日本保険・年金リスク学会誌 Vol.4 No.1 2009––––◊✲ㄽᩥ––––––––––––––––––––⤒῭⎔ቃࡢ୙☜ᐇᛶࢆ᭷ࡍࡿ㈇മ࡟ᑐᛂࡍࡿከᮇ㛫ືⓗ᭱㐺໬㛛⏣ࠉఙ୍2008 ᖺ 12 ᭶ 1 ᪥ᢞ✏
2009 ᖺ 9 ᭶ 3 ᪥ཷ⌮ᴫせ2009 ᖺ 6 ᭶⌧ᅾ, ࢃࡀᅜ࡛᥇⏝ࡉࢀ࡚࠸ࡿ఍ィᇶ‽࡟࠾࠸࡚ಖ㝤ไᗘࡢ㈇മ࡜ࡋ࡚ㄆ㆑ࡉࢀ
ࡿ㔠㢠ࡣಖ㝤ᩘ⌮࡟ᇶ࡙ࡃ㈐௵‽ഛ㔠㢠࡛࠶ࡿ. ࡓࡔࡋ, ๫వࡢᏳᐃᛶྥୖࢆពᅗࡍࡿ࡞ࡽࡤ,International Association of Insurance Supervisors[2006] ࡀᥖࡆࡿ ALM ࡢせ௳࡛ࡶせồࡉࢀࡿࡼ࠺࡟, ᙜヱಖ㝤఍♫࡟࠾ࡅࡿᢞ㈨ᡓ␎ࡢᵓ⠏࡟㝿ࡋ࡚ࡣ, ಖ㝤ᩘ⌮࡟ᇶ࡙ࡃ㈐௵‽ഛ㔠ࢆཧ↷ࡍࡿࡔࡅ࡛ࡣ࡞ࡃ, ಖ㝤ዎ⣙ࡢ౯್ࢆಖ㝤஦ᨾⓎ⏕⋡࣭฼Ꮚ⋡➼ィ⟬ᇶ♏⋡ࡢୖ࡟᭩࠿ࢀࡿࢹࣜ
ࣂࢸ࢕ࣈ࡜ࡋ࡚ホ౯ࡋ, ࣊ࢵࢪᡓ␎ࢆᐇ㊶ࡍࡿࡇ࡜ࡀᮃࡲࡋ࠸. ಖ㝤ዎ⣙ࡢ᭷ࡍࡿ᰿※ⓗ࡞୙☜ᐇᛶࢆᢕᥱࡋ, ᙜヱ୙☜ᐇᛶࡢ୍㒊ཪࡣ඲࡚ࢆ㈨⏘࡟ࡶᣢࡓࡏࡿࡇ࡜ࡀ࡛ࡁࢀࡤ, ࣜࢫࢡࡢ඲࡚ࢆ
ෆ㒊␃ಖ࡛ᢸಖࡏࡎ, ㈨ᮏᕷሙ࡟㌿᎑ࡍࡿࡇ࡜ࡀྍ⬟࡜࡞ࡿࡓࡵ࡛࠶ࡿ. ↛ࡿ࡟, മົホ౯ࡢィ⟬ᇶ♏ࡢ඲࡚ࡀᕷሙᛶࢆᣢࡓ࡞࠸ࡇ࡜, ᖹ‽ᡶࢆ๓ᥦ࡜ࡋࡓಖ㝤ᩘ⌮࡟ᇶ࡙ࡃ౯᱁タᐃ࡟ࡼࡾ࣊ࢵ
ࢪ㈈※ࡀᚲࡎࡋࡶ඘ศ࡛࡞࠸ࡇ࡜➼ࢆ㚷ࡳࢀࡤࠊᙜヱᢞ㈨⾜ືࡢ⌮ㄽᵓ⠏ཬࡧᐇ㊶ࡣᐜ᫆࡞ࡇ࡜
࡛ࡣ࡞࠸. ࡑࡇ࡛ᮏ✏࡛ࡣ, Brandt, Goyal, Santa-Clara and Stroud[2005] ࢆ ALM ࢔ࣉ࣮ࣟࢳ
࡟ᣑᙇࡋ, ᭱పಖドࢆ௜ࡋࡓኚ㢠ᖺ㔠ಖ㝤ࢆ౛࡜ࡋ࡚, ㈇മࡢ᰿※ⓗ࡞୙☜ᐇᛶࡢ୍㒊ࡀᕷሙᛶࢆ
᭷ࡍࡿࡇ࡜ࢆ๓ᥦ࡟, ௵ពࡢ᫬Ⅼ, ಖ㝤ᩱᡶ㎸᪉ἲ, ⤌ධ㈨⏘, ಖ㞀ཬࡧಖドෆᐜ, ┠ⓗ㛵ᩘ࡟ᑐᛂࡍࡿ୍⯡຺ᐃࡢ㏆ఝⓗ᭱㐺ゎࢆᑟฟࡋ, ࢩ࣑࣮ࣗࣞࢩࣙࣥᆺືⓗ᭱㐺໬ࡢᩘ್ゎᯒᡭἲࢆᥦ♧ࡍࡿ.
࣮࣮࢟࣡ࢻ㸸 㠀᏶ഛᕷሙ, ከᮇ㛫᭱㐺໬, ALM/LDIࠊኚ㢠ᖺ㔠ಖ㝤

1 ᗎㄽಖ㝤ᴗἲ➨ 97 ᮲ࠕᴗົࡢ⠊ᅖ➼ࠖ࡟ࡣ, ➨ 1 㡯࡛ಖ㝤఍♫ࡀಖ㝤ዎ⣙ࡢᘬࡁཷࡅࢆ࡛ࡁࡿࡇ࡜, ➨ 2 㡯࡛ಖ㝤ᩱ➼ࡢ㈨⏘㐠⏝ࢆ⾜࠺ሙྜ࡟ࡣಖ㝤ᴗἲ᪋⾜つ๎➨ 47 ᮲ࠕ㈨⏘ࡢ㐠⏝᪉ἲࡢไ㝈ࠖ࡟ࡼࡿࡇ࡜ࡀつᐃࡉࢀ࡚࠸ࡿ. ᨻ┬௧࡟ጤࡡࡿ⾲⌧࡜ࡉࢀ࡚࠸࡞࠸ࡇ࡜ࢆ㚷ࡳࢀࡤ, ㈨⏘㐠⏝ᴗົࡣಖ㝤఍♫ࡢᮏ᮶ᴗົ࡜ࡋ࡚, ಖ㞀ᥦ౪࡜ྠ㡰఩࡟఩⨨࡙ࡅࡽࢀ࡚࠸ࡿࡶࡢ࡜ゎ㔘࡛ࡁࡿ*1 . ࡇࢀࡣಖ㝤ᩘ⌮ཬࡧỴ⟬ࡢᵓ㐀ࢆ᣺ࡾ㏉ࢀࡤᙜ↛࡜࠸࠼ࡼ࠺.

㔝ᮧドๆᰴᘧ఍♫ࣇ࢕ࢹࣗࢩ࣮࣭ࣕࣜࢧ࣮ࣅࢫ◊✲ࢭࣥࢱ࣮ / ⟃Ἴ኱Ꮫ኱Ꮫ㝔ࣅࢪࢿࢫ⛉Ꮫ◊✲⛉௻ᴗ⛉Ꮫᑓᨷ༤ኈᚋᮇㄢ⛬ /ࠛ100-8130 ᮾி㒔༓௦⏣༊኱ᡭ⏫ 2-2-2 ࢔࣮ࣂࣥࢿࢵࢺ኱ᡭ⏫ࣅࣝE-mail: smonden@js6.so-net.ne.jp*1 ࡇࢀ࡟ᑐࡋ࡚☜ᐃ⤥௜௻ᴗᖺ㔠ἲ➨ 65 ᮲ཬࡧཌ⏕ᖺ㔠ἲ➨ 136 ᮲ࡢ 3 ࡛ࡣ㈨⏘㐠⏝ᴗົࢆእ㒊ጤクࢆ๓ᥦ࡜ࡋ࡚࠾ࡾ, ไᗘ㐠Ⴀ୺యࡓࡿᇶ㔠ⱝࡋࡃࡣ஦ᴗ୺ࡢᙺ๭ࡣ㐠⏝ᶵ㛵ࡢ㑅ᐃ࡟࠶ࡿ.

―67―⡆༢࡞౛࡛ぢ࡚ࡳࡼ࠺. ಖ㝤㔠㢠 1, ಖ㝤㔠ᮇᮎᡶࡢ㣴⪁ಖ㝤ዎ⣙࡟ᇶ࡙ࡃ t ᫬Ⅼᮎ㈐௵‽ഛ㔠⧞ධ㢠ࡣ,

lx+t+1 ·t+1 Vx: n  lx+t ·t Vx: n = (lx+t · Px: n  dx+t ) + i · lx+t · (t Vx: n + Px: n )(1)࡛࠶ࡿ. ࡓࡔࡋ, ಖ㝤ᩘᏛ࡟࠾ࡅࡿグྕ➼ࡢㄝ᫂ࡣ Appendix ࡟♧ࡋ, ࡇࡇ࡛ࡣ᩿ࡾ࡞ࡃ౑⏝ࡍࡿ. ྑ㎶➨ 1 㡯ཬࡧ➨ 2 㡯ࡣಖ㞀ᥦ౪ࡢᑐ౯࡜ࡋ࡚཰ཷࡍࡿಖ㝤ᩱ࡜ಖ㝤ዎ⣙࡟ᇶ࡙ࡃ⤥௜㔠㢠ࡢ཰ᨭṧ, ➨ 3 㡯ࡣᖺᗘጞ㈐௵‽ഛ㔠ཬࡧᙜヱᖺᗘ࡟཰ཷࡉࢀࡿಖ㝤ᩱ࡟ᑐࡍࡿணᐃ฼⋡┦ᙜࡢ฼ᜥ࡛࠶ࡿ. ᘧ࠿ࡽ᫂ࡽ࠿࡞ࡼ࠺࡟, ᢞ㈨⾜ື࡟ࡼࡾᙜヱ฼ᜥ┦ᙜ㢠ࢆ㈥࠺ࡇ࡜ࡀ࡛ࡁ࡞ࡅࢀࡤ, ᙜヱಖ㝤఍♫ࡣỴ⟬୙⬟࡟㝗ࡿ*2 ࡇ࡜࡟࡞ࡿ.
ࡇࡇ࡛ᐃᖖ≧ែࢆ௬ᐃࡍࡿ࡜, ཰ᨭṧPn1

t=0 (lx+t · Px: n dx+t ) ཬࡧ᝿ᐃඖᮏ

Pn1

t=0 lx+t (t Vx: n+ Px: n )ࡣ☜ᐃⓗ࡜࡞ࡿ. ᚑࡗ࡚, ㈨⏘㉸㐣ࡢ≧ែ࡟࠶ࡾ, ᅛᐃ㔠฼࡜ࡋ࡚ࡢணᐃ฼⋡ࡀ෇㔠฼㈨⏘཰┈⋡࡛㐩ᡂྍ⬟࡞Ỉ‽࡟タᐃࡉࢀࡿ࡞ࡽࡤ, മๆ࡟௦⾲ࡉࢀࡿ෇㔠฼㈨⏘ࢆ࢟ࣕࢵࢩࣗࣇ࣮ࣟ࡟ᑐᛂࡉࡏࡿᢞ㈨⾜ືࡀ୍ࡘࡢᡓ
␎࡜ࡋ࡚⪃࠼ࡽࢀࡿ*3 . ㈐௵‽ഛ㔠ホ౯࡟࠾࠸࡚ዎ⣙᫬ࡢィ⟬ᇶ♏⋡ࡀዎ⣙ᮇ㛫࡟ரࡾ㐺⏝ࡉࢀ, ‶ᮇಖ᭷┠ⓗࡢമๆ࡟ൾ༷ཎ౯ἲࡀㄆࡵࡽࢀࡿ࡞ࡽࡤ, ఍ィୖࡢᩚྜᛶࡶ☜ಖ࡛ࡁࡿ࡜࠸࠺ࡶࡢ࡛࠶ࡿ.
ࡉ࡚, ᙜヱᡓ␎ࡣືⓗ࡞ࣜࣂࣛࣥࢫࢆ๓ᥦ࡜ࡍࡿࡶࡢ࡛ࡣ࡞࠸. ࡇࡢࡼ࠺࡟㟼ⓗ࡞ᢞ㈨⾜ືࢆ๓ᥦ࡜࡛ࡁࡿ
⫼ᬒ࡟ࡣ, ᐃᖖ≧ែ࡟ᇶ࡙ࡃ㈈ᨻ཰ᨭࡢணぢྍ⬟ᛶࡀ࠶ࡿ. ᚑࡗ࡚, ዎ⣙⪅ࡢ⾜ື࣭⿕ಖ㝤⪅ࡢ≧ែ➼࡟౫Ꮡࡍࡿᑗ᮶࢟ࣕࢵࢩࣗࣇ࣮ࣟཬࡧ㈐௵‽ഛ㔠㢠ࡢᏳᐃᛶࡀᮇᚅ࡛ࡁࡿࡢ࡛࠶ࢀࡤ, 㟼ⓗ࣊ࢵࢪᡓ␎ࡣ᭷ຠ࡞ᡭẁ࡜࡞ࡿ࠿ࡶࡋࢀ࡞࠸. Pelsser[2003] ࡣṚஸ⋡ࡀỴᐃㄽⓗ࡛࠶ࡿ࡜࠸࠺௬ᐃࡢୗ࡛ಖドᖺ㔠࢜ࣉࢩࣙࣥ (GAO;

Guaranteed Annuity Option) ࡢබṇ౯್ࢆ㔠฼ࡢᮇ㛫ᵓ㐀ࡢୖ࡟᭩࠿ࢀࡿࢹࣜࣂࢸ࢕ࣈ࡜ࡋ࡚ホ౯ࡋ, ᝿ᐃඖᮏ࡜ዎ⣙ᮇ㛫ࡢ␗࡞ࡿ㔠฼ࢫ࣡ࢵࣉࢩࣙࣥࡢ࣏࣮ࢺࣇ࢛ࣜ࢜ࢆ㐺⏝ࡋࡓ㟼ⓗ࡞」〇ᡓ␎ࢆ♧ࡋ࡚࠸ࡿ*4 .
↛ࡿ࡟, ㈈ᨻ཰ᨭཬࡧ㈐௵‽ഛ㔠㢠࡟ࡣᚲࡎࡋࡶᏳᐃᛶࢆᮇᚅ࡛ࡁ࡞࠸ࡇ࡜ࢆ㚷ࡳࢀࡤ, ᙜヱᡓ␎ࡣ඲࡚ࡢಖ
㝤ၟရ࡟ࡣ㐺⏝࡛ࡁ࡞࠸ྍ⬟ᛶࡀ࠶ࡿ. ౛࠼ࡤ, ኚ㢠ಖ㝤࡟࠾ࡅࡿ㈈ᨻ཰ᨭཬࡧ㈐௵‽ഛ㔠㢠, බṇ౯್ホ౯ࡉࢀࡿಖ㝤ዎ⣙౯್ࡣ, ⤒῭⎔ቃ➼ࡢ୙☜ᐇᛶࢆ཯ᫎࡍࡿ. ᙜヱ஦౛࡟࠾࠸࡚ࡣ, ┠ᶆ᫬Ⅼ࡟࠾࠸࡚࣊ࢵࢪᑐ㇟࡜
ࡍࡿࢹࣜࣂࢸ࢕ࣈࡢ࣌࢖࢜ࣇࢆ」〇ࡍࡿྲྀᘬྍ⬟࡞㈨⏘ࡢ࣏࣮ࢺࣇ࢛ࣜ࢜ࢆ⌧᫬Ⅼ࡛ᵓ⠏ࡍࡿࡇ࡜ࡣᅔ㞴࡛࠶
ࢁ࠺. せồࡉࢀࡿ㐠⏝ᡂᯝࡀ≧ែ౫Ꮡࡍࡿࡇ࡜࡟క࠸, ᢞ㈨ᡓ␎ࡶྠᵝ࡟≧ែ౫Ꮡࡉࡏࡿືⓗ࡞ᑐᛂࡀᚲせ࡜࡞ࡿࡶࡢ࡜⪃࠼ࡽࢀࡿ*5 .

Hardy[2003] ࡣᰴ౯཰┈⋡ࣔࢹࣝ, ࣃ࣓࣮ࣛࢱࡢ᥎ᐃ᪉ἲ, ᩘ⌮ࣇ࢓࢖ࢼࣥࢫ, ࣜࢫࢡᑻᗘ➼, ኚ㢠ಖ㝤ࡢࣔࢹࣜࣥࢢ࡜ࣜࢫࢡ⟶⌮࡟㛵ࡍࡿᇶ♏ⓗ࡞▱㆑ࢆໟᣓࡋ, ゎㄝࡋ࡚࠸ࡿ. ືⓗ࣊ࢵࢪᡓ␎࡟ࡘ࠸࡚❶ࢆ๭ࡁ, ⤥௜ᙧែ࡟ᑐᛂࡍࡿゎㄝࢆ⾜ࡗ࡚࠸ࡿ. ࡓࡔࡋ, Ṛஸ⋡ཬࡧゎ⣙⋡࡟ࡘ࠸࡚ࡣỴᐃㄽⓗ࡜ࡋ࡚࠾ࡾ, ᮏ㉁ࡣ᏶ഛᕷሙ
ࢆ๓ᥦ࡜ࡋࡓࢹࣝࢱ࣭࣊ࢵࢪᡓ␎ࡢᥦ♧࡛࠶ࡿ. ᑠᏲᯘ [2002] ࡶྠᵝ࡟᏶ഛᕷሙࢆ๓ᥦ࡜ࡋࡓࢹࣝࢱ࣭࣊ࢵࢪ
ᡓ␎ࢆᥦ♧ࡍࡿࡀ, ಖ㝤ᩱᡶ㎸᪉ἲࡢᕪ␗ࢆ☜ㄆࡋ, ᖹ‽ᡶࡢሙྜ࡟ࡣ࣊ࢵࢪ⢭ᗘࡢྥୖࡀᅔ㞴࡛࠶ࡿࡇ࡜ࢆᣦ
᦬ࡋ࡚࠸ࡿ.
↛ࡿ࡟, ᏶ഛᕷሙࢆ๓ᥦ࡜ࡋࡓ࣊ࢵࢪᡓ␎ࡢᐇົ࡬ࡢ㐺⏝࡟ࡣ, ࠸ࡃࡘ࠿ࡢᠱᛕࢆᣦ᦬ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ. ➨୍࡟ࡣಖ㝤஦ᨾⓎ⏕⋡ࡢ㠀ᕷሙᛶ࡛࠶ࡿ. ෌ಖ㝤ࢆ㏻ࡌࡓὶ㏻ᕷሙࡣᏑᅾࡍࡿࡶࡢࡢ, ὶືᛶཬࡧ࢟ࣕࣃࢩ
ࢸ࢕ࡣ㝈ᐃⓗ࡛࠶ࡿ. ୍㒊ࡢᢞ㈨㖟⾜ࡀṚஸ⋡࢖ࣥࢹࢵࢡࢫࢆබ⾲ࡋ, Ṛஸ⋡ࢹࣜࣂࢸ࢕ࣈࡢྲྀᘬ஦౛ࡶᏑᅾ
ࡍࡿࡶࡢࡢ*6 , ᭷౯ドๆ➼࡜ẚ㍑ࡍࢀࡤつᶍࡣᑠࡉ࠸.
*2 ಖ㝤ᴗἲ➨ 116 ᮲ࠕ㈐௵‽ഛ㔠ࠖ➨ 1 㡯࡛ࡣ, ẖỴ⟬ᮇ࡟࠾ࡅࡿಖ㝤఍♫ࡢ㈐௵‽ഛ㔠✚ࡳ❧࡚ࢆせồࡋ࡚࠸ࡿ.
*3 ᖹᡂ 19 ᖺᗘỴ⟬᫬࡟࠾ࡅࡿ⏕࿨ಖ㝤஦ᴗᴫἣࡢ୺せ㈨⏘㐠⏝≧ἣࢆぢࡿ࡜, ෇㔠฼㈨⏘ࡢᵓᡂẚࡀ, ᅜമ (36.4%), ᆅ᪉മ(2.8%), ♫മ (9.1%), ୍⯡㈚௜㔠 (15.4%) ࡜ྜィࡋ࡚ 6 ๭ࢆ㉸࠼࡚࠸ࡿ. ᙜヱ⤫ィࡣ♫ᅋἲே⏕࿨ಖ㝤༠఍ࡢ࣮࣒࣮࣍࣌ࢪ
(http://www.seiho.or.jp/index.html) ࡢࠕ᝟ሗ࣭㈨ᩱࠖෆࠕ⏕࿨ಖ㝤஦ᴗᴫἣ࡛ࠖ☜ㄆ࡛ࡁࡿ.
*4 ᫬Ⅼ t ࡟࠾ࡅࡿ‶ᮇ T ࡢಙ⏝ࣜࢫࢡࡢ࡞࠸๭ᘬമๆ౯᱁ P (t, T ), x ṓ⿕ಖ㝤⪅ࡢṚஸ⋡ qx , ணࡵ⣙ᐃࡉࢀࡿ฼⋡ r G ࡟ᑐࡋ, ዎ⣙xGP
Ln+1 +rx
qx
P
(0,
t)

[1

P
(0,
n)]
ࡢ᝿ᐃඖᮏࢆ
L=ᮇ㛫 n ࡢᅛᐃཷኚືᡶ (ᅛᐃ㔠฼ Kn ) ࡢ㔠฼ࢫ࣡ࢵࣉ Sn = Kn n
n
t=1
1+Kn
P
࡜ࡋ,
n=1 Ln max(Sn , 0) ࡞ࡿࢫ࣡ࢵࣉࢩࣙࣥࡢ࣏࣮ࢺࣇ࢛ࣜ࢜ࢆᵓ⠏ࡍࡿࡇ࡜࡟ࡼࡾ, GAO ࡢ࣌࢖࢜ࣇࢆ」〇ࡍࡿ࡜࠸࠺ࡶ
ࡢ࡛࠶ࡿ.
*5 ᯇᒣ [2005] ࡣኚ㢠ᖺ㔠ಖ㝤ࡢ᭱ప⤥௜ಖドࢆᑐ㇟࡜ࡍࡿ㝿࡟࠾࠸࡚, ゎ⣙⋡ࣔࢹࣝࡢಙ㢗ᛶࡀ୙༑ศ࡛࠶ࡿሙྜ࡟ࡣ㟼ⓗ࣊ࢵࢪᡓ␎ࡣ⤒῭ⓗ࡛ࡣ࡞࠸࡜ᣦ᦬ࡋ࡚࠸ࡿ.
*6 ஦౛࡟ࡘ࠸࡚ࡣ Appendix ࡟♧ࡍ.

―68―

日本保険・年金リスク学会誌 Vol.4 No.1 2009➨஧࡟ࡣ㈈※ࡢၥ㢟࡛࠶ࡿ. ᏶ഛᕷሙࢆ๓ᥦ࡜ࡍࡿ࡞ࡽࡤ, ௵ពࡢ᫬Ⅼཬࡧ≧ែ࡟࠾ࡅࡿࢹࣜࣂࢸ࢕ࣈࡢ౯
್࡜࣊ࢵࢪ㈝⏝ࡣ୍⮴ࡍࡿ. ↛ࡿ࡟, ಖ㝤ᩱ࡜ࡋ࡚཰ཷࡉࢀࡿ㔠㢠ࡣබṇ౯್࡟ᑐࡍࡿ཰ᨭ┦ᙜࢆ᝿ᐃࡋ࡚࠸
࡞࠸. ౛࠼ࡤ, ฼Ꮚ⋡ࡢࡳ࡟୙☜ᐇᛶࢆ᝿ᐃࡍࡿ᫬, ಖ㝤㔠㢠 1, ಖ㝤㔠ᮇᮎᡶࡢ㣴⪁ಖ㝤ዎ⣙࡟ᇶ࡙ࡃಖ㝤ዎ⣙ࡢ t ᫬Ⅼᮎ࡟࠾ࡅࡿ⤒῭ⓗ౯್ࡣt Vx: n

+

n1
X
k=t

B(t, k + 1) ·lx+k
· ( k Vx: n + Px: n ) · [ i  Lt (k, k + 1) ]
lx+t

(2)

࡜࡞ࡿ. ࡍ࡞ࢃࡕ, ಖ㝤ዎ⣙࡟ᇶ࡙ࡃ㈐௵‽ഛ㔠㢠 (➨ 1 㡯) ࡟, ணᐃ฼⋡ࡢಖド౯್ (➨ 2 㡯) ࢆィୖࡋࡓ㔠㢠
࡛࠶ࡿ*7 . ࡜ࡇࢁࡀ, ಖ㝤ᩱࡣ➨ 1 㡯㒊ศ࡟ࡢࡳᑐᛂࡍࡿࡶࡢ࡛࠶ࡿ. ィ⟬ᇶ♏⋡࡟ࡣᏳ඲๭ቑࡀྵࡲࢀࡿࡶ
ࡢࡢ, ᙜヱᏳ඲๭ቑཬࡧෆ㒊␃ಖࡀ➨ 2 㡯㒊ศࢆ㈥࠺࡟༑ศ࡞Ỉ‽࡛࡞ࡅࢀࡤ, ㈈ᨻࡣ◚⥢ࡍࡿࡇ࡜࡜࡞ࡿࡔ
ࢁ࠺.
➨୕࡟ࡣ఍ィ࡜ࡢᕪ␗࡛࠶ࡿ. ⌧⾜఍ィไᗘୖ, ㈚೉ᑐ↷⾲ୖ࡟ィୖࡉࢀࡿ㔠㢠ࡣᘧ (2) ࡢ➨ 1 㡯ࡢࡳ࡛࠶ࡿ. ᚑࡗ࡚, ➨ 2 㡯ࢆ࣊ࢵࢪࡍࡿࡓࡵ࡟ᅛᐃཷኚືᡶࡢ㔠฼ࢫ࣡ࢵࣉዎ⣙ࢆ⥾⤖ࡍࡿ࡜, ࣊ࢵࢪຠᯝࡣㄆ㆑ࡉࢀ
ࡎᙜヱࢫ࣡ࢵࣉዎ⣙ࡢᕪᦆ┈ࡢࡳࡀホ౯ࡉࢀࡿࡇ࡜࡟࡞ࡿ. ࡲࡓ, ฼┈␃ಖࢆྍ⬟࡜ࡍࡿ຺ᐃࢆᣢࡓ࡞ࡅࢀࡤ,ዎ⣙㏵୰࡛࣊ࢵࢪ࣭࣏ࢪࢩࣙࣥ࡟฼┈ࡀⓎ⏕ࡍࡿሙྜ࡟, ᙜヱ฼┈ࢆ㓄ᙜ㈈※࡟඘࡚ࡿࡼ࠺࡞ㄗࡗࡓ⾜ືࡀ᥇ࡽࢀࡿྍ⬟ᛶࡶྰᐃ࡛ࡁ࡞࠸.
᭦࡟, ㏥⫋⤥௜఍ィ࡟࠾ࡅࡿ㏥⫋⤥௜മົ (PBO; Projected Beneht Obligation), ௻ᴗᖺ㔠ไᗘ࡟࠾ࡅࡿ
᭱ప✚❧ᇶ‽㢠ࡢࡼ࠺࡟Ⓨ⏕⤥௜ホ౯᪉ᘧ࡟ᇶ࡙ࡁホ౯ࡉࢀࡿሙྜ࡟ࡣ, ᑐ㇟࡜࡞ࡿ㈇മ౯᱁㐣⛬ࡀ୙㐃⥆࡜࡞ࡿ.
௨ୖࡼࡾ, ಖ㝤࣭ᖺ㔠ไᗘ࡟࠾ࡅࡿ࣊ࢵࢪᡓ␎ࡢᵓ⠏࡟㝿ࡋ࡚ࡣ, 㠀᏶ഛᕷሙࢆ๓ᥦ࡜ࡍࡿࡇ࡜ࡀᮃࡲࡋ࠸࡜
⪃࠼ࡽࢀࡿ. ࡇࡢሙྜ, ࠗࣜࢫࢡ᭱ᑠ໬࠘ࡢ௦⾲ⓗ࡞ᡓ␎࡜ࡋ࡚ࡣ, ࠕᖹᆒศᩓ࣊ࢵࢪࠖ࡜ࠕᒁᡤⓗࣜࢫࢡ᭱ᑠ
໬ࠖࢆᥖࡆࡿࡇ࡜ࡀ࡛ࡁࡼ࠺. ๓⪅ࡣ⮬ᕫ㈨㔠඘㊊ⓗ࡛࠶ࡿࡓࡵ, ✚❧㐣୙㊊ࡢⓎ⏕ࢆチᐜࡍࡿ. ᙜヱᕪ␗ࡢᢚไࢆពᅗࡋ, ‶ᮇ᫬Ⅼ࡟࠾ࡅࡿࠕ࣊ࢵࢪᑐ㇟౯್㸫 (ึᮇಖ᭷㈨⏘㸩ᢞ㈨฼ᚓ)ࠖࡢศᩓࢆ᭱ᑠ໬ࡍࡿᡓ␎࡛࠶ࡿ
୍᪉, ᚋ⪅ࡣ⮬ᕫ㈨㔠඘㊊ⓗ࡛ࡣ࡞ࡃ, ㈨㔠ㄪ㐩ᇶ‽ࢆ⦆࿴ࡍࡿ. ࡍ࡞ࢃࡕ, ✚❧୙㊊ࢆゎᾘࡍࡿࡓࡵࡢ㏣
ຍ㈨㔠ㄪ㐩ࢆチᐜࡋ, ᙜヱࢥࢫࢺࡢ᭱ᑠ໬ࢆᅗࡿᡓ␎࡛࠶ࡿ. ୧ᡓ␎ࡢヲ⣽࡟ࡘ࠸࡚ࡣ, Schweizer[2001], ሯཎ [2001], ⏣⏿ [2004] ➼ࢆཧ↷ࡉࢀࡓ࠸. ಖ㝤ศ㔝࡬ࡢ㐺⏝౛࡜ࡋ࡚ࡣ, ᒣ⏣࣭㣤⏣࣭ᳺ [2006], ᒣ⏣ [2008],

Dahl and Møller[2006] ➼ࡀཧ⪃࡜࡞ࡿࡔࢁ࠺.
ࡲࡓ, ⮬ᕫ㈨㔠඘㊊ᛶࢆព㆑ࡍࡿࡢ࡛࠶ࢀࡤ, ᚲࡎࡋࡶᦆኻࡢ᭱ᑠ໬ࡀ㐩ᡂࡉࢀࡿ࡜ࡣ㝈ࡽ࡞࠸ࡀ, ࠗᮇᚅຠ⏝᭱኱໬࠘࡟ࡼࡾᢞ㈨ᡓ␎ࢆᐃ⩏ࡍࡿࡇ࡜ࡶྍ⬟࡛࠶ࡿ. ෆᒣ [2005] ࡣᢞ㈨ᮇ㛫ᮎ᫬Ⅼ࡟࠾ࡅࡿ✚❧ẚ⋡ (㈨⏘¹㈇മ) ࡟ᑐࡍࡿ┦ᑐⓗ༴㝤ᅇ㑊ᗘ୍ᐃ (CRRA; Constant Relative Risk Aversion) ᆺຠ⏝ࡢᮇᚅຠ⏝᭱኱໬࡟ᇶ࡙ࡃ᭱㐺ᢞ㈨ᡓ␎ࢆᑟฟࡋ࡚࠸ࡿ.
ከᮇ㛫ືⓗ᭱㐺ᢞ㈨ᡓ␎࡟㛵ࡍࡿゎἲࡢ㐍ᒎ࡟ࡘ࠸࡚ࡣෆᒣ [2008] ࡟ヲࡋ࠸ࡀ, ୍⯡ⓗ࡟౯್㛵ᩘࡢゎᯒゎࢆᚓࡿࡇ࡜ࡣᅔ㞴࡛࠶ࡾ, ከࡃࡢሙྜ࡟㏆ఝⓗゎᯒゎⱝࡋࡃࡣᩘ್ゎ࡟㢗ࡿࡇ࡜࡟࡞ࡿ. ↛ࡿ࡟, ᮇ୰࡟࠾ࡅࡿ
࢟ࣕࢵࢩࣗࣇ࣮ࣟࡢ཯ᫎ➼, ㈨⏘ཬࡧ㈇മ㐣⛬ࡀ」㞧࡜࡞ࡿሙྜ, ⤌ධ㈨⏘ࡢቑຍ࡟క࠸≧ែኚᩘࡀከࡃ࡞ࡿሙ
ྜ➼࡟ࡣᩘ್ゎᯒࡉ࠼ࡶᅔ㞴࡛࠶ࡿ. ࡇࡢࡼ࠺࡞せᅉࡀಖ㝤ไᗘࡸᖺ㔠ไᗘࡢᐇົ࡟࠾࠸࡚ㄆ㆑ࡉࢀ࡚࠸ࡿၥ
㢟ࡢゎỴ࡟㝿ࡋ, ୍ࡘࡢ㞀ቨ࡜࡞ࡗ࡚࠸ࡿឤࡀ࠶ࡿࡀ, Brandt, Goyal, Santa-Clara and Stroud[2005]*8 ࡣ, ౯್㛵ᩘࢆゎࡃࡇ࡜࡞ࡃ, ┤᥋࡟㏆ఝⓗ᭱㐺㈨⏘㓄ศẚࢆồࡵࡿࡇ࡜࡟ࡼࡾ, ၥ㢟ゎỴࡢᐇ⾜ᛶࢆ㧗ࡵࡿᡭἲࢆᥦ
᱌ࡋࡓ.
ᮏ✏࡛ࡣ BGSS ἲࢆ ALM ࡟ᣑᙇࡋ, ᭱పಖドࢆ௜ࡋࡓኚ㢠ᖺ㔠ಖ㝤࡟ࡘ࠸࡚, ௵ពࡢ᫬Ⅼ, ಖ㝤ᩱᡶ㎸᪉ἲ, ⤌ධ㈨⏘, ಖ㞀ཬࡧಖドෆᐜ, ┠ⓗ㛵ᩘ࡟ᑐᛂࡍࡿ୍⯡຺ᐃࡢ㏆ఝⓗ᭱㐺ゎࢆᑟฟࡋ, ࢩ࣑࣮ࣗࣞࢩࣙࣥᆺ
ືⓗ᭱㐺໬ࡢᩘ್ゎᯒᡭἲࢆᥦ♧ࡍࡿ.
*7 ド᫂ཬࡧグྕࡢㄝ᫂ࡣ Appendix ࡟♧ࡍ.
*8 ௨ୗ࡛ࡣ⡆༢ࡢࡓࡵ࡟ BGSS ἲ࡜ࡍࡿ.

―69―

2 ኚ㢠ᖺ㔠ಖ㝤ᮏ✏࡛ࡣලయⓗ࡞ᑐ㇟࡜ࡋ࡚᭱పಖドࢆ௜ࡋࡓኚ㢠ᖺ㔠ಖ㝤ࢆ᥇⏝ࡍࡿ. ࡑࡢࡓࡵᢞ㈨ᡓ␎ࢆ㆟ㄽࡍࡿ๓࡟,ኚ㢠ᖺ㔠ಖ㝤ࡢᴫ␎ཬࡧ㛵㐃ࡍࡿ຺ᐃࡢ౯᱁㐣⛬ࢆᮏ⠇࡛♧ࡍ.

2.1 ኚ㢠ᖺ㔠ಖ㝤ᒣୗ [2005] ࡣ, ኚ㢠ಖ㝤 (Variable Insurance) ཬࡧኚ㢠ᖺ㔠ಖ㝤 (Variable Annuity) ࢆࠗணᐃ฼⋡, ணᐃṚஸ⋡ (ಖ㝤஦ᨾⓎ⏕⋡), ணᐃ஦ᴗ㈝⋡➼ࡢಖ㝤ᩱィ⟬ᇶ♏ࡢ࠺ࡕ, ணᐃ฼⋡ࢆಖ㝤⪅ࡀಖドࡏࡎ, ㈨⏘㐠⏝ᐇ⦼࡟ᛂࡌ࡚ಖ㝤㔠㢠, ᖺ㔠㢠ཬࡧゎ⣙㏉ᡠ㔠㢠ࢆኚືࡉࡏࡿࡇ࡜࡜ࡋࡓ⏕࿨ಖ㝤࠘࡜ᐃ⩏ࡋ࡚࠸ࡿ.
ಖ㝤ᴗἲ➨ 2 ᮲࡟࠾ࡅࡿࠕᐃ㢠⤥௜ࡢ⏕࿨ಖ㝤, ᐇᦆሸ⿵ࡢᦆᐖಖ㝤ࠖ࡜࠸࠺ᐃ⩏࡟ᙜヱಖ㝤✀㢮ࡣᩚྜࡍࡿࡢࡔࢁ࠺࠿࡜࠸࠺⣲ᮔ࡞␲ၥࢆᣢࡘࡀ, ᡃࡀᅜ࡟࠾ࡅࡿᑟධ࡟㝿ࡋ࡚ࡣࠗಖ㝤㔠㢠ࡀኚືࡍࡿࡶࡢ࡛࠶ࡗ࡚ࡶᐈほⓗ࡞ಖ㝤㔠㢠ࡢ⟬ฟᇶ‽ࡀዎ⣙ୖᐃࡵࡽࢀ࡚࠸ࢀࡤ⏕࿨ಖ㝤ዎ⣙࡛࠶ࡿ࠘࡜ᩚ⌮ࡉࢀ, ィ⟬ᇶ♏ࡢ඲࡚࡟ಖ㝤⪅ࡀಖドࢆᥦ౪ࡍࡿࡇ࡜ࡣᚲ㡲ࡢ᮲௳࡛ࡣ࡞࠸ࡶࡢ࡜ࡉࢀࡓࡼ࠺࡛࠶ࡿ. ⤖ᯝ࡜ࡋ࡚㈨⏘㐠⏝ᡂᯝ (ணᐃ฼⋡) ࡢಖドࡀእࢀࡓࡶࡢ࡛࠶ࡗ࡚ࡶ, ಖ㝤ዎ⣙࡜ࡋ࡚ㄆ㆑ࡉࢀࡓ. ࡍ࡞ࢃࡕ, ಖ㝤஦ᨾⓎ⏕⋡ (Ṛஸ⋡) ࡟ࡘ࠸࡚ಖドࡀ࠶ࡿࡇ࡜ࡀಖ㝤ࡢᚲ㡲せ௳࡜ᒣୗ [2005] ࡣᣦ᦬ࡋ࡚࠸ࡿ.
㈨⏘㐠⏝ࣜࢫࢡࡢᢸ࠸ᡭࡀಖ㝤⪅࡛࠶ࡿ࠿ಖ㝤ዎ⣙⪅࡛࠶ࡿ࠿࡟ࡼࡗ࡚, ᙜ↛࡟ಖ㝤⪅ࡢᢞ㈨⾜ືࡣኚ໬ࡍ
ࡿ. ࡇࡢࡓࡵ, ಖ㝤⪅ࡀ㈇࠺ሙྜࡣࠕ୍⯡຺ᐃࠖ, ಖ㝤ዎ⣙⪅ࡀ㈇࠺ሙྜࡣࠕ≉຺ูᐃࠖ࡜ࡋ຺࡚ᐃࡀศ㞳ࡉࢀ
࡚࠸ࡿ. ᭱పಖドࡢ௜ࡉࢀࡓኚ㢠ᖺ㔠ಖ㝤ࡢዎ⣙⟶⌮ࢆ᣺ࡾ㏉ࡿ࡜, ᖺ㔠㢠ࡢỴᐃᇶ‽ࡓࡿཧ↷㈨⏘࡜ࡋ࡚ࡢᛶ᱁ࢆ≉຺ูᐃࡀᣢࡘ. ᙜヱ຺ᐃ࡟࠾࠸࡚ࡣ㈨⏘㐠⏝ᡂᯝࢆ඲࡚ಖ㝤ዎ⣙⪅࡟ᖐᒓࡉࡏࡿࡢ࡛࠶ࡿ࠿ࡽ, ዎ⣙⪅ᣢศ (㈇മ) ࡣ㈨⏘࡜ྠ㢠࡟࡞ࡿ. ࡍ࡞ࢃࡕ, ㈐௵‽ഛ㔠ࡣ཰ᨭࡢṧ㧗࡟௚࡞ࡽ࡞࠸. ୍᪉, ᭱పಖド㒊ศࡣ㈨⏘㐠⏝ࣜࢫࢡࢆಖ㝤⪅ࡀ㈇࠺ࡇ࡜࡟࡞ࡾ, ᙜヱ㒊ศࡣ୍⯡຺ᐃࡀᑐᛂࡍࡿ. ㈐௵‽ഛ㔠㢠ࡣᖹᡂ 8 ᖺ 2 ᭶ 29 ᪥኱ⶶ┬࿌♧➨ 48 ྕࠕᶆ‽㈐௵‽ഛ㔠ࡢ✚❧᪉ᘧཬࡧィ⟬ᇶ♏⋡ࢆᐃࡵࡿ௳ࠖࢆཷࡅ, ಖ㝤ᩘ⌮࡟ᇶ࡙ࡁホ౯ࡉࢀࡿ㔠㢠ࡀィୖࡉࢀࡿࡇ࡜࡟࡞ࡿ. ḟᑠ⠇࡛ࡣ୧຺ᐃ࡟࠾ࡅࡿ㈨⏘ཬࡧ㈇മࡢ౯᱁㐣⛬ࢆゎㄝࡍࡿ.

2.2 ㈨⏘࡜㈇മࡢ౯᱁㐣⛬‶ᮇ᫬ཬࡧṚஸ᫬࡟᪤ᡶ㎸ಖ㝤ᩱྜィ┦ᙜ㢠ࢆ᭱పಖドࡍࡿ, ᖹ‽ᡶኚ㢠ᖺ㔠ಖ㝤ዎ⣙ࢆ⪃࠼ࡿ. ࡍ࡞ࢃࡕ,ᙜヱၟရ࡟࠾࠸࡚ࡣಖドỈ‽ (⾜౑౯᱁) ࡀẁ㝵ⓗ࡟ୖ᪼ࡍࡿ.
௨ୗ࡛ࡣಖ㝤ᮇ㛫ࢆ T , ணᐃ฼⋡ࢆ i, ᖹ‽ᡶಖ㝤ᩱࢆ P , ᫬Ⅼ t ࡟࠾ࡅࡿ≉຺ูᐃ㈨⏘㢠ࢆ S/At , ୍⯡຺ᐃ*9㈨⏘㢠ࢆ G/At , ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠ࢆ LG
min (t), [t  t, t] ᮇ㛫 ࡟㐺⏝ࡉࢀࡿ≉຺ูᐃ㐍ᒎ⋡
S/A

ࢆ Rt

G/A

, ୍⯡຺ᐃ㐍ᒎ⋡ࢆ Rt, ↓ࣜࢫࢡ฼Ꮚ⋡ࢆ rt , ಖドᩱ➼ࢆ⾲ 1 ࡟♧ࡍ࡜࠾ࡾ࡜ࡍࡿ.
⾲1

ಖドᩱ࣭ணᐃ஦ᴗ㈝࣭ಙクሗ㓘ಖ㝤ᩱẚ౛㒊ศ✚❧㔠ẚ౛㒊ศ

᭱పಖドᩱ

1P

1V

ணᐃ஦ᴗ㈝

2P

2V

ಙクሗ㓘

-

3V

ྜィ

P = 1P + 2P

V = 1V + 2V + 3V࡞࠾, ಖドᩱ➼࡟ࡘ࠸࡚ࡣᖺ㱋ཬࡧᛶูࡢタᐃࢆ᝿ᐃࡋ࡞࠸. ᖺ㱋ูࡢタᐃࢆ⾜ࡗࡓሙྜ࡟ࡣ, ዎ⣙᪥࣭ዎ⣙*9 ࡇࡇ࡛ࡣ㞳ᩓ⾲⌧ࢆ๓ᥦ࡜ࡋ࡚࠸ࡿ. t ࡣ᫬㛫㛫㝸࡛࠶ࡾ, ௨ୗ࡟࠾ࡅࡿᘧࡢᒎ㛤࡟࠾࠸࡚ࡣ t = 1 ࡜ࡋ࡚࠸ࡿ.

―70―

日本保険・年金リスク学会誌 Vol.4 No.1 2009ᖺ㱋࣭ᛶู࡟≉຺ูᐃࡢࣘࢽࢵࢺ౯᱁ࢆタᐃࡋ, ࢹ࢕ࢫࢡ࣮ࣟࢪ࣮ࣕ㈨ᩱࢆసᡂࡍࡿᚲせࡀ⏕ࡎࡿ. ᖺ㱋࣭ᛶูᩱ⋡ࡢ㐺⏝ࡣ஦ົ㈇Ⲵࡀ㔜ࡃ࡞ࡿࡔࡅ࡛ࡣ࡞ࡃ, ዎ⣙⪅ࡢΰ஘ࢆᘬࡁ㉳ࡇࡍࡇ࡜ࡉ࠼ࡶᠱᛕࡉࢀࡿ. ඹ㏻ᩱ⋡࡛࠶ࢀࡤ, タᐃࡉࢀࡿࣘࢽࢵࢺ౯᱁ࡣ၏୍࡛࠶ࡾ, ᙜヱၥ㢟ࡣᅇ㑊࡛ࡁࡿ. ᖺ㱋ཬࡧᛶู࡟ᇶ࡙ࡃᕪ␗ࡣ㈐௵‽ഛ㔠࡟ࡢࡳ཯ᫎࡍࢀࡤ༑ศ࡛࠶ࢁ࠺.

2.2.1 ಖ㝤ዎ⣙⪅࡜ಖ㝤఍♫㛫ཬࡧಖ㝤఍♫ෆ㒊ࡢ࢟ࣕࢵࢩࣗࣇ࣮ࣟ
୍⯡຺ᐃ, ≉຺ูᐃ, ㈐௵‽ഛ㔠ࡢ౯᱁㐣⛬ࢆᑟฟࡍࡿ๓࡟, ಖ㝤ዎ⣙⪅࡜ಖ㝤఍♫㛫, ಖ㝤఍♫ෆࡢ୍⯡຺ᐃ࡜≉຺ูᐃ㛫ࡢ㈨㔠ࡢὶࢀࢆᅗ 2.2.1 ࡟♧ࡍ.

ᅗ1ᙜヱಖ㝤ዎ⣙࡟ಀࡿ࢟ࣕࢵࢩࣗࣇ࣮ࣟࡢᵓ㐀ಖ㝤ዎ⣙ࡢ⥾⤖࡟క࠸, ಖ㝤ዎ⣙⪅ࡣᖹ‽ᡶಖ㝤ᩱ P ࢆಖ㝤఍♫࡟ᨭᡶ࠺. ᙜヱ㔠㢠ࡣࡲࡎ, ඲㢠ࡀ୺ዎ⣙ࡓࡿ୍⯡຺ᐃ࡛཰ධࡉࢀࡿ. ᖹ‽ᡶಖ㝤ᩱ P ࡢ࠺ࡕ, ಖ㝤ዎ⣙ࡢ⥔ᣢ⟶⌮➼࡟ᚲせ࡞ண⟬ࡀ஦ᴗ㈝࡜ࡋ࡚᥍㝖ࡉࢀ, ṧ㢠ࡀ㈐௵‽ഛ㔠ࡢᑐᛂ㈈※࡜ࡋ୍࡚⯡຺ᐃཬࡧ≉຺ูᐃ࡛✚ࡳ❧࡚ࡽࢀࡿ. ࡇࡢ࡜ࡁ, ୍⯡຺ᐃࡢ᭱పಖ
ド㈈※࡜ࡋ࡚␃ಖࡉࢀࡿ㔠㢠ࡣ P · 1P , ≉⣙ࡓࡿ≉຺ูᐃ࡟㐠⏝㈈※࡜ࡋ࡚᣺ࡾ᭰࠼ࡽࢀࡿ㔠㢠ࡣ P · (1  P )࡛࠶ࡿ.

஦ᴗᖺᗘᮎ (ࡇࡇ࡛ࡣ t ࡜ࡍࡿ) ࡟࠾࠸࡚ࡣ, ≉຺ูᐃ࡟ࡣᙜヱ஦ᴗᖺᗘጞࡢಖ᭷㈨⏘࡟๓ᥖ P · (1  P ) ࢆຍ࠼, ⤥௜㔠➼ࢆ᥍㝖ࡋࡓ㔠㢠ࡢ㐠⏝ᡂᯝ (ᙜヱ㔠㢠ࢆ౽ᐅⓗ࡟ V ࡜ࡍࡿ) ࡀ␃ಖࡉࢀ࡚࠸ࡿ. Ỵ⟬ฎ⌮࡜ࡋ࡚V ࠿ࡽ✚❧㔠ẚ౛ࡢ᭱పಖド㈈※ (V · 1V ), ஦ᴗ㈝ (V · 2V ) ཬࡧಙクሗ㓘 (V · 3V ) ࡀྲྀࡾᔂࡉࢀ, ᭱పಖド㈈※ࡀ୍⯡຺ᐃ㈨⏘࡟᣺ࡾ᭰࠼ࡽࢀࡿ. ṧࡉࢀࡓ S/At , ࡍ࡞ࢃࡕ, V · (1  V ) ࡀ⩣஦ᴗᖺᗘጞࡢ≉຺ูᐃ㈨⏘ṧ㧗࡜࡞ࡿ.

―71―୍⯡຺ᐃࡶྠᵝ࡟, ᙜヱ஦ᴗᖺᗘጞࡢಖ᭷㈨⏘࡟๓ᥖ P · 1P ࢆຍ࠼, ⤥௜㔠➼ࢆ᥍㝖ࡋࡓ㔠㢠ࡢ㐠⏝ᡂᯝཬࡧ≉຺ูᐃ࠿ࡽ᣺ࡾ㏉ࡽࢀࡓ᭱పಖド㈈※ࡀᖺᗘᮎ㈨⏘ṧ㧗 G/At ࡜ࡋ࡚␃ಖࡉࢀ࡚࠸ࡿ.

᭱⤊ⓗ࡟⿕ಖ㝤⪅࡟ಖ㝤஦ᨾࡀⓎ⏕ࡏࡎ‶ᮇࢆ㏄࠼ࡓሙྜ, ⱝࡋࡃࡣಖ㝤ᮇ㛫୰࡟ಖ㝤஦ᨾࡀⓎ⏕ࡋࡓሙྜ࡟ࡣ, ᭱పಖド㔠㢠ཬࡧ≉຺ูᐃ㈨⏘ṧ㧗ࡢఱࢀ࠿኱ࡁ࠸㔠㢠ࡀಖ㝤㔠ཷྲྀே࡟ᨭᡶࢃࢀࡿࡇ࡜࡟࡞ࡿ. ➨ +᫬Ⅼ࡛ಖ㝤஦ᨾࡀⓎ⏕ࡋࡓࡶࡢ࡜ࡍࢀࡤ, ᮏ౛࡟࠾ࡅࡿ᭱పಖド㔠㢠ࡣ᪤ᡶ㎸ಖ㝤ᩱྜィ┦ᙜ㢠 + · P ࡛࠶ࡿࡢ࡛, ಖ㝤㔠ཷྲྀே࡟ᨭᡶࢃࢀࡿ㔠㢠ࡣ max(S/A+ , + · P ) ࡛࠶ࡿ.

2.2.2 ≉຺ูᐃ㈨⏘㢠 (ᐇ㝿ࡢ㐠⏝ᡂᯝ)᫬Ⅼ (t  1) ࡟࠾ࡅࡿຍධ⪅ (⏕Ꮡ⪅) ᩘࡣ lx+t1 ࡛࠶ࡿࡢ࡛, ≉຺ูᐃࡢᮇጞ㈨⏘㢠ࡣ lx+t1 · S/At1 ࡛

࠶ࡿ. ᙜヱ㈨⏘㢠࡜ຍධ⪅ࡀᮇጞ࡟⣡௜ࡋࡓಖ㝤ᩱ P ࡢ࠺ࡕ, ≉຺ูᐃ࡟ᢞධࡉࢀࡿ P · (1  P ) ࢆᮇᮎ (᫬Ⅼ t) ࡲ࡛㐠⏝ࡋࡓ⤖ᯝࡣ

S/Alx+t1 · [S/At1 + P · (1  P )] · Rt

࡜࡞ࡿࡀ, ᙜヱᮇ㛫ෆ࡟࠾ࡅࡿṚஸ⪅࡟ᑐࡋ࡚⤥௜ࢆ⾜࠺ࡓࡵ, ⤥௜ᚋࡢṧ㢠ࡣ
S/A(lx+t1  dx+t1 ) · [S/At1 + P · (1  P )] · Rt

࡛࠶ࡿ. ᖺᗘᮎ࡟࠾࠸࡚ࡣᙜヱ㔠㢠࡟ᑐࡋ, ᭱పಖドᩱ࣭ணᐃ஦ᴗ㈝࣭ಙクሗ㓘ࡀ㈿ㄢࡉࢀ, ྲྀࡾᔂࡉࢀࡿࡢ࡛,ᮇᮎ (᫬Ⅼ t) ࡢ㈨⏘ṧ㧗ࡣS/A(lx+t1  dx+t1 ) · [S/At1 + P · (1  P )] · Rt

· (1  V )࡜࡞ࡿࡀ, ᙜヱ㔠㢠ࡣ᫬Ⅼ t ࡟࠾ࡅࡿຍධ⪅࡟ᖐᒓࡍࡿࡶࡢ࡛࠶ࡿ࠿ࡽ, lx+t · S/At ࡜⾲⌧ࡍࡿࡇ࡜ࡶ࡛ࡁࡿ.
ᚑࡗ࡚, ࣇ࢓ࢡ࣮ࣛࡢ෌ᖐᘧ*10 ࡣ,

S/Alx+t · S/At = (lx+t1  dx+t1 ) · [S/At1 + P · (1  P )] · Rt

· (1  V )(3)

࡜࡞ࡿ. ࡓࡔࡋ, S/A0 = 0 ࡛࠶ࡿ. ≉຺ูᐃ࡛ࡣ⤥௜㈈※ࡀᙜヱ≉຺ูᐃṧ㧗࡟㝈ᐃࡉࢀࡿࡓࡵ, ௵ពࡢ᫬Ⅼ࡟࠾࠸࡚ S/At Ӎ 0 ࡜࡞ࡿ. ࡲࡓ, ‶ᮇ฿㐩ᚋࡣṧవ㈈⏘ࢆዎ⣙⪅࡟⤥௜ࡋ, ᙜヱ຺ᐃࡣᾘ⁛ࡍࡿࡓࡵ, ᙜヱಖ㝤ዎ⣙ࢆᘬࡁཷࡅࡿಖ㝤఍♫ࡢෆ㒊␃ಖᙧᡂ࡟㈨ࡍࡿࡇ࡜ࡣ࡞࠸.
ࡇࡇ࡛⡆༢ࡢࡓࡵ࡟ಖ㝤ᮇ㛫୰࡟࠾ࡅࡿṚஸཬࡧゎ⣙ࡀ⏕ࡌ࡞࠸ࡶࡢ࡜ࡍࡿ. ࡇࡢ࡜ࡁ t ᫬Ⅼࡲ࡛ࡢ᝟ሗ࡟
ᇶ࡙ࡃ (t + k) ᫬Ⅼࡢ≉຺ูᐃ㈨⏘㢠 S/At+k | Ft ࡣᘧ (4) ࡟ࡼࡾᐃ⩏࡛ࡁࡿ. ࡓࡔࡋ, k > 0 ࡛࠶ࡿ.

S/At+k | Ft = S/At

t+k
Y

S/A

Rl

· (1  V )kt+kX

P · (1  P )

l=t+1

+

l=t+1

t+k
Y

m=l

S/A
Rm
· (1  V )(t+k)l+1(4)

2.2.3 ≉຺ูᐃ㈨⏘㢠 (㈐௵‽ഛ㔠ࡢཧ↷㈨⏘)
DEX
t ᫬Ⅼࡲ࡛ࡢ᝟ሗ࡟ᇶ࡙ࡃ (t + k) ᫬Ⅼࡢ㈐௵‽ഛ㔠ホ౯ࡢཧ↷㈨⏘ S/AIN| Ft ࡣ, ḟᘧ࡟ࡼࡾᐃ⩏ࡉt+k

ࢀࡿ. ࡓࡔࡋ, k > 0 ࡛࠶ࡿ.

DEX
S/AIN
| Ft = S/At · exp[ (  V 
t+k

+P · (1  P )

t+k1
X
l=t

2
) · k +  Wk ]2

exp[ (  V 

*10 Appensix ࢆཧ↷ࡉࢀࡓ࠸.

―72―

2
) · (t + k  l) +  Wt+kl ]2(5)

日本保険・年金リスク学会誌 Vol.4 No.1 2009ࡇࡇ࡛␃ពࡍ࡭ࡁⅬࡣ౯᱁㐣⛬ࡢࣃ࣓࣮ࣛࢱཬࡧ  ࡛࠶ࡿ.

ᖹᡂ 8 ᖺ 2 ᭶ 29 ᪥኱ⶶ┬࿌♧➨ 48 ྕ࡛ࡣ㈐௵‽ഛ㔠ࢆᶆ‽ⓗ᪉ᘧ࡟ࡼࡾ✚ࡳ❧࡚ࡿሙྜࡢࣃ࣓࣮ࣛࢱࡀつᐃࡉࢀ࡚࠸ࡿ. ࡍ࡞ࢃࡕ, ≉຺ูᐃ࡟࠾࠸࡚ಖ᭷ࡍࡿᢞ㈨㈨⏘౯᱁ࡢ࣎ࣛࢸ࢕ࣜࢸ࢕࡜ࡣᚲࡎࡋࡶ୍⮴ࡋ࡞࠸ྍ⬟ᛶࡀ࠶ࡿ. ↛ࡿ࡟, ➹⪅ࡣḟᥖࡢ⌮⏤࡟ࡼࡾ, ᙜヱࢺࣛࢵ࢟ࣥࢢ࣭࢚࣮ࣛࡢⓎ⏕࡟ࡣྰᐃⓗ࡛࠶ࡿ.
➨୍࡟ࡣᚲせෆ㒊␃ಖỈ‽ࡢୖ᪼࡟క࠺㈈※ࡢቑຍ࡛࠶ࡿ. ಖ㝤ᴗἲ➨ 116 ᮲➨ 1 㡯࡛ࡣಖ㝤ዎ⣙࡟ᇶ࡙ࡃᑗ᮶࡟࠾ࡅࡿമົࡢᒚ⾜࡟ഛ࠼ࡿࡓࡵ࡟, ẖỴ⟬ᮇ࡟࠾ࡅࡿ㈐௵‽ഛ㔠ࡢ✚❧ࢆせㄳࡋ࡚࠸ࡿ. ࡇࡇ࡛ᙜヱኚ㢠ಖ㝤ࡢᘬཷ࡟క࠺᭱పಖド⾜Ⅽ࡟࠾࠸࡚ࠕ㏻ᖖࡢண

ࠖ࡜ࡉࢀࡿ࣎ࣛࢸ࢕ࣜࢸ࢕ࡣ๓ᥖ࿌♧࡟つᐃࡉࢀࡿ್

࡛࠶ࡿ. ᚑࡗ࡚, ᙜヱ್࡜ᐇ⦼್ࡢ஋㞳ࡣせồࡉࢀࡿෆ㒊␃ಖỈ‽ࢆ㧗ࡵࡿࡇ࡜࡟࡞ࡿ. ㏣ຍⓗ࡟せồࡉࢀࡿෆ㒊␃ಖ⵳✚ࡢ㈈※ࡀ๫వ㔠࡟ồࡵࡽࢀ, ௚ၟရ⩌ᅋࡢ๫వ㔠ศ㓄㢠ࡀ๐ῶࡉࢀࡿࡼ࠺࡞஦ែࡀ⏕ࡎࢀࡤ, ࡇࢀࡣබᖹᛶࡢ㜼ᐖ࡟௚࡞ࡽ࡞࠸. ቑ㈨࡟ࡼࡿᑐᛂ➼, ௚ၟရ⩌ᅋ㛫ཬࡧᙜヱၟရ⩌ᅋෆࡢୡ௦㛫࡟࠾ࡅࡿබᖹᛶࢆ
☜ಖ࡛ࡁ࡞࠸ࡢ࡛࠶ࢀࡤ, ≉຺ูᐃࡢᮃࡲࡋ࠸ᢞ㈨ᡓ␎࡜ࡣ⪃࠼࡟ࡃ࠸.
➨஧࡟ࡣ㏫㑅ᢥࡢၥ㢟࡛࠶ࡿ. ࣎ࣛࢸ࢕ࣜࢸ࢕ࡀ㧗ࡃ࡞ࢀࡤ࢜ࣉࢩ࣭ࣙࣥࣉ࣑ࣞ࢔࣒ࡀ㧗ࡃ࡞ࡿࡇ࡜ࡣᖖ㆑
࡛࠶ࡿ. ᚑࡗ࡚, ཎ㈨⏘ࡓࡿ≉຺ูᐃࡢ࣎ࣛࢸ࢕ࣜࢸ࢕ࢆពᅗⓗ࡟ಖドᩱࡢィ⟬ᇶ♏ࡼࡾࡶ㧗ࡵࡿࡢ࡛࠶ࢀࡤ,ᙜヱ⾜Ⅽࡣ࣑ࢫ࣭ࣉࣛ࢖ࢫ࡜࠸࠺ࡼࡾࡣᑀࢁ, ㏫㑅ᢥࡢᥦ౪࡟௚࡞ࡽ࡞࠸. ᙜヱ≧ែࢆ┳㐣ࡍࡿࡼ࠺࡞ࡇ࡜ࡀ
࠶ࢀࡤ, ࢔ࢡࢳࣗ࢔࣮ࣜࡣಖ㝤ᴗἲ᪋⾜つ๎➨ 211 ᮲ࡢ 48 ࡟つᐃࡉࢀࡿ⫋㈐ࢆ඲࠺ࡋ࡚࠸ࡿ࡜ࡣ࠸࠼ࡲ࠸.

2.2.4 ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠ᙜヱ⛉┠ࡣ୍⯡຺ᐃ࡟࠾ࡅࡿ㈇മ࡛࠶ࡿ. ᫬Ⅼ k ࡲ࡛ࡢ᝟ሗࢆ཯ᫎࡋࡓ, ᑗ᮶ t ᫬Ⅼࡢ᭱పಖドࣜࢫࢡ࡟ಀ
ࡿ㈐௵‽ഛ㔠 LG
min (t) | Fk ࡣḟᘧ࡟ࡼࡾᐃ⩏ࡉࢀࡿ. ࡓࡔࡋ, t > k ࡛࠶ࡿ.

LG
min (t) | Fk =

Dx+T
DEX
· E[ max(T · P  S/AIN) | Fk ]
n
Dx+k
+

T
1
X
t=k

Cx+t
DEX
· E[ max((t + 1) · P  S/AIN) | Fk ]
t+1
Dx+k


T
1
X
t=k

Dx+t
· ( 1P · P + E[ āt | Fk ] )
Dx+k

(6)

ࡇࡇ࡛,

E[ āt | Fk ] =1V· e( V r) t · (e V r  1)
 V  rtXe( V ) l ]×[ S/Ak · e( V ) k + P · (1  P )

(7)

l=k

࡛࠶ࡿ.
࡞࠾, ᮏ✏࡛ࡣ᭱పಖドࣜࢫࢡ࡟ഛ࠼ࡿ༴㝤‽ഛ㔠 (ᡤㅝ༴㝤‽ഛ㔠Ϫ) ࡢ✚ࡳ❧࡚ࢆ⪃៖ࡋ࡞࠸ࡶࡢ࡜ࡍࡿ.

2.2.5 ୍⯡຺ᐃ㈨⏘㢠᫬Ⅼ (t  1) ࡟࠾ࡅࡿຍධ⪅ᩘࡣ lx+t1 ࡛࠶ࡿࡢ࡛, ୍⯡຺ᐃࡢᮇጞ㈨⏘㢠ࡣ lx+t1 · G/At1 ࡛࠶ࡿ. ᙜヱ㈨⏘㢠࡜ຍධ⪅ࡀᮇጞ࡟⣡௜ࡋࡓಖ㝤ᩱ P ࡢ࠺ࡕ, ୍⯡຺ᐃ࡟ᢞධࡉࢀࡿ P · 1P ࢆᮇᮎ (᫬Ⅼ t) ࡲ࡛㐠⏝ࡋ
ࡓ⤖ᯝࡣ

G/A

lx+t1 · [ P · 1P + G/At1 ] · Rt

࡜࡞ࡿ. ୍᪉, ᮇᮎ࡟࠾࠸࡚ࡣ≉຺ูᐃ㈨⏘㢠࠿ࡽྲྀࡾᔂࡉࢀࡓ᭱పಖドᩱ
S/A(lx+t1  dx+t1 ) · [S/At1 + P · (1  P )] · Rt―73―

· 1V =

1V
· lx+t · S/At
1  Vࡀ, ಖド㈈※࡜ࡋ୍࡚⯡຺ᐃ࡟⧞ࡾධࢀࡽࢀࡿ࡜ྠ᫬࡟, Ṛஸ⤥௜ࡢྲྀࡾᔂࡋࡶⓎ⏕ࡍࡿ. ᮇᮎ࡟ᨭᡶࡢ⏕ࡎࡿ

1 ே࠶ࡓࡾࡢṚஸ⤥௜㢠ࡣ, ≉຺ูᐃ㈨⏘㢠 S/At ࡀ᪤ᡶ㎸ಖ㝤ᩱ t · P ࡟୙㊊ࡍࡿ㔠㢠࡛࠶ࡿࡢ࡛, ຺ᐃ඲య࡛ࡣdx+t1 · max[ t · P  S/At , 0 ]࡛࠶ࡿ. ௨ୖࡼࡾ, ᮇᮎ࡟࠾ࡅࡿ㈨⏘ṧ㧗ࡣG/Alx+t1 · [ P · 1P + G/At1 ] · Rt+

1V
· lx+t · S/At  dx+t1 · max[ t · P  S/At , 0 ]1  V࡜࡞ࡿ. ᙜヱ㔠㢠ࡀ lx+t · G/At ࡛࠶ࡿࡢ࡛, t < T ࡟࠾ࡅࡿࣇ࢓ࢡ࣮ࣛࡢ෌ᖐᘧࡣ,1V· lx+t · S/At
1  Vdx+t1 · max[ t · P  S/At , 0 ]G/Alx+t · G/At = lx+t1 · [ P · 1P + G/At1 ] · Rt+

(8)࡜࡞ࡿ. ࡓࡔࡋ, G/A0 = 0 ࡛࠶ࡿ. ‶ᮇ᫬Ⅼ࡟࠾࠸࡚ࡣ⏕Ꮡ⪅࡟ᑐࡋ࡚Ṛஸ⤥௜㢠࡜ྠ㢠ࡢ‶ᮇಖ㝤㔠ࡀⓎ⏕ࡍࡿࡓࡵ, ࣇ࢓ࢡ࣮ࣛࡢ෌ᖐᘧࡣ,1V· lx+T · S/AT
1  V(dx+T 1 + lx+T ) · max[ T · P  S/AT , 0 ]G/Alx+T · G/AT = lx+T 1 · [ P · 1P + G/AT 1 ] · RT

+

(9)

࡛࠶ࡿ.
ᘧ (4) ࡜ྠᵝ࡟ࡋ࡚, ಖ㝤ᮇ㛫୰࡟࠾࠸࡚Ṛஸཬࡧゎ⣙ࡀ⏕ࡌ࡞࠸ࡶࡢ࡜ࡋࡓሙྜࡢ t ᫬Ⅼࡲ࡛ࡢ᝟ሗ࡟ᇶ
࡙ࡃ (t + k) ᫬Ⅼࡢ୍⯡຺ᐃ㈨⏘㢠 G/At+k | Ft ࡜ࡋ࡚

G/At+k | Ft = G/At
+

t+k
Y

l=t+1

t+k
X

l=t+1

=

t+k
X

S/A

Rl

l=t+1

=

t+k t+kXY

l=t+1 m=l

G/ARl

P · 1P· (1  V )k +

t+k
Y

m=l

t+k1
X

G/A
Rm
+ 1V { S/At ·  + P (1  P ) ·  }

l
Y

S/A
Rm

l=t+1 m=t+1

t+k
Y

o=l+1

(10)

RoG/A · (1  V )lt

S/A
Rm
· (1  V )(t+k)l+1

+

t+k1
X

l
X

l
Y

RnS/A

l=t+1 m=t+1 n=m

t+k
Y

o=l+1

RoG/A · (1  V )lm+1

ࢆᚓࡿ. ࡓࡔࡋ, k > 0 ࡛࠶ࡿ.
ᘧ (4) ࡜ᘧ (10) ࢆẚ㍑ࡍࡿ࡜, ≉຺ูᐃ࡜୍⯡຺ᐃࡢᮏ㉁ⓗ࡞ᵓ㐀ࡢᕪ␗ࢆ☜ㄆ࡛ࡁࡿ. ࡍ࡞ࢃࡕ, ๓⪅ࡀᙜヱ຺ᐃෆ࡛᏶⤖ࡍࡿࡇ࡜࡟ᑐࡋ, ᚋ⪅ࡣ௚຺ᐃࡢ㐠⏝ᡂᯝࡢᙳ㡪ࢆཷࡅࡿ࡜࠸࠺ࡇ࡜࡛࠶ࡿ.
⤒῭⎔ቃࡢప㏞ᮇࢆ᝿ᐃࡍࡿ࡜, ≉຺ูᐃ㈨⏘㢠ࡢῶ౯࡟క࠸᭱పಖド࡟ಀࡿ㈐௵‽ഛ㔠ࡀቑ౯ࡍࡿ୍᪉, ✚❧㔠ẚ౛ࡢಖドᩱ཰ධࡀῶᑡࡍࡿ. ᙜヱᵓ㐀ࡀኚ㢠ᖺ㔠ಖ㝤ࡢ୍⯡຺ᐃ࡟࠾ࡅࡿᢞ㈨ᡓ␎ࢆᅔ㞴࡞ࡶࡢ࡜ࡉࡏࡿせᅉࡢ୍ࡘ࡛࠶ࡿࡇ࡜ࡣᐜ᫆࡟⌮ゎ࡛ࡁࡿࡔࢁ࠺.

―74―

日本保険・年金リスク学会誌 Vol.4 No.1 20093 ㈇മࢆཧ↷ࡍࡿከᮇ㛫ືⓗ᭱㐺໬ၥ㢟ḟᑠ⠇࡛ࡣ౯್㛵ᩘࢆゎࡃࠕ୍⯡ⓗࠖ࡞ᡭἲࢆ♧ࡋ, ⥆ࡃᑠ⠇࡛ BGSS ἲࢆゎㄝࡋ, ୍⯡ⓗ࡞ᡭἲ࡜ࡢ┦㐪
Ⅼࢆ☜ㄆࡍࡿ.

3.1 ᇶᮏⓗ࡞⪃࠼᪉ (୍⯡ⓗ࡞ᡭἲ)ከᮇ㛫ືⓗ᭱㐺໬ࡢᯟ⤌࡟ࡼࡿᢞ㈨ᡓ␎࡟ࡘ࠸࡚ࡣ, Campbell and Viceira[2002] ➼࡟ヲࡋ࠸. ௨ୗ࡛ࡣࠕ㈨⏘ࠖࡢࡳ࡞ࡽࡎࠕ㈇മࠖࢆ⪃៖ࡋ, ஧ḟඖ࡟ᣑᙇࡋࡓሙྜࡢᇶᮏⓗ࡞⪃࠼᪉ࢆゎㄝࡍࡿ.
᫬Ⅼ t ࡟࠾ࡅࡿ㈨⏘㢠ࢆ At , ㈇മ㢠ࢆ Lt , ไᚚᑐ㇟࡜࡞ࡿ㈨⏘ᵓᡂẚࢆ x ࡜ࡍࡿ. ‶ᮇ T ᫬Ⅼࡢ㈨⏘㢠 AT
࡜㈇മ㢠 LT ࡟ࡼࡾᐃ⩏ࡉࢀࡿ㛵ᩘ u(AT , LT ) ࢆᑟධࡋ, ➨ t ᫬Ⅼࡢ౯್㛵ᩘ J (t, At , Lt ) ࢆJ (t, At , Lt ) =max{x+ }+ ;[t,T ]E[ u(AT , LT ) | Ft ](11)࡜ᐃ⩏ࡍࡿ. ࡓࡔࡋ, ᮏᑠ⠇࡛ࡣ௵ពࡢᑐ㇟࡟ᑐࡍࡿ ALM ࢆㄽࡌ࡚࠾ࡾ, ㈇മࡢᵓ㐀, ࡍ࡞ࢃࡕ, ಖ㞀ෆᐜࢆ≉ᐃࡋ࡚࠸࡞࠸.
ࡇࡇ࡛᭱㐺ᛶཎ⌮࡟ࡼࡾ, s > t ࡟࠾࠸࡚J (t, At , Lt ) =max{x+ }+ ;[t,s]E[ J (s, As , Ls ) | Ft ](12)ࡀᡂ❧ࡍࡿ. ࡘࡲࡾ, t ᫬Ⅼ࡟࠾ࡅࡿ౯್㛵ᩘࡣ, ࠕᑗ᮶ࡢ s ᫬Ⅼ࡟࠾ࡅࡿ౯್㛵ᩘࡢ᮲௳௜ᮇᚅ್ࢆ᭱኱໬ࡍ
ࡿࡶࡢ࡛ࠖ࠶ࡿ.
ఀ⸨ࡢ⿵㢟࡟ࡼࡾ,dJ (+, A+ , L+ ) =(((J · (dAt ) +J · (dLt ) + J · dt
(At
(Lt
(t
2
(
(2
1 (2
2
2
J
·
(dA
)
+
J
·
(dL
)
+
2
J · (dAt ) · (dLt ) ]+ [tt
2 (A2t(L2t(At (Lt(13)ࡲࡓ,

J (s, As , Ls ) = J (t, At , Lt ) +Z s

dJ (+, A+ , L+ )

(14)

t

࡛࠶ࡿࡢ࡛, ᘧ (13) ཬࡧᘧ (14) ࢆᘧ (12) ࡟㐺⏝ࡍࡿ࡜

max{x+ }+ ;[t,s]

E[

Z s
t

dJ (+, A+ , L+ ) | Ft ] = 0(15)ࢆᚓࡿ. ᭱㐺ᢞ㈨ᡓ␎ࢆồࡵࡿ࡟ࡣ, ㈨⏘ཬࡧ㈇മࡢண⟬ไ⣙ୗ࡟࠾࠸࡚ᙜヱせ௳ࢆ‶㊊ࡍࡿࡼ࠺࡟ಖ᭷㈨⏘㓄ศẚ x ࡢㄪᩚࢆᐇ㊶ࡍࢀࡤࡼ࠸. ࡇࢀࡀከᮇ㛫ືⓗ᭱㐺໬ၥ㢟ࡢᇶᮏⓗ࡞⪃࠼᪉࡛࠶ࡿ.
ၥ㢟ࢆ⡆༢࡟ࡍࡿࡓࡵ࡟, ㈨⏘ཬࡧ㈇മࡢ౯᱁㐣⛬ࢆḟᥖࡢࡼ࠺࡟タᐃࡋ࡚ࡳࡼ࠺.

dAt = (t, At ) At dt + (t, At ) At dWtA
dLt = (t, Lt ) Lt dt + (t, Lt ) Lt dWtL
dWtA · dWtL = ' dt―75―ᙜヱ᮲௳ࢆᘧ (13) ࡟཯ᫎࡋ, ࣈࣛ࢘ࣥ㐠ືࡢᮇᚅ್ࡀࢮࣟ࡜࡞ࡿࡇ࡜ࢆ฼⏝ࡋ࡚ᩚ⌮ࡍࢀࡤ

(((J · (t, At ) At +J · (t, Lt ) Lt + J
(At
(Lt
(t
2
1
1 (2
(
J · (t, At )2 A2t +J · (t, Lt )2 L2t+
2 (A2t2 (L2t(2+
J · (t, At ) (t, Lt ) ' At Lt = 0(At (Lt(16)ࢆᚓࡿ.
ࡇࡇ࡛➨ t ᫬Ⅼ࡟࠾ࡅࡿ㈨⏘㓄ศẚ࣋ࢡࢺࣝࢆ x, ㈇മᵓᡂẚ࣋ࢡࢺࣝ*11 ࢆ y, ࣜࢫࢡᛶ㈨⏘ࡢ཰┈⋡ࡢᮇᚅ್࣋ࢡࢺࣝࢆ M, ྠศᩓඹศᩓ⾜ิࢆࡔࡋ, x, y, e, M ࡣ (n × 1) ࣋ࢡࢺࣝ,, ඲࡚ࡢせ⣲ࡀ 1 ࡢ࣋ࢡࢺࣝࢆ e, ↓ࣜࢫࢡ฼Ꮚ⋡ࢆ rt ࡜ࡍࡿ. ࡓ

ࡣ (n × n) ⾜ิ࡛࠶ࡿ.

ࡇࡢ࡜ࡁ, ㈨⏘ཬࡧ㈇മࡢ౯᱁㐣⛬࡟࠾ࡅࡿಀᩘࡣ

(t, At ) = rt + (M  rt · e)T · x(t, Lt ) = rt + (M  rt · e)T · y(t, At )2 = xT x

(t, Lt )2 = yT y(t, At ) (t, Lt ) ' = xT y࡜᭩ࡁ᥮࠼ࡿࡇ࡜ࡀ࡛ࡁࡿࡢ࡛, 1 㝵ࡢ᮲௳࠿ࡽ᭱㐺㈨⏘㓄ศẚ x ࡣ(

x =  (At

J ·

1

2

(M  rt · e) + Lt · (A(t (Lt J · y

(17)

2

(
At · (A2Jt࡜࡞ࡾ, Merton[1971] ➼࡟ࡼࡾ♧ࡉࢀࡿ࡜࠾ࡾ,1(M  rt e) ࡟ᑐᛂࡍࡿ㏆ど║ⓗᡂศ࡜, y ࡟ᑐᛂࡍࡿ␗

᫬Ⅼ㛫ࡢ࣊ࢵࢪᡂศ (㈇മᵓ㐀) ࡢྜᡂ࡜ࡋ࡚ồࡵࡽࢀࡿ.

ᘧ (17) ࢆᘧ (16) ࡟௦ධࡍࡿ࡜, ౯್㛵ᩘ J ࡟ᑐࡍࡿ೫ᚤศ᪉⛬ᘧࢆᚓࡿ.
(2((
1 ( (At J )(J + Lt ·J)
· (M  rt · e)T
J + rt · (At ·
(2
(t
(At
(Lt
2 (A
J
2

1

(M  rt · e)t2(
J · (A(t (Lt J(+Lt · (J  (At) · (M  rt · e)T y
(2
(Lt
(A2t
2

( (A(t (Lt J )21
(2
+ L2t · [J
] · yT y = 0
(2
2
(L2t
J
(A2

(18)

t

2

2

(
(
(
ᘧ (18) ࢆゎࡁ, ᑟฟࡉࢀࡓ౯್㛵ᩘ J ࡟ᑐࡍࡿ೫ᚤศಀᩘ (AJ , (A
2 J ཬࡧ (A (L J ࢆ෌ࡧᘧ (17) ࡟㐺tttt⏝ࡍࡿࡇ࡜࡟ࡼࡾ, ᭱⤊ⓗ࡟᭱㐺㈨⏘㓄ศẚ x ࡀᑟ࠿ࢀࡿ. ᥮ゝࡍࢀࡤ, ᙜヱ౯್㛵ᩘࡢゎࡀᑟฟࡉࢀ࡞ࡅࢀࡤ, ᭱㐺ゎ x ࡣᚓࡽࢀ࡞࠸࡜࠸࠺ࡇ࡜࡛࠶ࡿ.

*11 ⧞ࡾ㏉ࡍࡀ, x ཬࡧ y ࡣ, ๓⠇➼࡟࠾ࡅࡿ୍⯡຺ᐃᵓᡂẚཬࡧ≉຺ูᐃᵓᡂẚ࡜ࡣูࡢࡶࡢ࡛࠶ࡿ.

―76―

日本保険・年金リスク学会誌 Vol.4 No.1 2009

3.2 BGSS ἲ [2005]
3.2.1 ᭱㐺໬ၥ㢟ࡢタᐃ
᫬Ⅼ t ࡟࠾ࡅࡿ୍⯡຺ᐃ㈨⏘㢠ࢆ G/At , ≉຺ูᐃ㈨⏘㢠ࢆ S/At ࡜ࡋ, ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠LGmin (t) ࢆ≉຺ูᐃ㈨⏘㢠ࡢୖ࡟᭩࠿ࢀࡿࢹࣜࣂࢸ࢕ࣈ࡜ぢ೴ࡋ, H[S/At ] ࡜グ㏙ࡍࡿ. ࡇࡇ࡛, ➨ t ᫬Ⅼࡢ≧ែኚᩘ࡜ࡋ࡚ḟᘧ࡟ࡼࡾᐃ⩏ࡉࢀࡿ✚❧ẚ⋡ FR(t) ࢆᑟධࡍࡿ.

FR(t) =

S/At + G/At
S/At + H[S/At ](19)ࡓࡔࡋ, t ; [0, T ] ࡛࠶ࡿ.

࡞࠾, ᪤࡟㏙࡭ࡓࡼ࠺࡟, ≉຺ูᐃ࡛ࡣ⤥௜㈈※ࡀᙜヱ≉຺ูᐃṧ㧗࡟㝈ᐃࡉࢀ, ཰ᨭṧࡀ㈇മ࡜ࡋ࡚ㄆ㆑ࡉ

ࢀࡿࡓࡵ㈨⏘࡜㈇മࡣྠ㢠࡜࡞ࡿ. ᚑࡗ࡚, ศᏊཬࡧศẕ࡟⌧ࢀࡿ≉຺ูᐃ㈨⏘㢠 S/At ࡣྠ㢠࡛࠶ࡿࡀ, ព࿡ࡀ␗࡞ࡿࡇ࡜࡟␃ពࡉࢀࡓ࠸. ศᏊࡢࡑࢀࡣࠕ≉຺ูᐃ࡟ᒓࡍࡿ㈨⏘㢠࡛ࠖ࠶ࡾ, ศẕࡢࡑࢀࡣࠕ≉຺ูᐃ࡟ᒓ
ࡍࡿ㈇മ㢠ࠖࢆព࿡ࡋ࡚࠸ࡿࡢ࡛࠶ࡿ.
ᚑࡗ࡚, ✚❧ẚ⋡ࡢศᏊࡣ୍⯡຺ᐃ㈨⏘㢠ཬࡧ≉຺ูᐃ㈨⏘㢠ࡢྜィ, ศẕࡣ≉຺ูᐃ㈇മ㢠ཬࡧ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠 (୍⯡຺ᐃ㈇മ㢠) ࡢྜィࢆព࿡ࡋ࡚࠾ࡾ, ✚❧ẚ⋡࡜ࡣᙜヱಖ㝤ዎ⣙ࡢ㈇മ࡟ᑐᛂࡍࡿ
㈨⏘඘㊊ᗘࢆホ౯ࡋࡓࡶࡢ࡜ゎ㔘࡛ࡁࡿ. ᙜヱẚ⋡ࡀᖖ࡟ 1.0 ࢆ㉸㐣ࡋ࡚࠾ࡾ, ࠿ࡘ, ಖ᭷㈨⏘ࡢὶືᛶࡶ☜ಖࡉࢀ࡚࠸ࢀࡤ, ⌮᝿ⓗ࡞≧ែ࡜࠸࠼ࡿࡔࢁ࠺.
୍᪉, ᖺᗘᮎỴ⟬᫬Ⅼ࡛ᙜヱẚ⋡ࡀ 1.0 ࢆୗᅇࢀࡤ, ᙜヱዎ⣙⩌ᅋࡣമົ㉸㐣ࡢ≧ែ࡟࠶ࡿ. ᭦࡟ᙜヱ≧ែ࡛⤥௜ࡀⓎ⏕ࡍࢀࡤᦆኻࡀ㢧ᅾ໬ࡋ, ࢯࣝ࣋ࣥࢩ࣮࣭࣐࣮ࢪࣥ (ᗈ⩏ࡢ⮬ᕫ㈨ᮏ) ࡀᲠᦆࡉࢀࡿࡓࡵ, ಖ㝤⪅ࡀ㐩ᡂࡍ࡭ࡁㄢ㢟ࡢ୍ࡘࡣ✚❧୙㊊ࡢᢚไⱝࡋࡃࡣᅇ㑊࡜⪃࠼ࡿࡇ࡜ࡀ࡛ࡁࡿ.
ࡑࡇ࡛ᮏ✏࡛ࡣ, ‶ᮇ T ᫬Ⅼࡢ✚❧ẚ⋡ FR(T) ࡟ᑐࡋ,u(FR(T)) = FR(T)  a · FR(T)2 ,( 0 < FR(T) <1, a>0)
2a(20)࡞ࡿ㛵ᩘ u ࢆᑟධࡋ, 㞳ᩓ᫬㛫ࣔࢹࣝ࡟ࡼࡾḟᥖࡢ᭱㐺໬ၥ㢟ࢆ⪃࠼ࡿ.

max E[u(FR(T)) | Ft ]

(21)

1
{x+ }+T=t

ࡓࡔࡋ, ಖ㝤ᮇ㛫୰࡟࠾ࡅࡿṚஸཬࡧゎ⣙ࢆ᝿ᐃࡏࡎ, ≉຺ูᐃ࡜୍⯡຺ᐃࡢ౯᱁㐣⛬ࢆ
S/A

S/At+1 = [ S/At + P (1  P ) ] · Rt+1 · (1  V )G/A

G/At+1 = [ G/At + P · 1P ] · Rt+1 +1V

1  V· S/At+1

(22)
(23)

࡜ࡍࡿ.

3.2.2 ㈨⏘౯᱁㐣⛬ࡢศゎ➨ t ᫬Ⅼࡢ୍⯡຺ᐃཬࡧ≉຺ูᐃࡢ㈨⏘ᵓᡂẚ࣋ࢡࢺࣝࢆ x ཬࡧ y, ࣜࢫࢡᛶ㈨⏘ࡢ཰┈⋡ࡢᮇᚅ್࣋ࢡࢺ
ࣝࢆ M, ྠศᩓඹศᩓ⾜ิࢆࢺࣝ,ࡣ (n × n) ⾜ิ࡛࠶ࡿ.

, ඲࡚ࡢせ⣲ࡀ 1 ࡢ࣋ࢡࢺࣝࢆ e, ࡜ࡍࡿ. ࡓࡔࡋ, x, y, e, M ࡣ (n × 1) ࣋ࢡࡇࡢ࡜ࡁ, [t, t + 1] ᮇ㛫࡟ᑐᛂࡍࡿ≉຺ูᐃ㐠⏝཰┈⋡ࢆᮇᚅ್ࡀ rt+1 + (M  rt+1 e)T · y, ศᩓࡀ yTyࡢṇつศᕸ࡟ᚑ࠺ࡶࡢ࡜ࡍࢀࡤ, ≉຺ูᐃ㐍ᒎ⋡ࡣ
S/A

Rt+1 = exp[ rt+1 + (M  rt+1 e)T y ―77―yT y p T
S/A+ y yW(t+1)t ]2(24)

࡛࠶ࡿ. ࡇࢀࢆࢸ࢖࣮ࣛᒎ㛤ࡋ, ఀ⸨࣮ࣝࣝࡢ‽⏝࡟ࡼࡾ
S/A

Rt+1  1 + rt+1 + (M  rt+1 e)T y +࡜㏆ఝࡍࡿ࡜, ᘧ (22) ࡣp
S/A
yT yW(t+1)t

(25)

S/A

S/At+1 = [ S/At + P (1  P ) ] · Rt+1 · (1  V )
 [ S/At + P (1  P ) ] · (1  V )×[ 1 + rt+1 + (M  rt+1 e)T y +࡜࡞ࡿ. ࡇࡇ࡛p
S/A
yT yW(t+1)t ]r

t+1
S/At+1= [ S/At + P (1  P ) ] · (1 + rt+1 ) · (1  V )p
S/A
S/Ayt+1 = [ S/At + P (1  P ) ] · [(M  rt+1 e)T y + yT yW(t+1)t ] · (1  V )

(26)

(27)

(28)

࡜ࡋ, ᘧ (22) ࢆ
r

(29)

r

(30)

t+1
S/At+1  S/At+1+ S/Ayt+1࡜⾲⌧ࡍࡿ.
ྠᵝ࡟ࡋ࡚, ᘧ (23) ࢆt+1
G/At+1  G/At+1+ G/Ax,y
t+1

࡜ࡍࡿ. ࡓࡔࡋ,r

t+1
G/At+1= [(G/At + P · 1P ) + 1V · {S/At + P (1  P )}] · (1 + rt+1 )
$
G/A
x,y
= (G/At + P · 1P ) · [(M  rt+1 e)T x + xT xW(t+1)t ]G/At+1p
S/A+1V · [S/At + P (1  P )] · [(M  rt+1 e)T y + yT yW(t+1)t ]

(31)

(32)

࡛࠶ࡿ. ࡞࠾, ᡃࠎࡀไᚚᑐ㇟࡜ࡍࡿࡶࡢࡣ୍⯡຺ᐃ࣏࣮ࢺࣇ࢛ࣜ࢜ x ࡢࡳ࡛࠶ࡿ. ≉຺ูᐃ࣏࣮ࢺࣇ࢛ࣜ࢜ yࡣไᚚᑐ㇟࡜ࡣ࡞ࡽ࡞࠸.

3.2.3

BGSS ἲ࡟ࡼࡿ ALM ࢔ࣉ࣮ࣟࢳ

ᘧ (21) ࡟ᇶ࡙ࡁ, ➨ t ᫬Ⅼ࡟࠾ࡅࡿ౯್㛵ᩘ J (t, G/At , S/At ) ࢆJ (t, G/At , S/At ) = max E[ u(FR(T)) | Ft ]

(33)

y
rt
J (t, G/At , S/At ) := f [ G/Art t + G/Ax,yt , S/At + S/At ]

(34)

1
{x+ }+T=t࡜ᐃ⩏ࡋ, ᭦࡟࡞ࡿ㛵ᩘ f ࢆᑟධࡍࡿ.

―78―

日本保険・年金リスク学会誌 Vol.4 No.1 2009

ḟ࡟
r

x,y
t+1
t
J (t + 1, G/At+1 , S/At+1 ) = f [ G/At+1+ G/At+1, S/Art+1+ S/Ayt+1 ]
r

r

t+1
t+1
ࢆᘧ (27) ཬࡧᘧ (31) ࡛ᐃ⩏ࡋࡓ G/At+1
ཬࡧ S/At+1ࡢ࿘ࡾ࡛ࢸ࢖࣮ࣛᒎ㛤ࡋ,J (t + 1, G/At+1 , S/At+1 )
rt+1
rt+1
, S/At+1]
 f [ G/At+1
(
rt+1
rt+1
f [ G/At+1, S/At+1]+G/Ax,yt+1 ·(G/At+1(
rt+1
rt+1+S/Ayt+1 ·f [ G/At+1, S/At+1](S/At+1
1
(2
rt+1
rt+1
2
+ (G/Ax,yf [ G/At+1, S/At+1]
t+1 ) ·
2
(G/A2t+1
1
(2
rt+1
rt+1
+ (S/Ayt+1 )2 ·f [ G/At+1, S/At+1]
2(S/A2t+1y+G/Ax,yt+1 · S/At+1 ·
r

(2
rt+1
rt+1
f [ G/At+1, S/At+1](G/At+1 S/At+1

(35)

r

t+1
t+1
࡜㏆ఝࡍࡿ. ௨ୗ࡛ࡣ⡆༢ࡢࡓࡵ࡟ f [ G/At+1, S/At+1] ࢆ f ࡜ࡢࡳグ㏙ࡍࡿ.

᭱㐺ᛶཎ⌮࡟ࡼࡾ

J (t, G/At , S/At ) = max E[J (t + 1, G/At+1 , S/At+1 ) | Ft ]
xt

(36)

ࡀᡂ❧ࡍࡿࡢ࡛, ᘧ (36) ࡢྑ㎶࡟ᘧ (35) ࢆ㐺⏝ࡍࢀࡤ, 1 㝵ࡢ᮲௳࠿ࡽ㏆ఝⓗ࡞᭱㐺ゎ x ࢆồࡵࡿࡇ࡜ࡀ࡛
ࡁࡿ.

x =

(
E[ (G/A
f | Ft ]1
t+1·· 1 (M  rt+1 e)
2
1(
G/At + P · P E[ (G/A2 f | Ft ]
t+1

2

(
E[ (G/At+1
S/At + P · (1  P )S/At+1 f | Ft ]
1

·
{

+
(1


)
·
}·y
V
2
V
(
G/At + P · 1PE[ (G/A
f | Ft ]
2
t+1

(37)
ࡓࡔࡋ,

=
x,y

+ (M  rt+1 e)(M  rt+1 e)T

(38)

y

࡛࠶ࡿ. ᘧ (35) ࡟࠾ࡅࡿ G/At+1 ཬࡧ S/At+1 ࡟㛵ࡍࡿ✚⋡ホ౯ࡣ Appendix ࡟♧ࡍ.
๓ᑠ⠇࡛♧ࡋࡓࠕ୍⯡ⓗ࡞ᡭἲ࡛ࠖࡣ, ᭱㐺ゎ x ࢆᚓࡿࡓࡵ࡟ࡣ౯್㛵ᩘࡢゎࢆᑟฟࡍࡿᚲせࡀ࠶ࡗࡓࡀ,BGSS ἲ࡛ࡣࡇࢀࢆせồࡋ࡞࠸. ౯್㛵ᩘ࡟ࢸ࢖࣮ࣛᒎ㛤ࢆ᪋ࡍࡇ࡜࡟ࡼࡾ, ㏆ఝⓗ᭱㐺ゎࢆ┤᥋࡟ồࡵࡿࡇ࡜ࢆྍ⬟࡜ࡋ࡚࠸ࡿࡢ࡛࠶ࡿ.
᭱㐺ゎ x ࡣ, ᘧ (17) ࡜ྠᵝ࡟, ㏆ど║ⓗᡂศ࡜␗᫬Ⅼ㛫ࡢ࣊ࢵࢪᡂศ (㈇മᵓ㐀) ࡟ࡼࡾᵓᡂࡉࢀࡿ. ࡓࡔ
ࡋ, ᘧ (17) ࡛ࡣ౯್㛵ᩘࡢ೫ᚤศ࡜ࡋ࡚୚࠼ࡽࢀࡓಀᩘࡀ, ᘧ (37) ࡛ࡣ᮲௳௜ᮇᚅ್࡟௦᭰ࡉࢀ, ㏆ど║ⓗᡂ
ศࡢせ⣲ࡀ

1

(M  rt e) ࠿ࡽ 1 (M  rt+1 e) ࡟ኚ໬ࡋ࡚࠸ࡿ.

(೫ᚤศ᪉⛬ᘧࢆゎࡁ౯್㛵ᩘࡢゎᯒゎࢆᑟฟࡍࡿࡇ࡜࡜ࡣ␗࡞ࡾ, ୕✀㢮ࡢ᮲௳௜ᮇᚅ್ E[ (G/A
f | Ft ],
t+1
2

2

(
(
f | Ft ], E[ (G/At+1
E[ (G/A2
S/At+1 f | Ft ] ࢆホ౯ࡍࡿࡇ࡜ࡣ Longsta& and Schwartz[2001] ࡢ᭱ᑠ஧஌ࣔ
t+1ࣥࢸ࢝ࣝࣟ (LSM; Least Square Monte-Carlo) ἲࡢ࢔࢖ࢹ࢔ࢆ฼⏝ࡍࢀࡤ, ẚ㍑ⓗᐜ᫆࡟⾜࠼ࡿ. ࡍ࡞ࢃࡕ,((2(2(G/At+1 f , (G/A2t+1 f , (G/At+1 S/At+1 f ࢆ┠ⓗኚᩘ, ᫬Ⅼ t ࡲ࡛ࡢ᝟ሗ࡟ᇶ࡙ࡃ≧ែኚᩘࡢ┤஺ᇶᗏࢆㄝ᫂ኚᩘ

࡜ࡋ࡚ᅇᖐࡍࡿࡢ࡛࠶ࡿ. ᮲௳௜ᮇᚅ್ࡢホ౯࡟ࡘ࠸࡚ࡣ, ᯌࠎᮌ [2007] ࡟ೌ࠸, ࢩ࣑࣮ࣗࣞࢩ࣭ࣙࣥࣃࢫࡢྛ
Ⅼ࠿ࡽ᱁Ꮚࢆᙇࡿࡇ࡜࡟ࡼࡿᑐᛂࡶ⪃࠼ࡽࢀࡿ. ࡓࡔࡋ, ᖹ‽ᡶಖ㝤ᩱࢆ๓ᥦ࡜ࡍࡿ࡜, ㈨⏘౯᱁㐣⛬ࡢ᱁Ꮚࡀ―79―෌⤖ྜࡋ࡞࠸ࡓࡵ࡟ィ⟬㈇Ⲵࡀ㧗ࡲࡿࡇ࡜࡟␃ពࡀᚲせ࡛࠶ࡿ. BGSS ἲࡣࢩ࣑࣮ࣗࣞࢩࣙࣥࢆᅵྎ࡟ࡍࡿࡇ
࡜࡛, 4 㝵ᚤศྍ⬟࡛࠶ࡿࡇ࡜ࢆ๓ᥦ࡜ࡋࡓᘧ (20) ࡬ࡢ௵ពࡢ㛵ᩘᙧࡢ㐺⏝ཬࡧ⤌ධᑐ㇟㈨⏘ࡢ௵ពࡢ཰┈⋡ศᕸ࡬ࡢᑐᛂ, ₯ᅾⓗ࡞⤒㊰౫Ꮡᛶࡢ཯ᫎ➼ࢆྍ⬟࡜ࡋ࡚࠸ࡿ.
ࡓࡔࡋ, ㏆ఝⓗᡭἲ࡛࠶ࡿࡓࡵ࡟, ㄗᕪ࡟␃ពࡍࡿᚲせࡶ࠶ࡿ. Binsbergen and Brandt[2006] ࡣ౯್㛵ᩘࢆ┤᥋࡟ゎ࠿ࡎ, ᮇᚅຠ⏝ࢆ᭱኱໬ࡍࡿ᭱㐺㈨⏘㓄ศẚࡢ㏆ఝゎࢆồࡵࡿ BGSS ἲࡣ, ≧ែኚᩘࡢከ㡯ᘧ㏆ఝ࡟ࡼࡾຠ⏝㛵ᩘࡀṇ☜࡟グ㏙࡛ࡁ࡞࠸ሙྜ࡟ࡣ, ౯್㛵ᩘࢆ㏆ఝࡍࡿ➼ࡢ௚ᡭἲ࡜ẚ㍑ࡋ࡚ㄗᕪࡢఏᦙ࡟ᕪ␗ࡀ
⏕ࡎࡿྍ⬟ᛶࢆᣦ᦬ࡋ࡚࠸ࡿ. ࡲࡓ, ᱵෆ [2008] ࡣ BGSS ἲࡢ㏆ఝㄗᕪࡀ౯್㛵ᩘࡢࢸ࢖࣮ࣛᒎ㛤࡟㉳ᅉࡍࡿࡶࡢ࡜ᣦ᦬ࡋ, ᙜヱㄗᕪࡢᢚไ⟇࡜ࡋ࡚㞳ᩓ㛫㝸ࡢ▷⦰, ࢸ࢖࣮ࣛᒎ㛤ࡢ㧗ḟ໬, ↓ࣜࢫࢡ㈨⏘཰┈⋡࡟௦᭰ࡍࡿ㏆ど║ⓗ࣏࣮ࢺࣇ࢛ࣜ࢜཰┈⋡࿘ࡾ࡛ࡢᒎ㛤➼ࢆᥦ᱌ࡋ࡚࠸ࡿ.
࡞࠾, ᘧ (37) ࠿ࡽ᫂ࡽ࠿࡞ࡼ࠺࡟, ㈇മ౯್ࡀ⤒῭⎔ቃ࡟࠾࠸࡚ఱࡽ୙☜ᐇᛶࢆཧ↷ࡏࡎ, ㈨⏘౯್࡜ࡣ⊂❧࡛࠶ࡿ࡞ࡽࡤ y ࡟ᑐᛂࡍࡿ࣊ࢵࢪᡂศࡣᏑᅾࡋ࠼࡞࠸. ูࡢ⾲⌧ࢆࡍࡿ࡞ࡽࡤ, ㏆ど║ⓗᡂศ, ࡍ࡞ࢃࡕ, ศᩓ࣏࣮ࢺࣇ࢛ࣜ࢜ࡣఱࡽ࣊ࢵࢪᶵ⬟ࢆ᭷ࡉ࡞࠸࡜࠸࠺ࡇ࡜࡛࠶ࡿ.

3.3 BGSS ἲ࡟ࡼࡿ᭱㐺㈨⏘ᵓᡂẚ๓ᥖฎ⌮ࢆ‶ᮇ᫬Ⅼ࠿ࡽ➨ 0 ᫬Ⅼࡲ࡛ᚋ㐍ᖐ⣡ⓗ࡟཯᚟ࡍࡿࡇ࡜࡛, ಖ㝤ᮇ㛫࡟ரࡿ᭱㐺ᢞ㈨ᡓ␎ࡀᑟฟࡉ
ࢀࡿ. ᘧ (37) ࡟ࡼࡾồࡵࡽࢀࡿ᭱㐺㈨⏘ᵓᡂẚ x ࢆ☜ㄆࡍࡿ.

3.3.1 (T-1) ᫬Ⅼᘧ (20) ࢆᚤศࡍࢀࡤ, ᘧ (37) ࡟࠾ࡅࡿ᮲௳௜ᮇᚅ್ࢆḟᥖࡢࡼ࠺࡟≉ᐃࡍࡿࡇ࡜ࡀ࡛ࡁࡿ.

(1
1
f | FT 1 ] = E[ 2a · { FR(T) }| FT 1 ](G/AT
2a
S/AT + H[S/AT ](22aE[f | FT 1 ] = E[ | FT 1 ]2(G/AT{ S/AT + H[S/AT ] }2

E[

E[

(39)
(40)

(2
f | FT 1 ](G/AT S/AT

= E[ 2a ·1(} · (1 + (S/A
H[S/AT ])  (1 + 1V )2{ FR(T)  4a
T

{ S/AT + H[S/AT ] }2

| FT 1 ](41)

ᮏ✏࡛ࡣ➨ t ᫬Ⅼ࡟࠾ࡅࡿ᮲௳௜ᮇᚅ್ࡢᅇᖐ࡟࠾ࡅࡿ≧ែኚᩘ࡜ࡋ࡚✚❧ẚ⋡ FR(t  t) ࢆ㑅ᢥࡋࡓ.

ࡇࢀࡣ௵ពࡢ᫬Ⅼ࡟࠾ࡅࡿ୍⯡຺ᐃ㈨⏘㢠, ≉຺ูᐃ㈨⏘㢠ཬࡧ㈐௵‽ഛ㔠㢠ࡀዎ⣙᫬Ⅼ࠿ࡽᙜヱ᫬Ⅼࡲ࡛ࡢ୙☜ᐇᛶ࡜᝟ሗࢆ⵳✚ࡋࡓ⤖ᯝ࡜ゎ㔘ࡋࡓࡇ࡜࡟ࡼࡿ.

ࡇࡇ୍࡛ࡘࡢ␲ၥࡀ⏕ࡎࡿࡔࢁ࠺. ᭱㐺ᢞ㈨ᡓ␎ࡣ‶ᮇ᫬Ⅼ࠿ࡽᚋ㐍ᖐ⣡ⓗ࡟ᑟฟࡉࢀࡿ࡟ࡶ࠿࠿ࢃࡽࡎ,ᙜヱ᫬Ⅼ࡟࠾ࡅࡿ㈨⏘౯್ࡀධຊ᝟ሗ࡜ࡋ࡚ᚲせ࡜࡞ࡗ࡚࠸ࡿ. ᙜヱ᫬Ⅼ࡟࠾ࡅࡿ㈨⏘౯್ࡣ, ዎ⣙᫬Ⅼ࠿ࡽᙜヱ᫬Ⅼࡲ࡛ࡢ᭱㐺ᢞ㈨⾜ື࡟ᇶ࡙ࡁỴᐃࡉࢀࡿࡢ࡛࠶ࡿ࠿ࡽ, ᚋ㐍ᖐ⣡ⓗ࡟࣏࣮ࢺࣇ࢛ࣜ࢜ࢆỴᐃࡍࡿ㝈ࡾ, ᙜヱ᫬Ⅼࡼࡾ㐣ཤ࡟࠾ࡅࡿ᭱㐺ᢞ㈨ᡓ␎ࡣỴᐃ࡛ࡁ࡞࠸ࡇ࡜࡟࡞ࡿ. Brandt, Goyal, Santa-Clara andStroud[2005] ࡛ࡣᙜヱၥ㢟ࢆゎᾘࡋ, ௵ពࡢ┠ⓗ㛵ᩘࢆᑐᛂྍ⬟࡜ࡍࡿࡓࡵ࡟, ㈨⏘౯್ࡢୖୗ㝈࡛᭱㐺㈨⏘ᵓᡂẚࢆồࡵࢀࡤࡼ࠸࡜ࡋ࡚࠸ࡿ.
㈨⏘౯್ࡢୖୗ㝈タᐃ࡟ࡣᗄࡘ࠿ࡢᡭἲࡀ⪃࠼ࡽࢀࡿࡀ, ୍⯡຺ᐃࡀ≉຺ูᐃ㈨⏘౯್࡟ᑐࡍࡿࣉࢵࢺ࣭
࢜ࣉࢩࣙࣥࡢᙺ๭ࢆせồࡉࢀࡿࡇ࡜࠿ࡽ, ᮏ✏࡛ࡣዎ⣙᫬௨㝆, ≉຺ูᐃ㈨⏘཰┈⋡ࡀ཯ᫎࡉࢀࡓ୍⯡຺ᐃ
S/A

G/At

r

, ྠ↓ࣜࢫࢡ฼Ꮚ⋡ࡀ཯ᫎࡉࢀࡓ୍⯡຺ᐃ G/At f , ᙜヱ᫬Ⅼ࡟࠾ࡅࡿ᭱పಖドỈ‽ t · P ࡟ᑐࡋ, ୖ㝈G/Amaxཬࡧୗ㝈 G/Aminࢆḟᥖࡢࡼ࠺࡟タᐃࡋࡓ.
t
t

S/A

G/Amax= max[ t · P , G/Att

, G/Atrr f ] · (1 + k)

S/A= min[ G/At , G/Atrr f ] ÷ (1 + k)
G/Amin
t

―80―

(42)
(43)

日本保険・年金リスク学会誌 Vol.4 No.1 2009ࡇࡇ࡛ k ࡣ⠊ᅖࢆಖᏲⓗ࡟タᐃࡍࡿࡓࡵࡢ࣐࣮ࢪ࡛ࣥ 10% ࡜ࡋ࡚࠸ࡿ.
ᙜヱ G/Amaxཬࡧ G/Amin࡟ᑐࡍࡿ௬ࡢ᭱㐺㈨⏘ᵓᡂẚࢆᚋ㐍ᖐ⣡ⓗ࡟ồࡵ࡚࠾ࡁ, 0 ᫬Ⅼ࠿ࡽ๓㐍ⓗ࡟ᢞtt㈨ᡓ␎ࢆᐇ⾜ࡍࡿࡇ࡜࡟ࡼࡾ஦ᚋⓗ࡟ุ᫂ࡍࡿ G/At ࡟ᑐࡋ࡚⿵㛫ฎ⌮ࢆ᪋ࡋ, ᭱⤊ⓗ࡞᭱㐺㈨⏘ᵓᡂẚ x ࢆỴᐃࡋ࡚࠸ࡿ*12 .

3.3.2 (T-2) ᫬Ⅼྠᵝ࡟ࡋ࡚,((
G/A
f | FT 2 ] = E[
f · RT | FT 2 ](G/AT 1
(G/AT
(2
(2
G/A
E[
f
|
F
]
=
E[
f · (RT )2 | FT 2 ]T
2
(G/A2T 1(G/A2T

(44)

E[

E[

(45)

(2
f | FT 2 ](G/AT S/AT
(2
G/A
S/A
f · RT · RT · 1V · (1  V )= E[
(G/A2T(2
G/A
S/A
f · RT · RT · (1  V ) | FT 2 ](G/AT (S/AT

+

2

(46)
2

(
(
(
ࢆᚓࡿ. ྛᚤศಀᩘࡢᒎ㛤ࡣ Appendix ࡟♧ࡍ. ྑ㎶࡟⌧ࢀࡿᚤศಀᩘ (G/A
f , (G/A
2 f ཬࡧ (G/A S/A f
T
T
T
T

G/A

ࡣ, ᘧ (39), ᘧ (40) ཬࡧᘧ (41) ࢆཧ↷ࡉࢀࡓ࠸. ࡞࠾, ୖᥖᘧ࡟࠾ࡅࡿ RT

ࡣ (T  1) ᫬Ⅼࡢฎ⌮࡛ᚓࡽࢀࡓ᭱㐺㈨⏘ᵓᡂẚ xT1 ࡟ᇶ࡙ࡃ୍⯡຺ᐃ㐍ᒎ⋡࡛࠶ࡿ. ᭱㐺㈨⏘ᵓᡂẚࢆᚋ㐍ᖐ⣡ⓗ࡟ゎ࠸࡚࠸ࡃࡓࡵ,(T  2) ᫬Ⅼ࡛ࡣ᪤▱࡜࡞ࡿ. ᥮ゝࡍࢀࡤ, ᑗ᮶࡟࠾࠸࡚᭱㐺ᢞ㈨⾜ືࢆ࡜ࡿࡇ࡜ࢆ๓ᥦ࡜ࡋ࡚, ᙜヱ᫬Ⅼࡢ᭱㐺ᢞ㈨ᡓ␎ࢆỴᐃࡋ࡚࠸ࡿࡢ࡛࠶ࡿ.

3.3.3 t ᫬Ⅼ (t ; [ 0, T  3 ])᮲௳௜ᮇᚅ್ࡣ

T

E[

Y G/A
(
(
f | Ft ] = E[f ·(Rl ) | Ft ](G/At+1
(G/AT

(47)

l=t+2
T

E[

Y G/A
(2
(2
f
|
F
]
=
E[
f
·
(
Rl )2 | Ft ]t
(G/A2t+1(G/A2T

(48)

l=t+2

E[

(2
f | Ft ](G/At+1 S/At+1
T

Y G/A
(2
f ·(
Rl ) · 1V
2
(G/AT

= E[

l=t+2

×[

T
Y

S/A

Rl

l=t+2

+

· (1  V )T (t+1) +(2(G/AT (S/AT

f ·(

T
Y

T
1
X

l
Y

l=t+2 m=t+2

l=t+2

G/A

Rl

)·(

T
Y

S/A
Rm
T
Y

l=t+2

o=l+1
S/A

Rl

RoG/A (1  V )l(t+1) ]) · (1  V )T (t+1) | Ft ](49)

࡛࠶ࡿ. ࡞࠾, 0 ᫬Ⅼ࡛ࡣ≧ែኚᩘ࡟ࡘ࠸ุ࡚᫂ࡋ࡚࠸ࡿ᝟ሗࡀᏑᅾࡋ࡞࠸. ᚑࡗ࡚ᮇᚅ್ࡣ၏୍࡜࡞ࡾ, ᭱㐺㈨⏘ᵓᡂẚ x0 ࡶ၏୍࡟≉ᐃࡉࢀࡿ. ࡇࢀ࡟ᑐࡋ, t ᫬Ⅼ (t ; [t, T  t]) ࡟࠾ࡅࡿ᭱㐺㈨⏘ᵓᡂẚࡣྛ≧ែ࡟ᑐᛂࡋ࡚タᐃࡉࢀࡿ.

*12 ஧Ⅼࡢ᭱㐺໬࣏࣮ࢺࣇ࢛ࣜ࢜ xmin ཬࡧ xmax ࡟ᣳࡲࢀࡿࠕ┿ࡢࠖ᭱㐺໬࣏࣮ࢺࣇ࢛ࣜ࢜ x

ࢆ, xmin ཬࡧ xmax ࡢ⿵㛫ฎ⌮࡛⾲⌧࡛ࡁࡿ࡜ࡍࡿࡇ࡜ࡣᙉ࠸௬ᐃ࡛࠶ࡾ, ࡜ࡾࢃࡅ᫬㛫㛫㝸ࡀ኱ࡁ࡞ሙྜ࡟ࡣㄗᕪࡢ⣼✚ࡍࡿྍ⬟ᛶࡀ࠶ࡿ.

―81―

2

(
ᘧ (41), ᘧ (46) ཬࡧᘧ (49) ࡢᵓᡂせ⣲࡛࠶ࡿ (G/AT( (S/AT f ࡢศᏊ࡟࠶ࡿ (S/A
H[S/AT ] ࡣ, ᭱పಖドࣜTࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠ࡢ≉຺ูᐃ㈨⏘౯᱁࡟ᑐࡍࡿឤᛂᗘ, ᡤㅝࢹࣝࢱ࡛࠶ࡿ. ௨ୖࡼࡾ୍⯡຺ᐃࡢ᭱㐺ᢞ㈨(H[S/AT ]) ࡢ㛵ಀࢆ୍ពᡓ␎ (x ) ࡜≉຺ูᐃᢞ㈨ᡓ␎ (y), ಖドᩱ (P ཬࡧ V ), ㈐௵‽ഛ㔠ࡢࢹࣝࢱ ( (S/AT࡟≉ᐃ࡛ࡁࡓࡇ࡜࡟࡞ࡿ.

4 ᩘ್ゎᯒ⤖ᯝ
4.1 ๓ᥦ᮲௳
4.1.1 ⤒῭๓ᥦ
ᢞ㈨ᑐ㇟㈨⏘ࡢ౯᱁㐣⛬࡟ࡣᑐᩘṇつ㐣⛬, ࡍ࡞ࢃࡕ, ཰┈⋡ศᕸ࡟ṇつศᕸࢆ௬ᐃࡋ, ࣃ࣓࣮ࣛࢱࡣ⾲ 2 ࡟♧ࡍ࡜࠾ࡾ࡜ࡋࡓ.
⾲2ᮇᚅ཰┈⋡, ᶆ‽೫ᕪཬࡧ┦㛵ಀᩘ

㈨⏘ࢡࣛࢫ

ᮇᚅ཰┈⋡

ᶆ‽೫ᕪ

▷ᮇ㈨⏘

0.57%

ᅜෆമๆ

┦㛵ಀᩘ
▷ᮇ㈨⏘

ᅜෆമๆ

ᅜෆᰴᘧ

እᅜമๆ

እᅜᰴᘧ

0.70%

1.000

0.143

2.47%

3.34%

0.143

1.000

40.06340.00940.009

ᅜෆᰴᘧ

4.42%

19.25%
10.20%

40.029

0.365

4.28%

40.053

1.000

እᅜമๆ

40.063

1.000

0.540

እᅜᰴᘧ

4.88%

16.65%

40.009

0.017

0.540

1.000

40.009(ฟ඾) ▼஭࣭኱ሯ [2008].

0.086

40.05340.029
0.365

0.086

0.017

ࡲࡓ, ᑗ᮶ࡢணᐃ฼⋡㐣⛬ࢆグ㏙ࡍࡿࡓࡵ࡟ᚲせ࡜࡞ࡿ฼Ꮚ⋡ࣔࢹࣝ࡟ࡣ 1 ኚ㔞ࡢ Vasicek Modeldr =[r̄  r] dt +  dWtࢆ㐺⏝ࡋ, 2008 ᖺ 8 ᭶ 1 ᪥ࡢ㔠฼࣭മๆࢹ࣮ࢱ࡟ࡼࡾࣃ࣓࣮ࣛࢱ᥎ᐃࢆ⾜ࡗࡓ. ⤖ᯝ࡜ࡋ࡚,࡟ 0.08210, r̄࡟ 0.03844,  ࡟ 0.00585 ࢆᚓࡓ. ࡞࠾, ⾲ 2 ࡟࠾ࡅࡿമๆ౯᱁ࡣᙜヱ฼Ꮚ⋡ࣔࢹࣝ࡟ࡼࡾグ㏙ࡉࢀࡿࡶࡢ࡛ࡣ
࡞࠸ࡀ, ᣑᩓ㡯ࢆᵓᡂࡍࡿࣈࣛ࢘ࣥ㐠ືࢆඹ᭷ࡉࡏࡿࡇ࡜࡟ࡼࡾ, ฼Ꮚ⋡࡜മๆ౯᱁ࡢ㛵ಀࢆᩚྜࡉࡏ࡚࠸ࡿ.

4.1.2 ≉຺ูᐃ㈨⏘ᵓᡂẚᯫ✵ࡢ⏕࿨ಖ㝤఍♫࡟࠾ࡅࡿ≉຺ูᐃ㈨⏘ᵓᡂẚ࡜ࡋ࡚, ᖺ㔠✚❧㔠⟶⌮㐠⏝⊂❧⾜ᨻἲேࡢ 2009 ᖺ 6 ᭶⌧ᅾ࡟࠾ࡅࡿᇶᮏ࣏࣮ࢺࣇ࢛ࣜ࢜ࢆ㐺⏝ࡋࡓ.
⾲3≉຺ูᐃࡢ㈨⏘ᵓᡂẚ

▷ᮇ㈨⏘

ᅜෆമๆ

ᅜෆᰴᘧ

እᅜമๆ

እᅜᰴᘧ

5%

67%

11%

8%

9%

―82―

日本保険・年金リスク学会誌 Vol.4 No.1 2009

4.1.3 ㈐௵‽ഛ㔠ホ౯ࡢィ⟬ᇶ♏
ዎ⣙㛤ጞ᫬Ⅼࡢணᐃ฼⋡ࡣ 1.5%, ᑗ᮶࡟࠾ࡅࡿணᐃ฼⋡ࡣ๓㏙ࡢ࡜࠾ࡾ, ᙜヱ᫬Ⅼ࡟࠾ࡅࡿ 10 ᖺ฼௜മๆ
ࡢࣃ࣮࣭ࢡ࣮࣏ࣥࢆ㐺⏝ࡋࡓ. ᭱పಖド࡟ಀࡿ㈐௵‽ഛ㔠ホ౯࡟㝿ࡋ㐺⏝ࡉࢀࡿ㈨⏘౯᱁ࡢ࣎ࣛࢸ࢕ࣜࢸ࢕ࡣ
⾲ 4 ࡢ࡜࠾ࡾ࡜ࡋ, ྛ㈨⏘཰┈⋡ࡣ⊂❧࡜ࡋࡓ. ᙜヱෆᐜࡣ♫ᅋἲே᪥ᮏ࢔ࢡࢳࣗ࢔࣮ࣜ఍ [2003], ᖹᡂ 8 ᖺ 2
᭶ 29 ᪥኱ⶶ┬࿌♧➨ 48 ྕ࡟‽ᣐࡋ࡚࠸ࡿ.
⾲4㈨⏘౯᱁ࡢண᝿ኚື⋡

ᅜෆമๆ

ᅜෆᰴᘧ

እᅜമๆ

እᅜᰴᘧ

3.5%

18.4%

12.1%

18.1%

⢒Ṛஸ⋡ࡣ 1 ᅉᏊࡢ࣮࣭࣮ࣜ࢝ࢱ࣮࣭ࣔࢹࣝ

ln( qx,t ) =x + #x t + x,t(50)

࡟ࡼࡾグ㏙, ㈐௵‽ഛ㔠ホ౯࡟㐺⏝ࡍࡿᶆ‽Ṛஸ⋡ࡣᙜヱ⢒Ṛஸ⋡࡟ᑐࡋᏳ඲๭ቑࢆィୖࡍࡿࡶࡢ࡜ࡋ࡚タᐃ
ࡋࡓ. ⢒Ṛஸ⋡࡟ࡘ࠸࡚ࡣ⡆᫆⏕࿨⾲ࢆධຊ᝟ሗ, 1987 ᖺ࠿ࡽ 2005 ᖺࢆほᮇ㛫࡜ࡋࡓࣃ࣓࣮ࣛࢱ᥎ᐃࢆ≉2␗್ศゎ࡟ࡼࡾ⾜ࡗࡓ. ㄗᕪ㡯ࡣᖺ㱋ཬࡧ᫬Ⅼ࡟౫Ꮡࡋ࡞࠸ṇつศᕸ N (0,  ) ࢆ㐺⏝ࡋ,  ࡢ᥎ᐃ⤖ᯝ࡜ࡋ࡚

0.03456 ࢆᚓࡓ.

࡞࠾, Ᏻ඲๭ቑࡣ 2005 ᖺ⡆᫆⏕࿨⾲࡟ᑐࡍࡿ⏕ಖᶆ‽⏕࿨⾲ 2007(Ṛஸಖ㝤⏝) ࡢẚࡀ୍ᐃ࡛࠶ࡿࡶࡢ࡜௬
ᐃࡋ, ᙜヱᖺ㱋ูಀᩘࢆ⢒Ṛஸ⋡࡟஌ࡎࡿࡇ࡜࡛ᶆ‽Ṛஸ⋡ࢆᐃ⩏ࡋࡓ. ᚑࡗ࡚, ᶆ‽⏕࿨⾲࡟࠾ࡅࡿᏳ඲๭ቑ
ࡣ⢒Ṛஸ⋡ࡢỈ‽࡟ẚ౛ࡍࡿࡇ࡜࡟࡞ࡾ, ⢒Ṛஸ⋡ࡀపୗࡍࢀࡤᏳ඲๭ቑࡀᅽ⦰ࡉࢀࡿࡇ࡜࡟࡞ࡿ.
ࡲࡓ, ฎ⌮࡟࠾࠸࡚ࡣ, Ṛஸ⋡࡜฼Ꮚ⋡ཬࡧᢞ㈨㈨⏘཰┈⋡ࡢ⊂❧ᛶࢆ๓ᥦ࡜ࡋ࡚࠸ࡿ.
⾲5

Lee-Carter Model ࡟࠾ࡅࡿࣃ࣓࣮ࣛࢱ

ᖺ㱋

0
10
20
30
40

#

ᖺ㱋

4 5.51010.020850

4 8.9957

0.022260

4 7.2709

0.016170

4 7.1823

0.001580

4 6.5086

0.0028

90

ᖺᩘ



ᖺᩘ

1987

11.8330

1997

1988

11.9430

1998

1989

10.2490

1999

1990

9.99792000

1991

7.3512

2001

1992

7.8563

2002

1993

5.7392

2003

1994

4.3411

2004

1995

7.0338

2005

1996

4

0.3937―83―#
4 5.5243

0.0055

4 3.6958

0.00814 1.6839

0.0105

4 4.5930

0.01014 2.6523

0.0126
42.474042.417549.79754

0.408846.38784

12.1780

414.80404

13.6380413.8480

4.1.4 ዎ⣙ෆᐜࣔࢹࣝ࡟࠾ࡅࡿಖ㝤ዎ⣙ࡢෆᐜࡣḟᥖࡢ࡜࠾ࡾ࡜ࡋࡓ.
⾲6ホ౯ᑐ㇟ࡢዎ⣙ෆᐜ⿕ಖ㝤⪅ᖺ㱋

30 ṓ

ᛶู

⏨ᛶ

ಖ㝤ᮇ㛫

10 ᖺ㛫

ಖ㝤ᩱ

10,000 ෇ (ᖺᡶ)᭱పಖドෆᐜ

⿕ಖ㝤⪅ࡢṚஸ᫬ⱝࡋࡃࡣ‶ᮇ฿㐩᫬࡟᪤ᡶ㎸ಖ㝤ᩱ┦ᙜ㢠ࢆಖド.

ಖドᩱ, ணᐃ஦ᴗ㈝ཬࡧಙクሗ㓘1P + 2P = 5%
1V + 2V = 2% , 3V = 0.2%

4.1.5 ࡑࡢ௚ࡢὀグ஦㡯㛵ᩘ u ࡢࣃ࣓࣮ࣛࢱ

2u(FR(T)) = FR(T)  a · FR(T) ࡟࠾࠸࡚, a ࡣ 0.5 ࡜ࡋࡓ.

࢟ࣕࢵࢩࣗࣇ࣮ࣟࡢࢱ࢖࣑ࣥࢢ

༢఩᫬㛫ࢆ n ศ๭ࡍࡿሙྜ, ಖ㝤ᩱࡢᡶ㎸ࡣᮇጞ, ✚❧㔠࠿ࡽࡢಖドᩱྲྀᔂࡣᮇᮎ࡛Ⓨ⏕ࡍࡿࡶࡢ࡜ࡋࡓ.
ᇶᗏ㛵ᩘ᮲௳௜ᮇᚅ್ホ౯ࢆᅇᖐ࡟ࡼࡾ⾜࠺ሙྜࡢㄝ᫂ኚᩘ࡜࡞ࡿᇶᗏ㛵ᩘࢆᵓᡂࡍࡿ┤஺ከ㡯ᘧ࡟ࡣ, 5ḟࡢ Laguerre ከ㡯ᘧࢆ㐺⏝ࡋࡓ.
༢఩᫬㛫࠶ࡓࡾࡢヨ⾜ᅇᩘಖ㝤ᮇ㛫඲య࡛ࡣࠕ(ಖ㝤ᮇ㛫)  (ศ๭ᩘ)  (≧ែᩘ)ࠖࡢࢢࣜࢵࢻࢆᣢࡘࡇ࡜࡟

࡞ࡿ. ᮏ✏࡛ࡣ༢఩᫬㛫࠶ࡓࡾࡢ≧ែᩘ, ࡍ࡞ࢃࡕ, ヨ⾜ᅇᩘࢆ 1,000 ᅇ࡜ࡋࡓ.
ࢯࣇࢺ࢙࢘࢔

ᩘ್ゎᯒࡣ Compaq Visual Fortran ࡛⾜࠸, ᅇᖐศᯒ࡟ࡣ IMSL ᩘ್ィ⟬ࣛ࢖ࣈࣛࣜࡢDLSBRR ࢆ㐺⏝ࡋࡓ.
ࣁ࣮ࢻ࢙࢘࢔

ࣉࣟࢭࢵࢧࡣ Intel Pentium4 CPU 3.40GHz, ࣓ࣔࣜࡣ 4.0GB, OS ࡣ Windows VISTA ࡛

࠶ࡿ.

―84―

日本保険・年金リスク学会誌 Vol.4 No.1 2009

4.2 ᩘ್ゎᯒ⤖ᯝ
4.2.1 ಖドᩱỈ‽ࡢ༑ศᛶಖドᩱ࡟ಖ㝤ᩱẚ౛㒊ศ࣭✚❧㔠ẚ౛㒊ศ➼ࡢࡼ࠺࡟」ᩘࡢỴᐃᇶ‽ࢆ୚࠼ࡿሙྜ, ྛᇶ‽࡟ᑐᛂࡍࡿಖドᩱࢆ୍ព࡟≉ᐃࡍࡿࡇ࡜ࡣᅔ㞴࡛࠶ࢁ࠺. ᐇ㊶ⓗ࡞ᑐᛂࡢ୍ࡘ࡜ࡋ࡚ࡣ, ணࡵᐃࡵࡽࢀࡿண⟬ไ⣙ࡢ⠊ᅖෆ࡛ྛᇶ‽࡟ᑐᛂࡍࡿಖドᩱࢆ୚࠼, ዎ⣙᫬Ⅼࡢ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠㢠ࢆ☜ㄆࡍࡿࡇ࡜ࡀ⪃࠼ࡽࢀࡿ.
ᙜヱ㔠㢠ࡀṇ್ࢆ࡜ࡿ࡞ࡽࡤ, ࡑࢀࡣ⤥௜⌧౯ࡀಖ㝤ᩱ཰ධ⌧౯ࢆୖᅇࡗ࡚࠸ࡿ࡜࠸࠺ࡇ࡜࡛࠶ࡾ, ಖドᩱࡣ༑ศ࡛ࡣ࡞࠸࡜⪃࠼ࡽࢀࡿࡓࡵ࡛࠶ࡿ. ᅗ 2 ࡣᶓ㍈࡟ಖ㝤ᩱẚ౛ࡢಖドᩱ, ⦪㍈࡟✚❧㔠ẚ౛ࡢಖドᩱࢆ࡜ࡾ, ዎ⣙᫬㈐௵‽ഛ㔠㢠ࡀṇ್࡜࡞ࡿ⠊ᅖࢆ⥙࠿ࡅࡋࡓࡶࡢ࡛࠶ࡿ.
DEX
ࡓࡔࡋ, ዎ⣙᫬Ⅼ࡟࠾ࡅࡿ㈐௵‽ഛ㔠ホ౯࡟࠾࠸࡚ࡣ, ཧ↷㈨⏘࡜࡞ࡿ≉຺ูᐃ S/AIN
ࢆᵓᡂࡍࡿせ⣲tࡣᖹᡂ 8 ᖺ 2 ᭶ 29 ᪥኱ⶶ┬࿌♧➨ 48 ྕ࡟つᐃࡉࢀࡿ㈨⏘౯᱁ࡢ࣎ࣛࢸ࢕ࣜࢸ࢕ཬࡧᮇᚅ཰┈⋡ࡢࡳ࡛࠶ࡾ,║๓ཬࡧᑗ᮶ࡢ⤒῭⎔ቃࢆ᝿ᐃࡋࡓࡶࡢ࡛ࡣ࡞࠸ࡇ࡜࡟␃ពࡀᚲせ࡛࠶ࡿ. タᐃࡉࢀࡓಖドᩱࡢ⤌ࡳྜࢃࡏࡀ
ᅗ 2 ࡢⓑ࠸㡿ᇦ࡟఩⨨࡙ࡅࡽࢀࡿࡶࡢ࡛࠶ࡗࡓ࡜ࡋ࡚ࡶ, ᢞ㈨㈨⏘ࡢ⌧ᐇࡢ཰┈⋡ཬࡧ࣎ࣛࢸ࢕ࣜࢸ࢕ࡀ㈐௵
‽ഛ㔠ࡢィ⟬ᇶ♏ࡼࡾ஋㞳ࡍࡿሙྜ, ࡲࡓ, 㐺ษ࡞ᢞ㈨ᡓ␎ࡀᇳ⾜ࡉࢀ࡞࠸ሙྜ➼࡟ࡣ, ᙜヱಖ㝤ዎ⣙ࡢ㈈ᨻ≧ἣࡣᑗ᮶࡟࠾࠸࡚ᮃࡲࡋࡃ࡞࠸≧ែ࡟㝗ࡿྍ⬟ᛶࢆྰᐃ࡛ࡁ࡞࠸࡜࠸࠺ࡇ࡜࡛࠶ࡿ.

ᅗ2ዎ⣙᫬㈐௵‽ഛ㔠࡜ಖドᩱỈ‽

4.2.2 ᭱㐺ᢞ㈨ᡓ␎ࡢィ⟬౛‶ᮇ᫬Ⅼࡢ༢఩᫬Ⅼ๓ (T  t ᫬) ࡟࠾࠸࡚, ᭱㐺㈨⏘ᵓᡂẚࢆᣢࡘ୍⯡຺ᐃ࡜≉຺ูᐃᇶᮏ࣏࣮ࢺࣇ࢛ࣜ

࢜ࢆࡑࢀࡒࢀࡢ㈨⏘つᶍ࡛ຍ㔜ᖹᆒࡋࡓ඲຺ᐃ㈨⏘ᵓᡂẚࢆᅗ 3 ࡟♧ࡍ. ࡓࡔࡋ, ༢఩᫬㛫ࡢศ๭ᩘࡣ 50 ࡛࠶ࡾ, ᶓ㍈ࡣ᭱పಖドỈ‽࡟ᑐࡍࡿ≉຺ูᐃࡢ඘㊊⋡ࡢ 5% ࢱ࢖್ࣝ, 10%-90% ࢱ࢖್ࣝ, 95% ࢱ࢖್ࣝࢆ㐺⏝
ࡋ࡚࠸ࡿ. ఱࢀࡢ㈨⏘ᵓᡂẚࡶྜィࡣ 1.0 ࡜࡞ࡿࡀ, ᭱పಖドỈ‽࡟ᑐࡍࡿ≉຺ูᐃࡢ඘㊊⋡ࡀ 1.0 ࢆ㉸࠼ࡿ⠊ᅖ (50% ࢱ࢖್ࣝ௨㝆) ࡛ࡣྛᢞ㈨㈨⏘ࡢᵓᡂẚࡀ [0, 1] ࡛࠶ࡿࡇ࡜࡟ᑐࡋ, ඘㊊⋡ࡀ 1.0 ࢆୗᅇࡿ⠊ᅖ࡛ࡣ▷ᮇ㈨⏘ࡢᵓᡂẚࡀ㈇್࡜࡞ࡾ, ௚㈨⏘ࡢᵓᡂẚࡀ 1.0 ࢆ㉸㐣ࡋ࡚࠸ࡿ. ࡘࡲࡾ, ▷ᮇ㈨⏘ࢆ኎ࡾᘓ࡚ㄪ㐩ࡋࡓ㈨

―85―

ᅗ3

඲຺ᐃࡢ㈨⏘ᵓᡂẚ (T  t ᫬Ⅼ)㔠ࢆ௚㈨⏘࡟㓄㈿ࡍࡿࡇ࡜࡛ࣞࣂࣞࢵࢪࢆ࠿ࡅ࡚࠸ࡿ. ᙜヱ㈨⏘ᵓᡂẚ࡟ᑐࡍࡿࣜࢫࢡᐤ୚ᗘ*13 ࡣᅗ 4 ࡢ࡜࠾ࡾ࡜࡞ࡿ. ᭱పಖドỈ‽࡟ᑐࡍࡿ≉຺ูᐃࡢ඘㊊⋡ࡀ㧗ࡃ࡞ࡿ࡟క࠸, ඲຺ᐃ࣏࣮ࢺࣇ࢛ࣜ࢜཰┈⋡ࡢᶆ‽೫

ᅗ4඲຺ᐃࡢࣜࢫࢡᐤ୚ᗘ (T  t ᫬Ⅼ)ᕪࡢపୗࡍࡿᵝᏊࡀ☜ㄆ࡛ࡁࡿ.

*13 ᙜヱ࣏࣮ࢺࣇ࢛ࣜ࢜཰┈⋡ࡢᶆ‽೫ᕪࢆᢞ㈨ᑐ㇟㈨⏘ẖ࡟ศゎࡋࡓࡶࡢ.

―86―

日本保険・年金リスク学会誌 Vol.4 No.1 2009

ᅗ5඲຺ᐃࡢࣜࢫࢡᵓᡂẚ (T  t ᫬Ⅼ)ࡲࡓ, ࣜࢫࢡᐤ୚ᗘࢆᇶ‽໬ࡋ࡚ぢࢀࡤ, ࡝ࡢᢞ㈨㈨⏘࡟ࣜࢫࢡ㓄ศࢆ⾜ࡗ࡚࠸ࡿ࠿ࢆ☜ㄆࡍࡿࡇ࡜ࡶ࡛ࡁ
ࡿ. ᭱పಖドỈ‽࡟ᑐࡍࡿ≉຺ูᐃࡢ඘㊊⋡ࡀప࠸⠊ᅖ࡛ࡣᅜෆമๆ࡬ࡢࣜࢫࢡ㓄ศࡀ㧗ࡃ࡞ࡿࡀ, ඘㊊⋡ࡢୖ᪼࡟క࠸, ෆእᰴᘧ࡬ࡢࣜࢫࢡ㓄ศࡀቑຍࡋ࡚࠸ࡿ.
඲຺ᐃ࣏࣮ࢺࣇ࢛ࣜ࢜཰┈⋡ࡢᶆ‽೫ᕪࢆᶓ㍈, ྠᮇᚅ್ࢆ⦪㍈࡟ࣉࣟࢵࢺࡋࡓ⤖ᯝࡀᅗ 6 ࡛࠶ࡿ. ✚❧ẚ
⋡ FR(t) ࡢపୗ࡟క࠸, ඲຺ᐃࡢᵓ㐀ࡣ≉຺ูᐃᇶᮏ࣏࣮ࢺࣇ࢛ࣜ࢜࡟ẚ㍑ࡋ, ᮇᚅ್࣭ᶆ‽೫ᕪ࡜ࡶ࡟㧗ࡃ࡞ࡗ࡚࠸ࡿ. ࡇࢀࡣ᪤㏙ࡢ࡜࠾ࡾ, ୍⯡຺ᐃ࡛ࣞࣂࣞࢵࢪࢆ࠿ࡅࡿࡓࡵ࡛࠶ࡿ.

ᅗ6࣏ࢪࢩࣙࣥࡢẚ㍑―87―

4.2.3 ᭱㐺ᢞ㈨ᡓ␎ࡢ⤖ᯝ✚❧㔠ẚ౛㒊ศཬࡧಖ㝤ᩱẚ౛㒊ศࡢಖドᩱࢆࡑࢀࡒࢀ 1% ࡜ࡋ, ୍⯡຺ᐃࢆከᮇ㛫ືⓗ᭱㐺໬ᡓ␎࡟ᇶ࡙
ࡁ㐠⏝ࡍࡿࡶࡢ࡜ࡍࡿ. ᙜヱಖドᩱࡢ⤌ࡳྜࢃࡏࡣ, ᅗ 2 ࡢ⥙࠿ࡅ㒊ศ࡟఩⨨ࡍࡿࡶࡢ࡛࠶ࡾ, ዎ⣙᫬㈐௵‽ഛ㔠ࢆࢮࣟ࡜࡛ࡁࡿỈ‽࡛ࡣ࡞࠸. ࡇࡢ࡜ࡁ, ‶ᮇ᫬Ⅼ࡟࠾ࡅࡿ✚❧ẚ⋡ࡀྛ≧ែ࡟ᑐᛂࡋ࡝ࡢࡼ࠺࡟ศᕸࡍࡿ࠿ࢆྍど໬ࡋࡓࡶࡢࡀᅗ 7 ࠿ࡽᅗ 12 ࡛࠶ࡿ.
ྛᅗࡣࠕ≉຺ูᐃ㈨⏘ᵓᡂẚ࡟ᇶ࡙ࡁ㐠⏝ࡋࡓ୍⯡຺ᐃ㈨⏘㢠࡜≉຺ูᐃ㈨⏘㢠ࡢྜィ㢠ࢆ᪤ᡶ㎸ಖ㝤ᩱ࡛ᇶ‽໬ࡋࡓ್ࠖࢆᶓ㍈࡟࡜ࡾ, ⦪㍈࡟ࠕከᮇ㛫ືⓗ᭱㐺໬ᡓ␎࡟ᇶ࡙ࡁ㐠⏝ࡋࡓ୍⯡຺ᐃ㈨⏘㢠࡜≉຺ูᐃ㈨⏘㢠ࡢྜィ㢠ࢆ᪤ᡶ㎸ಖ㝤ᩱ࡛ᇶ‽໬ࡋࡓ್ࠖࢆ ( ), ࠕ≉຺ูᐃ㈨⏘ᵓᡂẚ࡟ᇶ࡙ࡁ㐠⏝ࡋࡓ୍⯡຺ᐃ㈨⏘㢠࡜≉຺ูᐃ㈨⏘㢠ࡢྜィ㢠ࢆ᪤ᡶ㎸ಖ㝤ᩱ࡛ᇶ‽໬ࡋࡓ್ࠖཬࡧࠕ᭱పಖドỈ‽ࠖࢆ (·) ࡛ࣉࣟࢵࢺࡋ࡚࠸ࡿ.
ྛᅗࡢᕪ␗ࡣ༢఩᫬㛫࠶ࡓࡾࡢศ๭ᩘ࡛࠶ࡿ. ࡍ࡞ࢃࡕ, 1 ᖺ㛫࡟࠾ࡅࡿࣜࣂࣛࣥࢫᅇᩘࡢᕪ␗ࡀᢞ㈨ຠᯝ࡟ཬ
ࡰࡍᙳ㡪ࢆ☜ㄆࡍࡿࡇ࡜ࡀ࡛ࡁࡿ. 㧗ࠎᖺ 2 ᅇ⛬ᗘࡢࣜࣂࣛࣥࢫ (ᅗ 7 ཬࡧᅗ 8) ࡛ࡣ, ᭱పಖドỈ‽ࢆ㐠⏝ᡂᯝ࡛☜ಖ࡛ࡁࡑ࠺࡟ࡣ࡞࠸ࡀ, ᖺ 4 ᅇࡢࣜࣂࣛࣥࢫ (ᅗ 9) ௨㝆࡛ࡣᚎࠎ࡟᭱పಖドỈ‽࡟㏣㝶ࡋ, ᪥ḟᇶ‽ (ᅗ12) ࡛ࡣពᅗࡍࡿ࣌࢖࢜ࣇࢆᴫࡡ෌⌧࡛ࡁ࡚࠸ࡿ.
⾲ 7 ࡣ‶ᮇ᫬Ⅼ࡟࠾ࡅࡿ✚❧ẚ⋡ࡢᮇᚅ್, ᶆ‽೫ᕪ, ୗ᪉✚⋡ࢆࠕ୍⯡຺ᐃࢆከᮇ㛫ືⓗ᭱㐺໬ᡓ␎࡟ᇶ࡙
ࡁ㐠⏝ࡋࡓሙྜࠖࢆᕥഃ, ࠕ୍⯡຺ᐃࢆ≉຺ูᐃᇶᮏ࣏࣮ࢺࣇ࢛ࣜ࢜࡟ᇶ࡙ࡁ㐠⏝ࡋࡓሙྜ*14 ࠖࢆྑഃ࡟♧ࡋࡓࡶࡢ࡛࠶ࡿ. ୍⯡຺ᐃ࡟࠾ࡅࡿ᭱పಖドࡣ, ≉຺ูᐃ㈨⏘ࡀᡶ㎸ಖ㝤ᩱ㢠ࢆୗᅇࡿሙྜ࡟ࡢࡳ⾜ࢃࢀࡿࡢ࡛
࠶ࡿ࠿ࡽ, ᢞ㈨ᡓ␎ࡢຠᯝࡣୗ᪉✚⋡࡟ࡼࡾホ౯ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࡔࢁ࠺. ఱࡽ࣊ࢵࢪᡓ␎ࢆ᥇ࡽ࡞࠸ሙྜ࡟
ẚ㍑ࡋ࡚, ㏆ఝⓗ࡞ࡀࡽࡶ୍⯡຺ᐃ࡛᭱㐺໬ᡓ␎ࢆ᥇ࡗࡓ⤖ᯝࡣୗ᪉✚⋡, ࡍ࡞ࢃࡕ, ✚❧୙㊊ࡀⓎ⏕ࡋࡓሙྜࡢᮇᚅᦆኻ㢠ࡣᅽ⦰ࡉࢀ࡚࠸ࡿ. ࡲࡓ, ࣜࣂࣛࣥࢫࢆ᭶ḟᇶ‽࡛⾜࠺ሙྜ࡟ࡣ, ࣊ࢵࢪᡓ␎ࢆ᥇ࡽ࡞࠸ሙྜࡢ 4๭⛬ᗘ, 㐌ḟᇶ‽࡛⾜࠺ሙྜ࡟ࡣྠ 2 ๭⛬ᗘ, ᪥ḟᇶ‽࡛⾜࠺ሙྜ࡟ࡣྠ 1 ๭⛬ᗘࡲ࡛ୗ᪉✚⋡ࡣᅽ⦰ࡉࢀ࡚
࠾ࡾ, ࣜࣂࣛࣥࢫᅇᩘࡢቑຍࡀᮃࡲࡋ࠸⤖ᯝ࡜࡞ࡗ࡚࠸ࡿ. ࡓࡔࡋ, ᙜヱ⤖ᯝ࡟࠾࠸࡚ࡣྲྀᘬࢥࢫࢺࢆ⪃៖ࡋ࡚
࠸࡞࠸. ྲྀᘬࢥࢫࢺࢆ⪃៖ࡍࡿ࡜, ࢥࢫࢺࡀ┠ⓗ㛵ᩘࡢᨵၿᗘࢆୖᅇࡿ⠊ᅖ࡛ࡣࣜࣂࣛࣥࢫࡀ⚗Ṇࡉࢀࡿࡓࡵ,ᅇᩘࡢቑຍࡀᚲࡎࡋࡶẚ౛ࡋ࡚ୗ᪉✚⋡ࡢᅽ⦰ࢆࡶࡓࡽࡉ࡞࠸ሙྜࡶ࠶ࡿࡇ࡜࡟␃ពࡀᚲせ࡛࠶ࡿ.
ྲྀᘬࢥࢫࢺࡣಖ᭷㈨⏘ࡢつᶍ➼ࡢᙳ㡪ࢆཷࡅࡿࡶࡢ࡜᝿ᐃࡉࢀࡿࡀ, ࡇࡢ௚࡟࣊ࢵࢪຠᯝࢆపୗࡉࡏࡿせᅉ࡜ࡋ࡚ࡣ, ᢞ㈨㈨⏘ࡢಖ᭷ไ⣙, ᘬཷಖ㝤఍♫ࡢ᱁௜పୗ, ᝿ᐃࢆ㉸࠼ࡿዎ⣙⪅⾜ື➼ࡀ⪃࠼ࡽࢀࡿ.
⾲7

ᢞ㈨ᡓ␎ཬࡧ᫬㛫ศ๭ᩘࡢẚ㍑

ከᮇ㛫ືⓗ᭱㐺໬ᡓ␎ࡢሙྜ

≉຺ูᐃᢞ㈨ᡓ␎ࢆ‽⏝ࡢሙྜ

ᮇᚅ್

ᶆ‽೫ᕪ

ୗ᪉✚⋡

ᮇᚅ್

ᶆ‽೫ᕪ

ୗ᪉✚⋡

1

1.03770.04160.00651.03750.04510.007837.4712

1.03370.04980.00851.03200.04820.009941.0914

1.04690.04350.0052

1.03550.04730.008940.52912

1.04560.03970.00401.03240.04810.009868.82850

1.05150.03610.0022

1.0361

0.04830.00931189.601

100

1.05410.03400.00141.03900.04520.00758519.963

250

1.05200.03140.00081.03560.04710.008799274.403

ศ๭ᩘ(ὀ 1) ಖドᩱࡣ 1P = 1V = 1% ࡜ࡋ࡚࠸ࡿ.

*14 ᙜヱᢞ㈨⾜ືࡣᡶ㎸ಖ㝤ᩱࢆ඲࡚≉຺ูᐃ࡛㐠⏝ࡋ, ఱࡽ࣊ࢵࢪᡓ␎ࢆྲྀࡽ࡞࠸ሙྜ࡟┦ᙜࡍࡿ.

―88―ฎ⌮᫬㛫 (⛊)

日本保険・年金リスク学会誌 Vol.4 No.1 2009

ᅗ7༢఩᫬㛫ࡢศ๭ᩘ=1ᅗ8༢఩᫬㛫ࡢศ๭ᩘ=2

ᅗ9༢఩᫬㛫ࡢศ๭ᩘ= 4ᅗ 10༢఩᫬㛫ࡢศ๭ᩘ= 12ᅗ 11༢఩᫬㛫ࡢศ๭ᩘ= 50ᅗ 12༢఩᫬㛫ࡢศ๭ᩘ= 250―89―⾲ 8 ࡣ, ಖドᩱỈ‽ࡀዎ⣙᫬㈐௵‽ഛ㔠ࢆࢮࣟ࡜ࡍࡿሙྜ࡜, ᙜヱỈ‽࡟‶ࡓ࡞࠸ሙྜࢆẚ㍑ࡋࡓࡶࡢ࡛࠶
ࡿ. ಖドᩱࡢಖ㝤ᩱẚ౛㒊ศ (1P ) ࡜✚❧㔠ẚ౛ (1V ) ࡢ఩⨨ࡣᅗ 13 ࡛☜ㄆࡍࡿࡇ࡜ࡀ࡛ࡁࡿ.
ᅗ 13 ࡢྑୗࡀࡾࡢ┤⥺ࡼࡾୗࡢ㒊ศࡣᅗ 2 ࡢ⥙࠿ࡅ㒊ศ࡛࠶ࡾ, ┤⥺࠿ࡽࡢ஋㞳ࡣಖドᩱ཰ධࡢῶᑡࢆព࿡
ࡍࡿ. (1P , 1V ) = (1%, 0.5%) ⛬ᗘ࡟஋㞳ࡍࡿ࡜, ୗ᪉✚⋡ࡢᅽ⦰ࡣ࣊ࢵࢪᡓ␎ࢆ᥇ࡽ࡞࠸ሙྜࡢ 85% ⛬ᗘ࡛
࠶ࡿ. ┤⥺㏆ഐ࡟఩⨨ࡍࡿ (1P , 1V ) = (0.5%, 1.6%) ཬࡧ (4%, 0.8%) ࡛ࡣ࣊ࢵࢪᡓ␎ࢆ᥇ࡽ࡞࠸ሙྜ࡟ẚ㍑ࡋ
࡚ୗ᪉✚⋡ࢆ 2 ๭ᙉ࡟, ┤⥺ࡢୖ࡟఩⨨ࡍࡿ (1P , 1V ) = (3.5%, 1.9%) ࡛ࡣୗ᪉✚⋡ࢆ 1 ๭ᮍ‶࡟ᅽ⦰࡛ࡁ࡚
࠾ࡾ, ಖドᩱࡢỈ‽ࡶᢞ㈨ᡓ␎ࡢ๓࡟᳨ウࡉࢀࡿ࡭ࡁ࡜࠸࠺ࡇ࡜ࡀ෌☜ㄆ࡛ࡁࡿ.
⾲8ಖドᩱಖドᩱỈ‽ࡢẚ㍑

ከᮇ㛫ືⓗ᭱㐺໬ᡓ␎ࡢሙྜ

≉຺ูᐃᢞ㈨ᡓ␎ࢆ‽⏝ࡢሙྜ

ಖ㝤ᩱẚ౛

✚❧㔠ẚ౛

ᮇᚅ್

ᶆ‽೫ᕪ

ୗ᪉✚⋡

ᮇᚅ್

ᶆ‽೫ᕪ

ୗ᪉✚⋡

0.005

0.016

1.07670.03980.0011

1.06630.05000.0046

0.01

0.005

1.00230.03890.01441.00590.04620.0169

0.04

0.008

1.0676

0.03840.0012

1.05880.04930.0055

0.015

0.015

1.07840.04010.00071.070

0.04970.0036

0.035

0.019

1.11250.04030.00001.11930.04960.0004

0.045

0.009

1.0776

0.03850.00051.07290.04860.0035(ὀ) ༢఩᫬㛫ࡢศ๭ᩘࡣ 50 ᅇ࡜ࡋ࡚࠸ࡿ.

ᅗ 13

ಖドᩱỈ‽ࡢẚ㍑―90―

日本保険・年金リスク学会誌 Vol.4 No.1 20095 ⤖ㄽᮏ✏࡛ࡣከᮇ㛫ືⓗ᭱㐺໬ࡢᯟ⤌ࡳ࡟ࡼࡾ, ᭱పಖドࢆ௜ࡋࡓኚ㢠ᖺ㔠ಖ㝤ࡢᢞ㈨ᡓ␎ᵓ⠏ᡭἲࢆᥦ♧ࡋࡓ.
௵ពࡢ᫬Ⅼ, 4 㝵ᚤศྍ⬟࡛࠶ࡿࡇ࡜ࢆ๓ᥦ࡜ࡋࡓ┠ⓗ㛵ᩘ࡬ࡢ௵ពࡢ㛵ᩘᙧࡢ㐺⏝ཬࡧ௵ពࡢ⤥௜ᙧែ࡟࠾
ࡅࡿ୍⯡຺ᐃ࡜≉຺ูᐃ, ಖドᩱཬࡧ᭱పಖド㒊ศࡢ౯᱁ឤᛂᗘ (ᡤㅝࢹࣝࢱ), ಖ㝤ᩱᡶ㎸᪉ἲ, ᢞ㈨㈨⏘཰┈⋡ࡢᮇᚅ཰┈⋡࣭ᶆ‽೫ᕪ࣭┦㛵ಀᩘࢆ㏻ࡌࡓ⤒῭⎔ቃࡢ㛵ಀࢆᘧ (37) ࡟᫂♧࡛ࡁࡓࡇ࡜ࡀ୺ࡓࡿᡂᯝ࡛࠶ࡿ.
ᙜヱᢞ㈨ᡓ␎ࡣ᏶ഛᕷሙࢆ᝿ᐃࡍࡿࢹࣝࢱ࣭࣊ࢵࢪ࡜ࡣ␗࡞ࡾ, ಖ㝤஦ᨾⓎ⏕᫬࡟࠾ࡅࡿ฼ᚓࡢ」〇ࢆពᅗࡍࡿࡶࡢ࡛ࡣ࡞࠸. Ỵ⟬ฎ⌮࡟㔜Ⅼࢆ⨨ࡁ, ㈐௵‽ഛ㔠㢠ࡢ☜ಖࢆពᅗࡋࡓࡶࡢ࡛࠶ࡿ. ᮏ✏࡛ࡣᮇ୰ࡢಖ㝤஦
ᨾⓎ⏕ࢆ᝿ᐃࡏࡎ, ┠ⓗ㛵ᩘࡶ‶ᮇ᫬Ⅼࡢ✚❧ẚ⋡࡟ᑐࡋ࡚ࡢࡳᐃ⩏ࡋ࡚࠸ࡿࡀ, ⮬↛࡞ᣑᙇ࡜ࡋ࡚」ᩘࡢಖ㝤ᖺᗘᮎỴ⟬᫬࡟࠾ࡅࡿ✚❧ẚ⋡࡟ᑐࡍࡿ┠ⓗ㛵ᩘࢆ෌ᐃ⩏ࡍࡿࡇ࡜, ᮇ୰ࡢ࢟ࣕࢵࢩࣗࣇ࣮ࣟ࡟୙☜ᐇᛶࢆ୚࠼, ゎ⣙࣭㌿᥮➼ዎ⣙⪅⾜ື࡟≧ែ౫Ꮡࢆ᝿ᐃࡍࡿࡇ࡜➼ࡀ᳨ウ࡛ࡁࡼ࠺. ࡓࡔࡋ, ᙜヱᣑᙇ࡟㝿ࡋ࡚ࡣィ⟬㈇Ⲵࡢቑຍࡀᙜ↛࡟ண᝿ࡉࢀࡿ. ᐇົ࡬ࡢ㐺⏝࡟㝿ࡋ࡚ࡣ, ᐇ⾜ྍ⬟ᛶ࡟␃ពࡍࡿᚲせࡀ࠶ࢁ࠺.
ࡲࡓ, ࠕ┿ࠖࡢ᭱㐺໬࣏࣮ࢺࣇ࢛ࣜ࢜ࢆ㈨⏘౯᱁ࡢୖୗ㝈࡟ᑐᛂࡍࡿ࣏࣮ࢺࣇ࢛࡛ࣜ࢜⿵㛫ࡍࡿᡭἲ࡟ࡶ, ᣳࡳ㎸ࡴࢢࣜࢵࢻ㛫㝸ࢆࡼࡾ⊃ࡃࡍࡿ➼ࡢᨵၿࡀᮃࡲࢀࡿ.
౯᱁タᐃࡢㄗࡾ, ᢞ㈨ᡓ␎ࡢኻᩋࡑࡋ࡚᝿ᐃࢆ㉸࠼ࡿ⤒῭⎔ቃࡢኚ໬➼࡟ࡼࡾ, ୍⯡຺ᐃ࡟࠾࠸࡚㈈※୙㊊ࡀᠱᛕࡉࢀࡿሙྜ࡟ࡣ, ㈐௵‽ഛ㔠ࡢ㏣ຍ✚❧, ⮬ᕫ㈨ᮏ (ࢯࣝ࣋ࣥࢩ࣮࣭࣐࣮ࢪࣥ) ࡢ඘ᙜ, Ⴀᴗᡓ␎ࡢኚ᭦➼࡟ࡼࡾᑐᛂࡍࡿࡇ࡜࡜࡞ࡿ. ࢩ࣑࣮ࣗࣞࢩࣙࣥࢆ࣮࣋ࢫ࡜ࡍࡿ BGSS ἲࡢ㛗ᡤࡢ୍ࡘࡣ, ୍⯡຺ᐃ࡟࠾ࡅࡿ᭱㐺
ᢞ㈨ᡓ␎ࢆồࡵࡿ࡜ྠ᫬࡟, ձࢹࣇ࢛ࣝࢺ≧ែࡢⓎ⏕⋡ホ౯࡟ࡼࡿಖドᩱỈ‽ࡢ㐺ṇᛶࢆ☜ㄆࡋ, ղண᝿୙㊊㈈※㢠ࡢホ౯࡟ࡼࡿᚲせෆ㒊␃ಖỈ‽ࡢᢕᥱࢆྍ⬟࡜ࡍࡿࡇ࡜࡛࠶ࡿ. ᚑࡗ࡚, ᙜヱࣔࢹࣝࡢ☜ಖࡣಖ㝤఍♫࡟࠾ࡅࡿၟရタィཬࡧᢞ㈨ᡓ␎ࡢពᛮỴᐃࡢࡳ࡞ࡽࡎ, ┘╩ᐁᗇ࡟࠾ࡅࡿၟရᴫせཬࡧෆ㒊␃ಖỈ‽ࡢࢳ࢙ࢵࢡ,᱁௜ᶵ㛵ཬࡧ㈍኎ࢆ௦⌮ࡍࡿ㔠⼥ᶵ㛵࡟࠾ࡅࡿᙜヱၟရࡢࢹ࣮ࣗࢹࣜࢪ࢙ࣥࢫ࡟ࡶ᭷ຠ࡞ࡶࡢ࡜࡞ࢁ࠺.
ࡲࡓ, ᮏ✏࡛ࡣᑐ㇟ࢆኚ㢠ᖺ㔠࡟㝈ᐃࡋ࡚࠸ࡿࡀ, ௻ᴗᖺ㔠ไᗘ࣭බⓗᖺ㔠ไᗘ࡟㐺⏝ࡍࡿࡇ࡜ࡶᙜ↛࡟⪃࠼ࡽࢀࡿ. ౛࠼ࡤ☜ᐃ⤥௜௻ᴗᖺ㔠ไᗘ࡛ࡣ, ☜ᐃ⤥௜௻ᴗᖺ㔠ἲ➨ 61 ᮲, ྠ➨ 62 ᮲ཬࡧྠ➨ 63 ᮲࡛㈈ᨻ᳨ド
ไᗘࡀつᐃࡉࢀ, ㈐௵‽ഛ㔠ཬࡧ᭱ప✚❧ᇶ‽㢠࡟ᑐࡍࡿಖ᭷㈨⏘ࡢୗ㝈Ỉ‽ࡀ᫂♧ࡉࢀ࡚࠸ࡿ. ☜ᐃ⤥௜௻ᴗᖺ㔠ἲ᪋⾜つ๎➨ 84 ᮲࡟࠾࠸࡚ࠕ㛗ᮇ࡟ࢃࡓࡾ⥔ᣢࡍ࡭ࡁ㈨⏘ࡢᵓᡂ๭ྜࢆ㐺ษ࡞᪉ἲ࡟ࡼࡾᐃࡵࡿࡇ࡜ࠖࡀせồࡉࢀ࡚࠸ࡿࡓࡵ, ᇶᮏ࣏࣮ࢺࣇ࢛ࣜ࢜ࡑࡢࡶࡢࢆືⓗ࡟⟶⌮ࡍࡿࡇ࡜ࡣᮃࡲࡋ࠸ࡇ࡜࡛ࡣ࡞ࡃ, ࡑࡶࡑࡶ࡜ࡋ࡚ᐇ㊶ⓗ࡛࡞࠸ࡀ, ᇶᮏ࣏࣮ࢺࣇ࢛ࣜ࢜ (y) ࢆᡤ୚࡜ࡋ࡚, ࣊ࢵࢪᡓ␎ࡢᐇ㊶ (x) ࢆ㐠⏝ᶵ㛵࡟ጤクࡍࡿ➼ࡢᑐᛂࡀ࡛ࡁࡿࡔࢁ࠺. ᮏ✏࡛♧ࡋࡓࣔࢹࣝࢆ㐺⏝ࡍࢀࡤ, ᢞ㈨ᡓ␎࡟ຍ࠼, ࡝ࡢ⛬ᗘࡢ㈨㔠㓄㈿ࡀᚲせ࡜࡞ࡿ࠿ࡶ (P ) ࢆ㏻ࡌ࡚ᢕᥱࡍࡿࡇ࡜ࡀྍ⬟࡛࠶ࡿ.

―91―

A AppendicesA.1 ಖ㝤ᩘᏛ࡟࠾ࡅࡿグྕࡢㄝ᫂ᖺ㱋 x ࡢ⏕Ꮡ⪅ᩘࢆ lx , Ṛஸ⪅ᩘࢆ dx ࡜ࡍࡿ. ௵ពࡢ᫬Ⅼ t ࡟࠾࠸࡚, ⏕Ꮡ⪅ᩘ࡜Ṛஸ⪅ᩘࡢ㛵ಀࡣḟᥖࡢ࡜࠾ࡾ࡜࡞ࡿ.

lx+t  dx+t = lx+t+1(51)

ᖺ㱋 x ࡢ⿕ಖ㝤⪅࡟ᑐࡍࡿಖ㝤㔠㢠 1, ಖ㝤ᮇ㛫 n, ᖺᡶࡢ㣴⪁ಖ㝤ࡢಖ㝤ᩱ Px: n ࡣ, ḟࡢᡭ㡰࡟ࡼࡾồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ.
⤥௜⌧౯㣴⪁ಖ㝤࡜ࡣಖ㝤஦ᨾⓎ⏕᫬ (Ṛஸ᫬) ཬࡧಖ㝤஦ᨾᮍⓎ⏕ୗ࡛ࡢ‶ᮇ᫬࡟ಖ㝤㔠㢠ࢆ⤥௜ࡍࡿಖ㝤ၟရ࡛࠶ࡿ. ᚑࡗ࡚, ⤥௜㢠ࡢ⌧ᅾ౯್ Sx ࡣ, ಖ㝤㔠ࢆᮇᮎᡶ, ணᐃ฼⋡ࢆ i ࡜ࡍࡿ࡜ḟᘧ࡟ࡼࡾᐃ
⩏ࡉࢀࡿ.
n1

Sx =
=
1
ࡇࡇ࡛, v = 1+i,

1 X dx+u
lx+n+
u+1
lx u=0 (1 + i)lx (1 + i)nn1X

1 u+1 lx+u dx+u
1 n lx+n+())
1+i
lx lx+u
1+i
lx
u=0
(

(52)

lx+t
=t px , dlxx = qx , Dx = v x lx , Cx = v x+1 dx ࡜ࡍࡿ࡜, ᘧ (52) ࡣ
lx

Sx =
ࠉ=

࡜࡞ࡿ. ᭦࡟, Nx =P

Dx+u , Mx =

u=0

n1
X

v u+1 u px qx+u + v n n px

u=0
n1
X

Dx+n
Cx+u
+
Dx
Dx
u=0

P

(53)

(54)

Cx+u , ࡜ࡍࢀࡤ,

u=0

Sx =

(Mx  Mx+n ) + Dx+n:= Ax: n
Dx(55)

࡜グ㏙࡛ࡁࡿ.
཰ධ⌧౯ྠᵝ࡟ࡋ࡚ಖ㝤ᩱࡢ཰ධ⌧౯ Gx ࢆồࡵࡽࢀࡿ. ಖ㝤ᩱᣐฟࡀᖹ‽ⓗ࡟⾜ࢃࢀࡿࡶࡢ࡜ࡍࢀࡤ,n1

Gx = Px: n ·= Px: n ·= Px: n ·ಖ㝤ᩱࡢỴᐃ

1 X lx+u
lx u=0 (1 + i)u(56)

v u u px

(57)

n1
X
u=0

Nx  Nx+n:= Px: n · äx: n
Dx

(58)

⤥௜⌧౯࡜཰ධ⌧౯ࡀ➼౯࡜࡞ࡿࡼ࠺࡟, ࡍ࡞ࢃࡕ, ཰ᨭ┦ᙜࡢཎ๎࡟๎ࡾ, ಖ㝤ᩱࢆỴᐃࡍࢀࡤࡼ࠸.

Px: n · äx: n = Ax: n
Ax: n(Mx  Mx+n ) + Dx+n
Қ Px: n =:=äx: nNx  Nx+n

―92―

(59)
(60)

日本保険・年金リスク学会誌 Vol.4 No.1 2009㈐௵‽ഛ㔠㈐௵‽ഛ㔠ࡣᑗ᮶ἲⱝࡋࡃࡣ㐣ཤἲ࡟ࡼࡾホ౯ࡉࢀࡿ. ᑗ᮶ἲ࡛ࡣࠕᑗ᮶ࡢ⤥௜ࢆ㈥࠺ࡓࡵ࡟⌧᫬Ⅼ࡛␃ಖࡋ࡚࠾ࡃ࡭ࡁ㔠㢠ࠖ࡜ࡋ࡚, 㐣ཤἲ࡛ࡣࠕࡇࢀࡲ࡛࡟཰ཷࡉࢀࡓಖ㝤ᩱ࠿ࡽᨭᡶࢃࢀࡓ⤥௜㔠㢠ࢆ᥍㝖ࡋࡓṧ㢠ࠖ࡜ࡋ࡚㈐௵‽ഛ㔠ࡀホ౯ࡉࢀࡿ. ➨ t ᫬Ⅼࡢ㈐௵‽ഛ㔠 t Vx: n ࢆᑗ᮶ἲ࡟ࡼࡾグ㏙ࡍࢀࡤ,Sx+t = Ax+t: nt(61)(Mx+t  Mx+n ) + Dx+n=
Dx+t
Gx+t = Px: n · äx+t: nt
Nx+t  Nx+n
= Px: n ·Dx+tҚt Vx: n = Sx+t  Gx+t:= Ax+t: nt  Px: n · äx+t: nt

(62)
(63)
(64)
(65)
(66)

࡜࡞ࡿ.
ࣇ࢓ࢡ࣮ࣛࡢ෌ᖐᘧx ṓዎ⣙ࡢ௵ពࡢಖ㝤ዎ⣙࡟ࡘ࠸࡚, ➨ t ᫬Ⅼࡢ㈐௵‽ഛ㔠ࢆ Vt , ಖ㝤ᩱࢆ P ࡜ࡍࡿ. ➨t + 1 ᫬Ⅼࡢ㈐௵‽ഛ㔠ࡣ, ➨ t ᫬Ⅼࡢ㈐௵‽ഛ㔠ཬࡧᣐฟಖ㝤ᩱ࡟ 1 ᮇ㛫௜฼ࡋ, ᮇᮎ࡟ᙜヱᖺᗘ࡟Ⓨ⏕ࡍࡿ⤥௜㔠ࢆ᥍㝖ࡍࢀࡤࡼ࠸ࡢ࡛, ḟᥖࡢࡼ࠺࡟グ㏙ࡉࢀࡿ.

lx+t+1 · Vt+1 = lx+t · (Vt + P ) · (1 + i)  dx+t

(67)

ࡓࡔࡋ, ࡇࡇ࡛ࡣᮇ୰࡛ࡢゎ⣙ࢆ᝿ᐃࡋ࡚࠸࡞࠸.
ࡉ࡚, ᘧ (67) ࡢ୧㎶࡟ v x+t+1 ࢆ஌ࡎࢀࡤ,Dx+t+1 · Vt+1 = Dx+t · (Vt + P )  Cx+t

(68)

ࢆᚓࡿ. t = 0  (n  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤ,Dx+n · Vn = Dx · V0 + P
ҚP =

n1
X
t=0

Dx+t 

n1
X

Cx+t

t=0

Dx+n · Vn  Dx · V0 + (Mx  Mx+n )Nx  Nx+n(69)1ዎ⣙᫬㈐௵‽ഛ㔠 (V0 ) ࢆ 0, ‶ᮇ᫬㈐௵‽ഛ㔠 (Vn ) ࢆ 0 ࡜ࡍࢀࡤᐃᮇಖ㝤ࡢಖ㝤ᩱ Px:n ࢆ, 1 ࡜ࡍࢀࡤ㣴⪁ಖ㝤ࡢಖ㝤ᩱ Px: n ࡀᚓࡽࢀࡿ.
ࡲࡓ, t = 0  (t  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤ㐣ཤἲ㈐௵‽ഛ㔠ࡀ, t = t  (n  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤᑗ᮶ἲ㈐௵‽ഛ㔠ࡀᚓࡽࢀࡿ. ࡇࡢሙྜ࡟ࡶ, ‶ᮇ᫬㈐௵‽ഛ㔠 (Vn ) ࢆ 0 ࡜ࡍࢀࡤᐃᮇಖ㝤ࡢ➨ t ᖺᗘᮎ㈐௵

‽ഛ㔠 t Vx:1 n ࢆ, 1 ࡜ࡍࢀࡤ㣴⪁ಖ㝤ࡢ➨ t ᖺᗘᮎ㈐௵‽ഛ㔠 t Vx: n ࡀồࡵࡽࢀࡿ.

―93―

A.2 ␗࡞ࡿィ⟬ᇶ♏⋡ࡢୗ࡟࠾ࡅࡿዎ⣙౯್ࡢᕪ␗
㐺⏝ࡉࢀࡿィ⟬ᇶ♏⋡ࡀ␗࡞ࡿሙྜࡢ㈐௵‽ഛ㔠ࡢᕪ␗ࡣ, ᘧ (67) ࡟♧ࡋࡓࣇ࢓ࢡ࣮ࣛࡢ෌ᖐᘧࡢ㐺⏝࡟ࡼࡾホ౯ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ. Ṛஸ⋡ཬࡧணᐃ฼⋡࡟ᕪ␗ࢆ୚࠼ࡿࡶࡢ࡜ࡋ, ⏕Ꮡ⪅ᩘ lx , Ṛஸ⪅ᩘ dx ཬࡧ㈐௵‽ഛ㔠㢠 Vt ࡛ࡣࣉࣛ࢖࣒ࡢ᭷↓࡟ࡼࡾィ⟬ᇶ♏⋡ࡢᕪ␗ࢆ⾲⌧ࡍࡿ. ୍᪉, ಖ㝤ᩱ P ࡣ㐺⏝ࡉࢀࡿィ⟬ᇶ♏⋡࡟౫Ꮡࡏࡎྠ୍࡛࠶ࡿࡶࡢ࡜ࡋ*15 , ணᐃ฼⋡ࢆ i, ᫬Ⅼ t ࡛ᮇ㛫 [s, s + 1] ࢆ࢝ࣂ࣮ࡍࡿࣇ࢛࣮࣡ࢻ LIBOR ࣮ࣞࢺࢆ Lt (s, s + 1) ࡜ࡍࡿ. ࡓࡔࡋ, ಙ⏝ࣜࢫࢡࡢ↓࠸‶ᮇ T ᖺࡢ๭ᘬമࡢ᫬Ⅼ t ࡟࠾ࡅࡿ౯᱁ B(t, T ) ࡟ᑐࡋ,ࣇ࢛࣮࣡ࢻ LIBOR ࣮ࣞࢺࡣḟᘧ࡟ࡼࡾᐃ⩏ࡉࢀࡿ.

Lt (s, s + 1) =B(t, s)  B(t, s + 1)
B(t, s + 1)

(70)

௒, ␗࡞ࡿィ⟬ᇶ♏⋡ୗ࡟࠾࠸࡚, ḟᥖᘧࡀᡂ❧ࡋ࡚࠸ࡿ.

lx+t · (Vt + P ) · (1 + i)  dx+t = lx+t+1 · Vt+1
lx+t 0 · (Vt 0 + P ) · [1 + Lt (t, t + 1)]  dx+t 0 = lx+t+1 0 · Vt+1 0

(71)
(72)

ࡇࡇ࡛, lx 0 = lx + lx , Vt 0 = Vt + Vt ࡜ࡋ࡚, ᘧ (72) ࡼࡾᘧ (71) ࢆ᥍㝖ࡍࡿ࡜,lx+t · (Vt + P ) · [Lt (t, t + 1)  i] + lx+t 0 · Vt · [1 + Lt (t, t + 1)]  dx+t+ lx+t · (Vt + P ) · [1 + Lt (t, t + 1)] = lx+t+1 0 · Vt+1 + lx+t+1 · Vt+1
(73)
୧㎶࡟ B(t, t + 1) ࢆ஌ࡌᩚ⌮ࡍࡿ࡜(Vt · lx+t 0 + lx+t · Vt )  (Vt+1 · lx+t+1 0 + lx+t+1 · Vt+1 ) · B(t, t + 1)
= [lx+t · (Vt + P ) · [i  Lt (t, t + 1)] + dx+t ] · B(t, t + 1)  lx+t · P(74)࡜࡞ࡿ. ྠᵝࡢฎ⌮ࢆ (n-1) ᫬Ⅼࡲ࡛཯᚟ࡋ, ㎶ࠎࢆຍ⟬ࡍࢀࡤ(Vt · lx+t 0 + lx+t · Vt )  (Vn · lx+n 0 + lx+n · Vn ) · B(t, n)
=

n1
X
k=t

B(t, k + 1) · [lx+k · (Vk + P ) · [i  Lt (k, k + 1)] + dx+k ] 

n1
X
k=t

lx+k · P · B(t, k)
(75)

ࢆᚓࡿ.
୙☜ᐇᛶࢆ฼Ꮚ⋡࡟ࡢࡳ᝿ᐃࡍࡿࡢ࡛࠶ࢀࡤ, dx ཬࡧ lx ࡣࢮ࡛ࣟ࠶ࡾ, lx 0 = lx ࡜࡞ࡿࡢ࡛,Vt · lx+t  Vn · lx+n · B(t, n) =

n1
X
k=t

B(t, k + 1) · lx+k · (Vk + P ) · [i  Lt (k, k + 1)](76)ಖ㝤㔠㢠ཬࡧಖ㝤✀㢮ࡀྠ୍࡛࠶ࢀࡤ, ‶ᮇ᫬Ⅼࡢ㈐௵‽ഛ㔠㢠ࡣྠ୍, ࡍ࡞ࢃࡕ, Vn = 0 ࡛࠶ࡿࡢ࡛,

Vt =

n1
X
k=t

B(t, k + 1) ·lx+k
· (Vk + P ) · [i  Lt (k, k + 1)]
lx+t# Vt 0  Vt
௨ୖࡼࡾ, ᘧ (2) ࡀド᫂ࡉࢀࡓ.
*15 ಖ㝤ᩱ࡜㈐௵‽ഛ㔠ࡢィ⟬ᇶ♏⋡ࡀ␗࡞ࡿࡇ࡜࡟ఱࡽၥ㢟ࡣ࡞࠸. ಖ㝤ᴗἲ᪋⾜つ๎➨ 10 ᮲ࡣ, ➨ 1 ྕࢆィ⟬ᇶ♏⋡ࢆྵࡴಖ㝤ᩱࡢィ⟬᪉ἲ, ➨ 2 ྕࢆィ⟬ᇶ♏⋡ࢆྵࡴ㈐௵‽ഛ㔠ࡢィ⟬᪉ἲ࡟඘࡚࡚࠾ࡾ, ಖ㝤ᩱཬࡧ㈐௵‽ഛ㔠࡟ᑐࡋᚲࡎࡋࡶྠ୍ࡢィ⟬ᇶ♏
⋡㐺⏝ࢆせồࡋ࡚࠸࡞࠸. ᐇົ࡟࠾࠸࡚ࡶ, ኚ㢠ಖ㝤ࡢಖドᩱ࡟ࡣᖺ㱋࡟౫Ꮡࡏࡎ, ྠ୍ࡢᩱ⋡ࡀ㐺⏝ࡉࢀ࡚࠸ࡿ౛ࢆぢࡿࡇ࡜ࡀ࡛
ࡁࡿ.

―94―

日本保険・年金リスク学会誌 Vol.4 No.1 2009y
A.3 ᘧ (35) ࡢ G/Ax,yt+1 ཬࡧ S/At+1 ࡟㛵ࡍࡿ✚⋡ホ౯

x,yE[ G/At+1| Ft ]= (G/At + P · 1P ) · (M  rt+1 e)T x+[ S/At + P · (1  P ) ] · 1V · (M  rt+1 e)T y

(77)

x,y 2
E[ (G/At+1) | Ft ]= (G/At + P · 1P )2 · xT x + { S/At + P · (1  P ) }2 · (1V )2 · yT y+21V · (G/At + P · 1P ) · [ S/At + P · (1  P ) ] · xT y

E[ S/Ayt+1 | Ft ]= [ S/At + P · (1  P ) ] · (1  V ) · (M  rt+1 e)T y

E[ (S/Ayt+1 )2 | Ft ]= (1  V )2 · [ S/At + P · (1  P ) ]2 · yT y

(78)

(79)

(80)

y
E[ G/Ax,yt+1 · S/At+1 | Ft ]= (1  V ) · [ S/At + P · (1  P ) ] · (G/At + P · 1P ) · xT y+1V · (1  V ) · [ S/At + P · (1  P ) ]2 · yT yࡓࡔࡋ,  = (M  rt+1 e)(M  rt+1 e)T +

(81)

࡛࠶ࡿ.

A.4 t ᫬Ⅼࡢ┠ⓗ㛵ᩘࢆᵓᡂࡍࡿᚤศಀᩘ
ᘧ (44) ࠿ࡽᘧ (49) ࡢᚤศಀᩘࡢᒎ㛤ࡣḟᥖࡢ࡜࠾ࡾ࡛࠶ࡿ.

(
(
(G/AT
(
(S/AT
f=
f·
+
f·
(G/At+1(G/AT(G/At+1(S/AT(G/At+1(2
f(G/A2t+1
=

(82)

(2
(G/AT 2
(2
(G/AT
(S/AT
) +2·
f·
·
2 f ·((G/AT(G/At+1(G/AT (S/AT(G/At+1 (G/At+1+(2(S/AT 2(( 2 G/AT(( 2 S/AT
f
·
(
)
+
f
·
+
f
·
(S/A2T(G/At+1(G/AT(G/A2t+1(S/AT(G/A2t+1

(
(
(G/AT
(
(S/AT
f=
f·
+
f·
(S/At+1(G/AT(S/At+1(S/AT(S/At+1(2
f(S/A2t+1
=

(83)
(84)

(2
(G/AT 2
(2
(G/AT
(S/AT
f ·(
) +2·
f·
·
2
(G/AT
(S/At+1(G/AT (S/AT(S/At+1 (S/At+1+(2(S/AT 2(( 2 G/AT(( 2 S/AT
f ·(
) +
f·
+
f·
2
2
(S/AT
(S/At+1(G/AT(S/At+1(S/AT(S/A2t+1

―95―

(85)

(2
f
(G/AT (S/AT
(2
(G/AT
(G/AT
(2
(S/AT
(S/AT
=
f
·
·
+
f·
·
2
2
(G/AT
(G/At+1 (S/At+1(S/AT(G/At+1 (S/At+1
+

(2
(G/AT
(S/AT
(G/AT
(S/AT
f ·(
·
+
·
)
(G/AT (S/AT(G/At+1 (S/At+1(S/At+1 (G/At+1(( 2 G/AT(( 2 S/AT
+
f·
+
f·
(G/AT
(G/At+1 (S/At+1(S/AT(G/At+1 (S/At+1(86)

A.5 Mortality Index, Mortality Derivatives ࡢ஦౛
A.5.1

Mortality Index

2009 ᖺ 6 ᭶⌧ᅾ࡟࠾࠸࡚, ࡍ࡛࡟ 4 ࡘࡢ࢖ࣥࢹࢵࢡࢫࡀᕷሙ࡟ᢞධࡉࢀ࡚࠸ࡿ. ᚋᥖࡍࡿ SCOR Global
Life ࡢ SWAP ዎ⣙ࡣ JP Morgan ࡢ LifeMetrix ࢆཧ↷ࡍࡿࡶࡢ࡛࠶ࡿ.
⾲9ྲྀᘬྍ⬟࡞Ṛஸ⋡࢖ࣥࢹࢵࢡࢫ

ྡ⛠

ᥦ౪఍♫

ᑐ㇟ᅜ

ᥦ౪㛤ጞ᫬ᮇ

᭦᪂㢖ᗘCredit Suisse

Credit Suisse⡿ᅜ

2006 ᖺ 3 ᭶

ᖺḟJP Morgan⡿ᅜ2007 ᖺ 3 ᭶ᖺḟ࢖ࢠࣜࢫ࣭࢙࣮࢘ࣝࢬ

2007 ᖺ 3 ᭶࢜ࣛࣥࢲ2007 ᖺ 10 ᭶ࢻ࢖ࢶ2008 ᖺ 4 ᭶ࢻ࢖ࢶ2008 ᖺ 3 ᭶࢜ࣛࣥࢲ2008 ᖺ 4 ᭶⡿ᅜ2008 ᖺ 3 ᭶

Longevity Index
LifeMetrix

Xpect

Deutsche Börse

QxX

A.5.2

Goldman Sachs᭶ḟ᭶ḟ

Mortality Derivativesḟᥖࡢྲྀᘬ஦౛ࡣ, ཧ↷ࡍࡿṚஸ⋡࢖ࣥࢹࢵࢡࢫ࡟ᑐࡋ࢟ࣕࢵࢩࣗࣇ࣮ࣟࢆቑῶࡍࡿࡇ࡜࡟ࡼࡾ, Ṛஸ⋡ୖ᪼࡟క࠺ࣜࢫࢡࢆᢞ㈨ᐙⱝࡋࡃࡣྲྀᘬࡢ┦ᡭ᪉࡟⛣㌿ࡍࡿࡶࡢ࡜࡞ࡗ࡚࠸ࡿ. ࡍ࡞ࢃࡕ, ࢹࣜࣂࢸ࢕ࣈྲྀᘬࡀ෌
ಖ㝤ࡢ௦᭰ᶵ⬟ࢆᯝࡓࡋ࡚࠸ࡿ.
⾲ 10

Mortality Swap ࡢዎ⣙౛

ዎ⣙ᖺ᭶

ࢫ࣏ࣥࢧ࣮

ᮇ㛫

ࣀࢵࢡ࢖ࣥ/ୖ㝈࢟ࣕࢵࢩࣗࣇ࣮ࣟ

Jan-2008

SCOR Global Life4ᖺ

115%/125%

1 ൨ࢻࣝ +3600 ୓࣮ࣘࣟFeb-2008

Lucida㸫㸫㸫(ฟ඾ 1) ”Trading Risk”, Issue Number 2, March/April 2008.
(ฟ඾ 2) Johanna Symmons, ”Lucida guards against longevity”, Financial News, 19th February 2008.
(ฟ඾ 3) SCOR Press Release, ”SCOR enters fully collateralized USD 100 million and EUR 36 million
mortality risk swap”, 3rd March 2008.

―96―日本保険・年金リスク学会誌 Vol.4 No.1 2009

⾲ 11

Mortality Index Bond ࡢⓎ⾜౛

Ⓨ⾜ᖺ᭶

ࢫ࣏ࣥࢧ࣮

SPV

つᶍ

ᮇ㛫

๐ῶ/ἐ཰Ỉ‽ࢡ࣮࣏ࣥ

Dec-2003

Swiss Re

Vita Capital I

4 ൨⡿ࢻࣝ

3ᖺ

130%/150%

L+135bps

Apr-2005

Swiss Re

Vita Capital II

6200 ୓⡿ࢻࣝ5ᖺ

120%/125%

L+90bps2 ൨⡿ࢻࣝ

5ᖺ

115%/120%

L+140bps

1 ൨⡿ࢻࣝ

5ᖺ

110%/115%

L+190bps

7500 ୓⡿ࢻࣝ3ᖺ

115%/120%

L+19bps

8000 ୓⡿ࢻࣝ3ᖺ

110%/115%

L+300bps

1 ൨⡿ࢻࣝ

4ᖺ

114%/119%

E+20bps

5000 ୓⡿ࢻࣝ

4ᖺ

114%/119%

E+120bps

1.5 ൨⡿ࢻࣝ4ᖺ

110%/114%

L+285bps

1 ൨⡿ࢻࣝ

4ᖺ

106%/110%

L+500bps

9000 ୓⡿ࢻࣝ

4ᖺ

120%/125%

L+110bps

5000 ୓⡿ࢻࣝ5ᖺ

120%/125%

L+112bps

3000 ୓࣮ࣘࣟ

4ᖺ

120%/125%

E+110bps

1 ൨⡿ࢻࣝ

4ᖺ

120%/145%

L+21bps1 ൨⡿ࢻࣝ

5ᖺ

120%/145%

L+20bps

5000 ୓⡿ࢻࣝ5ᖺ

120%/125%

L+21bps

5500 ୓࣮ࣘࣟ

4ᖺ

120%/145%

E+21bps

5500 ୓࣮ࣘࣟ

4ᖺ

120%/125%

E+22bps

1 ൨࣮ࣘࣟ

5ᖺ

125%/145%

E+80bps1 ൨⡿ࢻࣝ

5ᖺ

120%/130%

L+135bps

May-2006
Nov-2006

Jan-2007

Feb-2008

Scottish Re
AXA

Swiss Re

Munich Re

Tartan Capital
Osiris Capital

Vita Capital III

Nathan(ὀ) ࢡ࣮࣏ࣥḍࡢ”L”ࡣ LIBOR, ”E”ࡣ EULIBOR ࡛࠶ࡿ.
(ฟ඾ 1) JP Morgan, ”LifeMetrics Technical Document”, 13th March 2007.
(ฟ඾ 2) ”Trading Risk”, Issue Number 2, March/April 2008.

ཧ⪃ᩥ⊩[1] ▼஭ᩥᙪ࣭኱ሯ୍჆ [2008], ࠕ࣐࣮ࢣࢵࢺ࣭ࣅࢪࣙࣥ (2008 ᖺ 6 ᭶)ࠖ, 㔝ᮧㆇๆ, 2008 ᖺ 7 ᭶ 25 ᪥.
[2] ෆᒣ᭸つ [2005], ࠕ㈇മࢆ᏶඲࡟࣊ࢵࢪ࡛ࡁ࡞࠸ሙྜࡢ☜ᐃ⤥௜ᖺ㔠ࡢືⓗ᭱㐺࣏࣮ࢺࣇ࢛ࣜ࢜ࠖ, ࠗ᪥ᮏಖ㝤࣭ᖺ㔠ࣜࢫࢡᏛ఍ㄅ࠘, Vol.1, No.1, 2005 ᖺ 9 ᭶, 23-43 㡫.

[3] ෆᒣ᭸つ [2008], ࠕ㛗ᮇᢞ㈨ࡢ࢔ࢭࢵࢺ࢔ࣟࢣ࣮ࢩࣙࣥࠖ, ࠗ᭱᪂㔠⼥ᕤᏛ࡟Ꮫࡪ㈨⏘㐠⏝ᡓ␎࠘, ᮾὒ⤒῭᪂ሗ♫, 2008 ᖺ 1 ᭶ 31 ᪥, 221-251 㡫.

[4] ᱵෆಇᶞ [2008], ࠕከᮇ㛫᭱㐺࣏࣮ࢺࣇ࢛ࣜ࢜ၥ㢟㸫 LSM ἲࢆ฼⏝ࡋࡓ㏆ఝⓗゎἲࡢ㏆ఝ⢭ᗘ㸫ࠖ, ࠗ㠀ᕷሙ㈨⏘ࡢ౯᱁௜ࡅ࡜ࣜ࢔ࣝ࢜ࣉࢩࣙࣥ ࢪࣕࣇ࢕࣮࣭ࢪ࣮ࣕࢼࣝ㸫㔠⼥ᕤᏛ࡜ᕷሙィ㔞ศᯒ㸫࠘, ᮅ಴᭩ᗑ, 2008 ᖺ 3 ᭶ 15 ᪥, 184-212 㡫.

[5] ᑠᏲᯘඞဢ [2002], ࠕᰴ౯㐃ືᆺಖ㝤ࡢၟရタィ࡜㐠⏝ᡓ␎ࠖ, ࠗ㈨⏘㐠⏝ࡢ᭱ඛ➃⌮ㄽ࠘, ᪥ᮏ⤒῭᪂⪺♫, 2002 ᖺ 3 ᭶ 18 ᪥, 201-227 㡫.

[6] ⏣⏿ྜྷ㞝 [2004], ࠗࣜࢫࢡ ᗘ࡜࣏࣮ࢺࣇ࢛ࣜ࢜⌮ㄽ࠘, ᮅ಴᭩ᗑ, 2004 ᖺ 8 ᭶ 31 ᪥.
[7] ሯཎⱥᩔ [2001], ࠕドๆᕷሙࡢ᏶ഛᛶ࡜୙᏶ഛᛶࠖ, ࠗ㔠⼥ᕤᏛࡢ᪂ᒎ㛤 ࢪࣕࣇ࢕࣮࣭ࢪ࣮ࣕࢼࣝ 2001࠘,ᮾὒ⤒῭᪂ሗ♫, 2001 ᖺ 7 ᭶ 3 ᪥, 121-158 㡫.

―97―[8] ♫ᅋἲே᪥ᮏ࢔ࢡࢳࣗ࢔࣮ࣜ఍ [2003], ࠕኚ㢠ᖺ㔠ಖ㝤➼ࡢ᭱పಖドࣜࢫࢡ࡟ಀࡿ㈐௵‽ഛ㔠ࡢ✚❧➼࡟ࡘ࠸࡚ࠖ, ᖹᡂ 15 ᖺ 12 ᭶ 17 ᪥.

[9] ᯌࠎᮌつ㞝 [2007], ࠕከᮇ㛫᭱㐺໬㈨⏘㓄ศࣔࢹࣝࠖ, ࠗࣜࢫࢡ࡜⛉Ꮫ㸫㔠⼥࡜ಖ㝤ࡢࣔࢹࣝศᯒ㸫࠘, ᮅ಴᭩ᗑ, 2007 ᖺ 11 ᭶ 15 ᪥, 1-31 㡫.

[10] ᯇᒣ┤ᶞ [2005], ࠕኚ㢠ᖺ㔠ಖ㝤ࡢࣜࢫࢡ⟶⌮ࠖ, ࠗ᪥ᮏಖ㝤࣭ᖺ㔠ࣜࢫࢡᏛ఍ㄅ࠘, Vol.1, No.1, 2005 ᖺ
9 ᭶, 69-82 㡫.
[11] ᒣୗ཭ಙ [2005], ࠗಖ㝤ἲ࠘, ᭷ᩫ㛶, 2005 ᖺ 3 ᭶ 10 ᪥.
[12] ᒣ⏣㞝஧࣭㣤⏣ឡ⨾࣭ᳺᗈィ [2006], ࠕࢺࣞࣥࢻண ࡟ᇶ࡙ࡃኳೃࢹࣜࣂࢸ࢕ࣈࡢ౯᱁௜ࡅ࡜஦ᴗࣜࢫࢡ
࣊ࢵࢪࠖ, ࠗ⤫ィᩘ⌮࠘, ➨ 54 ᕳ➨ 1 ྕ, 2006 ᖺ, 57-78 㡫.

[13] ᒣ⏣㞝஧ [2008],ࠕ㢼㏿ண ㄗᕪ࡟ᇶ࡙ࡃ㢼ຊࢹࣜࣂࢸ࢕ࣈࡢ᭱㐺໬タィࠖ, ࠗ㠀ὶືᛶ㈨⏘ࡢ౯᱁௜ࡅ࡜
ࣜ࢔ࣝ࢜ࣉࢩࣙࣥ ࢪࣕࣇ࢕࣮࣭ࢪ࣮ࣕࢼࣝ㸫㔠⼥ᕤᏛ࡜ᕷሙィ㔞ศᯒ㸫࠘, ᮅ಴᭩ᗑ, 2008 ᖺ 3 ᭶ 15 ᪥,

153-181 㡫.
[14] Pelsser, Antoon[2003], ”Pricing and Hedging Annuity Options via Static Option Replication”, Insurance: Mathematics and Economics, 33, 2003, P283-296.
[15] Binsbergen, Jules H. van and Michael W. Brandt[2007], ”Solving dynamic portfolio choice problems
by recursing on optimized portfolio weights or on the value function?”, Computational Economics,
Volume 29, Issue 3-4, 355-367.
[16] Brandt, Michael W., Amit Goyal, Pedro Santa-Clara and Jonathan R. Stroud[2005], ”A Simulation
Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability”,
Review of Financial Studies, 18, 831-873.
[17] Brandt, Michael W. and Pedro Santa-Clara[2006], ”Dynamic Portfolio Selection by Augmenting theAsset Space”, The Journal of Finance, Vol. LXI, No.5, 2187-2218, October 2006.
[18] Campbell, John Y. and Luis M. Viceira[2002], ”Strategic Asset Allocation: Portfolio Choice forLong-Term Investors”, [ヂ] 㔝ᮧㆇๆ㔠⼥⤒῭◊✲ᡤ, [┘ヂ] ᮌᓥṇ᫂, ࠗᡓ␎ⓗ࢔ࢭࢵࢺ࢔ࣟࢣ࣮ࢩࣙ
ࣥ: 㛗ᮇᢞ㈨ࡢࡓࡵࡢ㈨⏘㓄ศࡢ⪃࠼᪉࠘, ᮾὒ⤒῭᪂ሗ♫, 2005 ᖺ 2 ᭶ 10 ᪥.

[19] Dahl, Mikkel and Møller, Thomas[2006], “Valuation and hedging of life insurance liabilities withsystematic mortality risk”, Insurance: Mathematics and Economics, 39, 2006, P193-217.
[20] Hardy, Mary[2003], ”Investment Guarantees: Modeling and Risk Management for Equity-Linked Life
Insurance”, John Wiley & Sons Inc, 14th February 2003.
[21] International Association of Insurance Supervisors[2006], ”Standard on Asset-Liability Management”, Standard No.13, October 2006.
[22] Longsta&, Francis A. and Eduardo Schwartz[2001], ”Valuing American Option by Simulation: A
Simple Least-Squares Approach”, The Review of Financial Studies, Vol.14 No.1, 2001.
[23] Schweizer, Martin[2001], ”A Guided Tour through Quadratic Hedging Approaches”, Option Pricing,
Interest Rates and Risk Management, Cambridge University Press, 2001.
[24] Merton, R. C.[1971], ”Optimum Consumption and Portfolio Rules in a Continuous Time Model”,
Journal of Economic Theory, 3, 373-413.

―98―日本保険・年金リスク学会誌 Vol.4 No.1 2009

A dynamic optimal portfolio strategy for the insurance liability
exposed to uncertainty in the capital market
Shinichi Monden †Received 1st December 2008; received in revised form 3rd September 2009

Abstract
In current accounting system for japanese insurance company, the amount of the liability reserve basedon the insurance mathematical principle is recognized as value of the contract. However, when we plan theinvestment or hedge strategy, it is preferable to consider derivatives to which the value of the insurance
contract is written on the actuarial assumptions, and to refer to a fundamental uncertainty. But, thepractice of this action is not easy. The hrst cause is that all of actuarial assumptions are not tradable
in the capital market. For the second cause, in the case of level premium, hedge source are not always
enough. In this paper, we will improve Brandt, Goyal, Santa-Clara and Stroud[2005] to ALM-approach,and structure the hedging strategy for the variable annuity with minimum guarantee benehts under
the assumption that can trade a part of the fundamental risk in the capital market. As a result, an
approximation to derive the optimal solution of the general account ࠉ portfolio, to show a numericalanalysis of dynamic optimization.

Keyword㸸Incomplete Market, Multi-period Dynamic Optimization, Asset and Liability Management,
Liability Driven Investment, Variable Annuity†

Nomura Securities Co.,Ltd. / University of Tsukuba, Graduate School of Business Sciences
E-mail: smonden@js6.so-net.ne.jp―99―