Following
P. Jorion, Financial Risk Manager Handbook http://pluto.huji.ac.il/~mswiener/zvi.html
FRM 972-2-588-3049

Following P. Jorion 2001
Financial Risk Manager Handbook http://pluto.huji.ac.il/~mswiener/zvi.html
FRM 972-2-588-3049

Taking positions that lower the risk profile of the portfolio.
•
Static hedging
•
Dynamic hedging
Ch. 14, Handbook Zvi Wiener slide 3

A US exporter will receive Y125M in 7 months.
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.
Ch. 14, Handbook Zvi Wiener slide 4

Market data time to maturity
US interest rate
Yen interest rate
Spot Y/$
Futures Y/$
0
9
6%
5%
7m
2
6%
2%
125.00
150.00
124.07
149.00
Y 125 M

1
150

1
125
 
$ 166 , 667

10

Y 12 .
5 M

1
149

1
124 .
07

$ 168 , 621
Ch. 14, Handbook Zvi Wiener
P&L slide 5

Stacked hedge - to use a longer horizon and to revert the position at maturity.
Strip hedge - rolling over short hedge.
Ch. 14, Handbook Zvi Wiener slide 6

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
Ch. 14, Handbook Zvi Wiener slide 7

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.
Ch. 14, Handbook Zvi Wiener slide 8

What feature of cash and futures prices tend to make hedging possible?
A. They always move together in the same direction and by the same amount.
B. They move in opposite direction by the same amount.
C. They tend to move together generally in the same direction and by the same amount.
D. They move in the same direction by different amount.
slide 9 Ch. 14, Handbook Zvi Wiener

What feature of cash and futures prices tend to make hedging possible?
A. They always move together in the same direction and by the same amount.
B. They move in opposite direction by the same amount.
C. They tend to move together generally in the same direction and by the same amount.
D. They move in the same direction by different amount.
slide 10 Ch. 14, Handbook Zvi Wiener

Which statement is MOST correct?
A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract.
B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate.
C. Someone generally using futures contract for hedging does not bear the basis risk.
D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.
slide 11 Ch. 14, Handbook Zvi Wiener

Which statement is MOST correct?
A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract.
B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate.
C. Someone generally using futures contract for hedging does not bear the basis risk.
D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.
slide 12 Ch. 14, Handbook Zvi Wiener

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 hedge vehicle is less than one and their volatilities are unequal.
C. The underlying instrument and the hedge vehicle are dissimilar.
D. All of the above.
slide 13 Ch. 14, Handbook Zvi Wiener

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 hedge vehicle is less than one and their volatilities are unequal.
C. The underlying instrument and the hedge vehicle are dissimilar.
D. All of the above.
slide 14 Ch. 14, Handbook Zvi Wiener


S - change in $ value of the inventory

F - change in $ value of the one futures
N - number of futures you buy/sell

V
 
S

N
 
F

2

V
 
2

S

N
2

2

F

2 N


S ,

F
 

N
2

V

2 N

2

F

2


S ,

F
Ch. 14, Handbook Zvi Wiener slide 15

Ch. 14, Handbook
 

N
2

V

2 N

2

F

2


S ,

F
N opt
 



S ,

F
2

F
  

S ,

F


S


F
Minimum variance hedge ratio
Zvi Wiener slide 16

The optimal amount can also be derived as the slope coefficient of a regression
 s/s on
 f/f:
 s
    sf s
 f f
 sf

 sf
 f
2
  sf
 

 s f slide 17 Ch. 14, Handbook Zvi Wiener

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.
R
2 
(
 s
2

 
2
V *
)
2 s
 
2 sf

V *
  s
1

R
2
If R is low the hedge is not effective!
slide 18 Ch. 14, Handbook Zvi Wiener

At the optimum the variance is

2
V *
 
S
2 

2

SF
2
F
Ch. 14, Handbook Zvi Wiener slide 19

The hedge ratio is the ratio of the size of the position taken in the futures 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 slide 20 Ch. 14, Handbook Zvi Wiener

The hedge ratio is the ratio of the size of the position taken in the futures 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 slide 21 Ch. 14, Handbook Zvi Wiener

The 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 slide 22 Ch. 14, Handbook Zvi Wiener

The 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
slide 23 Ch. 14, Handbook Zvi Wiener

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.
Ch. 14, Handbook Zvi Wiener slide 24

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.
slide 25 Ch. 14, Handbook Zvi Wiener

•
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.
slide 26 Ch. 14, Handbook Zvi Wiener

The notional is Qs=$2,770,000, the SD in $ is

(
 s/s)sQ s
=0.2117

$277

10,000 = $586,409 the SD of one futures contract is

(
 f/f)fQ f
=0.1859

$0.6903

42,000 = $5,390 with a futures notional fQ f
= $0.6903

42,000 = $28,993.
slide 27 Ch. 14, Handbook Zvi Wiener

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
Q s
 s/Q f
 f = 89.7, or 90 contracts.
slide 28 Ch. 14, Handbook Zvi Wiener


2 unhedged
= ($586,409) 2 = 343,875,515,281
-

2
SF
/

2
F
= -(2,605,268,452/5,390) 2
 hedged
= $331,997
The hedge has reduced the SD from $586,409 to $331,997.
R 2 = 67.95% (= 0.8243
2 ) slide 29 Ch. 14, Handbook Zvi Wiener

In 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 longterm 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 slide 30 Ch. 14, Handbook Zvi Wiener

In 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 longterm 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 slide 31 Ch. 14, Handbook Zvi Wiener

dP
 
D *

P
 dy
Dollar duration

S
 
D
S
* 
S
  y

F
 
D
F
*



2
S
2
F
SF


D
S
* 
S
2

D
F
*
D
F
*
 
D
S
*
2
 y






F
F
2


 
2
 y

S

 
2
 y

F
  y slide 32 Ch. 14, Handbook Zvi Wiener

N *
 


SF
2
F
 
D
S
*
D
F
*

S

F
If we have a target duration D
V
* we can get it by using
N

D
V
* 
V
D
F
*


D
S
*
F

S slide 33 Ch. 14, Handbook Zvi Wiener

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.
Ch. 14, Handbook Zvi Wiener slide 34

The notional is:
(93+2/32)/100

$100,000 =$93,062.5
The optimal number to sell is:
N *
 
D
S
*
D
*
F

S

F
 
6 .
8

$ 10 , 000 , 000
9 .
2

$ 93 , 062 .
5
 
79 .
4
Note that DVBP of the futures is 9.2

$93,062

0.01%=$85 slide 35 Ch. 14, Handbook Zvi Wiener

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.
slide 36 Ch. 14, Handbook Zvi Wiener

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.
Ch. 14, Handbook Zvi Wiener slide 37

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:
N *
 
D
D
*
*
S
F


S
F
 
180

90

$ 4 , 520 , 000
$ 980 , 000
 
9 .
2
Note that DVBP of the futures is 0.25

$1,000,000

0.01%=$25 slide 38 Ch. 14, Handbook Zvi Wiener

What 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 slide 39 Ch. 14, Handbook Zvi Wiener

What 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 slide 40 Ch. 14, Handbook Zvi Wiener

If 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 slide 41 Ch. 14, Handbook Zvi Wiener

The DVBP of the portfolio is $1,100.
The DVBP of the futures is $25.
Hence the ratio is 1100/25 = 44
Ch. 14, Handbook Zvi Wiener slide 42

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 250
B. Short 3,200
C. Short 40,000
D. Long 250 slide 43 Ch. 14, Handbook Zvi Wiener

The dollar duration of a 5-year 6% par bond is about 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 slide 44 Ch. 14, Handbook Zvi Wiener

R it
  i
  i
R mt
  it
 represents the systematic risk,

- the intercept (not a source of risk) and

- residual.

S
S
 

M
M
A stock index futures contract

F
F

1

M
M slide 45 Ch. 14, Handbook Zvi Wiener


V
 
S

N

F
The optimal N is
 
S

M
N *
 
M

S
F

NF

M
M
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.
Ch. 14, Handbook Zvi Wiener slide 46

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.
slide 47 Ch. 14, Handbook Zvi Wiener

The notional of the futures contract is
$250

1,400 = $350,000
The optimal number of contracts for hedge is
N *
 
 
S
F
 
1 .
5

$ 10 , 000 , 000
1

$ 350 , 000
 
42 .
9
The quality of the hedge will depend on the size of the residual risk in the portfolio.
Ch. 14, Handbook Zvi Wiener slide 48

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
(1-0.5
2 ) 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.
Ch. 14, Handbook Zvi Wiener slide 49

A 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 slide 50 Ch. 14, Handbook Zvi Wiener

The optimal hedge ratio is
N = -1.8

$50,000,000/(0.623

$500,000)=289
Ch. 14, Handbook Zvi Wiener slide 51