FRMJorion14HedgingLinearRisk

8/2/2019 FRMJorion14HedgingLinearRisk

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http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049FRM

Zvi WienerFollowing

P. Jorion,Financial Risk Manager Handbook

Financial Risk Management


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http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049FRM

Chapter 14

Hedging Linear RiskFollowing P. Jorion 2001

Financial Risk Manager Handbook


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Ch. 14, Handbook Zvi Wiener slide 4

Unit Hedging with Currencies

A US exporter will receive Y125M in 7months.

The perfect hedge is to enter a 7-months

forward contract.Such a contract is OTC and illiquid.

Instead one can use traded futures.

CME lists yen contract with face value

Y12.5M and 9 months to maturity.

Sell 10 contracts and revert in 7 months.


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Market data 0 7m P&L

time to maturity 9 2US interest rate 6% 6%

Yen interest rate 5% 2%

Spot Y/$ 125.00 150.00Futures Y/$ 124.07 149.00

667,166$125

1

150

1

125

MY

621,168$07.124

1

149

15.1210

MY


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Stacked hedge - to use a longer horizon and

to revert the position at maturity.

Strip hedge - rolling over short hedge.


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Basis Risk

Basis risk arises when the characteristics of

the futures contract differ from those of the

underlying.For example quality of agricultural product,

types of oil, Cheapest to Deliver bond, etc.

Basis = Spot - Future



Cross hedging

Hedging with a correlated (but different) asset.

In order to hedge an exposure to Norwegian

Krone one can use Euro futures.

Hedging a portfolio of stocks with index future.



FRM-00, Question 78

What feature of cash and futures prices tend to makehedging possible?

A. They always move together in the same directionand by the same amount.

B. They move in opposite direction by the sameamount.C. They tend to move together generally in the same

direction and by the same amount.D. They move in the same direction by differentamount.



FRM-00, Question 17Which statement is MOST correct?

A. A portfolio of stocks can be fully hedged bypurchasing a stock index futures contract.

B. Speculators play an important role in the futuresmarket by providing the liquidity that makeshedging possible and assuming the risk that hedgersare trying to eliminate.C. Someone generally using futures contract forhedging does not bear the basis risk.

D. Cross hedging involves an additional source ofbasis risk because the asset being hedged is exactlythe same as the asset underlying the futures.



FRM-00, Question 79

Under which scenario is basis risk likely to exist?A. A hedge (which was initially matched to the

maturity of the underlying) is lifted before expiration.

B. The correlation of the underlying and the hedgevehicle is less than one and their volatilities are

unequal.

C. The underlying instrument and the hedge vehicleare dissimilar.

D. All of the above.


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FRM-00, Question 79

Under which scenario is basis risk likely to exist?A. A hedge (which was initially matched to the

maturity of the underlying) is lifted before expiration.

B. The correlation of the underlying and the hedgevehicle is less than one and their volatilities are

unequal.

C. The underlying instrument and the hedge vehicleare dissimilar.

D. All of the above.


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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

FNSV

FSFSV NN ,2222 2

FSFV N

N

,

22

22


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The Optimal Hedge Ratio

FSFV N

N

,

22

22

F

SFS

F

FS

optN

,2

,

Minimum variance hedge ratio


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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:

f

f

s

ssf

f

ssf

f

sf

sf

2


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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

VsR

2

* 1 RsV

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

SV


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FRM-99, Question 66

The hedge ratio is the ratio of the size of the position taken in thefutures contract to the size of the exposure. Denote the standard

deviation of change of spot price by 1, the standard deviation of

change of future price by 2, the correlation between the changes in

spot and futures prices by . What is the optimal hedge ratio?

A. 1/1/2

B. 1/2/1

C. 1/2

D. 2/1


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FRM-99, Question 66

The hedge ratio is the ratio of the size of the position taken in thefutures contract to the size of the exposure. Denote the standard

deviation of change of spot price by 1, the standard deviation of

change of future price by 2, the correlation between the changes in

spot and futures prices by . What is the optimal hedge ratio?

A. 1/1/2

B. 1/2/1

C. 1/2

D. 2/1


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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 tobuy/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

(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.


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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* = sfQss/Qff = 89.7, or 90 contracts.


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Solution

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

- 2SF/2

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

hedged = $331,997The hedge has reduced the SD from $586,409

to $331,997.

R2 = 67.95% (= 0.82432)


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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 thatMG 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


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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 thatMG 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


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Duration Hedging

dyPDdP *

Dollar duration

yFDFySDS FS **

2**

22*2

22*2

ySFSF

yFF

ySS

SDFD

FD

SD


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Duration Hedging

FD

SDN

F

S

F

SF

*

*

2*

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

FD

SDVD

NF

SV

*

**


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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 optimalprotection.


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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*

*

*

FD

SDN

F

S

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


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Example 2

On February 2, a corporate treasurer wants tohedge 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 tradesat 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.


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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*

*

*

FD

SDN

F

S

Note that DVBP of the futures is 0.25$1,000,0000.01%=$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 amountB. 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 amountB. 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 basispoint, 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. 11D. 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 LIBOREurodollar 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 40,000D. 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

$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


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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 contractMM

FF 1


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Beta Hedging

M

MNF

M

MSFNSV

The optimal N isFSN *

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

$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.


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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. 289

C. 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

FRMJorion14HedgingLinearRisk

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