Appendix D Some Portfolio Theory Math for Water Supply

Transcription

1 DESALINATION, WITH A GRAIN OF SALT A CALIFORNIA PERSPECTIVE 9 Appedix D Some Portfolio Theory Math for Water Supply Costat-Reliability-Beefit Uit Costs The reliability ad cost of differet water-supply optios ca vary, makig comparisos betwee differet optios difficult. To create a level playig field, the Pacific Istitute developed a method for adjustig estimated uit costs of water-supply optios (icludig coservatio ad ed-use efficiecy) so as to keep the reliability for all optios the same. The method borrows ad adapts tools from fiacial portfolio theory. 3 It leads to costat-reliability-beefit uit costs that provide a more fair compariso betwee supply optios with differet ucertaity characteristics. Fidig costat-reliability-beefit uit costs ivolves a two-step process. First, a plaer must specify a costat-reliability-beefit stadard. For example, the water plaer might say that water supply (or coservatio measures) must equal drought year demad 97.5% of the time. Mathematically, this meas that the aual average of the supply portfolio,, mius two times 4 the stadard deviatio (SD) of the supply portfolio,, must be equal to future (plaed for) drought-year demad, D F : A ( () D F Other reliability stadards ca be chose accordig to a table preset i ay statistics textbook that shows the percetage of time a radom variable will be more tha a chose multiple of the stadard deviatio from the average. For example, specifyig a i Equatio rather tha a yields a reliability stadard of about 84 percet. Stated differetly, a ormally distributed radom variable will be less tha the average mius oe stadard deviatio about 6% of the time, or oe i six years. The average supply of a portfolio is the sum of the average supplies of each of its parts. I our example, oe compares combiatios of the existig supply,, with a ew supply, : A ( + () where Q xi i Number of years of aual flow data Q xi Aual flow i year i from Source x The stadard deviatio of a portfolio of sources depeds o the stadard deviatio ad average of each source, the correlatio betwee the sources, ad the percetage of water from each source. The stadard deviatio of a portfolio is the square root of the variace of the portfolio. 3 This work was supported i part by the U.S. Bureau of Reclamatio. See Wolff ad Kasower (006). 4 Or if expressed with a additioal sigificat figure, as is commo i statistics textbooks,.96.

2 0 SOME PORTFOLIO THEORY MATH FOR WATER SUPPLY The appropriate formula (modified by the author from Tucker et al. 994) whe two sources are ivolved is: SD ( W ( + W ( N ) N ) + W ( W ( N ) Rho( E, N ) N ) (3) where W ( + W ( N ) W ( Rho(E, is the correlatio coefficiet betwee E ad N Formulas for the stadard deviatio ad correlatio coefficiet (Rho) are provided i ay statistics textbook, ad oe ca calculate these summary statistics usig a spreadsheet program. Combiig Equatios,, ad 3 yields: ( A + D + Rho( E, F (4) where A ( + N ) If oe kows the average existig supply, the stadard deviatios of the existig ad ew sources of supply, ad the correlatio coefficiet betwee supplies, Equatio 4 will cotai oly oe ukow,. This is the average ew supply required to esure that the chose reliability stadard (97.5% i this case) 5 will be achieved. ca be foud by assumig a value for, seeig how close or far apart the left ad right had sides of the equatio are, ad iteratively adjustig the assumed value util the value of that solves the equatio is foud. Table D- presets the solutios foud i the body of this report (ew surface water supply, desaliatio, ad outdoor water coservatio). Fially, the costat-reliability-beefit uit price for each optio differs from the average uit price for each optio by the ratio of /D N. Whe equals growth i drought demad (D N ) 6, as with desaliatio ad similar optios, the average uit price for that water supply optio is also the costat-reliability-beefit uit price. Whe is greater tha or less tha D N, as with the surface water ad outdoor coservatio examples i Table D-, the costat-reliability-beefit uit price for each optio is higher or lower tha the average uit price for that optio, respectively. 5 Replacig the i the deomiator o the right had side with the appropriate value, as discussed above, yields the appropriate equatio for other reliability stadards. 6 Recall that D N equals D F -D E.

3 DESALINATION, WITH A GRAIN OF SALT A CALIFORNIA PERSPECTIVE Table D-: Uit Cost Reliability Premiums Uder Various Assumptios Water Supply Optios Coefficiet of Variace (SD/A) Correlatio of Supply Optios (Rho(E,) Surface Water 0%.0 3,333 AFY Desaliatio 0% 0.0,000 AFY Outdoor Water Coservatio 0% -.0,667 AFY Notes: AFY acre-feet per year D F future drought-year demad Assumes coefficiet of variace of the existig source of 0%; 0,000 AFY; D F 0,000 AFY; reliability level of about 97.5 percet. Mathematics of Bledig Whe Water Quality Is Ucertai As i the reliability mathematics, a two-step process is used to determie the appropriate bledig of water supply sources eeded to obtai a specified water-quality objective. First, a plaer must specify a water-quality stadard ad probability of achievig that stadard. For example, the plaer might specify that water quality must be 500 parts per millio (ppm) total dissolved solids (TDS) at least 99.5% of the time. Mathematically, this meas that the average quality of the supply portfolio, Q, mius three times the stadard deviatio of the portfolio s quality, Q, must equal the water quality target (500 ppm): A ( Q 3 Q 500 (5) Other probabilities of achievig the target stadard ca be chose usig a table preset i ay statistics textbook that shows the percetage of time a radom variable will be more tha a chose multiple of the stadard deviatio from the average. For example, a reliability stadard of about 84% requires specifyig a i Equatio 5 rather tha a 3. Specifyig a 0 rather tha 3 would mea water quality will be worse tha 500 ppm 50% of the time. I this case, bleded quality is simply the arithmetic average of the quality of the water sources. The average quality of a portfolio is the weighted sum of the average qualities of the bleded water sources. I our example, oly two sources are bleded at a time: A ( Q W () Q) + W ( Q (6) where W () Percet of the portfolio from Source W ( Percet of the portfolio from Source X X Source or 3 W ( ) + W (

4 SOME PORTFOLIO THEORY MATH FOR WATER SUPPLY AQy ( ) q yi i Average quality of Source y y Source,, or 3 umber of years of aual average quality data q yi aual average quality i year i from Source y The stadard deviatio of the quality of a portfolio of sources, Q, depeds o the stadard deviatio ad average quality of each source, the correlatio betwee the source qualities, ad the percetage of water from each source. The stadard deviatio of a portfolio is the square root of the variace of the portfolio. The appropriate formula (modified by the author from Tucker et al. 994) whe two sources are ivolved is: SD ( Q W () ) + W ( + W () W ( Rho(, ) (7) where S ( y) Stadard deviatio of the quality of Source y Rho(,X) correlatio coefficiet betwee the quality of Source ad the quality of Source X Formulas for the stadard deviatio (SD) ad correlatio coefficiet (Rho) are provided i ay statistics textbook ad oe ca calculate these summary statistics usig a spreadsheet program. Combiig Equatios 5, 6, ad 7 yields: Q 500 ( W ( ) ) + W ( + ( W ( ) W ( Rho(, ) (8) 3 Q where A ( Q ( W ( ) Q) + W ( Q ad S ( y) As with the reliability example, there is oly oe ukow i Equatio 8 if oe kows the summary statistics related to water quality for the water-supply optios (average quality, stadard deviatio of quality, ad correlatio coefficiet betwee quality measures). The ukow is W(X), the fractio of the bled with Source that must come from Source X i order to maitai 500 ppm or better 99.5% of the time. As before, oe must solve for W(X) by

5 DESALINATION, WITH A GRAIN OF SALT A CALIFORNIA PERSPECTIVE 3 iteratio. Oe the fids the fractio of the bled from source by subtractig W(X) from. The cost of each bled that satisfies the quality specificatio is the weighted average cost usig these fractios. Refereces Tucker, A.L., K.G. Becker, M.J. Isambabi, ad J.P.Ogde Cotemporary Portfolio Theory ad Risk Maagemet. West Publishig Compay. St. Paul, Miesota.