HUJI-03

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Financial Risk Management

• Zvi Wiener
• mswiener_at_mscc.huji.ac.il
• 02-588-3049

2
Financial Risk Management

• Following P. Jorion, Value at Risk, McGraw-Hill
• Chapter 7
• Portfolio Risk, Analytical Methods

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Portfolio of Random Variables
4
Portfolio of Random Variables
5
Product of Random Variables

• Credit loss derives from the product of the
  probability of default and the loss given default.

When X1 and X2 are independent
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Transformation of Random Variables

• Consider a zero coupon bond

If r6 and T10 years, V 55.84, we wish to
estimate the probability that the bond price
falls below 50. This corresponds to the yield
7.178.
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Example

• 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

8
Marginal VaR

• How risk sensitive is my portfolio to increase in
  size of each position?
• - calculate VaR for the entire portfolio VaRPX
• - 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...?

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

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Incremental VaR by Risk Type...
with minor corrections
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Incremental VaR by Currency....
with minor corrections
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VaR decomposition
VaR
Incremental VaR
Marginal VaR
Portfolio VaR
Component VaR
Position in asset A
100
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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
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Barings Example

• Long 7.7B Nikkei futures
• Short of 16B JGB futures
• ?NK5.83, ?JGB1.18, ?11.4

VaR951.65??P 835M VaR992.33
??P1.18B Actual loss was 1.3B
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The Optimal Hedge Ratio
P. Jorion Handbook, Ch 14

• ?S - change in value of the inventory
• ?F - change in value of the one futures
• N - number of futures you buy/sell

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The Optimal Hedge Ratio
P. Jorion Handbook, Ch 14
Minimum variance hedge ratio
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Hedge Ratio as Regression Coefficient
P. Jorion Handbook, Ch 14

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

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Optimal Hedge
P. Jorion Handbook, Ch 14

• 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!
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Optimal Hedge
P. Jorion Handbook, Ch 14

• At the optimum the variance is

22
FRM-99, Question 66
P. Jorion Handbook, Ch 14

• 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

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FRM-99, Question 66
P. Jorion Handbook, Ch 14

• 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

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Example
P. Jorion Handbook, Ch 14

• 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.

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Example
P. Jorion Handbook, Ch 14

• 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
P. Jorion Handbook, Ch 14

• 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.

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Solution
P. Jorion Handbook, Ch 14

• 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.

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Solution
P. Jorion Handbook, Ch 14

• 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.

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Solution
P. Jorion Handbook, Ch 14

• ?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)

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FRM-99, Question 67
P. Jorion Handbook, Ch 14

• 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

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FRM-99, Question 67
P. Jorion Handbook, Ch 14

• 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

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Duration Hedging
P. Jorion Handbook, Ch 14
33
Duration Hedging
P. Jorion Handbook, Ch 14
If we have a target duration DV we can get it by
using
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Example 1
P. Jorion Handbook, Ch 14

• 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.

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Example 1
P. Jorion Handbook, Ch 14

• 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
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Example 2
P. Jorion Handbook, Ch 14

• 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.

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Example 2
P. Jorion Handbook, Ch 14

• 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
P. Jorion Handbook, Ch 14

• 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
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FRM-00, Question 73
P. Jorion Handbook, Ch 14

• 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

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FRM-00, Question 73
P. Jorion Handbook, Ch 14

• 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

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FRM-99, Question 61
P. Jorion Handbook, Ch 14

• 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

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FRM-99, Question 61
P. Jorion Handbook, Ch 14

• 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
P. Jorion Handbook, Ch 14

• 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

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FRM-99, Question 109
P. Jorion Handbook, Ch 14

• 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

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Beta Hedging
P. Jorion Handbook, Ch 14

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

A stock index futures contract
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Beta Hedging
P. Jorion Handbook, Ch 14
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.
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Example
P. Jorion Handbook, Ch 14

• 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.

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Example
P. Jorion Handbook, Ch 14

• 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.
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P. Jorion Handbook, Ch 14

• 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.

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FRM-00, Question 93
P. Jorion Handbook, Ch 14

• 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

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FRM-00, Question 93
P. Jorion Handbook, Ch 14

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

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Financial Risk Management

• Following P. Jorion, Value at Risk, McGraw-Hill
• Chapter 8
• Forecasting Risks and Correlations

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Volatility

• Unobservable, time varying, clustering
• Moving average rt daily returns

Implied volatility (smile, smirk, etc.)
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GARCH Estimation

• Generalized Autoregressive heteroskedastic
• Heteroskedastic means time varying

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EWMA

• Exponentially Weighted Moving Average

? - is decay factor
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Home assignment
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VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of risk factors
VaR method
Exposures
VaR
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Ideas

• Monte Carlo for financial assets
• Stress testing
• VaR OG
• Collar example
• ESOP hedging
• Swaps Credit Derivatives
• Linkage
• Your personal financial Risk