures on the Nikkei 225

($7 billion position)

• beginning of 1995 – value fell more than 15%

• Barings was a conservative bank – so this was a

wake-up call

Chapter 11 5

• Problems:

– lack of control: Leeson was in charge of trading

desk and back office

– Singapore and Osaka exchanges did not notice the

size of the positions

– problem was not peculiar to derivatives

Chapter 11 6

Origins of Value-at-Risk

From Jorion:

• Till Guldimann was head of global research at J.P.

Morgan in late ’80s

• risk-management needed to decide between

– long bonds ⇒ stable earnings

– cash ⇒ stable market value

• decided that “value risks” were more important that

“earnings risk”

Chapter 11 7

• concern at this time about managing derivatives risks

• Group of 30 (with representative from J.P. Morgan):

G-30 report in July 1993 used term “value at risk”

• October 1994: J.P Morgan introduces Riskmetrics

– data represent an elaborate covariance-matrix of

risk and correlation measures that evolve through

time

Chapter 11 8

Introduction

Loss is the loss (−revenue) from some portfolio.

• It is a random variable

Two parameters:

• T = horizon

• 1− α = confidence level

P (Loss > VaR) = α.

or

P (Loss ≤ VaR) = 1− α.

Chapter 11 9

From previous slide:

P (Loss ≤ VaR) = 1− α.

(−∞, VaR) is a 1− α confidence interval for the loss

Stated differently:

(−VaR, ∞) is a 1− α confidence interval for the revenue

Chapter 11 10

VaR for One Asset

Nonparametric method

Data: n returns

• Take the Kth smallest return where K = α n,

rounded.

– call it R(K)

• S is the size of the initial investment

• VaR = −S ×R(K)

– R(K) will be negative so that VaR is positive

Chapter 11 11

VaR for One Asset

Parametric method

• Assume that the returns are normally distributed

• X and s are the sample mean and standard deviation

• µ + Φ−1(α)σ is the α-quantile of the return

distribution

• S is the value of the portfolio

• VaR = −S × {X + Φ−1(α)s}

Chapter 11 12

VaR for a Portfolio of Stocks

• Assume that all returns are normally distributed

• X i and s2i are the sample mean and variance for the

ith stock

• sij is the sample covariance

• w1, . . . , wN are the portfolio weights

• µP = wiµ1 + . . . + wNµN

• σP =√∑N

i=1

∑Nj=1 wiwjσij

• µ̂P and σ̂P substitute sample means, variances, and

covariance

Chapter 11 13

• µ̂P + Φ−1(α)σ̂P is the α-quantile of the return

distribution

• VaR = −S × {µ̂P + Φ−1(α)σ̂P}

If the portfolio holds Mi shares of the ith stock which is

selling at Pi, then

S =N∑

j=1

MiPi

and

wi =MiPi

S

Chapter 11 14

VaR for a Portfolio of One Stock and An Option

on that Stock

Review

change in price of option ≈ ∆× change in price of asset.

Or, using math notation,

P optiont − P option

t−1 ≈ ∆(P assett − P asset

t−1 )

Therefore,

P optiont − P option

t−1

P optiont−1

≈

(∆

P assett−1

P optiont−1

)P asset

t − P assett−1

P assett−1

Chapter 11 15

From last slide:

P optiont − P option

t−1

P optiont−1

≈

(∆

P assett−1

P optiont−1

)P asset

t − P assett−1

P assett−1

Define:

L =

(∆

P assett−1

P optiont−1

)Then:

R option ≈ L×R stock

Chapter 11 16

From previous slide:

R option ≈ L×R stock

If w and 1− w are the weights on the stock and option,

then

• RP ≈ wR option + (1− w)L×R option or

• RP ≈ {w + (1− w)L} ×R stock

• VaRportfolio ≈−S × {w + (1− w)L} × α-quantile of stock return

• S = value of portfolio = value of stock + value of

options

Chapter 11 17

Example:

• S = 75 (stock price), σ = 0.3/year, µ = 0.04/year,

r = 0.01/year, for option: T = 0.5 year, K = 80

• for calculating VaR: T = 1 month

Portfolio 1: own 100 shares and call options on 200

shares –

C = 4.4835 and ∆ = Φ(d1) = 0.4307

L = 0.430775

4.4835= 7.2046

S = (100)(75) + (200)(4.4835) = 8, 396.70

Chapter 11 18

w =(100)(75)

(100)(75) + (200)(4.4835)= 0.8932, 1−w = 0.1068

w + (1− w)L = 1.6626

(0.1-quantile of R stock) =

(0.04

12− 1.645

0.03√12

)= −0.1391

VaRportfolio ≈ −S{w + (1− w)L}(0.1-quantile of R stock)

= −(8, 396.70)(1.6626)(−0.1391) = 1, 942.30

Chapter 11 19

Notes:

• Does not account for time-erosion of option value

• Standard deviation of one-month return is 1/√

12

times the standard deviation of one-year return

Chapter 11 20

Portfolio 2: own 100 shares and put options on 150

shares –

P = 9.0845 and ∆ = Φ(d1)− 1 = 0.4307− 1 = −0.5693

L = −0.569375

9.0845= −4.7001

w =(100)(75)

(100)(75) + (150)(9.0845)= 0.8462, 1−w = 0.1538

w + (1− w)L = 0.1236

S = (100)(75) + (150)(9.0845) = 8, 862.70

Chapter 11 21

VaRportfolio = −S{w + (1− w)L}(0.1-quantile of R stock)

= −(8, 862.70)(0.1236)(−0.1391) = 152.39

Note: Does not account for time-erosion of option value

Chapter 11 22

C(S, T, t, K, σ, r) = price of an option

∆ =∂

∂SC(S, T, t, K, σ, r) “Delta”

Θ =∂

∂tC(S, T, t, K, σ, r) “Theta”

R =∂

∂rC(S, T, t, K, σ, r) “Rho”

V =∂

∂σC(S, T, t, K, σ, r) “Vega”

Chapter 11 23

Chapter 11 24

0 0.1 0.20

1

2

3

4

σ

Pric

e of

cal

l

90 100 1100

5

10

15

E

Pric

e of

cal

l

90 100 1100

5

10

15

S0

Pric

e of

cal

l

0 0.04 0.08 0.121

1.5

2

2.5

3

r

Pric

e of

cal

l

0 0.04 0.08 0.120

0.5

1

1.5

T

Pric

e of

cal

l

Chapter 11 25

Differentiating Black-Scholes: for call

Θ =−Sσφ(d1)

2√

T − t−Kre−r(T−t)Φ(d2)

Get Θ for the put from put-call parity:

P = C + e−r(T−t) − S

Chapter 11 26

In examples:

• Θ for call = −6.5293

• Θ for put: −5.7333

• So price of both put and call will drop by about $0.50

in one-month (1/12 year) due to time erosion

• These drops could be incorporated into the expected

return

Chapter 11 27

Expected shortfall

ES = E{Loss|Loss > VaR}.

Example 1:

Buy $2000 of bonds from a corportation.

Loss is

• 0 with probability 0.96

• 2000 with probability 0.04

If α = 0.05, then VaR = 0.

ES = 2000

Chapter 11 28

Example 2 :

Buy $1000 of bonds from each of two corportations.

Loss is

• 0 with probability 0.962 = 0.9216

• 1000 with probability 2(0.96)(0.04) = 0.0768

• 2000 with probability 0.042 = 0.0016

If α = 0.05, then VaR = 1000.

ES =(1000)(0.0768) + (2000)(0.0016)

0.0768 + 0.0016= 1020

Chapter 11 29

Comparison

Portfolio VaR ES

Non-diversified 0 2000

Diversified 1000 1020

Chapter 11 30

Subadditivity

• P1, P2, . . ., Pm are portfolios

• R(·) is a risk measure

R(·) is subadditive if

R(P1 + · · ·+ Pm) ≤ R(P1) + · · ·+ R(Pm).

Subadditivity encourages diversification and has

other benefits

• VaR is not subadditive

• ES is subadditive

Chapter 11 31

Confidence Intervals for VaR by Resampling

There is a “true VaR” but it is unknown

• The VaR we report is just an estimate

• How accurate is it?

• Need a confidence interval for VaR

– Easily done by resampling

Chapter 11 32

Example: VaR for one asset – confidence interval

using the nonparametric estimator

• S, α, and T are known.

• What is unknown is the α-quantile of the return

distribution.

• Assume a sample of n returns

• K = α n, rounded.

For the bth of B resample, compute:

• Rb1, . . . , R

bn, a resample from the sample

• VaRb = −S ×Rb(K)

Chapter 11 33

Suppose we want a 1− γ confidence interval for VaR. For

concreteness, take γ = 0.02 and B = 50, 000.

• Then γB/2 = (0.02)(50, 000)/2 = 500

• The confidence interval is the 500th smallest to the

500th largest value of VaRb

Chapter 11 34

Example

$20,000 position in an S&P 500 index fund.

• 1000 daily returns for S&P 500

• α = 0.05 ⇒ K = 50

• 50th smallest return was −0.0227

• VaR = $454 = 20,000× 0.0227

• 50,000 resamples

• 98% confidence interval for VaR is $412 to $495