PJVaRch78

7/31/2019 PJVaRch78

http://slidepdf.com/reader/full/pjvarch78 1/58

RMhttp://pluto.mscc.huji.ac.il/~mswi

ener/zvi.htmlIDC

Zvi Wiener [email protected]

02-588-3049

Financial Risk Management

mailto:[email protected]

mailto:[email protected]

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RMhttp://pluto.mscc.huji.ac.il/~mswi

ener/zvi.htmlIDC

Following P. Jorion, Value at Risk, McGraw-HillChapter 7

Portfolio Risk, Analytical Methods


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Zvi Wiener VaR-PJorion-Ch 7-8 slide 3

Portfolio of Random Variables

X w X wY T N

i

ii == ∑=1

∑=

==== N

i

ii X

T T

p ww X E wY E 1

)()( µ µ µ

∑∑= =

=Ω= N

i

N

j

jiji

T wwwwY 1 1

2 )( σ σ

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Portfolio of Random Variables

[ ]

=

N

NN N N

N

N

w

w

w

www

Y

2

1

21

11211

21

2

,,,

)(

σ σ σ

σ σ σ

σ

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Product of Random Variables

Credit loss derives from the product of the

probability of default and the loss given

default. ),()()()( 212121 X X Cov X E X E X X E +=

When X1 and X2 are independent

)()()( 2121 X E X E X X E =

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Transformation of Random Variables

Consider a zero coupon bond

T r V

)1(

100

+=

If r=6% and T=10 years, V = $55.84,

we wish to estimate the probability that the

bond price falls below $50.

This corresponds to the yield 7.178%.

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http://slidepdf.com/reader/full/pjvarch78 7/58 Zvi Wiener VaR-PJorion-Ch 7-8 slide 7

The probability of this event can be derived

from the distribution of yields.

Assume that yields change are normallydistributed with mean zero and volatility 0.8%.

Then the probability of this change is 7.06%

Example

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Marginal VaR

How risk sensitive is my portfolio to increase in size of

each position?- calculate VaR for the entire portfolio VaR P=X

- increase position A by one unit (say 1% of the portfolio)- calculate VaR of the new portfolio: VaR Pa= Y

- incremental risk contribution to the portfolio by A: Z = X-Y

i.e. Marginal VaR of A is Z = X-Y

Marginal VaR can be Negative; what does this mean...?

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Exposure vs. Risk

F/XHedging

Present Value vs VaR

GroupedbyPositionMonte CarloSimulation, 1-Month, 0.94Decay, GBP

Present Value VaR, 95.00%

EUR/USDOption: 20030915 -558,920 186,407

AUD/USDForward: 20020405 -162,449 126,461

NZD/USDOption: 20030220 -10,801 11,417

CAD/USDForward: 20021115 -5,183 28,550

EUR/JPYForward: 20010715 1,148 84,335

USD/ESPOption: 20011125 22,911 8,065

AUD/NZDForward: 20020310 144,612 51,004USD/ITLForward: 20010906 173,161 66,613

JPY/DEMForward: 20011007 227,307 74,090

EUR/USDForward: 20010907 306,975 311,886

EUR/GBPForward: 20021209 354,239 149,577

DEMCash 648,139 31,069

JPYCash 775,317 35,104

Details:

Report Type Scattergram

Number of Positions 13

Iterations 1,000

Seed 1234567

Business Date 1/8/2001

PricingDate 1/8/2001

TimeSeries Start 1/8/1999

TimeSeries End 1/8/2001

with minor corrections

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Marginal VaR

F/X Hedging

Marginal VaR by Curre ncy

Grouped by Position

Parametric 95.00%, 1-Month, 0.94 Decay, GB

Total AUD CAD DEM ESP EUR GBP ITL JPY NZD USD

Total 339,981 161,716 9,973 -13,987 -6,673 285,797 -3,451 -50,895 -1,837 -43,284 2,621

AUD/NZD Forward: 20020310 20,422 58,754 -38,332

AUD/USD Forward: 20020405 90,488 102,962 -12,474

CAD/USD Forward: 20021115 833 9,973 -9,141

DEM Cash 28,682 28,682

EUR/GBP Forward: 20021209 139,084 142,535 -3,451

EUR/JPY Forward: 20010715 59,753 55,995 3,758

EUR/USD Forward: 20010907 242,489 251,968 -9,480

EUR/USD Option: 20030915 -134,979 -164,701 29,722

JPY Cash -2,310 -2,310JPY/DEM Forward: 20011007 -45,954 -42,669 -3,285

NZD/USD Option: 20030220 -3,781 -4,952 1,171

USD/ESP Option: 20011125 -6,175 -6,673 498

USD/ITL Forward: 20010906 -48,571 -50,895 2,324

Marginal VaR by currency..... with minor corrections

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Incremental VaR

F/X Hedging

Incremental VaRby Risk Type

Grouped by Position

Parametric 95.00%, 1-Month, 0.94 Decay, GBP

Total FX Risk Interest Rate Risk

Total 339 ,981 307 ,997 10,072

AUD /NZD Forward : 20020310 16,917 15,127 1,738

AUD /USD Forward : 20020405 74,373 78,967 -5,119

CAD /USD Forward : 20021115 -353 3,720 -4,165

DEM Cash 28,398 28,398EUR /GBP Forward : 20021209 128 ,805 121 ,131 9,285

EUR /JPY Forward : 20010715 53,738 52,545 1,222

EUR /USD Forward : 20010907 139 ,317 141 ,262 -4,714

EUR /USD Option : 20030915 -145 ,964 -154 ,427 9,273

JPY Cash -4,436 -4,436

JPY /DEM Forward : 20011007 -49,879 -48,996 -833

NZD /USD Option : 20030220 -3,859 -4,200 342

USD /ESP Option : 20011125 -6,222 -6,526 305

USD /ITL Forward : 20010906 -50,942 -52,264 1,295

Details :

Report Type Table

Number of Positions 13

Business Date 1/8/2001

Pricing Date 1/8/2001

Time Series Start 1/8/1999

Time Series End 1/8/2001

Incremental VaR by Risk Type... with minor corrections

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Incremental VaR by Currency.... with minor corrections

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VaR decomposition

Position in asset A

VaR

100

Portfolio VaR

Incremental VaR

Marginal VaR

Component VaR

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Example of VaR decomposition

Currency Position Individual Marginal Component Contribution

VaR VaR VaR to VaR in %

CAD $2M $165,000 0.0528 $105,630 41%

EUR $1M $198,000 0.1521 $152,108 59%

Total $3M

Undiversified $363K

Diversified $257,738 100%

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Barings Example

Long $7.7B Nikkei futuresShort of $16B JGB futures

σ NK =5.83%, σJGB=1.18%, ρ=11.4%

0118.0114.00583.0167.720118.0160583.07.7 22222⋅⋅⋅⋅⋅+⋅+⋅= P σ

VaR 95%=1.65⋅ σP = $835M

VaR 99%=2.33 ⋅ σP=$1.18M

Actual loss was $1.3B

P J i H db k Ch 14

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The Optimal Hedge Ratio

∆S - change in $ value of the inventory

∆F - change in $ value of the one futures

N - number of futures you buy/sell

F N S V ∆×+∆=∆

F S F S V N N ∆∆∆∆∆ ++= ,

2222 2 σ σ σ σ

F S F V N

N ∆∆∆

∆ +=∂

∂,

22

22 σ σ σ

P. Jorion Handbook, Ch 14

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P J i H db k Ch 14

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Hedge Ratio as Regression Coefficient

The optimal amount can also be derived as the

slope coefficient of a regression ∆s/s on ∆f/f:

ε β α +∆+=∆ f

f

s

s sf

f

s sf

f

sf

sf σ σ ρ

σ σ β ==

2


P Jorion Handbook Ch 14

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Optimal Hedge

One can measure the quality of the optimal

hedge ratio in terms of the amount by which

we have decreased the variance of the original

portfolio.2

2

2

*

22 )(

sf

s

V s R ρ σ

σ σ =

−=

2

* 1 R sV −=σ σ

If R is low the hedge is not effective!



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Optimal Hedge

At the optimum the variance is

2

222

*

F

SF

S V σ

σ

σ σ −=



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FRM-99, Question 66

The hedge ratio is the ratio of derivatives to a spot position (viceversa) that achieves an objective such as minimizing or eliminating

risk. Suppose that the standard deviation of quarterly changes in the

price of a commodity is 0.57, the standard deviation of quarterly

changes in the price of a futures contract on the commodity is 0.85,

and the correlation between the two changes is 0.3876. What is theoptimal hedge ratio for a three-month contract?

A. 0.1893

B. 0.2135C. 0.2381

D. 0.2599



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FRM-99, Question 66

The hedge ratio is the ratio of derivatives to a spot position (viceversa) that achieves an objective such as minimizing or eliminating

risk. Suppose that the standard deviation of quarterly changes in the

price of a commodity is 0.57, the standard deviation of quarterly

changes in the price of a futures contract on the commodity is 0.85,

and the correlation between the two changes is 0.3876. What is theoptimal hedge ratio for a three-month contract?

A. 0.1893

B. 0.2135C. 0.2381

D. 0.2599



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Example

Airline company needs to purchase 10,000tons of jet fuel in 3 months. One can use

heating oil futures traded on NYMEX.

Notional for each contract is 42,000 gallons.We need to check whether this hedge can be

efficient.



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Example

Spot price of jet fuel $277/ton.

Futures price of heating oil $0.6903/gallon.

The standard deviation of jet fuel price rate of

changes over 3 months is 21.17%, that of

futures 18.59%, and the correlation is 0.8243.



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Compute

The notional and standard deviation f the

unhedged fuel cost in $.

The optimal number of futures contracts to buy/sell, rounded to the closest integer.

The standard deviation of the hedged fuel cost

in dollars.



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Solution

The notional is Qs=$2,770,000, the SD in $ is

σ(∆s/s)sQs=0.2117× $277 × 10,000 =

$586,409

the SD of one futures contract is

σ(∆f/f)fQf

=0.1859× $0.6903× 42,000 =

$5,390

with a futures notional

fQ = $0.6903× 42,000 = $28,993.



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Solution

The cash position corresponds to a liability

(payment), hence we have to buy futures as a

protection.

βsf = 0.8243 × 0.2117/0.1859 = 0.9387

σsf = 0.8243 × 0.2117 × 0.1859 = 0.03244

The optimal hedge ratio is

N* = βsf Qs× s/Qf × f = 89.7, or 90 contracts.



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Solution

σ2unhedged = ($586,409)2 = 343,875,515,281

- σ2SF/ σ2

F = -(2,605,268,452/5,390)2

σhedged = $331,997

The hedge has reduced the SD from $586,409

to $331,997.R 2 = 67.95% (= 0.82432)

,


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FRM-99, Question 67

In the early 90s, Metallgesellshaft, a German oil company, suffered aloss of $1.33B in their hedging program. They rolled over short dated

futures to hedge long term exposure created through their long-term

fixed price contracts to sell heating oil and gasoline to their customers.

After a time, they abandoned the hedge because of large negative

cashflow. The cashflow pressure was due to the fact that MG had tohedge its exposure by:

A. Short futures and there was a decline in oil price

B. Long futures and there was a decline in oil priceC. Short futures and there was an increase in oil price

D. Long futures and there was an increase in oil price

,

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Duration Hedging

dy P DdP ××−= *

Dollar duration

y F D F yS DS F S ∆××−=∆∆××−=∆ **

( )

( )( )( ) 2**

22*2

22*2

yS F SF

y F F

yS S

S D F D

F D

S D

∆

∆

∆

×××=

××=

××=

σ σ

σ σ

σ σ

,


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Duration Hedging

F D

S D N

F

S

F

SF

××−=−=

*

*

2*

σ

σ

If we have a target duration DV* we can get it by using

F D

S DV D

N F

S V

×

×−×

=*

**


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Example 1

A portfolio manager has a bond portfolio worth$10M with a modified duration of 6.8 years, to

be hedged for 3 months. The current futures

prices is 93-02, with a notional of $100,000.

We assume that the duration can be measured

by CTD, which is 9.2 years.

Compute:

a. The notional of the futures contract

b.The number of contracts to by/sell for optimal protection.


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Example 1

The notional is:(93+2/32)/100× $100,000 =$93,062.5

The optimal number to sell is:

4.795.062,93$2.9

000,000,10$8.6*

*

*

−=×

×−=

×

×−=

F D

S D N

F

S

Note that DVBP of the futures is 9.2× $93,062× 0.01%=$85


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Example 2

On February 2, a corporate treasurer wants to hedge a July17 issue of $5M of CP with a maturity of 180 days, leading

to anticipated proceeds of $4.52M. The September

Eurodollar futures trades at 92, and has a notional amountof $1M.

Compute

a. The current dollar value of the futures contract. b. The number of futures to buy/sell for optimal hedge.


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Example 2

The current dollar value is given by

$10,000× (100-0.25(100-92)) =

$980,000

Note that duration of futures is 3 months,

since this contract refers to 3-month LIBOR.


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Example 2

If Rates increase, the cost of borrowing will

be higher. We need to offset this by a gain, or

a short position in the futures. The optimalnumber of contracts is:

2.9000,980$90

000,520,4$180*

*

*

−=×

×−=

×

×−=

F D

S D N

F

S

Note that DVBP of the futures is 0.25× $1,000,000× 0.01%=$25


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FRM-00, Question 73

What assumptions does a duration-based hedgingscheme make about the way in which interest rates

move?

A. All interest rates change by the same amount

B. A small parallel shift in the yield curve

C. Any parallel shift in the term structure

D. Interest rates movements are highly correlated


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FRM-00, Question 73

What assumptions does a duration-based hedgingscheme make about the way in which interest rates

move?

A. All interest rates change by the same amount

B. A small parallel shift in the yield curve

C. Any parallel shift in the term structure

D. Interest rates movements are highly correlated


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FRM-99, Question 61

If all spot interest rates are increased by one basis point, avalue of a portfolio of swaps will increase by $1,100. How

many Eurodollar futures contracts are needed to hedge the

portfolio?

A. 44

B. 22

C. 11

D. 1100


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FRM-99, Question 61

The DVBP of the portfolio is $1,100.

The DVBP of the futures is $25.

Hence the ratio is 1100/25 = 44


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FRM-99, Question 109

Roughly how many 3-month LIBOR Eurodollar futures contracts are needed to hedge a position in a

$200M, 5 year, receive fixed swap?

A. Short 250B. Short 3,200

C. Short 40,000

D. Long 250


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FRM-99, Question 109

The dollar duration of a 5-year 6% par bond isabout 4.3 years. Hence the DVBP of the fixed

leg is about

$200M× 4.3× 0.01%=$86,000.

The floating leg has short duration - small

impact decreasing the DVBP of the fixed leg.

DVBP of futures is $25.

Hence the ratio is 86,000/25 = 3,440. Answer

A


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Beta Hedging

β represents the systematic risk, α - the

intercept (not a source of risk) and ε - residual.

it mt iiit R R ε β α ++=

M

M

S

S ∆≈

∆β

A stock index futures contract M M

F F ∆≈∆ 1


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Beta Hedging

M

M NF

M

M S F N S V

∆+

∆=∆+∆=∆ β

The optimal N is F S N β −=*

The optimal hedge with a stock index futures

is given by beta of the cash position times its

value divided by the notional of the futures

contract.


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Example

A portfolio manager holds a stock portfolio

worth $10M, with a beta of 1.5 relative to

S&P500. The current S&P index futures

price is 1400, with a multiplier of $250.

Compute:

a. The notional of the futures contract b. The optimal number of contracts for hedge.


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Example

The notional of the futures contract is

$250× 1,400 = $350,000

The optimal number of contracts for hedge is

9.42

000,350$1

000,000,10$5.1* −=

×

×−=

×−=

F

S N

β

The quality of the hedge will depend on the

size of the residual risk in the portfolio.


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A typical US stock has correlation of 50% with S&P.

Using the regression effectiveness we find that the

volatility of the hedged portfolio is still about

(1-0.52)0.5 = 87% of the unhedged volatility for a

typical stock.

If we wish to hedge an industry index with S&P

futures, the correlation is about 75% and the

unhedged volatility is 66% of its original level.

The lower number shows that stock market hedging is

more effective for diversified portfolios.


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FRM-00, Question 93

A fund manages an equity portfolio worth $50Mwith a beta of 1.8. Assume that there exists an

index call option contract with a delta of 0.623 and

a value of $0.5M. How many options contracts are

needed to hedge the portfolio?

A. 169

B. 289C. 306

D. 321


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FRM-00, Question 93

The optimal hedge ratio is

N = -1.8× $50,000,000/

(0.623× $500,000)=289

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RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

IDC

Following P. Jorion, Value at Risk, McGraw-HillChapter 8

Forecasting Risks and Correlations


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Volatility

Unobservable, time varying, clustering

Moving average r t daily returns: ∑=

−=

M

i

it t r M 1

22 1σ

Implied volatility (smile, smirk, etc.)

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GARCH Estimation

Generalized Autoregressive heteroskedastic

Heteroskedastic means time varying

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EWMA

Exponentially Weighted Moving Average

2

11 )1( −− −+⋅= t t t r hh λ λ

λ - is decay factor

λ

λ λ

−

+++= −−−

1

2

3

22

2

2

1 t t t t

r r r h

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Home assignment

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VaR system

Risk factorsHistorical data

Model

Distribution of

risk factorsVaR

method

Portfolio positions

Mapping

Exposures

VaR

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Ideas

Monte Carlo for financial assetsStress testing

VaR – OG

Collar example

ESOP hedging

Swaps + Credit DerivativesLinkage

Your personal financial Risk

Documents

PJVaRch78