地域安全学会論文集 No.27, 地震リスクを考慮したリアルオプションによる公共不動産管理の最適化 : 公共教育施設の耐震補強計画を事例として Optimization of Public Real Estate Management Using Real Option Consid

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1 地域安全学会論文集 No.7, 地震リスクを考慮したリアルオプションによる公共不動産管理の最適化 : 公共教育施設の耐震補強計画を事例として Optimization of Public Real Estate Management Using Real Option Considering Seismic Risk: Case study of a retrofit project scheduling of public schools 1 Maki DAN and Masayuki KOHIYAMA 1 School of Science for Open and Environmental Systems, Graduate School of Science and Technology, Keio University Department of System Design Engineering, Faculty of Science and Technology, Keio University This study focuses on the scheduling in public real estate management (PREM) and proposes a method to optimize the scheduling using real option approach considering seismic risk. The seismic risk is represented by risk curves using Is value, which is a structural performance index. The risk curve is evaluated with a loss curve derived from a fragility curve, which represents the seismic vulnerability of a building, and a hazard curve, which represents the seismic risk at the site location. The damage due to earthquakes and its occurrence probability are modelled using the Monte Carlo simulation. Through the case study of a retrofitting problem of school facilities, it is confirmed that the proposed method can derive the optimal retrofit schedule with a reduced cost and risk of damage due to earthquakes. Keywords : real option, operations research, genetic algorithm, public real estate management, seismic risk assessment, retrofit scheduling problem 1. 研究背景と目的 ) PRE PRE (Public Real Estate) PRE ) PRE Real Estate Management ) NPV 3)-5) Ekern 6) Smit 7) Micalizzi 8) 313

2 表 1 単純なリアルオプションの種類 Stonier 9) Deng 10) 11)-1) 13) 14) 14) PRE. リスクカーブによる地震リスクのモデル化 (1) Is 値に基づく建物性能モデル Is 15)-16) Is Is [1] Is = E 0 SD T [1] E 0 SD 1.0 T 1.0 Is < Is < Is Is 0.7 () Is 値を用いたリスクカーブによる地震リスクの評価 Is 17) Is a) ハザードカーブの作成 18) Web J-SHIS 19) YashiroAnnaka 0) J-SHIS 1) 00 km J-SHIS b) フラジリティカーブとロスカーブの作成 c m ζ z [] 314

3 1 F( y) = πζ z x exp 1 ln x ln c y m ζ dx [] 0 z ) RC PGV Is Is [] t Is Is(t) V i [3] V 1 F ( V, Is(t), PGV i ) = 0 πζ z (Is(t))x exp 1 ln x ln(pgv i (Is(t) Is 0 )) [3] ζ z (Is(t)) dx PGV i Is 0 ζ z (Is(t)) [4], [5][4] PGV 1 ζ z (Is(t)) Is PGV 1 = 0.5 m/s PGV =1.0 m/s PGV 3 =1.5 m/s [4] PGV 4 =.0 m/s PGV 5 =.5 m/s Is 0 = 0.4, ζ z (Is(t)) = 0.6 [5] 100 %[6] [4] " C 1 = 0.05R C $ 0 C = 0.1R C 0 $ # C 3 = 0.R C 0 [6] $ $ C 4 = 0.3R C 0 $ % C 5 = R C 0 R C 0 C [4] [4] ~ [6] t Is Is(t) V i [7] n 1 { } L(V, Is(t)) = C i F(V, Is(t), PGV i ) F(V, Is(t), PGV i+1 ) [7] +C n F(V, Is(t), PGV n ) n c) リスクカーブの作成 t Is Is(t) Δt 1 C D P R [8] P R (C D,t,Δt) = P H (Y > y;t,δt) [8] C D = L( y, Is(t)) [9] y C D C D [7][9] P H (Y > y; t, t) t t 1 y (3) モンテカルロ法による地震リスクのモデル化 t (t = 0, 1,..., T)Δt C D (t, Δt) [9][10] C D (t,δt) = P 1 ( p ) [10] R i p i (i = 1,,..., N mc )[0,1] N mc (i = 1,,..., N mc ) ) [3] Is(t)[7] C i (i = 1,,..., n) 地震リスクを考慮したリアルオプションの評価手法の提案 (1) リアルオプション評価手法 BSM 3)-4) 5) Cox 6) BSM a) 項モデルの格子法 t (t = 0, 1,..., T) S(t) S(t)[11] Z t [1] u, p Z t = [11] d, 1 p t S(t) = S(0) Z i [1] u d u p d u d 1 p S(t)[13] 315

4 S(t) = S(0)u i d t i (i = 0, 1,..., t) [13] t = T T 1 T b) スイッチングオプションの評価 1 1 t (t = 0, 1,..., T) m S(t) s B (s B = 1,,..., m) s A (s A = 1,,..., m) C sb, s A (t,s(t)) s B s A C sb =, s A t S(t) s (t, S(t), s B ) V A (t, S(t), s A ) B A Before After t S(t) s A CF(t, S(t), s A ) t = T [14] V A (T,S(T ),s A ) = CF(T,S(T ),s A ) [14] T [15] (T,S(T ),s B ) = max V s A (T,S(T ),s A ) C sb (T,S(T )), s A A { } [15] t = 0, 1,..., T 1 t t [16] V A (T,S(t),s A ) = CF(t,S(t),s A ) r E [16] Q V t B (t +1,S(t +1),s B ) r E Q t s A t = s B t+1 r E Q t [11][17] E Q t (t +1,S(t +1),s B ) = qv t +1,uS(t),s B ( B) [17] +(1 q) (t +1,dS(t),s B ) q [18] (1+ r) d q =,0 < q <1 [18] u d t T [19] (t,s(t),s B ) = max V s A (t,s(t),s A ) C sb (t,s(t)), s A A { } [19] t = 0 s B (0, S(0), s B ) c) NPV 法との比較 NPV NPV [15][19] (T,S(T ),s B ) =V A (T,S(T ),s A ) C sb (T,S(T )) [0],s A (t,s(t),s B ) =V A (t,s(t),s A ) C sb,s A (t,s(t)) [1] NPV NPV () 地震リスクを考慮したリアルオプションの評価手法 a) リスクカーブに基づく格子法の提案 [10] C D (t, Δt) P R (t, Δt)[11] 1 p [11] u d t [] u(t), 1 P Z t = R (C D,t,Δt) [] d(t), P R (C D,t,Δt) S(t)[13] 316

5 b) 地震リスクを考慮したスイッチングオプションの評価 t (t, S(t), s B ) V A (t, S(t), s A ) 3(1)b) [14] ~ [19] t = 0 s B (0, S(0), s B ) 4. 公的不動産管理における最適化手法の提案 (1) 公的不動産管理の目的 PRE N R T t (t = 0, 1,..., T) i S i (t) i m i t S all (t) C all (t, S all (t))[3] ~ [5] N R S all (t) = S i (t) [3] N R C all (t)) = C (t,s (t)) sb, [4] i (t ), s A, i (t ) i s B,i (t),s A,i (t) =1,,,m i [5] s B,i (t) s A,i (t) t i C (t,s (t)) s sb, i (t ), s A, i (t ) i B,i(t) s A,i (t) T [14], [15][1], [3] [6], [7] V A (T,S all (T ),s A (T )) = CF(T,S all (T ),s A (T )) [6] (T,S all (T ),s B (T )) = [7] max V A (T,S all (T ),s A (T )) C all (T,S all (T )) s A (T ) { } s B (t) =! " s B,1 (t),s B, (t),,s B,NR (t)# $ [8] s A (t) = s A,1 (t),s A, (t),,s A,NR (t) [9] CF [30] N R CF (t),s A (t)) = CF(t,S i (t),s A,i (t)) [30] [16], [19][6][7] t (t = 0, 1,..., T) [31][3] V A (t),s A (t)) = CF (t),s A (t)) r E [31] Q! V t " B (t +1,S all (t +1),s B (t))# $ (t),s B (t)) = max{v s A (t ) A (t),s A (t)) [3] C all (t))} t = 0 (0, S all (0), s B (0)) 表公的不動産管理におけるリアルオプション 317

6 [7][3] (0, S all (0), s B (0)) t (t = 0, 1,..., T) PRE t (t = 0, 1,..., T) i S i (t) [] Z t i Z i,t [11] J [33] Maximize J = E (0,S all (0),s B (0)) = 1 N mc [33] V N B (0,S all (0),s B (0)) mc n=1 N mc [10] () 公的不動産管理にかかる制約 t (t = 0, 1,..., T) C b (t) [34] Subject to C all (t)) C b (t) +CF (t),s A (t)) [34] (t = 0,1,,T ) (3) 最適化問題のための決定変数 T [8][9] s B (t) s A (t) NP t (t = 0, 1,..., T) i C b,i (t) x [35] x = C b,1,c b,,,c b,nr [35] C b,i = C b,i (0),C b,i (1),,C b,i (T ) [36] t (t = 0, 1,..., T) i s B,i (t) C b,i (t)[37] s A,i (t) = argmax{ C (t) C (t,s (t)) < C (t) sb, s s i (t ) s B, i (t ), s(t ) i b,i } [37] s 5. 事例による提案手法の有効性の検証 (1) 解析モデル 0 A B C 表 3 対象地域にある施設の概要 [m ] Is A RC B RC RC C RC RC B C Is Is Is 図 1 各施設のフラジリティカーブ ( 中央値 ) 黒実線 :A 校, 青一点鎖線 :B 校, 緑破線 :C 校 5 /m 318

7 図各施設のロスカーブ ( 中央値 ) B Δt t = 0 t = 0 0 t = 図 3 各施設のハザードカーブ Is B C Is A 図 4 各施設のリスクカーブ ( 中央値 ) A A C Is 0.7 Is Is 0.9 Is Is 0.1 /m 表 5 本事例におけるステージとスイッチングコスト C sb, s A [/m ] s A s B Is Is () 最適化問題の設定と最適化手法 0 319

8 [36] [38] x = C b,1,c b,,c b,3 [38] C b,i = C b,i (0),C b,i (1),,C b,i (0) [39] C b,i i [33] [40] Maximize J = 1 N mc V N B (0,S all (0),s B (0)) [40] mc n=1 N mc 1000 s B (t) [8] [41] s A (t)[4] s B (t) = s B,1 (t),s B, (t),s B,3 (t) [41] s A (t) = s A,1 (t),s A, (t),s A,3 (t) [4] [14] ~ [19][16] r 0.04 t S all (t)[3] 3 S all (t) = S i (t) [43] Is 0.9 Genetic Algorithm GAGA (3) 数値解析結果と考察 N mc = 図 5 提案手法により算出された耐震補強の最適計画 NPV A C 図 6 NPV 法により算出された耐震補強の最適計画 NPV B 4 3 Is 0.9 NPV B B NPV B A Is 0.74 A A C A 1 C 3 1 N mc = NPV NPV NPV B B 4 30

9 表 6 提案手法および NPV 法による最適計画のプロジェクト価値および地震被害 (N mc = 10000) [] [] A B C A B C NPV A B C A B C NPV N mc = 100 N mc = プロジェクト価値 [ 億円 ] モンテカルロサンプル数図 7 モンテカルロサンプル数ごとのプロジェクト価値 6. 結論と今後の課題提案手法 NPV 法参考文献 1) ) PRE ) Myers, S. C.: Finance Theory and Financial Strategy, Interfaces, Vol. 14, No. 1, pp , ) Trigeorgis, L.: Real Options : Managerial Flexibility and Strategy in Resource Allocation, MIT Press., ) Amram, M. and Kulatilaka, N.: Real Options: Managing Strategic Investment in an Uncertain World, Harvard Business School Press, ) Ekern, S.: An Option Pricing Approach to Evaluating Petroleum Projects, Energy Economics, pp , ) Smit, Han T. J.: Investment Analysis of Offshore Concessions in The Netherlands, Financial Management, Vol. 6, No., pp. 5-17, ) Micalizzi, A.: The Flexibility for Discontinuing Product Development and Market Expansion: The Glaxo Wellcome Case, Real Options and Business Strategy: Applications to Decision Making, edited by Trigeorgis, L., Risk Books, pp , ) Stonier, J.: Airline Long-Term Planning under Uncertanity: The 31

10 Benefits of Asset Flexibility Created Through Product Commonality and Manufacturer Lead Time Reductions, Real Options and Business Strategy: Applications to Decision Making, edited by Trigeorgis, L., Risk Books, pp , ) Deng, S. J., Johnson, B., and Sogomonian, A.: Exotic Electricity Options and the Valuation of Electricity Generation and Transmission Assets, Decision Support System, Vol. 30, No. 3, pp , ) BVol. 130No. 6pp ) BVol. 13No. 1 pp ) Vol. 7pp , ) Vol. 9pp ) ) ) No. 11 pp ) J-SHIS Maphttp:// ) No. 55pp ) Annaka, T. and Yashiro, H.: Uncertainties in a probabilistic model for seismic hazard analysis in Japan, Risk Analysis II, WIT Press, pp , ) ) RC No. pp ) Black, F. and Scholes, M.: The Pricing of Options and Corporate Liabilities, Journal of Political Economy, Vol. 81, No. 3, pp , ) Merton, R. C.: Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, pp , ) Sharpe, W. F., Gordon J. A., and Jeffery V. B.: Investments, Vol. 6, Prentice Hall, ) Cox, J. C., Ross, S. A., and Rubinstein, M.: Option Pricing: A Simplified Approach, Journal of Financial Economics, Vol. 7, No. 3, pp. 9-63,

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