Introduction

Model and Data

Results

Downside Risk in Reservoir Management Catarina Roseta-Palma1 and Yi˘ git Sa˘ glam2 1

Instituto Universit´ ario de Lisboa (ISCTE-IUL), 2 Victoria University of Wellington (VUW)

September 10, 2016

Roseta-Palma, and Sa˘ glam Downside Risk

Conclusion

Introduction

Model and Data

Results

Conclusion

Introduction

Motivation

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I

Demand Side: In various papers so far, we have seen (components of) demand rising thanks to population growth, industrialization, higher standards of living, etc. Supply Side: Large deviations in seasonal and inter-annual precipitation patterns often bring about serious problems for water users. ? For electricity, despite being non-storable, markets and trading mechanisms are well established to ensure shortages do not occur (frequently). ? Meanwhile, for water, although being storable, user rights, pricing rules, user groups’ reaction (i.e. awareness) to pricing, still need attention.

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Introduction

Motivation

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The most challenging issue is dealing with water scarcity and droughts, which represent the downside of natural variability in such areas. ? Wada et al. (2011) provide a global assessment of water stress, highlighting that population growth has heightened pressures on what is essentially a finite resource.

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Climate change is expected to increase supply volatility (via lower precipitation and higher temperature), which can only worsen the situation.

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Introduction

Motivation

I

We need to consider two aspects: (1) the definition of relevant risk measures and (2) the discussion and modelling of different attitudes to uncertainty. ? Risk measurement ranges from the simple calculation of variance to stochastic dominance. ? If there is concern over the bad outcomes (rather than the good ones), we could utilize skewness, semivariance, other lower partial moments (LPMs), and value-at-risk (VAT).

I

Meanwhile, water user groups adapt, which should be part of the analysis. ? In agriculture, lack of rainfall can be compensated by irrigation or changes in crop choice. ? In urban water supply, measures can be taken to tap alternative sources or to reduce demand.

Roseta-Palma, and Sa˘ glam Downside Risk

Conclusion

Introduction

Model and Data

Results

Conclusion

Introduction

Lower Partial Moments I

Lower Partial Moments (LMPs) is our main focus in this paper: they are one-sided measures that considers outcomes below a target level T : Z T LPM (a, T ) = (T − x)a dF (x), a ≥ 0. (1) −∞

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The value of a determines the risk profile: ? a Q 1 indicates risk-seeking, risk neutrality, and risk aversion behaviour. ? a = ∞ indicates only the worst possible outcomes considered, ? a = 2 is the special case of mean semivariance.

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Fishburn (1977) shows that there is a utility function consistent with LPM measures. It is an asymmetric function, as follows: ( x, x ≥ T, U (x) = (2) x − k(T − x)a , x < T. where k is a positive scaling term.

Roseta-Palma, and Sa˘ glam Downside Risk

Go to Results

Introduction

Model and Data

Results

Conclusion

Introduction

Goals I

This paper aims to: 1. analyze the dynamic water management problem in the presence of scarcity/shortages due to uncertainty in supply. 2. evaluate how user groups (i.e. agriculture) are affected in the extreme cases where the threshold level for a user group cannot be met. 3. investigate how different environmental constraints/regulations impact the optimal savings and usage.

I

Main results: ? Our simulation results indicate that average irrigation use does not change much across different cases. ? Total supply and savings are affected positively by stricter environmental restrictions (regulating minimum water stock in the reservoir) and negatively by increasing thresholds (defining penalty levels for shortages in irrigation). ? Although the impact on irrigation use is not reflected in the mean, it does change the shape of the distribution. ? Increasing thresholds lower agricultural profits, but help avoid extreme shortages.

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Empirical Setup

Reservoir Model

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w: carryover stock, which the flow of water between periods.

I

R: stochastic recharge (inflows). S: the stock of water supply available in a period, which is a function of the carryover stock from last period w, and stochastic recharge R.

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1. Some water is supplied for urban water use U (constant), 2. Water is released for flood control F (to avoid overflows), only utilized during periods with high inflows. Therefore, no economic return. 3. Irrigation water Q is for agriculture (control variable). 4. Rest of the stock is saved as carryover stock w (control variable).

Roseta-Palma, and Sa˘ glam Downside Risk

Conclusion

Introduction

Model and Data

Results

Conclusion

Empirical Setup

Reservoir Model I

Profit maximization problem: 
 V (w, R) = max Π(Q) + β ER0 |R V w0 , R0 0

(3)

w ,Q

S(w, R) ≥w0 + Q + U + F, w0 ≥E(S), I

(4) (5)

E(S) represents the environmental constraints, which may depend on the current stock of water. ? Absolute restrictions: there is a fixed threshold of carryover stock: any volume above it can be used to irrigate (or simply released to avoid overflows). ? Relative restrictions: a proportion of available volume to environmental uses as water levels increase.

I

In a situation where w0 < E(S), the regulator prorates the irrigation water use until the constraint is met.

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Empirical Setup

Reservoir Model

I

The payoff function ( P π(Q) = N c=1 (pc αc Lc ) min(Qc /γc , 1), Π(Q) = π(Q) − κ1 (Q − Q)κ2 , ? ? ? ? ? ?

I

(6)

c: index for crop, p: the price of crops, α: the land productivity, γ: the crop water requirement, L: the land allocation across crops, Q: the water allocation across crops.

The threshold Q can be viewed as the minimum amount of water below which drastic measures have to be taken (such as leaving the land fallow).

Roseta-Palma, and Sa˘ glam Downside Risk

Q ≥ Q, Q < Q.

Introduction

Model and Data

Results

Empirical Setup

Data: Southeastern Turkey

Legend

Details

¯

Dams

Kartalkaya Dam Other dams

KARTALKAYA DAM

Rivers Aksu River Other rivers GAZ0ANTEP

Counties Pazarcik County

River Basins Ceyhan Basin

0

40

80

160 km

Figure 1: Map of the Region

Roseta-Palma, and Sa˘ glam Downside Risk

Conclusion

Introduction

Model and Data

Results

Conclusion

Solving the Dynamic Problem

Solving the Dynamic Problem To solve the dynamic problem, we assume that the exogenous stochastic shocks in this economy stem from two components: inflows to the reservoir and crop prices. I

Inflows: ? We estimate the inflows with an auto-regressive process, but reject the test for autocorrelation. ? We fit the inflows data with the gamma distribution.

I

Crop Prices: ? We assume a log-normal distribution for the crop price of cotton, estimate an AR(1) process, and derive the transition matrix using the algorithm described by Tauchen (1986). ? We assume the year–2006 prices for the other crops, and the year–2007 values for the land productivities.

More on Data

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Solving the Dynamic Problem

Solving the Dynamic Problem

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Carryover stock is bounded below by the environmental constraints, and above by the reservoir capacity w ¯ = 173.173 hm3 . 1. 2. 3. 4.

I

E(S) E(S) E(S) E(S)

is is is is

constant at w = 5.65, constant at w = 11.30, proportional to total stock at 5%, proportional to total stock at 10%.

Procedure: 1. Use grid-search method to compute the value function in MATLAB. 2. Simulate the model 1000 times for 25 years. 3. Calculate total supply S, carryover stock w0 , irrigation water Q over the periods to see how these variables vary over time, with different penalty thresholds and environmental constraints.

Roseta-Palma, and Sa˘ glam Downside Risk

Conclusion

Introduction

Model and Data

Results

Conclusion

Solving the Dynamic Problem

Solving the Dynamic Problem Type

Parameter

Code

Value

Computational

Water grid points Stochastic recharge grid points Cotton price grid points Periods in each simulation Number of simulations

Nw NR Np NT M

100 8 2 25 1000

LPM

Penalty scalar Order of partial moment Penalty thresholds (hm3 )

κ1 κ2 Q1–Q3

Economic

Discount rate (%) Residential demand (hm3 ) Maximum carryover stock (hm3 ) Constant environmental constraints (hm3 ) Percent environmental constraints (% of total supply)

rβ U w ¯ EC1–EC2 EC3–EC4

20 2 {109, 121, 152}

Table 1: Parameter Values for the Empirical Illustration

Roseta-Palma, and Sa˘ glam Downside Risk

1% 95.32 173.173 (5.65, 11.30) (5%, 10%)

Introduction

Model and Data

Results

Conclusion

Solving the Dynamic Problem

Results: Simulated Irrigation Water across Periods

Details

Period 5

1 0.5 0 40

60

80

100

120

140

160

120

140

160

120

140

160

120

140

160

Period 10

1 0.5 0 40

60

80

100

Period 15

1 0.5 0 40

60

80

100

Period 25

1 0.5 0 40

60

80

100

Average

0.4 0.2 0 115

120

125

130

135

140

145

150

155

Figure 2: Histogram of irrigation use (in hm3) for the case Q=152 and E(S) = 5.65 Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Monte-Carlo Simulations: Summary Statistics

Summary Statistics Irrigation Use - Mean

Total Supply - Mean

Period

EC1

EC2

EC3

EC4

EC1

EC2

EC3

EC4

5 10 Benchmark 15 25 Overall

86.95 128.01 134.48 136.91 129.02

87.19 122.16 127.60 135.39 128.94

91.34 124.43 127.65 136.59 128.97

135.39 126.71 129.67 129.05 128.49

267.65 292.72 292.55 295.93 288.87

269.26 291.15 293.08 297.49 292.99

272.83 293.22 293.93 299.15 294.28

310.89 301.70 304.18 303.71 303.69

5 10 15 25 Overall

87.81 132.83 139.85 134.71 129.13

100.82 128.57 131.36 133.57 129.04

114.06 128.78 130.44 132.69 128.99

140.15 126.78 127.13 131.88 128.83

269.01 295.19 294.07 292.91 287.73

278.05 293.52 293.49 295.37 292.08

288.70 294.60 294.13 296.13 293.75

309.32 301.24 299.81 302.97 301.24

5 10 15 25 Overall

115.00 121.30 125.95 128.82 130.98

125.36 126.99 129.21 129.48 130.74

129.34 129.28 129.62 130.22 130.70

141.94 135.63 134.12 132.58 130.05

251.28 255.60 257.79 260.68 262.10

263.94 265.92 267.46 267.56 269.83

267.30 267.66 268.12 269.89 271.01

296.54 291.66 290.16 290.05 288.13

5 10 15 25 Overall

132.79 131.01 130.78 132.06 132.30

131.59 130.80 130.87 131.53 132.04

131.46 130.77 130.91 131.47 132.02

131.06 130.01 130.64 131.39 131.49

260.30 255.49 253.70 252.14 256.89

261.28 259.17 259.02 258.03 260.88

261.53 259.88 259.88 258.49 261.45

268.13 269.66 271.53 271.48 270.97

Q1= 109

Q2= 121

Q3= 152

Table 2: Summary Statistics of the Key Variables (in hm3 ) in the Simulation Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Monte-Carlo Simulations: Summary Statistics

Agricultural Profits Averagea

Std. Dev.

% change (rel. to EC1)

EC1b

EC2

EC3

EC4

EC1

EC2

EC3

EC4

EC1

EC2

EC3

Benchmarkc

4.11

4.06

4.03

3.83

0.28

0.28

0.28

0.26

0

-1.26

-1.92

-6.8

Q1= 109d

4.11

4.03

3.97

3.62

0.28

0.28

0.28

0.29

0

-1.86

-3.43

-11.76

Q2= 121

2.66

2.62

2.58

2.43

0.34

0.35

0.35

0.36

0

-1.7

-2.88

-8.74

Q3= 152

2.22

2.2

2.2

2.16

0.19

0.18

0.18

0.18

0

-0.72

-0.92

-2.79

Period

EC4

Table 3: Average Discounted Lifetime Agricultural Profits (without Penalties) a The term “Average” indicates the average, across simulations, of the sum of the discounted lifetime agricultural profits, measured in real terms of the domestic currency. This measure does not account for the penalty if fallen below threshold. b The notation “EC1–EC4” refer to the environmental constraints for the minimum carryover stock: (EC1) constant at 5.65hm3 , (EC2) constant at 11.30hm3 , (EC3) variable with 5% of the stock, (EC4) variable with 10% of the stock. c The “Benchmark” model refers to the case where there is no penalty threshold. d The notation “Q1–Q3” refer to the threshold levels of the irrigation use that lead to {25%, 20%, 10%} of the land left fallow, respectively.

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Monte-Carlo Simulations: Summary Statistics

Lower Partial Moments

LPM Equation

Shortfall Probability (κ2 = 0)

Q1= 109

Q2= 121

Q3= 152

Expected Shortfall (κ2 = 1)

Period

EC1

EC2

EC3

EC4

EC1

EC2

EC3

EC4

5 10 15 25 Overall

0.68 0.31 0.26 0.30 0.34

0.58 0.36 0.33 0.31 0.35

0.49 0.37 0.35 0.33 0.36

0.30 0.44 0.43 0.39 0.42

40.77 18.56 15.26 17.81 20.53

33.90 20.05 18.88 17.96 20.03

26.62 19.45 18.91 17.89 19.43

12.36 16.78 17.13 14.69 15.92

5 10 15 25 Overall

0.91 0.82 0.74 0.71 0.67

0.73 0.72 0.69 0.69 0.67

0.68 0.68 0.69 0.68 0.68

0.51 0.61 0.64 0.66 0.71

10.96 9.11 8.43 7.02 6.88

9.40 8.52 7.60 7.18 6.80

8.07 7.96 7.37 7.07 6.56

3.98 5.06 5.33 5.56 5.91

5 10 15 25 Overall

1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00

19.18 20.97 21.20 19.91 19.68

20.39 21.17 21.11 20.44 19.94

20.52 21.21 21.07 20.51 19.96

20.91 21.96 21.33 20.58 20.48

Table 4: Lower Partial Moments for Irrigation Use in the Simulation

Roseta-Palma, and Sa˘ glam Downside Risk

Introduction

Model and Data

Results

Conclusion

Main Conclusion

Conclusions I

We analyze the risk profiles of different assignment rules and environmental constraints in a water reservoir that serves the agricultural demand. ? We present a model that allows us to model a risk averse reservoir manager and to see how he adapts his behavior and actions to different assignment rules and increasingly demanding environmental constraints.

I

Main results are: 1. While thresholds (for LPM) and environmental constraints do not impact the average irrigation use, total supply and carryover stock are affected positively by stricter environmental constraints and negatively by increasing thresholds. 2. The reservoir manager’s utility goes down with higher thresholds and stricter environmental constraints. On the one hand, thresholds put more emphasis on avoiding shortages, at the expense of lower average utility. 3. As the threshold increases, the shortfall probability increases. Meanwhile, for a given threshold, stricter environmental constraints slightly raises the shortfall probability.

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Possible extensions include incorporating more uncertainty in the environment and endogenizing residential water use or hydro-power use to reevaluate the effects of thresholds and environmental constraints.

Roseta-Palma, and Sa˘ glam Downside Risk

Appendix: Data

Appendix: Results

Data: Reservoir Flows

Back 60 Irrigation Water Use

Tap Water Use

10 8 6 4

50 40 30 20 10 0

1

2

3

4

5

6 7 8 Month

9 10 11 12

1

2

3

4

5

6 7 8 Month

9 10 11 12

1

2

3

4

5

6 7 8 Month

9 10 11 12

250

200

200

Inflows

Flood Control Use

300 250

150

150

100

100

50

50

0 1

2

3

4

5

6 7 8 Month

9 10 11 12

Figure 3: Boxplot of the reservoir flows Roseta-Palma, and Sa˘ glam Downside Risk

Appendix: Data

Appendix: Results

Data: Reservoir Flows

Back

I

Since precipitation is mostly during the winter, reservoirs are necessary.

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Residential use amounts to around 100 hm3 annually and is mostly stable.

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Irrigation use is slightly higher than residential use, ranging between 130 − 150 hm3 .

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Water released for flood control avoids losses, but it does not have any other economic benefit.

Roseta-Palma, and Sa˘ glam Downside Risk

Appendix: Data

Appendix: Results

Data: Crop Prices

Back Irrigation Water Prices (1994=100) Cotton Maize Sugar Beet Wheat

0.16

Real Prices

0.14 0.12 0.1 0.08 0.06 0.04 1985

−3

x 10

1990

1995 Year

2000

Crop Prices (1994=100)

7

Prices

6 5 4 3 2 1 Year

Figure 4: Irrigation and Crop Output Prices Roseta-Palma, and Sa˘ glam Downside Risk

2005

Appendix: Data

Appendix: Results

Data: Changes in Crop Composition

Back

Crop Composition 100

Cotton Maize Wheat Sugarbeet Fallow

90

80

Percent Land Allocation

70

60

50

40

30

20

10

0

1985

1990

1995 Year

2000

2005

Figure 5: Changes in crop composition Roseta-Palma, and Sa˘ glam Downside Risk

Appendix: Data

Appendix: Results

Data: Inflows vs. Flood Control

Back

300 flood control reservoir capacity

250

Flood Control

200

150

100

50

0 50

100

150

200

250

300

350

Stock before Release

Figure 6: Inflows vs. Flood Control Roseta-Palma, and Sa˘ glam Downside Risk

400

450

Appendix: Data

Appendix: Results

Results: Simulated Water Stock across Periods

Back

Period 5

0.2 0.1 0 160

180

200

220

240

260

280

300

320

260

280

300

320

260

280

300

320

260

280

300

320

Period 10

0.1 0.05 0 160

180

200

220

240

Period 15

0.1 0.05 0 160

180

200

220

240

Period 25

0.2 0.1 0 160

180

200

220

240

Average

0.2 0.1 0 230

240

250

260

270

280

290

300

Figure 7: Histogram of total water supply (in hm3) for the case Q=152 and E(S) = 5.65 Roseta-Palma, and Sa˘ glam Downside Risk

Appendix: Data

Appendix: Results

Results: Simulated Carryover Stock across Periods

Back

Period 5

0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

60

70

80

90

60

70

80

90

60

70

80

90

Period 10

0.2 0.1 0 0

10

20

30

40

50

Period 15

0.2 0.1 0 0

10

20

30

40

50

Period 25

0.4 0.2 0 0

10

20

30

40

50

Average

0.1 0.05 0 10

20

30

40

50

60

70

Figure 8: Histogram of carryover stock (in hm3) for the case Q=152 and E(S) = 5.65 Roseta-Palma, and Sa˘ glam Downside Risk