Hedging Exposures with Forward and Futures Contracts:
Hedging Linear Risk – Chapter 14
Forward Contracts:
Underlying
Hedge/Contract date
Futures Contracts:
Underlying
Hedge/Contract date
Managing using Forward/Futures Contracts
0
t=T
Risk measured
Outcome realized
Inception
Unwind
0
Risk measured
Inception
t
Outcome realized
Unwind
T
Delivery Date
Futures Hedging pages 311-320
Hedge positions utilizing forward or futures contracts:
Long underlying and Short derivative (LS)
Short underlying and Long derivative (SL)
Objective of pairing derivative (forward/futures) position with underlying risk exposure:
To reduce variance of the hedged cash flow.
When using forward contracts that exactly match the underlying position the variance of
dollar value of terminal cash flow can be reduced to zero.
When using futures contracts, because the critical terms of the underlying position will
not necessarily exactly match those of the futures contract, basis risk will remain in the
hedged position. It is not necessarily possible to reduce the variance of dollar value of
terminal cash flow to zero using futures contracts.
Basisd = Sd - Gd
Basis is the contemporaneous difference between spot (S) and futures (G) prices.
Note – Basis is sometimes expressed as Gd - Sd , in particular for commodity markets.
When basis is discussed you must determine which definition is being used.
Basis is a random variable. The absolute size of basis and the sign of basis depend on
the underlying, market conditions, supply-demand and time to contract delivery date.
1
The law of one price guarantees that at the contract delivery date BT = ST – GT = 0.
However, on date t < T, the size of basis is a random variable.
LS hedge positions are long basis. LS hedge positions benefit when basis increases.
LS position value and
0
S
$0.713
G
0.714
B
-0.001
S
-G
S - G
impact of basis volatility
t
t
t
$0.710 $0.710 $0.710
0.713
0.711
0.707
-0.003
-0.001
0.003
-0.003
-0.003
-0.003
0.001
0.003
0.007
-0.002
0.000
0.004
SL hedge positions are short basis. SL hedge positions benefit when basis decreases.
SL position value and impact of basis volatility
0
t
t
t
S
$0.713 $0.710 $0.710 $0.710
G
0.714
0.713
0.711
0.707
B
-0.001
-0.003
-0.001
0.003
0.003
0.003
0.003
-S
-0.001
-0.003
-0.007
G
0.002
0.000
-0.004
-S + G
Hedged cash flow LS positions: S0 + S – h* G
Hedged cash flow SL positions: -S0 - S + h* G
Where;
S = St – S0
G = Gt – G0
h = hedge ratio, i.e. quantity contracted in futures market per unit of underlying.
Assume h = 1:
Hedged cash flow LS positions: S0 + S – h* G = (St – Gt) + G0 = Bt + G0
Hedged cash flow SL positions: -S0 - S + h* G = -(St – Gt) – G0 = -Bt + G0
2
Variance minimizing hedge ratio: Same in absolute value for both LS and SL hedge
positions.
Hedged cash flow LS positions: S0 + S – h* G
Variance(Hedge cash flow) = Variance(S – h* G)
Variance(Hedge cash flow) = Variance(S)+h2Variance(G)-2*h*Covariance(S, G)
To find the value of h that minimizes Variance(Hedge cash flow):
 Take first derivative Variance(Hedge cash flow) with respect to h.
 Set this derivative equal to zero and solve for h.
The variance minimizing hedge ratio is;
h
Co var iance(S , G)
Variance(G)
The variance minimizing hedge ratio may be estimated from a regression of S on G.
S  a  h  G  
The estimated slope coefficient from such a regression, ĥ , is the volatility minimizing
hedge ratio over the estimation period.
The R2 statistic from the regression results is the proportion of the variance S explained
by the linear relationship, â + ĥ*G.
Consequently 1-R2 is the proportion of the variance of S remaining in the hedged
position.
The volatility remaining in the hedge position can be found from the original naked
volatility and the result above.
 H   N  1 R2
If implementing the iid-Normal methodology the risk reduction produced by the hedge
position can be found directly.
VaRH = normsinv(c)*H < VaRN = normsinv(c)*N
notes5.xls contains an evaluation for application to price risk exposure resulting from
positions in Swiss Franc.
3
Data sources for example:
Daily, weekly, monthly and annual timeseries: Interest rates, foreign exchange rates,
monetary aggregates available at:
http://www.federalreserve.gov/releases/
Daily Update
The weekly release is posted on Monday. Daily updates of the weekly release are posted
Tuesday through Friday on this site.
H.10 DAILY UPDATE: WEB RELEASE ONLY
For immediate release
FOREIGN EXCHANGE RATES
January 29, 2004
The Board of Governors of the Federal Reserve System is advised that the Federal Reserve
Bank of New York has certified for customs purposes the following noon buying rates in
New York City for cable transfers payable in foreign currencies:
(Rates in currency units per U.S. dollar except as noted)
MONETARY
COUNTRY
UNIT
Jan. 26
Jan. 27
Jan. 28
Jan. 29
*AUSTRALIA
DOLLAR
0.773
0.7805
0.7788
0.7585
BRAZIL
REAL
2.843
2.865
2.88
2.945
CANADA
DOLLAR
1.313
1.3048
1.3218
1.334
CHINA, P.R.
YUAN
8.2771
8.2771
8.2771
8.277
DENMARK
KRONE
5.9345
5.893
5.898
6.02
*EMU MEMBERS
EURO
1.2552
1.2643
1.262
1.2389
HONG KONG
DOLLAR
7.767
7.7644
7.768
7.775
INDIA
RUPEE
45.36
45.38
45.36
45.29
JAPAN
YEN
106.15
105.56
105.52
106.09
MALAYSIA
RINGGIT
3.8
3.8
3.8
3.8
MEXICO
PESO
10.934
10.88
10.897
11.097
*NEW ZEALAND
DOLLAR
0.6727
0.6795
0.6785
0.6673
NORWAY
KRONE
6.864
6.813
6.9197
7.073
SINGAPORE
DOLLAR
1.6943
1.695
1.6914
1.7035
SOUTH AFRICA
RAND
7.1926
7.08
6.92
7.0735
SOUTH KOREA
WON
1181
1180
1172
1173
SRI LANKA
RUPEE
98.2
97.5
97.6
97.48
SWEDEN
KRONA
7.308
7.245
7.249
7.41
4
SWITZERLAND
FRANC
1.2484
1.2389
1.243
1.2621
TAIWAN
DOLLAR
33.73
33.51
33.33
33.36
THAILAND
BAHT
39.32
39.13
39.12
39.28
*UNITED KINGDOM POUND
1.82
1.8277
1.837
1.8112
VENEZUELA
1600
1600
1600
1600
BOLIVAR
MEMO:
Jan. 26 Jan. 27 Jan. 28 Jan. 29
UNITED STATES
DOLLAR
1)BROAD
JAN97=100
2)MAJOR CURRENCY MAR73=100
JAN97=100
3)OITP
112.99
84.99
112.5 112.74 113.88
84.45
84.79
85.88
142.63 142.39 142.36 143.31
For more information on exchange rate indexes for the U.S. dollar, see "New Summary Measures
of the Foreign Exchange Value of the Dollar," Federal Reserve Bulletin, vol. 84 (October 1998),
pp. 811-18 (http://www.federalreserve.gov/pubs/bulletin/). Weights for the broad index can be
found at http://www.federalreserve.gov/releases/H10/Weights; weights for the major currencies
index and the other important trading partners (OITP) index are derived from the broad index
weights. The most recent annual revision of the currency weights and dollar indexes took effect
with the December 16, 2003, release of this report.
* U.S. dollars per currency unit.
1) A weighted average of the foreign exchange value of the U.S. dollar against the currencies
of a broad group of major U.S. trading partners.
2) A weighted average of the foreign exchange value of the U.S. dollar against a subset of
the broad index currencies that circulate widely outside the country of issue.
3) A weighted average of the foreign exchange value of the U.S. dollar against a subset of
the broad index currencies that do not circulate widely outside the country of issue.
The euro is reported in place of the individual euro-area currencies. These currency rates can
be derived from the dollar/euro rate by using the fixed conversion rates (in currencies per euro)
given below:
1 EURO = 13.7603 AUSTRIAN SCHILLINGS
= 40.3399 BELGIAN FRANCS
= 5.94573 FINNISH MARKKAS
= 6.55957 FRENCH FRANCS
= 1.95583 GERMAN MARKS
= .787564 IRISH POUNDS
5
= 1936.27 ITALIAN LIRE
= 40.3399 LUXEMBOURG FRANCS
= 2.20371 NETHERLANDS GUILDERS
= 200.482 PORTUGUESE ESCUDOS
= 166.386 SPANISH PESETAS
= 340.750 GREEK DRACHMAS
http://www.cme.com/
Contract terms Chicago Mercantile Exchange Swiss Franc Futures contract:
Ticker Symbol
Sample Quote
Contract Size
Minimum Price Fluctuation
(Tick)
Price Limit
Contract Month Listings
Trading Hours
Last Trading Day
Final Settlement Rule
Position Limits/
Accountability
Trading Venue
Clearing=E1; Ticker=SF; GLOBEX=6S; AON=LS; (20 Threshold)&
Sample Quote = .6004 USD/SF
125,000 Swiss francs
Floor: Regular - 0.0001=$12.50; Calendar Spread - 0.00005=$6.25; All
or None - 0.00005=$6.25
GLOBEX®: Regular - 0.0001=$12.50; Calendar Spread 0.00005=$6.25
Floor:No limits
GLOBEX®:No limits
Six months in the March Quarterly cycle, Mar, Jun Sep, Dec. See notes
+++, **
Floor: 7:20 a.m.-2:00 p.m. LTD(9:16 a.m.)^
GLOBEX®: Mon/Thurs 5:00 p.m.-4:00 p.m. Sun & Hol 5:30 p.m.-4:00
p.m.
Trading ceases at 9:16 a.m. Central Time on the second business day
immediately preceding the third Wednesday of the contract month
(usually Monday).
Final settlement price is determined by the Trading Floor Pit Committee.
Contract is physically delivered.
A person owning or controlling more than 10,000 contracts net long or
net short in all contract months combined shall provide, in a timely
fashion, upon request by the Exchange, information regarding the
nature of the position, trading strategy, and hedging information if
applicable. For positions involving options on Swiss franc futures, this
rule is superseded by the option position accountability rule.
Floor, GLOBEX®
Futures data contained in the file notes5.xls were obtained at:
http://www.econstats.com/fut/xcb__d73.htm
http://www.econstats.com/econstats.htm
6