メイン寄せリスクと貸出債権価値の評価:ゲーム論的リアルオプションによるアプローチ
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1 2 i ii RIETI tshibata@tmu.ac.jp tetsuya.yamada@boj.or.jp / /
2 DCF DCF Leland [1994] Mella-Barral and Perraudin [1997] DCF DCF 2 /
3 1 1 3
4 2 Mella-Barral and Perraudin [1997] 2 i ii 2 3 Mella-Barral and Perraudin [1997] /
5 = = , ROA 1990 ROA 3 2 5
6 COO 5 5% 2 6 = = = /
7 Grenadier [1996] Weeds [2002] R&D 7
8 Leland [1994] Mella-Barral and Perraudin [1997] MP Leland [1994] 8 8 Debt Equity Swap 8 /
9 MP 9 MP MP X t dx t D μx t dt C ffx t dz t ; X 0 D ; (1) z t w X t w M C C c D C =M 100% C» M 10 μ r r > μ 11 b b > r t X t X t F t fi b fi c F t F t % 11 r > μ R 0;1/ e rt.x 0 e μt / dt X 0 =.r μ/ r» μ 9
10 t T t F t E t. / D E. j F t / Z fib E.X t / D ma E t e r.s t /.X s w bm / ds fi b 2T t t ; (2) E./ (2) t < fi b fi b» t (2) fi Λ 2 T b t D.X t / D ma Z fi Λ b E t fi c 2T t t e r.s t / bm dsc Z fic fi Λ b e r.s t /.X s w/dsce r.fi c t / C fi Λ 2 T c t D./ (3) 1 t < fi Λ b bm 2 fi Λ b < t < fi c X s w 3 t D fi c C (3) ; 2 (2) E./ E./ D r w C bm C b μ r r C w C bm fl ; b < ; μ r b (4) b D fl w C bm.r μ/; (5) fl 1 r 2 fl (3) ff 2 =2 fl.fl 1/ C μfl D r fl D 1=2 μ=ff 2 10 /
11 p.μ=ff 2 1=2/ 2 C 2r=ff 2 < 0 (4) 1 12 μ ff 2 1 ff 13 b (5) b b fl r > μ (4) 1 Λ b ΛΛ Λ < ΛΛ < Λ b < < Λ 1 2 (4) 2.= b / fl b 14! 1.= b / fl! 0 D b.= b / fl D 1 E./! 1 1 D b E./ D b E 0./ D b w C bm r r μ (1) 13 (4) ff de=dff D f.@e=@ b /.@ b =@fl / =@ff D.@E=@fl /.@fl =@ff b D D f b =.r μ/ C.w C bm /=r g.= b / fl log.= b / > 0.@fl =@ff / > 0 de=dff > 0 14 t; t C dt/ P.t» fi b < t C dt/ R 0;1/ e rtp.t» fi b < t Cdt/ R 0;1/ e rtp.t» fi b < t Cdt/ D E.ep. r fi b // D.= b / fl Diit and Pindyck [1994] 11
12 1 (3) b (5) c D./ D bm C Dj»b. b / bm fl ; b < ; (6) r r b Dj»b./ D c D r w C cm c μ r r C w μ r fl fl 1 cm C w r c fl ; (7).r μ/; (8) 3 (6) 1 2 D./! 1 1 b 2 D b (7) Dj»b./ 12 /
13 (7) 1 2 (4) 1 3 C (D cm).= c / fl Dj»b./ c D c C D 0. c / D 0 (6) (7) b < < b c 100% b > r c < b C (7) 3 c
14 1 3 A B M A M B M A C M B D M m A m B b 100% 15 2 L F fi Λ b D L i./ i D A ; B fi m 2 T t D L i.x t / D ma Z fi Λ b E t fi m 2T t t Z fim C fi Λ b e r.s t / m i bm ds e r.s t / m i.x s w/dsce r.fi m t / m i M ; (9) (9) 1 t < fi Λ b 2 fi Λ» b t < fi m 1 t fi Λ b 2 3 t D fi m m i M 15 M 6 14 /
15 fi Λ m D F j.x t / D ma fi c 2T E t Z fi Λ b t e r.s t / m j bm dsc e r.fi Λ m t / m i M C Z fic fi Λ m Z fi Λ m fi Λ b e r.s t / m j.x s w/ds e r.s t /.X s w/dsce r.fi c t / C (10) ; fi Λ m (10) (10) 1 t < fi Λ b 2 fi Λ» b t < fi Λ m 3 fi Λ m 4 fi Λ m < t < fi c 5 fi c C (10) 3 e r.fi Λ m t / m i M D D Z 1 e r.s t / r m i M ds fi Λ m Z fic fi Λ m e r.s t / r m i M ds e r.fi c t / m i M: (11) fi Λ m (11) fi Λ» m t < fi c (10) D F j.x t /D ma Z fi Λ b E t fi c 2T t t Z fic C e r.s t / m j bm dsc fi Λ m Z fi Λ m fi Λ b e r.s t / m j.x s w/ds e r.s t /.X s w r m i M / dsce r.fi c t /.C m i M / (12) ; 15
16 2 D L i./ D L i./ D m i bm C D Lj i» r b. b / m fl i bm ; b < ; (13) r b D L i j» b./ D m i m D fl fl 1 r μ w r M C w r C m i m M r C w μ r m fl ; (14).r μ/; (15) 4 (13) 1 2 (6) 2 D Lj i» b./ (14) 1 2 M m.= m / fl (15) m m D m D L i j» b./ (14) D L0 i j»b. m / D 0 (5) b (15) m b > r b > m 16 /
17 2 2 1 m 1 (13) 1 M (13) (6) 2 m 2 D F j./ b m D F j./ D m j bm C D Fj j» r b. b / m fl j bm ; b < ; (16) r b 17
18 D Fj j» b D m j r w μ r D Fj j» m D r w μ r m i M C C D Fj m j» m. m / m j r w ; μ r m fl (17) cm c r C w fl ; (18) μ r c 5 (16) D F j j» b./ (17) 2 D F j j» m./ (18) 1 2 cm c.= c / fl 100% c» m 100% c m 3 18 /
19 A B A B 16 19
20 4 D F A./ DF B./ 4 80% 20% DA F./ DF B./ DB F./ 4 1 B P B P /
21 DB F./ DA F./ P A < < P B A P < < P B o < o M B 4 1 o < b=r M B o 18 2 o A P < < P B o D A P» A P DF A./ A P 19 > A P DA F./ 1 2 D A P < A P DA F./ c c < < A P» c P 18 (15) 2 DB L./ DL B./ A P < < P B D A 21
22 20 X t P fi P D minft j X t < P g 4 m P 4 2 A B 5 1 DAB L./ DF AB./ DAB F./ 1 P > P < P P 1 P pre-emption 21 Fudenberg and Tirole [1985] D P 2 22 Huisman and Kort [1999] (10) /
23 5 1 pre-emption 23 Kijima and Shibata [2005] n pre-emption P m 5 2 b < P i D Fj i» b. b /» m i M 4 23 Huisman [2001] Weeds [2002] 1=2 23
24 5 2 3 P A DA main./ B DB sub (10) fi Λ m./ fi P DA main./ fi P fi P DB sub./ (9) fi Λ m fi P (9) (10) fi b < fi Λ m fi b < fi P 24 /
25 fi P < fi b. fi b < fi P B DB main./ (9) D sub B Z fi Λ.X b t / D E t t Z fi P C fi Λ b e r.s t / m B bm ds e r.s t / m B.X s w/dsce r.fi P t / m B M ; (19) (9) C j» b. b / m BbM DB sub./ D m BbM r D sub B DB subj» b./ D m B r w μ r C m B M P r b fl ; b < ; r μ C w r P fl ; (20) 2 DB subj» b./ (21) 2 A DA main./ DA main.x t / D ma Z fi Λ b E t fi c 2T t t Z fi P C C e r.s t / m A bm ds fi Λ b Z fic fi P e r.s t / m A.X s w/ds e r.fi P t / m B M e r.s t /.X s w/ds C e r.fic t / cm (20) (21) ; (22) 25
26 (10) fi Λ m fi P fi c DA main./ D m AbM C DA mainj» r b. b / m fl AbM ; b < ; r b (23) DA mainj» b D m A w r μ r C D main A j» P.P / m A P r μ w r P fl ; (24) DA mainj» P D w r μ r m BM C cm c C w fl ; (25) r μ r c (25) DA mainj» P 2. fi P < fi b B DB main./ (9) D sub B Z fi P.X t / D E t t e r.s t / m B bm ds C e r.fi P t / m B M DB sub./ D m BbM r C m B M m BbM r ; (26) P fl ; P < ; (27) A DA main./ DA main.x t / D ma Z fi P E t fi c 2T t t e r.s t / m A bm ds e r.fi P t / m B M C C Z fic fi b Z fib fi P e r.s t / bm ds e r.s t /.X s w/ ds C e r.fic t / cm ; (28) 26 /
27 DA main./ D m AbM r DA mainj» P D bm r C D main A m B M C j» P.P / m AbM r D main A DA mainj» b D r w μ r m BM C cm P fl ; P < ; (29) b fl ; j» b. b / bm C m B M r (30) c r C w fl ; (31) μ r c P P P 27
28 6 28 /
29 6. P X t μ 0% X t ff 10% w 1 b 5% r 2% M 10 M A 7 C 6 60% 1 C 7 c c D 60% 24 i P 6 P 6 P ii c 24 7 c c c 29
30 7 iii b c 7 c D % /
31 P 2 b P 8 i P ii 6% P iii 2002, a, b
32 8 3 ff P 9 i 9 P ii P iii 2 Shibata [2005] Kijima and Shibata [2005] i i 32 /
33 9 (15) (17) D C C 9 ff 25% P ff 25% 33
34 μ ff c 26 μ ff ff 9 P i ii iii /
35 μ ff μ μ w μ 35
36 1. 1 A B A M A M B M c b p r A-1 A 00.p/ B 00.p/ 1 1 p M A.1 C b/ p M A c 1 M A..1 p/.1 C b/ C pc/ M A A 00.p/ A 00.p/ D M A..1 p/.1 C b/ C pc/=.1 C r / M A ; B 00.p/ D M B..1 p/.1 C b/ C pc/=.1 C r / M B ; (A-1) B A A B M A M A 00.p/ M B A-1.A 00.p/; B 00.p//.A 01.p/; B 01.p//.A 10.p/; B 10.p//.A 11.p/; B 11.p// 36 /
37 A 01.p/ D M..1 p/.1 C b/ C pc/=.1 C r / M ; B 01.p/ D 0: (A-2) A B A 10 D 0; B 10.p/ D M..1 p/.1 C b/ C pc/=.1 C r / M; (A-3) cm A M A A 11.p/ D cm A M A ; B 11.p/ D cm B M B : (A-4) 2 p p 3 P 0 P 1 P 2 0 < M B < M A P 0 < P 1 < P 2 P 0 b r 1 c C b ; (A-5) P 1 P 2 b r 1 c C C M B b M b r 1 c C C M A b M.1 C r /.1 c/ ; (A-6) 1 c C b.1 C r /.1 c/ : (A-7) 1 c C b. p < P 0 A 00 > A 10 A 01 > A 11 B 00 > B 10 B 01 > B 11 A B. ; /. P 0 < p < P 1 A 00 < A 10 A 01 > A 11 B 00 < B 10 B 01 > B 11 A B. ; /. ; / 37
38 . P 1 < p < P 2 A 00 < A 10 A 01 > A 11 B 00 < B 10 B 01 < B 11 B A B A A B. ; /. P 2 < p A 00 < A 10 A 01 < A 11 B 00 < B 10 B 01 < B 11 A B. ; / 38 /
39 2. (2) E.X t / E.X tcdt / E.X t / D e rdt maf.x t w bm / dt C E t E.X tcdt / ; 0g: (A-8) E./ re./ D w bm C 1 2 ff 2 2 E 00./ C μe 0./; 8 < : E.! 1/! =.r μ/.w C b/=r; E. b / D 0; E 0. b / D 0; (A-9) (A-10) (A-10) b b b (A-10) E./ D 8 < : r w C bm μ r C b r C w C bm fl ; b < ; μ r b 0;» b ; b D fl w C bm fl 1 r (A-11).r μ/: (A-12) fl (6) p ff 2 =2 fl.fl 1/ C μfl D r fl D 1=2 μ=ff 2.μ=ff 2 1=2/ C 2r=ff 2 39
40 3. 2 b 1 (3) D.X t / D ( e rdt.bm dt C E t D.X tcdt / /; t < fi Λ b ; e rdt ma f.x t w/ dt C E t D.X tcdt / ; C g; fi Λ b» t; (A-13) D./ rd./ D bm C 1 2 ff 2 2 D 00./ C μd 0./; b < ; (A-14) ( D.! 1/! 0; D. b / D Dj»b. b /; (A-15) rd./ D. w/ C 1 2 ff 2 2 D 00./ C μd 0./;» b ; (A-16) 8 < : D.! 1/! =.r μ/ w=r; D. c / D C; D 0. c / D 0; (A-17) (A-16) (A-14) c (A-17) 8 bm C Dj»b. b / bm fl ; b < ; r r b D./ D < : c D fl fl 1 r w C C c μ r r C w μ r c fl ; c <» b ; C;» c ; (A-18) cm C w.r μ/: (A-19) r 40 /
41 4. D L i./ (17) D L i.x t / D L i.x tcdt/ D L i.x t / D ( e rdt.m i bm dt CE t D L i.x tcdt/ /; t < fi Λ b ; e rdt mafm i.x t w/dt CE t D L i.x tcdt/ ; m i M g; fi Λ b» t: (A-20) D L i./ rd L i./ D m i bm C 1 2 ff 2 2 D L i 0./ C μd L00 i./; b < ; (A-21) ( D L i.! 1/! 0; D L i. b/ D D Lj i» b. b /; (A-22) rd L i./ D m i. w/ C 1 2 ff 2 2 D L i 8 < : D L i.! 1/! m i.=.r μ/ w=r /; D L i. m/ D m i M; D L i 0.m / D 0; 00./ C μd L0 i./;» b ; (A-23) (A-24) i m (A-24) D L i./ D 8 < : m D fl fl 1 m i bm C D Lj i m <» r b. b / m i bm r m i r w Cm i M m μ r r C w μ r b fl ; b < ; m fl ; m <» b ; m i M;» m ; M C w r (A-25).r μ/: (A-26) 41
42 5. D F j./ (18) D F j.x t / D 8 < : e rdt.m i bm dtce t D F j.x tcdt/ /; t < fi Λ b ; e rdt.m j.x t w/dtce t D F j.x tcdt/ /; fi Λ b» t < fi Λ m ; e rdt maf.x t w r m i M / dtce t D F j.x tcdt/ ; C m i M g; fi Λ m» t: (A-27) D F j./ rd F j./ D m j bm C 1 2 ff 2 2 D F j 00./ C μd F0 j./; b < ; (A-28) ( D F j.! 1/! 0; D F j. b/ D D Fj j» b. b /; (A-29) rd F j./ D m j. w/ C 1 2 ff 2 2 D F j 00./ C μd F0 j./; m <» b ; ( D F j.! 1/! m j.=.r μ/ w=r /; D F j. m/ D D Fj j» m. m /; rd F j./ D. w r m i M / C 1 2 ff 2 2 D F j 8 < : (A-30) (A-31) 00./ C μd F0 j./;» m ; D F j.! 1/!.=.r μ/ w=r / m i M; D F j. c/ D cm m i M; D F j 0.c / D 0; (A-32) (A-33) j c (A-33) 42 /
43 D F j./ D 8 < : c D fl fl 1 m i bm r m w j r μ r C D Fj j m <» b. b / m i bm r C r w μ r m i M C D F j j c <» m. m / m j m c cm r C w μ r fl ; b < ; b r μ w r m fl, m <» b ; m fl ; c <» m ; cm m i M;» c ; cm C w r (A-34).r μ/: (A-35) 43
44 VaR a VaR VaR b CARF CARF-J Diit, A., and R. Pindyck, Investment under Uncertainty, Princeton University Press, Fudenberg, D., and J. Tirole, Pre-emption and Rent Equalisation in the Adoption of New Technology, Review of Economic Studies, 55, 1985, pp Grenadier, R., The Strategic Eercise of Options: Development Cascades and Overbuilding in Real Estate Markets, Journal of Finance, 51, 1996, pp Huisman, K., and P. Kort, Effect of strategic interactions of the option value of waiting, CentER Discussion Paper 9922, 1999., Technology Investment: The game theoretic Real Option Approach, Kluwer Academic Publishers, Kijima, M., and T. Shibata, Real Options in An Oligopoly Market, Kyoto Economic Review, 74, 2005, pp Leland, H., Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, Journal of Finance, 49, 1994, pp /
45 Mella-Barral, P., and W. Perraudin, Strategic Debt Services, Journal of Finance, 52, 1997, pp Shibata, T., On the Value-Volatility Relationship in a Real Options Model, Kyoto Economic Review, 74, 2005, pp Weeds, H., Strategic Delay in a Real Options Model of R&D Competition, Review of Economic Studies, 69, 2002, pp
46 46 /
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