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Transcript of RM mswiener/zvi.html HUJI-03 Zvi Wiener [email protected] 02-588-3049 Financial Risk...
RMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlHUJI-03
Zvi Wiener
02-588-3049
Financial Risk Management
RMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlHUJI-03
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 7
Portfolio Risk, Analytical Methods
Financial Risk Management
Zvi Wiener VaR-PJorion-Ch 7-8 slide 3
Portfolio of Random Variables
XwXwY TN
iii
1
N
iiiX
TTp wwXEwYE
1
)()(
N
i
N
jjiji
T wwwwY1 1
2 )(
Zvi Wiener VaR-PJorion-Ch 7-8 slide 4
Portfolio of Random Variables
NNNNN
N
N
w
w
w
www
Y
2
1
21
11211
21
2
,,,
)(
Zvi Wiener VaR-PJorion-Ch 7-8 slide 5
Product of Random Variables
Credit loss derives from the product of the
probability of default and the loss given default.
),()()()( 212121 XXCovXEXEXXE
When X1 and X2 are independent
)()()( 2121 XEXEXXE
Zvi Wiener VaR-PJorion-Ch 7-8 slide 6
Transformation of Random Variables
Consider a zero coupon bond
TrV
)1(
100
If r=6% and T=10 years, V = $55.84,
we wish to estimate the probability that the
bond price falls below $50.
This corresponds to the yield 7.178%.
Zvi Wiener VaR-PJorion-Ch 7-8 slide 7
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%
Example
Zvi Wiener VaR-PJorion-Ch 7-8 slide 8
Marginal VaR
How risk sensitive is my portfolio to increase in size of each position?- calculate VaR for the entire portfolio VaRP=X- 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...?
Zvi Wiener VaR-PJorion-Ch 7-8 slide 9
Exposure vs. RiskF/X Hedging
Present Value vs VaR
Grouped by Position
Monte Carlo Simulation, 1-Month, 0.94 Decay, GBP
Present Value VaR, 95.00%EUR/USD Option: 20030915 -558,920 186,407AUD/USD Forward: 20020405 -162,449 126,461NZD/USD Option: 20030220 -10,801 11,417CAD/USD Forward: 20021115 -5,183 28,550EUR/JPY Forward: 20010715 1,148 84,335USD/ESP Option: 20011125 22,911 8,065AUD/NZD Forward: 20020310 144,612 51,004USD/ITL Forward: 20010906 173,161 66,613JPY/DEM Forward: 20011007 227,307 74,090EUR/USD Forward: 20010907 306,975 311,886EUR/GBP Forward: 20021209 354,239 149,577DEM Cash 648,139 31,069JPY Cash 775,317 35,104
Details:
Report Type Scattergram
Number of Positions 13
Iterations 1,000
Seed 1234567
Business Date 1/8/2001
Pricing Date 1/8/2001
Time Series Start 1/8/1999
Time Series End 1/8/2001
with minor corrections
Zvi Wiener VaR-PJorion-Ch 7-8 slide 10
Marginal VaRF/X Hedging
Marginal VaR by Currency
Grouped by Position
Parametric 95.00%, 1-Month, 0.94 Decay, GBP
Total AUD CAD DEM ESP EUR GBP ITL JPY NZD USDTotal 339,981 161,716 9,973 -13,987 -6,673 285,797 -3,451 -50,895 -1,837 -43,284 2,621
AUD/NZD Forward: 20020310 20,422 58,754 -38,332AUD/USD Forward: 20020405 90,488 102,962 -12,474CAD/USD Forward: 20021115 833 9,973 -9,141DEM Cash 28,682 28,682EUR/GBP Forward: 20021209 139,084 142,535 -3,451EUR/JPY Forward: 20010715 59,753 55,995 3,758EUR/USD Forward: 20010907 242,489 251,968 -9,480EUR/USD Option: 20030915 -134,979 -164,701 29,722JPY Cash -2,310 -2,310JPY/DEM Forward: 20011007 -45,954 -42,669 -3,285NZD/USD Option: 20030220 -3,781 -4,952 1,171USD/ESP Option: 20011125 -6,175 -6,673 498USD/ITL Forward: 20010906 -48,571 -50,895 2,324
Marginal VaR by currency..... with minor corrections
Zvi Wiener VaR-PJorion-Ch 7-8 slide 11
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...?
Zvi Wiener VaR-PJorion-Ch 7-8 slide 12
Incremental VaRF/X Hedging
Incremental VaR by Risk Type
Grouped by Position
Parametric 95.00%, 1-Month, 0.94 Decay, GBP
Total FX Risk Interest Rate RiskTotal 339,981 307,997 10,072
AUD/NZD Forward: 20020310 16,917 15,127 1,738AUD/USD Forward: 20020405 74,373 78,967 -5,119CAD/USD Forward: 20021115 -353 3,720 -4,165DEM Cash 28,398 28,398EUR/GBP Forward: 20021209 128,805 121,131 9,285EUR/JPY Forward: 20010715 53,738 52,545 1,222EUR/USD Forward: 20010907 139,317 141,262 -4,714EUR/USD Option: 20030915 -145,964 -154,427 9,273JPY Cash -4,436 -4,436JPY/DEM Forward: 20011007 -49,879 -48,996 -833NZD/USD Option: 20030220 -3,859 -4,200 342USD/ESP Option: 20011125 -6,222 -6,526 305USD/ITL Forward: 20010906 -50,942 -52,264 1,295
Details:
Report Type Table
Number of Positions 13
Business Date 1/8/2001
Pricing Date 1/8/2001
Time Series Start 1/8/1999
Time Series End 1/8/2001
Incremental VaR by Risk Type... with minor corrections
Zvi Wiener VaR-PJorion-Ch 7-8 slide 14
VaR decomposition
Position in asset A
VaR
100
Portfolio VaR
Incremental VaR
Marginal VaR
Component VaR
Zvi Wiener VaR-PJorion-Ch 7-8 slide 15
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%
Zvi Wiener VaR-PJorion-Ch 7-8 slide 16
Barings Example
Long $7.7B Nikkei futures
Short of $16B JGB futures
NK=5.83%, JGB=1.18%, =11.4%
0118.0114.00583.0167.720118.0160583.07.7 22222 P
VaR95%=1.65P = $835M
VaR99%=2.33 P=$1.18B
Actual loss was $1.3B
Zvi Wiener VaR-PJorion-Ch 7-8 slide 17
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
,2
2
22
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 18
The Optimal Hedge Ratio
FSFV N
N
,2
2
22
F
SFS
F
FSoptN
,2
,
Minimum variance hedge ratio
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 19
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
sfsf
2
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 20
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.
22
2*
22 )(
sfs
VsR
2* 1 RsV
If R is low the hedge is not effective!
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 21
Optimal Hedge
At the optimum the variance is
2
222
*F
SFSV
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 22
FRM-99, Question 66The 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 23
FRM-99, Question 66The 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 27
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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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 Qss/Qff = 89.7, or 90 contracts.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 29
Solution
2unhedged = ($586,409)2 = 343,875,515,281
- 2SF/ 2
F = -(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)
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 30
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 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 31
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 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 32
Duration Hedging
dyPDdP *
Dollar duration
yFDFySDS FS **
2**
22*2
22*2
ySFSF
yFF
ySS
SDFD
FD
SD
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 33
Duration Hedging
FD
SDN
F
S
F
SF
*
*
2*
If we have a target duration DV* we can get it by using
FD
SDVDN
F
SV
*
**
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 34
Example 1A 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 contractb.The number of contracts to by/sell for optimal protection.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 35
Example 1The 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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:
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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 39
FRM-00, Question 73What 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 40
FRM-00, Question 73What 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 41
FRM-99, Question 61If 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 43
FRM-99, Question 109Roughly 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 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
$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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 45
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 contractM
M
F
F
1
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 46
Beta Hedging
M
MNF
M
MSFNSV
The optimal N is F
SN
*
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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 47
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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 48
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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 49
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
(1-0.52)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.
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 50
FRM-00, Question 93A 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
P. Jorion Handbook, Ch 14
Zvi Wiener VaR-PJorion-Ch 7-8 slide 51
FRM-00, Question 93
The optimal hedge ratio is
N = -1.8$50,000,000/(0.623$500,000)=289
P. Jorion Handbook, Ch 14
RMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlHUJI-03
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 8
Forecasting Risks and Correlations
Financial Risk Management
Zvi Wiener VaR-PJorion-Ch 7-8 slide 53
Volatility
Unobservable, time varying, clustering
Moving average rt daily returns:
M
iitt r
M 1
22 1
Implied volatility (smile, smirk, etc.)
Zvi Wiener VaR-PJorion-Ch 7-8 slide 54
GARCH Estimation
Generalized Autoregressive heteroskedastic
Heteroskedastic means time varying
Zvi Wiener VaR-PJorion-Ch 7-8 slide 55
EWMA
Exponentially Weighted Moving Average
211 )1( ttt rhh
- is decay factor
1
23
222
21 ttt
t
rrrh
Zvi Wiener VaR-PJorion-Ch 7-8 slide 57
VaR system
Risk factors
Historical data
Model
Distribution ofrisk factors
VaRmethod
Portfolio
positions
Mapping
Exposures
VaR