1 Systems Analysis Advisory Committee (SAAC) Thursday, October 24, 2002 Michael Schilmoeller John...

1

Systems Analysis Advisory Committee (SAAC)

Thursday, October 24, 2002Michael Schilmoeller

John Fazio

Northwest Power Planning Council

2

Original Agenda

• Metrics– Stakeholders– Risk measures– Timing

• Representations in the portfolio model– thermal generation– hydro generation– conservation and renewables– loads– contracts– reliability– ** Plan Issues **


3

Plan Issues

• incentives for generation capacity• price responsiveness of demand• sustained investment in efficiency• information for markets• fish operations and power• transmission and reliability• resource diversity• role of BPA• global change


4

Revised Agenda

• Approval of the Oct 4 meeting minutes• Price Processes• Representations in the portfolio model

– thermal generation


• Representations in the portfolio model– ** Plan Issues ** : price responsive demand


5

Revised Agenda






6

Price Processes

• Problem: There are mathematical difficulties with describing prices and price processes statistically.

• To show: The natural logarithm of prices (or price ratios) provides a solution to the problem

Price Processes


7

Price Processes

• Problem: Naïve attempts to describe prices statistically lead to nonsense. For example, a symmetric distribution, unbounded on the high side, must be unbounded on the low side

Price Processes

Probability density

0

0.05

0.1

0.15

0.2

0.25

-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Price

den

sity

Oops


8

Price Processes

• Problem: More seriously, if we try to describe variation in a price process from historical data, we run into seasonality problems. Suppose we wanted to estimate the daily variation in prices from a price series over 90 days:

Price Processes

Standard deviation of the price curve (black) would be quite

large and would not describe the daily variation (red)

Clearly, we want something that more

closely resembles daily price returns


9

Price Processes

• Problem: Returns themselves, however, have bad statistical properties. For example, what is the meaning of the average of a 50% increase in prices and a 50% decrease in prices?

Price Processes

75.5.05.1


10

Price Processes

• Solution: What does work is the logarithm of returns (and the inverse transformation, exponentiation)

Price Processes

period over thereturn the/)/ln(

)ln()ln(

)ln()ln()ln()ln()ln()ln(

)/ln()/ln()/ln(

:returns of Sum

11

1

12312

12312

pppp

pp

pppppp

pppppp

nn

n

nn

nn

• If price ends where it started out, the ratio of the prices is one, and the logarithm of returns is zero.


11

Price Processes

• Other nice properties of the logarithm of returns

Price Processes

rategrowth period the/))/ln((

)/ln(1

)/ln()/ln()/ln(1

:returns of Average

1

1

1

1

12312

n pppp

ppn

ppppppn

nn

n

n

nn

returnregular the,1~/)~/ln(

,~ As

pppp

pp


12

GBM• Geometric Brownian Motion (GBM)

– independent draws of ln(p) are made from a normal distribution

– grows as

-12

-8

-4

0

4

8

12

0 2 4 6 8 10 12 14 16 18 20

19

16

T

Time T-->

• Makes sense, because the standard deviation of the sum of T draws from a

distribution is

),0( 2N

TT

T

2

222

21

Price Processes


13

GBM• And the normal distribution of ln(p) gives us a reasonable

distribution of prices

Lognormal Prices

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150 200 250 300 350 400

Prices $/MWh

Pro

b d

ensi

ty

Price Processes


14

Conclusion

• The natural logarithm of prices (or price ratios) overcome the mathematical difficulties with describing prices and price processes statistically

• Normal log returns produce lognormal price distributions, which have desirable distribution

• GBM is perhaps the simplest description of a stochastic process, where draws are independent, random, normal. May describe processes like stock prices well, where daily returns should be about normal

Price Processes


15

Revised Agenda






16

Thermal Generation

• Objectives– We need a way to quickly estimate the dispatch factor for

thermal generation, so that we can calculate variable cost.– Should have certain basic properties

• If average monthly prices ($/MWh) for gas are about the same as average monthly prices for electricity, the dispatch factor should be about 50 percent.

• If average monthly prices ($/MWh) for gas are well above the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should dispatch, albeit a small amount

• If average monthly prices ($/MWh) for gas are well below the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should run close to, but not quite 100% capacity factor (disregarding maintenance and forced outage)


17

Thermal Generation

• Objectives– To show: Thermal dispatch can be reasonably well using a

spread call option on electricity and gas– To show: The monthly capacity factor of the thermal unit is

provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread

– To show: The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments:

• Discount rate r = 0• Volatility incorporates terms for the uncertainty in and the

expected variation of the spread over the specified time frame


18

Typical Dispatchable

• Example of 1 MW Single Cycle Combustion Turbine (no dispatch constraints)

• Natural Gas price, $3.33/MBTU

• Heat rate, 9000 BTU/kWh

• Price of electricity generated from gas, $30/MWh

Representations - thermal

0

5

10

15

20

25

30

35

40

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

Mar

ket

Pri

ce $

/MW

h

Value:note: we assuming the

entire capacity is switched on when the turbine runs

case)our in MW (1 turbine theofcapacity theis

($/MWh) rateheat fixed a assuming

hour, in this gas of price theis )(

($/MWh)hour in thisy electricit of price theis )(

case) in this (672 hours ofset theis

where

)))()((,0max(

C

hp

hp

H

hphpCV

g

e

Hhge


19

Price Duration Curve

• If we assume each hour’s dispatch is independent, we can ignore the chronological structure. Sorting by price yields the market price duration curve (MCD)

0

5

10

15

20

25

30

35

40

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

Mar

ket

Pri

ce $

/MW

h

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

Count of Hours

$/M

Wh

Value V is this area


operatorn expectatio theis (672) period in the hours ofnumber theis

where])()(,0[max

or

)()(,0max

)()(,0max

EN

hphpECNV

N

hphpCN

hphpCV

H

geH

H

Hhge

H

Hhge


20

Variability viewed as CDF

• Turning the MCD curve on its side, we get something that looks like a cumulative probability density function (CDF)

Value V is this area

factorcapacity or the CDF theof value theis )(where

)(

Calculus) of Thm (Fund

)(

e

eHe

e

P

eH

pf

pfNdp

dV

dppfNVg

Cumulative Frequency

0

100

200

300

400

500

600

37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20

$/MWh

Co

un

t o

f h

ou

rs

0%10%20%30%40%50%60%70%80%90%100%

Cap

acit

y F

acto

r



21

Cumulative FrequencySingle, fixed hour

0

0.2

0.4

0.6

0.8

1

40.00 38.00 36.00 34.00 32.00 30.00 28.00 26.00 24.00 22.00 20.00

$/MWh

Pro

b o

f p

ric

e

ex

ce

ed

ing

p

Uncertainty

• To this point, we have assumed we know what the hourly electricity price will be in each hour. However, we could similarly calculate the expected capacity factor for fixed hour using a CDF that described our uncertainty about prices within that hour. The preceding results still apply.


??


22

Transformation of variables

• Everything we have done to this point still holds if we used transformed prices


price base fixedbut arbitrary, some is ~where

)~/ln(

)~/ln(

p

ppzp

ppzp

ggg

eee

g

ge

eHH

e

z

f

zz

V

dp

dVfNfN

dz

dV

for

CDF theof value theis '

ofleft the tofor

CDF under the area theis '

where

''

z is dual to p, for positive p

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5


23

Simplification #1

• The expected capacity factor over the time period will be a function of expected variation over the period and the uncertainty associated with each hour. It will be determined by the CDF of


0

5

10

15

20

25

30

35

40

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

Mar

ket

Pri

ce $

/MW

h

)(hpe

)()()( hhph ee

)(h

)(he


24

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

ln(pe/

pg

)

Simplification #1• Assume distributions of uncertainty in , say ,

are identical across all hours of the time period (e.g., month). Note is still a vector and has covariance structure


)(hze

)()()( hhzh eee

)(he

)(he)(hze

)(he


25

Simplification #1

•

• Note that constant uncertainty for z implies greater price uncertainty during times of high prices than during times of low prices (cool)


with themnscalculatio perform easier tomuch be it will

t,independen now are (h) and (h) Because eez

0

5

10

15

20

25

30

35

40

1 25

Hour

$/M

Wh

)(hze


26

Digression to Options

• European call option: Confers the right, but not the obligation to purchase a specified quantity of an instrument or commodity at a fixed price at a specified time in the future


• Example of a call option on a stock with a strike price of $30

• Below $30, the option is worthless

• For each dollar over $30 that the price of the stock reaches, the value of the option increases a dollar

European Call Option

0

2

4

6

8

10

12

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Price of underlying

Val

ue

of

Op

tio

n (

$) Intrinsic value


27


• Because the option will expire at some specified time in the future and because we are uncertain what the value of the stock will be when the option expires, the value of the option is greater than the intrinsic value



0.00

2.00

4.00

6.00

8.00

10.00

12.00

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Price of underlying

Val

ue

of

Op

tio

n (

$)

max(0,p-X)


28


• The value of an option is the expected value of the discounted payoff (See Hull, 3rd ed., p. 295)


price strike theis stock theof price theis

(years) expiration to time theis ratediscount annual theis

operatorn expectatio theis where

)],0max([

XpTrE

XpeEV rT


29


• The challenge in solving for the value explicitly (in closed form) is determining the discount rate r. The problem was essentially solved otherwise much earlier by A.J. Boness in his Ph.D. thesis.

• Fischer Black and Myron Scholes in 1973 showed that if certain assumptions held, the discount rate r should be the risk-free discount rate:– the stock pays no dividends, markets are efficient, interest rates are

known– returns are normally distributed (Geometric Brownian Motion), so

prices are lognormally distributed– prices change continuously so the option can be hedged



30


• The Black Scholes pricing formula


TddrTXpd

NN

dNXedpNV rT

12

21

21

)2/()/ln()p~ln(p/ ofdeviation standard is variablerandom )1,0( afor CDF theis

where

)()(


31

Simplification #2

• We will discount the payoff externally, so let r = 0


price strike theis stock theof price theis operatorn expectatio theis

where

)],0[max(

XpE

XpEV

• This now closely resembles our calculation for the value of a thermal plant, assuming the CDF is the CDF of

)()()( hhzh eee


32

Simplification #3

• We assume that the distributions of


can be adequately approximated by normal distributions.

• Seems reasonable that could be described by a multivariate normal, because our uncertainty is largely symmetric, continuous, and unbounded above and below

)( and )( hhz ee

)(he


33

Frequency Distribution

0

1

2

3

4

-0.1

2306

54

-0.1

0653

61

-0.0

9027

55

-0.0

7427

52

-0.0

5852

68

-0.0

4302

26

-0.0

2775

52

-0.0

1271

73

0.00

2097

79

0.01

6696

59

0.03

1085

32

0.04

5269

96

0.05

9256

2

0.07

3049

52

0.08

6655

18

0.10

0078

2

0.11

3323

42

0.12

6395

5

Price

Co

un

t

0

0.05

0.1

0.15

0.2

0.25

Idea

l n

orm

al

Simplification #3

• Case of


)(hze

Prices

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Hours on peak

pri

ce l

n($

/MW

h/p

)

30

31

32

33

34

35

36

ln(p/p') prices


34

Almost there...

• Then the B-S formula for the value the plant is


TddTXpd

NN

dXNdpNV

12

21

21

2/)/ln()p~ln(p/ ofdeviation standard is variablerandom )1,0( afor CDF theis

where

)()(

with the variance of playing the role of)()()( hhzh eee

2T


35

Option “Delta”

• The change in value of the option with respect to the price of the underlying is the option “Delta.” It is just the slope of the price curve at a specified price



0.00

2.00

4.00

6.00

8.00

10.00

12.00

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Price of underlying

Val

ue

of

Op

tio

n (

$)


36

The payoff

• The B.S. formula for the capacity factor the plant is


2/)()/ln(2/)/ln(

where

)(

22

21

1

ezge epp

TXpd

dNp

Vf


37

• Two issues remain– Gas prices are not constant (X is not fixed)

– Most of what we may think we know about future price uncertainty might be expressed in terms of average monthly prices

• Solution– Use a European “spread” option instead of a standard

European call option

– Try to estimate the volatility of the hourly spread from the monthly volatilities and correlations

Ah, Darn It



38

• Use a European spread call option instead of a standard European call option

European spread option


European Spread Option

0.00

2.00

4.00

6.00

8.00

10.00

12.00

-10 -8 -6 -4 -2 0 2 4 6 8 10

Spread in Price (pe-pg)

Val

ue

of

Op

tio

n (

$)0

5

10

15

20

25

30

35

40

12

54

97

39

71

21

14

51

69

19

32

17

24

12

65

28

93

13

33

73

61

38

54

09

43

34

57

48

15

05

52

95

53

57

76

01

62

56

49

Hour

$/M

Wh

-30

-25

-20

-15

-10

-5

0

5

10


39


• The Margrabe pricing formula for the value of a spread option, assuming no yields


212,12

22

12

12

212

1

2112

2

2/)/ln(where

)()(

TddT

Tppd

dNpdNpV


40


• The delta for the Margrabe spread option, assuming no yields


212,12

22

12

12

212

1

1

2

2/)/ln(where

)(

TddT

Tppd

dNf


41

Hourly Volatilities from Monthly

• We are dealing with expected variation of electricity and gas price over the specific time period and with uncertainties in these, as well. Using our assumption that the hourly uncertainties are constant and independent of the temporal variations in the respective commodities,


)2(

)2(

implies assumption ceindependenour by which

)()()()()(

,22

)()(,2

)(2

)(2

gegege

gegege hzhzzzhzhz

ggee hhzhhzh


42


• The first term is determined by the expected temporal covariance in commodity prices over the period, due to normal “seasonality”


)2( )()(,2

)(2

)( hzhzzzhzhz gegege

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

ln(pe/

pg

)


43


• Question: Is unlikely new information will become available that would influence our view of temporal structure?

• If not, we do not expect uncertainties in monthly averages to be affected too much by assumptions about expected temporal variations in price.



44


• The second term is determined by the hourly uncertainties surrounding the hourly values for gas and electricity and their expected covariance.

• Clearly, uncertainty factors can swamp temporal variations


)2( ,22

gegege

05

101520253035404550556065707580859095

100105110115

1 25 49 73 97 121

145

169

193

217

241

265

289

313

337

361

385

409

433

457

481

505

529

553

577

601

625

649

Hour

$/M

Wh )( ee )( gg


45


• Assumptions about hourly covariance in uncertainties is important here. They will affect the uncertainty in the monthly average price.

• If the hourly uncertainties for electricity (gas) are perfectly correlated, the relationship between monthly variance and hourly variance is

• If the hourly uncertainties for electricity (gas) are uncorrelated, the relationship between monthly variance and hourly variance is


)( in the covariance theis where

/112

h

N

e

He

22

ee

HNee

/22


46


• Assignment for the next class:

– If we know the correlations between the monthly average returns for electricity and gas, what can we conclude about the correlation between the hourly returns?



47

Example

• If we use the example we started out with, and the BS equation for delta, we use a normal distribution on ln(prices) that looks like

• But exact CF is 50% and B-S calculated CF is 49.7% (not bad)

• 100% uncertainty in the log returns (prices could be higher or lower by a factor of 2.7) gives us a 69% CF


Distribution of ln(p)

0

10

20

30

40

50

60

70

80

90

0.3

0.2

8

0.2

6

0.2

4

0.2

2

0.2

0.1

8

0.1

6

0.1

4

0.1

2

0.1

0.0

8

0.0

6

0.0

4

0.0

2 0 -0 -0

-0.1

-0.1

-0.1

-0.1

-0.1

-0.2

-0.2

-0.2

-0.2

-0.2

-0.3

-0.3

-0.3

-0.3

-0.3

Fre

qu

en

cy

0

0.01

0.02

0.03

0.04

0.05

0.06


48

Conclusion

– Thermal dispatch can be reasonably well using a spread call option on electricity and gas

– The monthly capacity factor of the thermal unit is provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread

– The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments:

• Discount rate r = 0• Volatility incorporates terms for the uncertainty in and the

expected variation of the spread over the specified time frame

– We need to better understand not only the expected correlation of uncertainties in electricity and gas prices, but how hourly prices are self-correlated over the time period of interest.



49

Revised Agenda






50

Objective

• Objectives of this section:– Develop a risk metric for the region

– To show: Risk metric should be minimum total power cost, subject to an annual CVaR constraint

Metrics


51

Stakeholders

• Candidate Stakeholders– Load Serving Entity (Investor-Owned Utility

or Public Utility District)– Customer– Regulatory Agency (Public Utility

Commission)– BPA

Metrics


52

Stakeholders

• Proposed Stakeholder Perspective– Total societal costs, to include– Capital costs, including those of

transmission– Variable costs,– Internalized emission costs

Metrics


53

Metric Candidates

• Candidates– Value at Risk (VaR)

– Standard deviation

– Expected shortfall

– Conditional VaR (CVaR)

– Van Neumann utility functions

– Block maxima

Metrics


54

V@R

Metrics

95% one-day V@R

Frequency Distribution

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Price

Co

un

t

V@R=4

costcost


55

Example of Power Plants

Metrics

Consider an ensemble of 1MW power plants, each with a forced outage rate of 0.10, equal to that of

the 10MW plant.

A Paradox, because we know a system of smaller

plants are better

V@R is not “subadditive”0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0 1 2 3 4 5 6 7 8 9 10

Var85=2

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

0 1

Var85=0


56

Desirable Properties

Metrics

• Subadditivity – For all random losses X and Y,

(X+Y) (X)+(Y)

• Monotonicity – If X Y for each scenario, then

(X) (Y)

• Positive Homogeneity – For all 0 and random loss X

(X) = (Y)

• Translation Invariance – For all random losses X and constants

(X+) = (X) +


57

A List of Loss Scenarios

Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.00

10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00

Define a measure of risk (X) = Maximum{Xi}Metrics


58

Subadditivity


10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00

(X+Y) (X)+(Y)Metrics


59

Monotonicity

Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.0010 0.00 1.00 1.00 0.00 1.00

Maximum Loss 4.00 4.00 5.00 8.00 5.00

If X Y for each scenario, then (X) (Y)Metrics


60

Positive Homogeneity

Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.0010 0.00 1.00 1.00 0.00 1.00

Maximum Loss 4.00 4.00 5.00 8.00 5.00

For all 0 and random loss X, (X) = (Y)Metrics


61

Translation Invariance


10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00

For all random losses X and constants (X+) = (X) + Metrics


62

Axioms for Coherent Measures

• Subadditivity – For all random losses X and Y,(X+Y) (X)+(Y)

• Monotonicity – If X Y for each scenario, then(X) (Y)

• Positive Homogeneity – For all 0 and random loss X

(X) = (Y)• Translation Invariance – For all random losses X and

constants (X+) = (X) +

Metrics


63

Value at Risk/Probability of Ruinviolates subadditivity

Scenario X1 X2 X1+X2

1 0.00 0.00 0.002 0.00 0.00 0.003 0.00 0.00 0.004 0.00 0.00 0.005 0.00 0.00 0.006 0.00 0.00 0.007 0.00 0.00 0.008 0.00 0.00 0.009 0.00 1.00 1.0010 1.00 0.00 1.00

VaR@85% 0.00 0.00 1.00

1 2 1 20 X X X X 1 Metrics


64

Standard Deviationviolates monotonicity

Scenario X1 X2

1 1.00 5.002 2.00 5.003 3.00 5.004 4.00 5.005 5.00 5.006 5.00 5.007 4.00 5.008 3.00 5.009 2.00 5.00

10 1.00 5.00E[Loss] 3.00 5.00

StDev[Loss] 1.41 0.00E[Loss]+2*StDev[Loss] 5.83 5.00

Metrics


65

Conditional Value at Risk - (CVaR)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Subject Loss

Cu

mu

lati

ve P

rob

abili

ty

Value At Risk

CVaR is the average of all losses above the Value at Risk

Metrics


66

What aboutthe upside potential?

CVaR is coherent

But ! variation from year to year can be large! What about minimizing variation from year to year?

We expect that upside will be sold to minimize cost, and variation will be automatically reduced

Metrics


67

Timing

Study? No

Annual? Yes: Rates are often recalculated annually

Metrics


68

Conclusion

Risk metric is

CVAR<X

Objective function is

Min costs

More on coherent measures:See Artzerner, Delbaen, Eber, Heath, “Coherent Measures of Risk,” July 22, 1998, preprint

http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf

Metrics

Date post:	26-Dec-2015
Category:	Documents
Upload:	andrew-hardy
View:	213 times
Download:	0 times

1 Systems Analysis Advisory Committee (SAAC) Thursday, October 24, 2002 Michael Schilmoeller John...

Documents