Zvi Wiener

1
Financial Risk Management

• Zvi Wiener
• Following
• P. Jorion, Financial Risk Manager Handbook

2
Chapter 14Hedging Linear Risk

• Following P. Jorion 2001
• Financial Risk Manager Handbook

3
Hedging

• Taking positions that lower the risk profile of
  the portfolio.
• Static hedging
• Dynamic hedging

4
Unit Hedging with Currencies

• 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.

5

• Market data 0 7m PL
• time to maturity 9 2
• US interest rate 6 6
• Yen interest rate 5 2
• Spot Y/ 125.00 150.00
• Futures Y/ 124.07 149.00

6

• Stacked hedge - to use a longer horizon and to
  revert the position at maturity.
• Strip hedge - rolling over short hedge.

7
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

8
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.

9
FRM-00, Question 78

• 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.

10
FRM-00, Question 78

• 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.

11
FRM-00, Question 17

• 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.

12
FRM-00, Question 17

• 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.

13
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
  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.

14
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
  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.

15
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

16
The Optimal Hedge Ratio
Minimum variance hedge ratio
17
Hedge Ratio as Regression Coefficient

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

18
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.

If R is low the hedge is not effective!
19
Optimal Hedge

• At the optimum the variance is

20
FRM-99, Question 66

• 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

21
FRM-99, Question 66

• 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

22
FRM-99, Question 66

• 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

23
FRM-99, Question 66

• 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

24
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.

25
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.

26
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.

27
Solution

• The notional is Qs2,770,000, the SD in is
• ?(?s/s)sQs0.2117?277 ?10,000 586,409
• the SD of one futures contract is
• ?(?f/f)fQf0.1859?0.6903?42,000 5,390
• with a futures notional
• fQf 0.6903?42,000 28,993.

28
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 Qs?s/Qf?f 89.7, or 90 contracts.

29
Solution

• ?2unhedged (586,409)2 343,875,515,281
• - ?2SF/ ?2F -(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)

30
FRM-99, Question 67

• 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 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

31
FRM-99, Question 67

• 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 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

32
Duration Hedging
33
Duration Hedging
If we have a target duration DV we can get it by
using
34
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.

35
Example 1

• The notional is
• (932/32)/100?100,000 93,062.5
• The optimal number to sell is

Note that DVBP of the futures is
9.2?93,062?0.0185
36
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.

37
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.

38
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

Note that DVBP of the futures is
0.25?1,000,000?0.0125
39
FRM-00, Question 73

• 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

40
FRM-00, Question 73

• 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

41
FRM-99, Question 61

• 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

42
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

43
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 250
• B. Short 3,200
• C. Short 40,000
• D. Long 250

44
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
• 200M?4.3?0.0186,000.
• 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

45
Beta Hedging

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

A stock index futures contract
46
Beta Hedging
The optimal N is
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.
47
Example

• A portfolio manager holds a stock portfolio worth
  10M, with a beta of 1.5 relative to SP500. The
  current SP 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.

48
Example

• The notional of the futures contract is
• 250?1,400 350,000
• The optimal number of contracts for hedge is

The quality of the hedge will depend on the size
of the residual risk in the portfolio.
49

• A typical US stock has correlation of 50 with
  SP.
• 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 SP
  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.

50
FRM-00, Question 93

• 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

51
FRM-00, Question 93

• The optimal hedge ratio is
• N -1.8?50,000,000/(0.623?500,000)289