Financial Risk Management
Zvi Wiener
mswiener@mscc.huji.ac.il
02-588-3049
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 7
Portfolio Risk, Analytical Methods
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Portfolio of Random Variables
N
Y   wi X i  w X
T
i 1
N
E (Y )   p  w E ( X )  w  X   wi  i
T
T
i 1
N
N
 (Y )  w w   wi ij w j
2
T
i 1 j 1
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 3
Portfolio of Random Variables
 (Y ) 
2
  11  12

w1 , w2 ,, wN  
 N 1  N 2
Zvi Wiener
VaR-PJorion-Ch 7-8
 w1 
  1N   
  w2 
  
  NN   
 wN 
slide 4
Product of Random Variables
Credit loss derives from the product of the
probability of default and the loss given default.
E( X 1 X 2 )  E( X 1 ) E( X 2 )  Cov( X 1 , X 2 )
When X1 and X2 are independent
E( X 1 X 2 )  E( X 1 ) E( X 2 )
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 5
Transformation of Random Variables
Consider a zero coupon bond
100
V 
T
(1  r )
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 6
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%
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 7
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 8
with minor corrections
Exposure vs. Risk
F/X Hedging
Pr e s e nt V alue vs V aR
Gr oupe d by Pos ition
M onte Car lo Sim ulation, 1-M onth, 0.94 De cay, GBP
EUR/USD Option: 20030915
AUD/USD Forward: 20020405
NZD/USD Option: 20030220
CAD/USD Forward: 20021115
EUR/JPY Forward: 20010715
USD/ESP Option: 20011125
AUD/NZD Forward: 20020310
USD/ITL Forward: 20010906
JPY/DEM Forward: 20011007
EUR/USD Forward: 20010907
EUR/GBP Forward: 20021209
DEM Cash
JPY Cash
Pre se nt Va lue Va R, 95.00%
-558,920
186,407
-162,449
126,461
-10,801
11,417
-5,183
28,550
1,148
84,335
22,911
8,065
144,612
51,004
173,161
66,613
227,307
74,090
306,975
311,886
354,239
149,577
648,139
31,069
775,317
35,104
De tails :
Report Type
Scattergram
13
Number of Positions
1,000
Iterations
Zvi Wiener
Seed
1234567
Business Date
1/8/2001
Pricing Date
1/8/2001
Time Series Start
1/8/1999
Time Series End
1/8/2001
VaR-PJorion-Ch 7-8
slide 9
Marginal VaR by currency.....
with minor corrections
Marginal VaR
F/X Hedging
Marginal VaR by Currency
Grouped by Position
Param etric
.
%, -Month, .
Decay, GBP
Total
,
Total
AUD/NZD Forward:
AUD/USD Forward:
CAD/USD Forward:
DEM Cash
EUR/GBP Forward:
EUR/JPY Forward:
EUR/USD Forward:
EUR/USD Option:
JPY Cash
JPY/DEM Forward:
NZD/USD Option:
USD/ESP Option:
USD/ITL Forward:
Zvi Wiener
,
,
AUD
,
CAD
,
DEM
- ,
ESP
- ,
EUR
,
GBP
- ,
ITL
- ,
JPY
- ,
,
,
NZD
- ,
- ,
- ,
- ,
,
-
,
,
,
,
,
,
,
,
,
,
USD
,
,
-
,
,
,
,
- ,
,
- ,
,
- ,
- ,
- ,
- ,
,
- ,
- ,
VaR-PJorion-Ch 7-8
,
slide 10
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 11
Incremental VaR by Risk Type...
with minor corrections
Incremental VaR
F/X Hedging
Increm ental VaR by Risk Type
Grouped by Position
Param etric
.
%,
-Month,
Total
AUD/NZD Forward:
AUD/USD Forward:
CAD/USD Forward:
DEM Cash
EUR/GBP Forward:
EUR/JPY Forward:
EUR/USD Forward:
EUR/USD Option:
JPY Cash
JPY/DEM Forward:
NZD/USD Option:
USD/ESP Option:
USD/ITL Forward:
.
Decay, GBP
Total
,
,
,
,
,
,
,
,
- ,
- ,
- ,
- ,
- ,
FX Risk Interest Rate Risk
,
,
,
,
,
,
,
,
,
,
- ,
- ,
- ,
- ,
- ,
,
- ,
- ,
,
,
- ,
,
-
,
Details :
Report Type
Table
Number of Positions
Zvi Wiener
Business Date
/ /
Pricing Date
/ /
Time Series Start
/ /
Time Series End
/ /
VaR-PJorion-Ch 7-8
slide 12
Incremental VaR by Currency....
Zvi Wiener
VaR-PJorion-Ch 7-8
with minor corrections
slide 13
VaR decomposition
VaR
Incremental VaR
Marginal VaR
Portfolio VaR
Component VaR
100
Zvi Wiener
VaR-PJorion-Ch 7-8
Position in asset A
slide 14
Example of VaR decomposition
Currency Position Individual Marginal Component
VaR
VaR
Contribution
VaR
to VaR in %
CAD
$2M
$165,000 0.0528
$105,630
41%
EUR
$1M
$198,000 0.1521
$152,108
59%
Total
$3M
$257,738
100%
Undiversified
Diversified
Zvi Wiener
$363K
VaR-PJorion-Ch 7-8
slide 15
Barings Example
Long $7.7B Nikkei futures
Short of $16B JGB futures
NK=5.83%, JGB=1.18%, =11.4%
 P2  7.7 2  0.05832  162  0.01182  2  7.7 16  0.0583  0.114  0.0118
VaR95%=1.65P = $835M
VaR99%=2.33 P=$1.18B
Actual loss was $1.3B
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 16
P. Jorion Handbook, Ch 14
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
V  S  N  F

2
V

2
S
N 
2
2
F
 2 N S ,F

2
 2 N F  2 S ,F
N
2
V
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 17
P. Jorion Handbook, Ch 14
The Optimal Hedge Ratio

2
 2 N F  2 S ,F
N
2
V
N opt
 S ,F
 S
  2    S ,F
 F
 F
Minimum variance hedge ratio
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 18
P. Jorion Handbook, Ch 14
Hedge Ratio as Regression Coefficient
The optimal amount can also be derived as the
slope coefficient of a regression s/s on f/f:
s
f
    sf

s
f
 sf
s
 sf  2   sf
f
f
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 19
P. Jorion Handbook, Ch 14
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.
2
2
R 
2
( s   V * )

2
s
V*   s 1 R

2
sf
2
If R is low the hedge is not effective!
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 20
P. Jorion Handbook, Ch 14
Optimal Hedge
At the optimum the variance is

Zvi Wiener
2
V*

 

2
S
VaR-PJorion-Ch 7-8
2
SF
2
F
slide 21
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 22
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 23
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 24
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 25
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 26
P. Jorion Handbook, Ch 14
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$0.690342,000 = $5,390
with a futures notional
fQf = $0.690342,000 = $28,993.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 27
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 28
P. Jorion Handbook, Ch 14
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%
Zvi Wiener
(= 0.82432)
VaR-PJorion-Ch 7-8
slide 29
P. Jorion Handbook, Ch 14
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 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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 30
P. Jorion Handbook, Ch 14
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 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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 31
P. Jorion Handbook, Ch 14
Duration Hedging
dP   D * P  dy
Dollar duration
S   DS*  S  y
F   DF*  F  y
  D  S   
2
S
2
*
S
2
y
  D  F   
2
F
2
*
F
2
y
 SF  D  F D  S  
*
F
Zvi Wiener
*
S
VaR-PJorion-Ch 7-8
2
y
slide 32
P. Jorion Handbook, Ch 14
Duration Hedging
 SF
D S
N*   2  
F
D F
*
S
*
F
If we have a target duration DV* we can get it by using
D V  D  S
N
*
DF  F
*
V
Zvi Wiener
*
S
VaR-PJorion-Ch 7-8
slide 33
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 34
P. Jorion Handbook, Ch 14
Example 1
The notional is:
(93+2/32)/100$100,000 =$93,062.5
The optimal number to sell is:
D S
6.8  $10,000,000
N*  

 79.4
D F
9.2  $93,062.5
*
S
*
F
Note that DVBP of the futures is 9.2$93,0620.01%=$85
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 35
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 36
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 37
P. Jorion Handbook, Ch 14
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:
D S
180  $4,520,000
N*  

 9.2
D F
90  $980,000
*
S
*
F
Note that DVBP of the futures is 0.25$1,000,0000.01%=$25
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 38
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 39
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 40
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 41
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 42
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 43
P. Jorion Handbook, Ch 14
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0.01%=$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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 44
P. Jorion Handbook, Ch 14
Beta Hedging
Rit   i   i Rmt   it
 represents the systematic risk,  - the
intercept (not a source of risk) and  - residual.
S
M

S
M
A stock index futures contract
Zvi Wiener
VaR-PJorion-Ch 7-8
F
M
1
F
M
slide 45
P. Jorion Handbook, Ch 14
Beta Hedging
M
M
V  S  NF  S
 NF
M
M
S
The optimal N is N *  
F
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 46
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 47
P. Jorion Handbook, Ch 14
Example
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

 42.9
1 $350,000
The quality of the hedge will depend on the
size of the residual risk in the portfolio.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 48
P. Jorion Handbook, Ch 14
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.
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 49
P. Jorion Handbook, Ch 14
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
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 50
P. Jorion Handbook, Ch 14
FRM-00, Question 93
The optimal hedge ratio is
N = -1.8$50,000,000/(0.623$500,000)=289
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 51
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 8
Forecasting Risks and Correlations
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Volatility
Unobservable, time varying, clustering
Moving average rt daily returns:
1
2
t 
M
M
2
r
 t i
i 1
Implied volatility (smile, smirk, etc.)
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 53
GARCH Estimation
Generalized Autoregressive heteroskedastic
Heteroskedastic means time varying
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 54
EWMA
Exponentially Weighted Moving Average
ht    ht 1  (1   )r
2
t 1
 - is decay factor
ht 
Zvi Wiener
r  r
2
t 1
  r 
1 
2
t 2
2 2
t 3
VaR-PJorion-Ch 7-8
slide 55
Home assignment
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 56
VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of
risk factors
VaR
method
Exposures
VaR
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 57
Ideas
Monte Carlo for financial assets
Stress testing
VaR – OG
Collar example
ESOP hedging
Swaps + Credit Derivatives
Linkage
Your personal financial Risk
Zvi Wiener
VaR-PJorion-Ch 7-8
slide 58