rm mswiener/zvi.html huji-03 zvi wiener [email protected] 02-588-3049 financial risk...

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~mswiener/zvi.htmlHUJI-03

Zvi Wiener

[email protected]

02-588-3049

Financial Risk Management

mailto:[email protected]









Following P. Jorion, Value at Risk, McGraw-Hill

Chapter 7

Portfolio Risk, Analytical Methods


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

Portfolio of Random Variables

XwXwY TN

iii

1

N

iiiX

TTp wwXEwYE

1

)()(

N

i

N

jjiji

T wwwwY1 1

2 )(


Portfolio of Random Variables

NNNNN

N

N

w

w

w

www

Y

2

1

21

11211

21

2

,,,

)(


Product of Random Variables

Credit loss derives from the product of the

probability of default and the loss given default.

),()()()( 212121 XXCovXEXEXXE

When X1 and X2 are independent

)()()( 2121 XEXEXXE


Transformation of Random Variables

Consider a zero coupon bond

TrV

)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%.


The probability of this event can be derived

from the distribution of yields.

Assume that yields change are normally

distributed with mean zero and volatility 0.8%.

Then the probability of this change is 7.06%

Example


Marginal VaR

How risk sensitive is my portfolio to increase in size of each position?- calculate VaR for the entire portfolio VaRP=X- increase position A by one unit (say 1% of the portfolio)- calculate VaR of the new portfolio: VaRPa= 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...?


Exposure vs. RiskF/X Hedging

Present Value vs VaR

Grouped by Position

Monte Carlo Simulation, 1-Month, 0.94 Decay, GBP

Present Value VaR, 95.00%EUR/USD Option: 20030915 -558,920 186,407AUD/USD Forward: 20020405 -162,449 126,461NZD/USD Option: 20030220 -10,801 11,417CAD/USD Forward: 20021115 -5,183 28,550EUR/JPY Forward: 20010715 1,148 84,335USD/ESP Option: 20011125 22,911 8,065AUD/NZD Forward: 20020310 144,612 51,004USD/ITL Forward: 20010906 173,161 66,613JPY/DEM Forward: 20011007 227,307 74,090EUR/USD Forward: 20010907 306,975 311,886EUR/GBP Forward: 20021209 354,239 149,577DEM Cash 648,139 31,069JPY Cash 775,317 35,104

Details:

Report Type Scattergram

Number of Positions 13

Iterations 1,000

Seed 1234567

Business Date 1/8/2001

Pricing Date 1/8/2001

Time Series Start 1/8/1999

Time Series End 1/8/2001

with minor corrections


Marginal VaRF/X Hedging

Marginal VaR by Currency

Grouped by Position

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

Total AUD CAD DEM ESP EUR GBP ITL JPY NZD USDTotal 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 -38,332AUD/USD Forward: 20020405 90,488 102,962 -12,474CAD/USD Forward: 20021115 833 9,973 -9,141DEM Cash 28,682 28,682EUR/GBP Forward: 20021209 139,084 142,535 -3,451EUR/JPY Forward: 20010715 59,753 55,995 3,758EUR/USD Forward: 20010907 242,489 251,968 -9,480EUR/USD Option: 20030915 -134,979 -164,701 29,722JPY Cash -2,310 -2,310JPY/DEM Forward: 20011007 -45,954 -42,669 -3,285NZD/USD Option: 20030220 -3,781 -4,952 1,171USD/ESP Option: 20011125 -6,175 -6,673 498USD/ITL Forward: 20010906 -48,571 -50,895 2,324

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


Incremental VaR

Risk contribution of each position in my portfolio.- calculate VaR for the entire portfolio VaRP= X- remove A from the portfolio- calculate VaR of the portfolio without A: VaRP-A= Y- Risk contribution to the portfolio by A: Z = X-Y

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

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


Incremental VaRF/X Hedging

Incremental VaR by Risk Type

Grouped by Position

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

Total FX Risk Interest Rate RiskTotal 339,981 307,997 10,072

AUD/NZD Forward: 20020310 16,917 15,127 1,738AUD/USD Forward: 20020405 74,373 78,967 -5,119CAD/USD Forward: 20021115 -353 3,720 -4,165DEM Cash 28,398 28,398EUR/GBP Forward: 20021209 128,805 121,131 9,285EUR/JPY Forward: 20010715 53,738 52,545 1,222EUR/USD Forward: 20010907 139,317 141,262 -4,714EUR/USD Option: 20030915 -145,964 -154,427 9,273JPY Cash -4,436 -4,436JPY/DEM Forward: 20011007 -49,879 -48,996 -833NZD/USD Option: 20030220 -3,859 -4,200 342USD/ESP Option: 20011125 -6,222 -6,526 305USD/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


Incremental VaR by Currency.... with minor corrections


VaR decomposition

Position in asset A

VaR

100

Portfolio VaR

Incremental VaR

Marginal VaR

Component VaR


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%


Barings Example

Long $7.7B Nikkei futures

Short of $16B JGB futures

NK=5.83%, JGB=1.18%, =11.4%

0118.0114.00583.0167.720118.0160583.07.7 22222 P

VaR95%=1.65P = $835M

VaR99%=2.33 P=$1.18B

Actual loss was $1.3B


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

FNSV

FSFSV NN ,2222 2

FSFV N

N

,2

2

22

P. Jorion Handbook, Ch 14


The Optimal Hedge Ratio

FSFV N

N

,2

2

22

F

SFS

F

FSoptN

,2

,

Minimum variance hedge ratio



Hedge Ratio as Regression Coefficient

The optimal amount can also be derived as the slope coefficient of a regression s/s on f/f:

f

f

s

ssf

f

ssf

f

sfsf

2



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.

22

2*

22 )(

sfs

VsR

2* 1 RsV

If R is low the hedge is not effective!



Optimal Hedge

At the optimum the variance is

2

222

*F

SFSV



FRM-99, Question 66The hedge ratio is the ratio of derivatives to a spot position (vice versa) 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 the optimal hedge ratio for a three-month contract?

A. 0.1893

B. 0.2135

C. 0.2381

D. 0.2599



Example

Airline company needs to purchase 10,000 tons 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.



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.



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.



Solution

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

(s/s)sQs=0.2117$277 10,000 = $586,409

the SD of one futures contract is

(f/f)fQf=0.1859$0.690342,000 = $5,390

with a futures notional

fQf = $0.690342,000 = $28,993.



Solution

The cash position corresponds to a liability

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

protection.

sf= 0.8243 0.2117/0.1859 = 0.9387

sf = 0.8243 0.2117 0.1859 = 0.03244

The optimal hedge ratio is

N* = sf Qss/Qff = 89.7, or 90 contracts.



Solution

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

- 2SF/ 2

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

hedged = $331,997

The hedge has reduced the SD from $586,409

to $331,997.

R2 = 67.95% (= 0.82432)



FRM-99, Question 67In the early 90s, Metallgesellshaft, a German oil company, suffered a loss 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 to hedge its exposure by:

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

B. Long futures and there was a decline in oil price

C. Short futures and there was an increase in oil price

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



Duration Hedging

dyPDdP *

Dollar duration

yFDFySDS FS **

2**

22*2

22*2

ySFSF

yFF

ySS

SDFD

FD

SD



Duration Hedging

FD

SDN

F

S

F

SF

*

*

2*

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

FD

SDVDN

F

SV

*

**



Example 1A 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 contractb.The number of contracts to by/sell for optimal protection.



Example 1The 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*

*

*

FD

SDN

F

S

Note that DVBP of the futures is 9.2$93,0620.01%=$85



Example 2

On February 2, a corporate treasurer wants to hedge a July 17 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 amount of $1M.

Compute

a. The current dollar value of the futures contract.

b. The number of futures to buy/sell for optimal hedge.



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.



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 optimal

number of contracts is:

2.9000,980$90

000,520,4$180*

*

*

FD

SDN

F

S

Note that DVBP of the futures is 0.25$1,000,0000.01%=$25



FRM-00, Question 73What assumptions does a duration-based hedging scheme 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



FRM-99, Question 61If all spot interest rates are increased by one basis point, a value 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



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



FRM-99, Question 109Roughly how many 3-month LIBOR Eurodollar futures contracts are needed to hedge a position in a $200M, 5 year, receive fixed swap?

A. Short 250

B. Short 3,200

C. Short 40,000

D. Long 250



FRM-99, Question 109

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

$200M4.30.01%=$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



Beta Hedging

represents the systematic risk, - the intercept (not a source of risk) and - residual.

itmtiiit RR

M

M

S

S

A stock index futures contractM

M

F

F

1



Beta Hedging

M

MNF

M

MSFNSV

The optimal N is F

SN

*

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.



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.



Example

The notional of the futures contract is

$2501,400 = $350,000

The optimal number of contracts for hedge is

9.42000,350$1

000,000,10$5.1*

F

SN

The quality of the hedge will depend on the size of the residual risk in the portfolio.



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.



FRM-00, Question 93A fund manages an equity portfolio worth $50M with 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. 289

C. 306

D. 321



FRM-00, Question 93

The optimal hedge ratio is

N = -1.8$50,000,000/(0.623$500,000)=289




Following P. Jorion, Value at Risk, McGraw-Hill

Chapter 8

Forecasting Risks and Correlations



Volatility

Unobservable, time varying, clustering

Moving average rt daily returns:

M

iitt r

M 1

22 1

Implied volatility (smile, smirk, etc.)


GARCH Estimation

Generalized Autoregressive heteroskedastic

Heteroskedastic means time varying


EWMA

Exponentially Weighted Moving Average

211 )1( ttt rhh

- is decay factor

1

23

222

21 ttt

t

rrrh


Home assignment


VaR system

Risk factors

Historical data

Model

Distribution ofrisk factors

VaRmethod

Portfolio

positions

Mapping

Exposures

VaR


Ideas

Monte Carlo for financial assets

Stress testing

VaR – OG

Collar example

ESOP hedging

Swaps + Credit Derivatives

Linkage

Your personal financial Risk

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