The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management The Oxford Guide to Financial Modeling Thomas S. Y. Ho and Sang Bin Lee

The Oxford Guide to Financial Modeling by Ho & Lee

Chapter 15. Risk Management

The Oxford Guide to Financial Modeling

Thomas S. Y. Ho and Sang Bin Lee

Copyright © 2004 by Thomas Ho and Sang Bin Lee. All rights reserved.

Chapter 15. Risk Management 2


15.1 Risk Measurement -Value at Risk (VaR) • Definition: a measure of potential loss at a level (99% or

95% confidence level) over a time horizon

53.473 67.1029 100Portfolio Value : $million

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0.015

0.02

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53.473 67.1029 100Portfolio Value : $million

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15.2 Market Risk

• Market risk is the losses that arise from the mark to market of the trading securities

• “Prices” for tradable securities of a portfolio are marked to market that are often derived from the fair values of the valuation models



15.3 VaR for single securities

Definition: VaR time factor volatility • is called the critical value which determines the one-

tail confidence level of standard normal distribution.

• Time factor is defined as where t is the time horizon in measuring the VaR.

• Volatility is the standard deviation of the stock measured in $ over one year.

t




[ ] ( [ ] )P f P M fE R r E R r

$P Duration r

( ) $

.

VaR bond time factor Duration r

rStd

r

- portfolio return distribution

- the price of the bond

- the critical value for a particular interval of a normal distribution



1

$n

i

P KRD i r i

0.5

1 1

( ) ( $ $ )n n

iji j

VaR bond time factor KRD i KRD j

OAS OASVaR time factor P Duration

- the VaR of the bond

- the bond price uncertain value is a multivariate normal distribution




15.3 Delta Normal Methodology (2)• VaR for a Portfolio (I)

- The portfolio value

- The portfolio uncertain value

- The VaR of the portfolio

1

n

i ii

P x P

1

$n

ii

P Duration i

0.5

1 1

( )

( $ $ )n n

iji j

VaR portfolio time factor

Duration i Duration j



15.3 Portfolio VaR • VaR for a Portfolio (II)

- Component VaR

1

0.5

1 1

( ) $ $

( $ $ )

n

i ijj

n n

iji j

VaR portfolio time factor Duration i Duration j

Duration i Duration j

1

n

ii

VaR VaR



15.3 Three Stocks Case

• A Numerical Example

- Calculating the VaR of a portfolio of three different

stocks (GM, WMT, and IBM)

- Calculating the daily rates of return and the variance-covariance matrix

, , 1,

, 1

22,

1

, , ,1

GM, WMT, and IBM

0

1

1

i t i ti t

i t

i

m

i i t it

m

i j i t i j t it

S Sr i

S

r

r rm

r r r rm

where m is the number of days in the estimation period.



15.3 Correlations of the Stock Returns• Calculating the daily rates of return and the variance-

covariance matrix

0.00050827 0.000154099 0.000179167

0.000154099 0.00373365 0.00013894

0.000179167 0.00013894 0.00054746

Ω1 0.353741 0.339255

0.353741 1 0.306955

0.339255 0.306955 1

Σ

1 1 1, ,

3 3 3

Tw

2

0.00050827 0.000154099 0.000179167 1/ 3

1/ 3 1/ 3 1/ 3 0.000154099 0.00373365 0.00013894 1/ 3

0.000179167 0.00013894 0.00054746 1/ 3

0.000263866

Portfolio

Tw Ω w



15.3 VaR Derivations• The detailed derivation of the individual VaR as well

as the portfolio VaR is given as follows.

Where,

i i i

P P

VaR total invest w days

VaR total invest days

,

2 2,

{ , , }

2

P

i j i ji j

i i i j i ji i j i

i GM WMT IBM

Tw Ω w



15.3 VaR Derivation

100 0.00050827 2.32635 5 3.9091

3100

0.00373365 2.32635 5 3.35053

100 0.00054746 2.32635 5 4.

3

GM GM GM

WMT WMT WMT

IBM IBM IBM




0619

100 0.000263866 2.32635 5 8.44989P PVaR total invest days



15.3 Component VaR

1/ 3

1/ 3

1/ 3

1/ 3

1/ 3 1/ 3 1/ 3 1/ 3

1/ 3

1.0631

0.8418

1.0951

GM

Delta Normal Method WMT

IBM

Beta

T

Ω

Ω w

w Ω w

Ω

, , i i i PortfolioComponent VaR VaR i GM WMT and IBM

1 = 1.0630 8.4498 2.9943

3GM GM GM PortfolioComponent VaR VaR



5-day VaR GM WMT IBM Total

Weight 1/3 1/3 1/3 1

Individual stock

VaR 3.9091 3.3505 4.0619 11.3215

Portfolio VaR - - - 8.4499

Beta 1.0631 0.8418 1.0951 -

Beta*Weight 0.3544 0.2806 0.3650 1

Component VaR 2.9943 2.3711 3.0844 8.4499

Portfolio Effects 0.9148 0.9793 0.9775 2.8716

15.3 VaR Calculation

VaR calculation output by Delta-Normal Method



15.4 Historical Simulation Methodology

…

return return return return return return … return return return return

sorting the data and finding x% percentile

Today

The Historical Simulation methodology



15.4 Historical Returns

Historical Return data set

8.1626[1]4.03623.22644.32921% VaR

-8.1626-4.0362-3.2264-4.32921% percentile

0.1094-0.1740-0.08800.37142002,05,02

1.48250.21490.56090.70672002,05,01

0.2531-0.0517-0.20170.50642002,04,30

-0.0720-0.17840.5217-0.41522001,10,31

-1.53840.0092-0.8306-0.71702001,10,30

-3.6504-0.7636-0.9508-1.93612001,10,29

-1.0441-0.15700.0000-0.88712001,01,08

-2.86460.2915-1.3321-1.82402001,01,05

-0.3901-0.5069-1.28651.40322001,01,04

8.35943.85572.82191.68172001,01,03

(1)+(2)+(3)Portfolio

(3)IBM

(2)WMT

(1)GM

Date

8.1626[1]4.03623.22644.32921% VaR

-8.1626-4.0362-3.2264-4.32921% percentile

0.1094-0.1740-0.08800.37142002,05,02

1.48250.21490.56090.70672002,05,01

0.2531-0.0517-0.20170.50642002,04,30

-0.0720-0.17840.5217-0.41522001,10,31

-1.53840.0092-0.8306-0.71702001,10,30

-3.6504-0.7636-0.9508-1.93612001,10,29

-1.0441-0.15700.0000-0.88712001,01,08

-2.86460.2915-1.3321-1.82402001,01,05

-0.3901-0.5069-1.28651.40322001,01,04

8.35943.85572.82191.68172001,01,03

(1)+(2)+(3)Portfolio

(3)IBM

(2)WMT

(1)GM

Date





Weight 1/3 1/3 1/3 1

Individual stock VaR

4.3292 3.2264 4.0362 11.5917


Beta 1.0631 0.8418 1.0950 -

Beta*Weight 0.3544 0.2806 0.3650 1

Component VaR 2.8925 2.2906 2.9795 8.1626


VaR calculation output by Historical Simulation Method



15.5 Monte Carlo Simulation Methodology

• Random numbers generated from Multi Normal Distribution

-3.7629**-2.0318-1.5446-1.72111% percentile

-0.0832/3-0.0279-0.0108-0.0445Scenario 5

0.0078/30.0073-0.00340.0039Scenario 4

0.0047/3-0.00810.00220.0106Scenario 3

-0.0544/3-0.0017-0.0148-0.0379Scenario 2

-0.0149/3-0.0165-0.00740.0090Scenario 1

Portfolio((1)+(2)+(3))/3

IBM (3)WMT (2)GM (1)

Random Number = Return data

Scenario

-3.7629**-2.0318-1.5446-1.72111% percentile

-0.0832/3-0.0279-0.0108-0.0445Scenario 5

0.0078/30.0073-0.00340.0039Scenario 4

0.0047/3-0.00810.00220.0106Scenario 3

-0.0544/3-0.0017-0.0148-0.0379Scenario 2

-0.0149/3-0.0165-0.00740.0090Scenario 1

Portfolio((1)+(2)+(3))/3

IBM (3)WMT (2)GM (1)

Random Number = Return data

Scenario



15.5 Simulations based on the Correlation Matrix• The variance-covariance matrix of stock returns generated by Monte Carlo

simulation

Monte-Carlo

0.00050792 0.00015374 0.00017815

0.00015374 0.00037336 0.00013749

0.00017815 0.00013749 0.00054419

Ω

0.01

100 0.052593 5 3.9200

3

0.01

GM GM GM

WMT WMT WMT

MonteCarlo VaR Percentile of Scenario total invest w day


100 0.044945 5 3.3504

3

0.01

100 0.054149 5 4.0361

3

0.01

IBM IBM IBM

P


MonteCarlo VaR Percent

0.037629 100 5 8.4141

Pile of Scenario total invest day




VaR calculation output by Monte Carlo Simulation Method


Weight 1/3 1/3 1/3 1

Individual stock

VaR 3.9200 3.3504 4.0361 11.3065


Beta 1.0656 0.8433 1.0910 -

Beta*Weight 0.3552 0.2811 0.3637 1

Component VaR 2.9888 2.3652 3.0601 8.4141




15.6 Extreme Value Theory

• multiply historical returns by –1 to convert them into positive values.

• choose a threshold (): a parametric distribution of the tail beyond the threshold.

• The ratio: count how many observations are beyond the threshold in the actual data and divide it by the total observation.

• Parameters () estimation • VaR calculation



-0.1 -0.05 0 0.05GM daily stock return

0

5

10

15

20ytisned

15.6 Extreme Value Theory - Historical return data vs. Standard Normal Distribution

1/1 (1 ) 0( )

1 exp( ) 0

u

u

Ny when

NF yN

y whenN

- cumulative distribution functions



15.6 calculate the VaR by the extreme value theory

0.2086

0.20860.6509 12.5 (1 0.9995) 1 5 9.600

170.2086332

EVTVaR

0.2086, 0.6509, 0.9995confidence level

ˆ(1 ) 1EVT

u

NVaR u c day

N

1

11

( ) 1uNf x x u

N

- The formula to calculate the VaR based on the Extreme Value Theory

- probability density function



15.7 Credit Risk

• Definition

the loss of principal or interest or any promised payments from the borrow for bonds or loans of any securities



15.7 Credit Risk and Market Risk Model

• VaR of a Bond - Firm value process (Merton)

• Integrating Credit Risk and Market Risk in a Portfolio Context (I)

- Firm value process (Merton, Longstaff and Schwarzt)

- interest rate model (Hull and White)

( )dV V C dt dz

( )dV rV C dt Vdz

( ( ) )dr t ar dt dz



15.7 Portfolio Credit and Market Risk

• Integrating Credit Risk and Market Risk in a Portfolio Context

- The stock risk

( ( ) )i f i M fk r E R r z



15.7 Portfolio Credit Risk • Specify a set of macro-economic factors that would

affect the credit risk of the firms. • Define the default index by measuring default rate. The

macro economic factors are used as the independent variables to explain the default rate.

• Measure the rating migrations against the speculative default rates: change of the speculative default rate determines the change in the rating migrations.

• The simulations can then be used to simulate the change in value of a bond portfolio.



15.7 Credit VaR - a Numerical Example by

CreditMetrics

Initial

rating

Rating at year-end (%)

AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0.00 0.00 0.00

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0.00

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0.00 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0.00 0.22 1.30 2.38 11.24 64.86 19.79

- One-year Transition Matrix



Category 1 2 3 4

AAA 3.60 4.17 4.73 5.12

AA 3.65 4.22 4.78 5.17

A 3.72 4.32 4.93 5.32

BBB 4.10 4.67 5.25 5.63

BB 5.55 6.02 6.78 7.27

B 6.05 7.02 8.03 8.52

CCC 15.05 15.02 14.03 13.52

Bond number Credit Grade Face value Maturity Coupon RateRecovery

Rate

1 A 100 5 zero 0.60

2 BBB 100 5 0.06 0.55

3 BB 100 5 0.03 0.40

- Bond Data set

15.7 cont.. - Example one -year forward zero curves by crediting rating category



Grade AAA AA A BBB BB B CCC Default

Price 95.6241 95.456 94.959 93.928 88.7654 85.0951 71.9208 40

Profit/Loss 6.8587 6.6909 6.1936 5.1626 0 -3.6703 -16.8446 -48.7654

Probability 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

Cumulative Probab

ility100.00 99.97 99.83 98.16 91.43 10.90 2.06 1.06

15.7 cont..

- 1year forward bond Price, Profit/Loss and Probability for the BB-grade bond



-50 -40 -30 -20 -10 0ProfitLoss $0

0.2

0.4

0.6

0.8

1FDC

15.7 cont.. - Cumulative Probability of BB-Grade Bond’s Profit/Loss



Thresholds A BBB BB

AAZ 3.12139 3.54008 3.43161

AZ 1.9845 2.69684 2.92905

BBBZ -1.50704 1.53007 2.39106

BBZ -2.30085 -1.49314 1.36772

BZ -2.71638 -2.17808 -1.23186

CCCZ -3.19465 -2.74778 -2.04151

DZ -3.23888 -2.91124 -2.30440

Covariance

Matrix A BBB BB

A 0.9 0.7 -0.3

BBB 0.7 2 0.5

BB -0.3 0.5 1

- Estimate the correlation matrix or variance-covariance matrix among the bond returns

15.7 cont.. - Z-threshold



-120 -100 -80 -60 -40 -20 0Portfolio LossProfit0

2000

4000

6000

8000

10000

12000

14000

ycneuqerF

Number of Scenario 100000

15.7 cont.. - Bond Portfolio Default Risk Distribution

A-Grade

Bond

BBB-Grade

Bond

BB-Grade

Bond Portfolio

10% VaR 0.9143 5.3193 5.01155 9.1948

Portfolio effect 11.2448-9.1948=2.05



15.8 Risk Reporting Aggregation of Risks to Equity ($mil.) (the VaR Table)

10.590.9810.591,078Equity

15.161.7319.851,146Long term market funding

36.890.8544.625,250Demand deposits

9.552.6411.69443Fixed rate time deposits

0.980.541.56289Prime rate time deposits

3.240.305.831,959Base rate time deposits

-28.21.1733.462,854Bonds

-22.52.5030.491,231Fixed rate loans

-4.80.875.47625Variable rate mortgages

-4.30.234.922,170Base rate loans

4.50.3411.313,286Prime rate loans

Component VaRVaR/MV (%)VaRMarket ValueItems

10.590.9810.591,078Equity

15.161.7319.851,146Long term market funding

36.890.8544.625,250Demand deposits

9.552.6411.69443Fixed rate time deposits

0.980.541.56289Prime rate time deposits

3.240.305.831,959Base rate time deposits

-28.21.1733.462,854Bonds

-22.52.5030.491,231Fixed rate loans

-4.80.875.47625Variable rate mortgages

-4.30.234.922,170Base rate loans

4.50.3411.313,286Prime rate loans

Component VaRVaR/MV (%)VaRMarket ValueItems



15.9 Risk Monitoring • Back testing

600 650 700 750 800Sample Period

-20

-10

0

10

tiforP?ssoL



15.10 Risk Measurement • Strategic Risk Management

– Smith and Smithson (1998) determines the economic factors affecting the equity value of a firm

– Hedging against these economic factors is strategic risk management

• Business Process – Build a model of the firm as a system of

processes– Manage the processes by monitoring and

controlling the risks in each phase



15.10 Risk Measurement (2)• Investment Cycle

▶ Implementation Phase :

Execute trades, and reportpositions

▶ Test Phase :

PerformanceEvaluation

▶ Requirement Phase :

Monitor portfolio returns andpositions, and establish goals tomeet client's needs

▶ Design Phase :

Forcast market dynamics, adjust forconstraints, and set directions forportfolio managers

InvestmentObjective

InvestmentStrategies

Take directions from marketoutlook, evaluate portfolioposition, and set trades for traders

MarketOutLook

Documents

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 15. Risk Management The Oxford Guide to Financial Modeling Thomas S. Y. Ho and Sang Bin Lee