日本保険・年金リスク学会誌 Vol.3 No.1 2008––––◊✲ㄽᩥ––––––––––––––––––––
ዎ⣙᮲௳ࡢኚ᭦࡜◚⥢ฎ⌮ࡢẚ㍑
㛛⏣ࠉఙ୍
2007 ᖺ 09 ᭶ 24 ᪥ᢞ✏2008 ᖺ 02 ᭶ 25 ᪥ཷ⌮

ᴫせ➨ 156 ᅇᅜ఍࡟࠾࠸࡚ಖ㝤ᴗἲࡢ୍㒊ᨵṇࡀ⾜ࢃࢀࡓ. ᚑ᮶ࡣ⤒Ⴀ◚⥢ฎ⌮ࡢ㐣⛬࡟࠾࠸࡚ࡢࡳチᐜࡉࢀ࡚࠸ࡓዎ⣙᮲௳ࡢኚ᭦ࢆ, ⥅⥆௻ᴗࢆ๓ᥦ࡜ࡋ࡚ྍ⬟࡜ࡍࡿࡶࡢ࡛࠶ࡿ. ◚⥢ฎ⌮ࡣ
㏱᫂ᛶࡀ㧗࠸ࡶࡢࡢ, ௻ᴗ౯್ࡀΎ⟬౯್࡟ࡼࡾホ౯ࡉࢀࡿࡓࡵ࡟ᦆኻࡀᣑ኱ࡍࡿྍ⬟ᛶࡀ㑊ࡅࡽࢀ࡞࠸. ฎ⌮᫬Ⅼࡢ඲ዎ⣙⪅࡟㢮ࡀཬࡧ, ᪩ᮇ㐺⏝࡟ࡣ㝈⏺ࡀ࠶ࡿࡇ࡜➼ࡀᣦ᦬ࡉࢀ࡚࠸ࡿ. ୍᪉, ዎ⣙᮲௳ࡢኚ᭦ࡣᶒ฼ࡢ⦰ῶ࡟ୗ㝈ࡀタࡅࡽࢀࡿࡓࡵ, ᚲࡎࡋࡶ඲ዎ⣙⪅ࡀᑐ㇟࡜ࡣ࡞ࡽ࡞࠸
ࡇ࡜, ኚ᭦ᑐ㇟ࡀಖ㝤ዎ⣙⪅࡟㝈ᐃࡉࢀࡿ➼ᗄࡘ࠿ࡢⅬ࡛, ◚⥢ฎ⌮࡜᰿ᮏⓗ࡟┦㐪ࡍࡿࡶࡢ࡜
࡞ࡗ࡚࠸ࡿ. ᮏ✏࡛ࡣ, ◚⥢ฎ⌮࡟㝈ᐃࡉࢀ࡚࠸ࡓᚑ๓࡟ᑐࡋ, ᙜヱไᗘࡢ᪂タࡀಖ㝤⪅ࡢ⾜ືཬࡧಖ㝤ዎ⣙⪅ࡢ฼┈࡟ዴఱ࡞ࡿᙳ㡪ࢆཬࡰࡋࡓࡢ࠿ࢆࣜ࢔࣭ࣝ࢜ࣉࢩࣙࣥࡢどⅬࡼࡾ☜ㄆࡍࡿ.
࣮࣮࢟࣡ࢻ㸸 㠀᏶ഛᕷሙ, ὴ⏕ドๆ౯᱁ホ౯, ࣜࢫࢡ⟶⌮, ಖ㝤ᩘ⌮1 ᗎㄽ
ᖹᡂ 15 ᖺ 7 ᭶ 18 ᪥, ➨ 156 ᅇᅜ఍࡟࠾࠸࡚ಖ㝤ᴗἲࡢ୍㒊ᨵṇࡀ⾜ࢃࢀ, ࠕዎ⣙᮲௳ࡢኚ᭦ࠖࡀ᪂タࡉࢀࡓ. 㐠⏝฼ᅇࡾࡀணᐃ฼⋡ࢆୗᅇࡿᙜ᫬ࡢ≧ἣࢆ㋃ࡲ࠼࡚, ⏕࿨ಖ㝤఍♫ࡀ᏶඲࡞◚⥢࡟㝗ࡿ๓ࡢẁ㝵࡛ணᐃ฼⋡ࢆᘬࡁୗࡆ, ࡑࢀ࡟ࡼࡾಖ㝤㔠㢠ࡢ๐ῶ➼ಖ㝤ዎ⣙ୖࡢಖ㝤ዎ⣙ࡢᶒ฼⦰ῶࡍࡿࡇ࡜ࢆྍ⬟࡜ࡍࡿࡶࡢ࡛
࠶ࡿ*1 .
ዎ⣙᮲௳ࡢኚ᭦ࡣዎ⣙⪅࡟ᑐࡋ࡚୙฼┈࡞ኚ᭦ࢆチᐜࡍࡿࡶࡢ࡛࠶ࡾ, ಖ㝤⪅࡟ࡼࡿᶒ฼⾜౑ࡣỴࡋ࡚ᮃࡲࡋ࠸஦ែ࡜ࡣ⪃࠼ࡽࢀ࡞࠸. ↛ࡿ࡟, ◚⥢ฎ⌮࡟㝈ᐃࡉࢀ࡚࠸ࡓᚑ๓࡜ẚ㍑ࡍࢀࡤ, ᙜヱไᗘࡢタ❧ࡣྰᐃࡉࢀ
ࡿࡶࡢ࡛ࡣ࡞࠸. ᖹᡂ 15 ᖺ 6 ᭶ 10 ᪥ࡢ➨ 156 ᅇ㈈ົ㔠⼥ጤဨ఍࡟࠾ࡅࡿṇ⏣ᩥ⏨ཧ⪃ேࡢⓎゝࡀཧ⪃࡜࡞ࢁ࠺.
➨୍ࡢᣦ᦬ࡣ㈨⏘ホ౯ᇶ‽ࡢᕪ␗࡛࠶ࡿ. ◚⥢ฎ⌮࡛ࡣΎ⟬౯್ࡀᇶ‽࡜࡞ࡿࡇ࡜࡟ᑐࡋ, ⥅⥆௻ᴗࢆ๓ᥦ
࡜ࡍࡿዎ⣙᮲௳ࡢኚ᭦࡛ࡣฎ⌮࡟క࠺ホ౯ῶࡀⓎ⏕ࡋ࡞࠸. ➨஧ࡢᣦ᦬ࡣዎ⣙⪅ࡢ㈇ᢸࡢᕪ␗࡛࠶ࡿ. ◚⥢ฎ⌮࡛ࡣ඲࡚ࡢዎ⣙⪅ࡀ㈐௵‽ഛ㔠ࢆ 10% ๐ῶࡉࢀࡿࡇ࡜࡟ᑐࡋ, ዎ⣙᮲௳ࡢኚ᭦࡛ࡣ㈐௵‽ഛ㔠ࡢ๐ῶࡣ⾜ࢃࢀ࡞࠸. ຍ࠼࡚, ணᐃ฼⋡ࡢᘬୗࡆỈ‽࡟ࡶୗ㝈ࡀタࡅࡽࢀ࡚࠸ࡿࡇ࡜࠿ࡽ, ㈇ᢸࢆᙉ࠸ࡽࢀࡿዎ⣙⪅ࡣ㝈ᐃࡉࢀࡿࡇ࡜࡟࡞ࡿ. ࡓࡔࡋ, ዎ⣙᮲௳ࡢኚ᭦࡟࠾࠸࡚ࡣ, ⤒῭ⓗ࡞㈇ᢸࡀዎ⣙⪅࡟㝈ᐃࡉࢀࡿ࡜࠸࠺ࡇ࡜࡟ࡣ␃ព
ࡍ࡭ࡁ࡛࠶ࢁ࠺. ➨୕ࡢᣦ᦬ࡣ᫬㛫ⓗไ⣙࡛࠶ࡿ. ◚⥢ฎ⌮ࡣྖἲ௓ධࢆ๓ᥦ࡜ࡍࡿ◚⏘ไᗘ࡛࠶ࡿࡓࡵ࡟, ᪩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@ics.hit-u.ac.jp*1 ᒣୗ཭ಙ [2005].

―1―ᮇ㐺⏝࡟ࡣ㝈⏺ࡀ࠶ࡿ࡜࠸࠺ࡇ࡜࡛࠶ࡿ.
࡜ࡇࢁ࡛, 㑅ᢥ⫥ࡢ㏣ຍࡣᚑ๓࡜ẚ㍑ࡋ࡚, ಖ㝤⪅ࡢ⾜ືཬࡧዎ⣙⪅฼┈࡟ዴఱ࡞ࡿᙳ㡪ࢆཬࡰࡋࡓࡢ࡛࠶ࢁ
࠺࠿. ࡇࢀࡣࣜ࢔࣭ࣝ࢜ࣉࢩࣙࣥࡢ୍౛࡜ࡋ࡚⪃࠼ࡿࡇ࡜ࡀ࡛ࡁࡿࡔࢁ࠺. ᮏ✏࡛ࡣ, ᶒ฼⾜౑๓ᚋࡢಖ㝤ዎ⣙࡟ಀࡿබṇ౯್ࢆ஺᥮ࡍࡿ࢜ࣉࢩࣙࣥ, ࡍ࡞ࢃࡕ, ᘬཷࡓಖ㝤ዎ⣙ࡢබṇ౯್ࢆ๐ῶࡍࡿᶒ฼࡜ࡋ࡚ዎ⣙᮲௳ࡢኚ᭦࡜◚⥢ฎ⌮ࢆࣔࢹࣝ໬ࡋ, ୧⪅ࡢ౯್ཬࡧᶒ฼⾜౑᫬ᮇࢆ☜ㄆࡋࡓ.

2 ไᗘࡢᴫせ⏕࿨ಖ㝤ዎ⣙࡛ࡣ, ⣙Ḱ࡟࠾࠸࡚ᇶ♏⋡ኚ᭦ᶒࢆ␃ಖࡋ࡞࠸㝈ࡾ, ዎ⣙⥾⤖᫬ࡢィ⟬ᇶ♏⋡ࡀಖ㝤ᮇ㛫࡟ரࡾ㐺⏝ࡉࢀࡿ. ⤒㦂ಖ㝤஦ᨾⓎ⏕⋡ࡢୖ᪼, ㈨⏘㐠⏝཰┈⋡ࡢపୗ➼࡟ࡼࡾணᐃࡋࡓィ⟬ᇶ♏⋡࡜ࡢ஋㞳ࡀ⏕ࡌ࡚ࡶ, ಖ㝤ዎ⣙⪅ࡢ୙฼┈࡜࡞ࡿࡼ࠺࡞ዎ⣙ෆᐜኚ᭦ࡣཎ๎࡜ࡋ࡚⾜ࢃࢀ࡞࠸. ๫వ㔠ศ㓄⋡ࡢ୍㒊ⱝࡋࡃࡣ඲࡚ࡀ๐ῶࡉࢀࡿࡔࡅ࡛࠶ࡿ.
↛ࡿ࡟, ィ⟬ᇶ♏⋡ಖド࡟ಀࡿᦆኻࡀ๫వࡢ⠊ᅖෆ࡟཰᮰ࡍࡿ࡜ࡣ㝈ࡽ࡞࠸. ๫వࢆ㉸࠼ࡓᦆኻࡀⓎ⏕ࡍࢀࡤ⮬ᕫ㈨ᮏ࡟ࡼࡾ⿵ሸࡏࡊࡿࢆᚓ࡞࠸. ᭦࡟ᦆኻࡀ⮬ᕫ㈨ᮏࢆ㉸㐣ࡍࢀࡤ, ᙜヱಖ㝤఍♫ࡣ⤒Ⴀ◚⥢࡟ࡼࡾᕷሙ࠿ࡽࡢ᧔㏥ࢆవ൤࡞ࡃࡉࢀࡿࡇ࡜࡜࡞ࡿ. ṧᛕ࡞ࡀࡽ 1996 ᖺࡢಖ㝤ᴗἲᨵṇᚋ, ᪤࡟ 7 ௳ࡢ◚⥢஦౛ࢆࢃࡀ
ᅜࡢ⏕࿨ಖ㝤ᕷሙ࡛ࡣ⤒㦂ࡋ࡚࠸ࡿ.
ࡉ࡚, ⤒Ⴀ◚⥢᫬࡟ࡣ⏕࿨ಖ㝤ዎ⣙⪅ಖㆤᶵᵓ࠿ࡽ㈨㔠᥼ຓࡀ⾜ࢃࢀࡿࡀ, ࡇࢀࡣ୙㊊㢠࡟ᑐࡋ‶㢠ࡀ඘ᙜࡉࢀࡿࢃࡅ࡛ࡣ࡞࠸. ㈨㔠᥼ຓࡣᙜヱ◚⥢఍♫ࡢ㈐௵‽ഛ㔠ࡢ 90% ࡟ᑐࡋ, ୙㊊ࡍࡿ㈨⏘㢠ࢆᑐ㇟࡜ࡍࡿࡶࡢ࡛
࠶ࡿ*2 . ࡍ࡞ࢃࡕ, ㈐௵‽ഛ㔠ࡢ 10% ๐ῶࢆ๓ᥦ࡜ࡋ࡚࠾ࡾ, ඲࡚ࡢಖ㝤ዎ⣙⪅ࡣ୍ᚊ࡟⤒῭ⓗ㈇ᢸࢆᙉไࡉ
ࢀࡿࡇ࡜࡟࡞ࡿ. ᙜ↛࡟, ୍⯡മᶒ⪅, ᇶ㔠ᣐฟ⪅ⱝࡋࡃࡣᰴ୺ࡶ⤒῭ⓗ㈇ᢸࢆ㈇࠺ࡇ࡜࡟࡞ࡿ.
ࡇࡢࡓࡵ, ᴗົཪࡣ㈈⏘ࡢ≧ἣ࡟↷ࡽࡋ࡚ಖ㝤ᴗࡢ⥅⥆ࡀᅔ㞴࡜࡞ࡿ⵹↛ᛶࡢ࠶ࡿሙྜ࡟ࡣዎ⣙᮲௳ࡢኚ᭦
ࢆྍ⬟࡜ࡍࡿࡼ࠺࡟, ➨ 156 ᅇᅜ఍࡟࠾࠸࡚ಖ㝤ᴗἲࡢᨵṇࡀ⾜ࢃࢀࡓ*3 . ᙜヱᨵṇࡣ◚⥢ฎ⌮ࢆ⾜ࢃࡎ, ⥅⥆௻ᴗࡢࡲࡲಖ᭷ዎ⣙ࡢணᐃ฼⋡ࢆᘬࡁୗࡆࡿࡇ࡜࡛ಖ㝤ዎ⣙⪅࡟ㄢࡏࡽࢀࡿ㈇ᢸࢆ㍍ῶࡋࡼ࠺࡜ࡍࡿࡇ࡜ࡀ┠ⓗ࡜ࡉࢀ࡚࠸ࡿ. ࡇࡢࡓࡵ, ዎ⣙᮲௳ࡢኚ᭦࡟ࡣḟᥖࡢไ㝈ࡀタࡅࡽࢀ࡚࠸ࡿ*4 .
㈐௵‽ഛ㔠๐ῶࡢ⚗Ṇୗ㝈ணᐃ฼⋡ࡢ☜ಖዎ⣙᮲௳ኚ᭦ࡢ᪥ࡲ࡛࡟✚ࡳ❧࡚ࡽࢀࡓ㈐௵‽ഛ㔠ࡣ๐ῶࡉࢀ࡞࠸.
ᨻ௧࡟ࡼࡾୗ㝈ணᐃ฼⋡ࡀᐃࡵࡽࢀࡿ*5 .

ᶆ‽㈐௵‽ഛ㔠ࡢィ⟬ᇶ♏*6 ࢆ᣺ࡾ㏉ࡿ࡜, ᖹᡂ 8 ᖺ 4 ᭶ 1 ᪥௨㝆ᖹᡂ 11 ᖺ 3 ᭶ 31 ᪥ࡲ࡛࡟㐺⏝ࡉࢀࡿணᐃ฼⋡ࡣ 2.75% ࡜ࡉࢀ࡚࠾ࡾ, ๓ᥖୗ㝈ணᐃ฼⋡࡟ࡣ㐩ࡋ࡚࠸࡞࠸. ᚑࡗ࡚, ኚ᭦ᑐ㇟ࡀᖹᡂ 8 ᖺ௨๓࡟࠾ࡅ
ࡿ 4% ࠿ࡽ 6% ྎࡢணᐃ฼⋡ࢆ㐺⏝ࡍࡿዎ⣙࡛࠶ࡿࡇ࡜ࡣ᫂ࡽ࠿࡛࠶ࡿ.
࡞࠾, ᙜヱไᗘ࡟࠾࠸࡚ࡣ◚⥢ฎ⌮࡜␗࡞ࡾ, ዎ⣙⪅ࡢ୍㒊ࡢࡳࡀ⤒῭ⓗ㈇ᢸࢆ⾜࠺ࡇ࡜࡜࡞ࡿࡓࡵ*7 , ᑗ᮶࡟࠾࠸࡚⤒Ⴀ⎔ቃࡀᨵၿࡋ, ⤒῭⎔ቃࡀዲ㌿ࡋࡓሙྜ࡟ࡣ, ዎ⣙᮲௳ࡢኚ᭦ᑐ㇟࡜࡞ࡗࡓዎ⣙⪅࡟ᑐࡋዎ⣙⪅㓄ᙜ, ๫వ㔠ࡢศ㓄ࡑࡢ௚ࡢ㔠㖹ࡢᨭᡶ࠸࡟ࡼࡾ฼┈㑏ඖࢆ⾜࠺ࡇ࡜ࡀྍ⬟࡜ࡉࢀ࡚࠸ࡿ. ࡓࡔࡋ, ࡇࡢሙྜ࡟ࡣ, ฼┈㑏ඖ᪉㔪ෆᐜࢆ஦๓࡟㏻▱ࡋ, ᐃḰ࡟グ㍕ཪࡣグ㘓ࡍࡿࡇ࡜࡜࡞ࡿ*8 .
*2 ಖ㝤ᴗἲ➨➨ 270 ᮲ࡢ 3(ಖ㝤ዎ⣙ࡢ⛣㌿➼࡟࠾ࡅࡿ㈨㔠᥼ຓ), ಖ㝤ዎ⣙⪅➼ࡢಖㆤࡢࡓࡵࡢ≉ูࡢᥐ⨨➼࡟㛵ࡍࡿ࿨௧➨ 50 ᮲ࡢ

5(ἲ➨ 270 ᮲ࡢ 3 ➨ 2 㡯➨ 1 ྕ࡟つᐃࡍࡿ⥲⌮ᗓ௧࣭ෆ㛶ᗓ௧࣭㈈ົ┬௧࡛ᐃࡵࡿ⋡). ࡞࠾, ᙜヱෆᐜࡣಖ㝤㔠㢠ࡢ 90% ࢆಖ㞀ࡍࡿࡶࡢ࡛ࡣ࡞࠸ࡇ࡜࡟␃ពࡀᚲせ࡛࠶ࡿ.
*3 ᖹᡂ 15 ᖺ 7 ᭶ 18 ᪥ᡂ❧. ྠᖺ 8 ᭶ 24 ᪥᪋⾜.
*4 ಖ㝤ᴗἲ➨ 240 ᮲ࡢ 4(ዎ⣙᮲௳ࡢኚ᭦ࡢ㝈ᗘ)*5 ಖ㝤ᴗἲ᪋⾜௧➨ 36 ᮲ࡢ 3(ዎ⣙᮲௳ࡢኚ᭦ࡢ㝈ᗘ) ࡟ࡼࡾ, ᖺ 3% ࡜ࡉࢀ࡚࠸ࡿ.
*6 ኱ⶶ┬࿌♧➨ 48 ྕ.
*7 ᶒ฼⦰ῶᑐ㇟ࡀዎ⣙⪅࡟㝈ᐃࡉࢀ࡚࠸ࡿࡀ, ዎ⣙⪅࡟㝈ᐃࡍࡿෆᐜ࡛ࡣ␗㆟⏦❧௳ᩘࡀಖ㝤ᴗἲ➨ 240 ᮲ࡢ 12 ➨ 4 㡯࡟つᐃࡍࡿチᐜᩘ࡛࠶ࡿࠕኚ᭦ᑐ㇟ዎ⣙⪅ᩘࡢ 1/10ࠖࡢ⠊ᅖ࡟཰᮰ࡍࡿ࡜ࡣ⪃࠼࡟ࡃ࠸. ಖ㝤ᴗἲ➨ 240 ᮲ࡢ 5 ➨ 3 㡯࡛ࡣᰴ୺⥲఍➼ࡢኚ᭦Ỵ㆟࡟㝿ࡋ, ኚ᭦ෆᐜ࡟ຍ࠼, ᇶ㔠ཬࡧ୍⯡മᶒ⪅ࡢമົ࡟ᑐࡍࡿྲྀࡾᢅ࠸, ⤒Ⴀ㈐௵➼➼ࢆ㏻▱ࡍࡿࡇ࡜ࢆ㚷ࡳࢀࡤ, ⌧ᐇⓗ࡞ᑐᛂ࡜ࡋ࡚ዎ⣙⪅௨እ࡟ࡶᦆኻࡢศᢸࢆồࡵࡿࡇ࡜ࡀ๓ᥦ࡜ࡉࢀ࡚࠸ࡿࡶࡢ࡜᥎ᐹࡉࢀࡿ.
*8 ಖ㝤ᴗἲ➨ 240 ᮲ࡢ 5(ዎ⣙᮲௳ࡢኚ᭦ࡢỴ㆟)―2―

日本保険・年金リスク学会誌 Vol.3 No.1 2008

ࡑࡋ࡚, ෆ㛶⥲⌮኱⮧࡟ࡼࡾዎ⣙᮲௳ࡢኚ᭦ࡢ⏦ࡋฟࡀᢎㄆࡉࢀࢀࡤ, ಖ㝤ዎ⣙࡟ಀࡿゎ⣙ࡢᴗົ೵Ṇࡀ⾜ࢃࢀࡿ*9 . ࡇࢀࡣฎ⌮ࢆ෇⁥࡟⾜࠺ࡓࡵ࡟ࡣᙜ↛ࡢࡇ࡜࡛࠶ࢁ࠺.

3 ホ౯ࣔࢹࣝࡢᴫせ3.1 ಖ㝤ᩘ⌮࡟ࡼࡿグ㏙
ಖ㝤ᩱ Px: n , ㈐௵‽ഛ㔠  Vx: n , ⤥௜⌧౯ Ax: n , ᖺ㔠⌧౯ äx: n ➼, ಖ㝤ᩘᏛୖ㐺⏝ࡉࢀࡿグྕࡢゎㄝࡣ

Appendix ࡟♧ࡋ, ௨ୗ࡛ࡣ᩿ࡾ↓ࡃᙜヱグྕࢆ㐺⏝ࡍࡿ.
3.1.1 ዎ⣙᮲௳ࡢኚ᭦
ᮏ✏࡛ࡣዎ⣙᮲௳ࡢኚ᭦ฎ⌮ࢆḟᥖࡢࡼ࠺࡟タᐃࡍࡿ.

(a) ኚ᭦ᑐ㇟࡜࡞ࡿィ⟬ᇶ♏⋡ࡣṚஸ⋡ཬࡧணᐃ฼⋡࡜ࡍࡿ.
(b) ィ⟬ᇶ♏⋡ࡣዎ⣙᮲௳ኚ᭦᪥᫬Ⅼࡢࡶࡢࢆ㐺⏝ࡋ, ୗ㝈ࡣタࡅ࡞࠸.
(c) ዎ⣙᮲௳ኚ᭦᪥᫬Ⅼࡲ࡛࡟✚ࡳ❧࡚ࡽࢀࡓ㈐௵‽ഛ㔠ࡣ๐ῶࡋ࡞࠸.
(d) ዎ⣙᮲௳ࡢኚ᭦࡟ࡼࡾ, ಖ㝤ᩱࡣቑ㢠ࡉࢀ࡞࠸.
(e) ዎ⣙᮲௳ࡢኚ᭦࡟ࡼࡾ, ಖ㝤㔠ࡣῶ㢠ࡉࢀࡿ.
ᖺ㱋 x ࡛ຍධࡋࡓಖ㝤ᮇ㛫 n, ಖ㝤㔠㢠 1 ࡢ㣴⪁ಖ㝤࡟ࡘ࠸࡚➨  ಖ㝤ᖺᗘᮎ࡟ዎ⣙᮲௳ኚ᭦ࡀ⾜ࢃࢀࡿࡶࡢ࡜ࡍࡿ. ࡇࡢ࡜ࡁ, ㈐௵‽ഛ㔠㢠ࡣ๐ῶࡉࢀ࡞࠸ࡢ࡛࠶ࡿ࠿ࡽ, ᡶ῭ಖ㝤㔠㢠ࡣḟᥖ࡟ࡼࡾồࡵࡽࢀࡿ. ࡓࡔࡋ, ⫪ࡢῧᏐ”start”ࡣ᮲௳ኚ᭦๓, ”rst”ࡣ 1 ᅇ┠᮲௳ኚ᭦ᚋࡢィ⟬ᇶ♏⋡ࡀ㐺⏝ࡉࢀ࡚࠸ࡿࡇ࡜ࢆព࿡ࡋ࡚
࠸ࡿ.
ኚ᭦ᚋᡶ῭ S1 =

 Vx: n

start

(1)

Ax+ : n f irst

ḟ࡟, ዎ⣙᮲௳ࡢኚ᭦๓ᚋ࡟࠾࠸࡚ಖ㝤ᩱࡀኚ໬ࡋ࡞࠸ࡶࡢ࡜ࡍࢀࡤ,ኚ᭦ᚋᖹ‽ S1 =Px: n start(2)

Px+ : n f irst

ࢆᚓࡿ. ᚑࡗ࡚, ➨ 1 ಖ㝤ᖺᗘᮎ࡟ዎ⣙᮲௳ኚ᭦ࡋࡓሙྜࡢ, ➨ 2 ಖ㝤ᖺᗘᮎ (1 < 2 ) ࡢ㈐௵‽ഛ㔠㢠ࡣḟᥖ࡟ࡼࡾᐃ⩏ࡉࢀࡿ.
2 V

f irst=1 Vx: nstart

Ax+1 : n1

f irst

irst
Afx+
+
: n
2

2

Px: n start
Px+1 : n1 f irst

2 1 Vx+1 : n1

f irst(3)

3.1.2 ◚⥢ฎ⌮ࡢሙྜ๓ᥖࡢ࡜࠾ࡾ, ⏕࿨ಖ㝤఍♫ࡢ⤒Ⴀ◚⥢᫬࡟ࡣ, ಖ㝤ዎ⣙⪅ಖㆤᶵᵓ࠿ࡽ㈐௵‽ഛ㔠ࡢ 90% ࢆᇶ‽࡜ࡋࡓ㈨㔠᥼ຓࡀ⾜ࢃࢀࡿ. ᚑࡗ࡚, ᮏ✏࡛ࡣ◚⥢ฎ⌮ࢆḟᥖࡢࡼ࠺࡟タᐃࡍࡿ. ࡲࡓ, 㐣ཤࡢ◚⥢ฎ⌮࡟ぢࡽࢀࡓ㛗ᮇ࡟ரࡿゎ⣙᥍㝖ࡢቑ㢠➼ࡣ⪃៖ࡋ࡞࠸.

(a) ኚ᭦ᑐ㇟࡜࡞ࡿィ⟬ᇶ♏⋡ࡣṚஸ⋡ཬࡧணᐃ฼⋡࡜ࡍࡿ.
(b) ィ⟬ᇶ♏⋡ࡣ◚⥢᫬Ⅼࡢࡶࡢࢆ㐺⏝ࡋ, ୗ㝈ࡣタࡅ࡞࠸.
(c) ◚⥢᫬Ⅼࡲ࡛࡟✚ࡳ❧࡚ࡽࢀࡓ ㈐௵‽ഛ㔠ࡢ 10% ࡀ๐ῶࡉࢀࡿ.
(d) ◚⥢࡟ࡼࡾ, ಖ㝤ᩱࡣቑ㢠ࡉࢀ࡞࠸.
*9 ಖ㝤ᴗἲ➨ 240 ᮲ࡢ 3(ᴗົࡢ೵Ṇ➼)―3―

(e) ◚⥢࡟ࡼࡾ, ಖ㝤㔠ࡣῶ㢠ࡉࢀࡿ.
ࡍ࡞ࢃࡕ, ◚⥢ᚋᡶ῭ಖ㝤㔠㢠ࡣ◚⥢ᚋᡶ῭ S1 =0.9 Vx: n start
Ax+ : n f irst

(4)࡜࡞ࡾ, ◚⥢ᚋᖹ‽ಖ㝤㔠㢠ࡣ◚⥢ᚋᖹ‽ S1 =Px: n start
Px+ : n f irst

(5)࡜࡞ࡿ. ᚑࡗ࡚, ࢖࣋ࣥࢺⓎ⏕᫬Ⅼ࡟࠾ࡅࡿ㈐௵‽ഛ㔠๐ῶࡢ᭷↓࡟ࡼࡾ, ◚⥢ฎ⌮࡜ዎ⣙᮲௳ኚ᭦ฎ⌮ࢆᕪู໬ࡍࡿࡇ࡜࡟࡞ࡿ.

3.2 ホ౯ࣔࢹࣝࡢᯟ⤌3.2.1 Ṛஸ⋡ಖ㝤ዎ⣙ཬࡧಖ㝤ዎ⣙࡟ໟྵࡉࢀࡿᶒ฼ࢆ฼Ꮚ⋡ཬࡧṚஸ⋡ࡢୖ࡟᭩࠿ࢀࡿࢹࣜࣂࢸ࢕ࣈ࡜ࡋ࡚⾲⌧ࡍࡿ.
➨୍ࡢㄢ㢟ࡣཎ㈨⏘ࡓࡿṚஸ⋡ࢆ࠸࠿࡟⾲⌧ࡍࡿ࠿࡛࠶ࡿ.
ඛ⾜◊✲ࢆᴫほࡍࢀࡤ, Milvesky and Promislow[2001] ࡛ࡣ, Gompertz ἲ๎ࢆᣑᙇࡋ, ᫬Ⅼ t, ᖺ㱋 x ࡢṚຊ x,t ࡟ࡘ࠸࡚, x,t = 0 et x+Yt , dYt = Yt dt + dWt ࡜ࡍࡿࡇ࡜࡟ࡼࡾᖹᆒᅇᖐᛶࢆ୚࠼࡚࠸
ࡿ. ᒣᮏಙ୍࣭ୖ⏣Ὀ୕ [2003], Dahl[2004] ➼࡛ࡣ, Hull-White, Cox-Ingersoll-Ross ➼ࡢࢫ࣏ࢵࢺ࣭࣮ࣞ
ࢺ࣭ࣔࢹࣝࢆṚຊ⾲⌧࡟௦᭰㐺⏝ࡋ࡚࠸ࡿ. Schrager[2006] ࡶྠᵝࡢ”Gaussian Stochastic Thiele ࣔࢹࣝ”,”Gaussian Stochastic Makeham ࣔࢹࣝ”ࢆᥦၐࡋ࡚࠸ࡿ. ࡲࡓ, Bis[2005] ࡣ Gompertz-Makeham ἲ๎t = m(t) + t · Yt ࢆᣑᙇࡋ, ≧ែኚᩘ Y ࡟ࢪࣕࣥࣉᣑᩓ㐣⛬ dYt = {ȳ(t)  Yt }dt + dWt  dJt ࢆ㐺⏝ࡋ࡚࠸ࡿ.
↛ࡿ࡟, Cairns, Blake and Dowd[2006] ࡀᣦ᦬ࡍࡿࡼ࠺࡟Ṛஸ⋡࡟ᖹᆒᅇᖐᛶࢆᣢࡓࡏࡿࡇ࡜࡟ࡣၥ㢟ࡀ࠶ࡿ࠿ࡶࡋࢀ࡞࠸. ఱᨾ࡞ࡽࡤ⌧ᅾୡ⏺ⓗ࡟ほࡉࢀࡿṚஸ⋡ࡢపୗࢺࣞࣥࢻ*10 ࡀᑗ᮶࡟࠾࠸࡚཯㌿ࡋ, ᝏ໬ࢺࣞࣥࢻࢆᣢࡘࡇ࡜ࢆ᝿ᐃࡍࡿࡓࡵ, ་Ꮫࡢ㐍Ṍ࡟ᑐࡋ࡚ṇᙜ໬ࡍࡿࡇ࡜ࡀᅔ㞴࡛࠶ࢁ࠺࡜⪃࠼ࡽࢀࡿࡓࡵ࡛
࠶ࡿ.
ࢪࣕࣥࣉࢆ㐺⏝ࡍࡿ௚ࡢ◊✲౛࡛ࡣ, Lin and Coxc[2006] ࡀቯ⁛ⓗ࡞Ṛஸ⋡ࡢኚືࢆ⾲⌧ࡍࡿࡓࡵ, ᑐᩘṇつ㐣⛬࡟ᇶ࡙ࡃᶆ‽Ṛஸ⋡ qt ࡟ࢪࣕࣥࣉ࣭࢖࣋ࣥࢺ࡟ࡼࡿࢩࣙࢵࢡࢆィୖࡋ,

½

q˜t =

qt Y[Nt Nth ]qt(ࢪࣕࣥࣉⓎ⏕; ☜⋡ p)(ࢪࣕࣥࣉᮍⓎ⏕; ☜⋡ 1  p)(6)࡜ࡋ࡚࠸ࡿ. ࡇࡇ࡛ࡣᖹᆒᅇᖐᛶࡣ཯ᫎࡉࢀ࡚࠸࡞࠸.
ࡲࡓ, Lin and Cox[2004a] ࡀᣦ᦬ࡍࡿࡼ࠺࡟, Ṛஸ⋡ᨵၿண࡟ࡶᅔ㞴ࡀక࠺. ῰㇂ᨻ᫛࣭⳹ᒣᐉ⬍ [2004]ࡣᴟ್⌮ㄽࢆ㐺⏝ࡍࡿࡇ࡜࡟ࡼࡿ᪥ᮏே⏨ᛶᑑ࿨ศᕸࡢୖ㝈್ࢆ᥎ᐃࡋ࡚࠸ࡿࡀ, Ṛஸ⋡⾲⌧ࣔࢹࣝ࡟࠾ࡅࡿ
ୖ㝈್タᐃࡣ௒ᚋࡢㄢ㢟࡛࠶ࡿ. ࡜ࡾࢃࡅ, ⏕Ꮡಖ㞀ࣜࢫࢡホ౯ࡢࡓࡵ࡟ࡣᙜヱࢸ࣮࣐ࡢゎỴࡀᚲせ࡜⪃࠼ࡽࢀࡿ.
ᮏ✏࡛ࡣ, Lee and Carter[1992] ࡢᥦၐࡍࡿ Lee-Carter ࣔࢹࣝ*11 ࢆ᥇⏝ࡍࡿ. Lee-Carter ࣔࢹࣝࡣ๓ᥖṚஸἲ๎࡟౫Ꮡࡏࡎ, ᫬Ⅼ t, ᖺ㱋 x ࡢṚஸ⋡ qx,t ࡢᑐᩘ್ࢆ,ln(qx,t ) = x +X

x t + x,t*10 ཌ⏕ປാ┬ࡀ 2007 ᖺ 7 ᭶ 26 ᪥࡟Ⓨ⾲ࡋࡓᖹᡂ 18 ᖺ⡆᫆⏕࿨⾲࡛☜ㄆ࡛ࡁࡿ.

*11 ᅜ❧♫఍ಖ㞀࣭ேཱྀၥ㢟◊✲ᡤ [2006] ࡛ࡣ, ಟṇ Lee-Carter ࣔࢹࣝࡀ㐺⏝ࡉࢀ࡚࠸ࡿ.

―4―

(7)

日本保険・年金リスク学会誌 Vol.3 No.1 2008

X
X

x = 1

t = 0࡜ศゎࡍࡿᡭἲ࡛࠶ࡿ.
ඛ࡟⤂௓ࡋࡓ Parametric ࣔࢹ࡛ࣝࡣ୍㒊ࡢᖺ㱋࡟࠾ࡅࡿ㐺ྜᗘࡀ≛≅࡜࡞ࡿࡀ, Lee-Carter ࣔࢹ࡛ࣝࡣᙜ
ヱၥ㢟Ⅼࢆ⦆࿴࡛ࡁࡿ. ࣃ࣓࣮ࣛࢱ , ,  ࡣ≉␗್ศゎ࡟ࡼࡾ᭱ᑠ஧஌᥎ᐃࢆ⾜࠼ࡤࡼ࠸.
ࡓࡔࡋ, Brouhns, Denuit and Vermunt[2002] ࡣᘧ (7) ࡟ࡼࡿண⤖ᯝ࡜⤒㦂್ࡢ஋㞳せᅉࡀ≉␗್ศゎࢆ

㏻ࡌࡓ᭱ᑠ஧஌᥎ᐃࡢ๓ᥦࢆ➼ศᩓ࡜ࡋ࡚࠸ࡿࡇ࡜࡜ᣦ᦬ࡋ, ࣃ࣓࣮ࣛࢱ᥎ᐃࢆ Newton-Raphson ἲ࡛⾜࠸,ㄗᕪ㡯࡟ Poisson ศᕸࢆ࠶࡚ࡣࡵ࡚࠸ࡿ. ࡲࡓ, Lee-Carter ࣔࢹࣝࡢ␃ពⅬ࡜ࡋ࡚, ་Ꮫࡢ㐍Ṍཬࡧ⎔ቃኚ໬࡟㛵ࡍࡿ㐣⛬ࢆ⪃៖ࡏࡎ, ᪤ Ṕ௨እࡢ᝟ሗࡣ཯ᫎࡉࢀ࡞࠸ࡇ࡜࡟ὀពࢆಁࡋ࡚࠸ࡿ. ᑠᬽཌஅ࣭㛗㇂ᕝ▱ᘯ

[2005], ᑠᬽཌஅ [2005] ࡣㄗᕪ㡯࡟ṇつศᕸࢆ㐣⛬ࡍࡿᶆ‽ⓗ࡞ Lee-Carter ࣔࢹࣝ࡜ Poisson ཮⥺ᙧᅇᖐࣔࢹࣝࢆẚ㍑ࡋ, ᖺ㱋ࣃ࣓࣮ࣛࢱ (, ) ࡣ୧⪅ࡢ᥎ᐃ್ࡀ୍⮴ࡍࡿࡶࡢࡢ, ᬺᖺࣃ࣓࣮ࣛࢱ () ࡟ࡣ┦ᙜ⛬ᗘࡢㄗᕪࡀ⏕ࡎࡿࡇ࡜, ᶆ‽ⓗ࡞ Lee-Carter ࣔࢹࣝࡢṧᕪศᯒ࠿ࡽ, 㧗㱋⪅ᒙ࡟࠾ࡅࡿㄗᕪ㡯ࡢศᩓ୙ᆒ୍ᛶࢆ☜ㄆࡋ, ศᩓ୙ᆒ୍ᛶࢆ᫂♧ⓗ࡟⪃៖ࡍࡿ Poisson ཮⥺ᙧᅇᖐࣔࢹࣝࡢ᪉ࡀಙ㢗ᗘࡀ㧗࠸ࡔࢁ࠺࡜ண᝿ࡋ࡚࠸ࡿ.
*12

3.2.2

ᗘኚ᥮

ᕷሙᛶཬࡧ௦᭰ᛶࢆ᭷ࡉ࡞࠸ಖ㝤ၟရ࡟ࡘ࠸࡚ࡢබṇ౯್ࡣ, ࣜࢫࢡㄪᩚࢆ཯ᫎࡋࡓᑗ᮶࢟ࣕࢵࢩࣗࣇ࣮ࣟ
ࡢᮇᚅ⌧ᅾ౯್࡟ࡼࡾᐃ⩏ࡉࢀࡿ. ࣜࢫࢡㄪᩚࡣࠕ๭ᘬ⋡ࠖࠕ☜⋡ᗘࠖࠕ࢟ࣕࢵࢩࣗࣇ࣮ࣟࠖ࡟ࡼࡾ⾜ࢃࢀ,Casualty Actuarial Task Force[2000] ࡛ࡣ๭ᘬ⋡ࢆㄪᩚࡍࡿ CAPM ἲ, ྲྀᘬ౯᱁ࢆᇶ‽࡜ࡍࡿ෌ಖ㝤ἲ, ㈨⏘࡜㈨ᮏࡢබṇ౯್ᕪ㢠ࡼࡾ㈇മࡢබṇ౯್ࢆホ౯ࡍࡿ㛫᥋ἲ, ᦆኻศᕸࡢ☜⋡ࢹ࢕ࣥࢢࢆ⾜࠺ᗘኚ᥮࡟ࡼࡾࣜࢫࢡ࣭࣮ࣟ

ᗘኚ᥮ἲ➼ࡀ౛♧ࡉࢀ࡚࠸ࡿ. ࡇࡇ࡛᪥ᮏᅜࡢ⏕࿨ಖ㝤ᕷሙࢆ᣺ࡾ㏉ࡿ࡜, 2008 ᖺ 2 ᭶⌧ᅾႠᴗᒎ㛤ࡍࡿ඲࡚ࡢಖ㝤ᘬཷ఍♫ࡀ㠀ୖሙ௻ᴗ࡛࠶ࡿ*13 . ࡇࡢࡓࡵ CAPM ἲཬࡧ㛫᥋ἲࡣᑐ㇟࠿ࡽ㝖እࡉࢀ
ࡿ. ෌ಖ㝤౯᱁࡟ࡣฟ෌࣭ཷ෌఍♫ࡢಙ⏝ຊཬࡧཷ෌఍♫ࡢᐜ㔞ࡀᙳ㡪ࡍࡿࡓࡵ, බṇ౯್ホ౯ࡢࡓࡵ࡟ࡣᙜヱ㒊ศࢆ᥍㝖ࡋ࡞ࡅࢀࡤ࡞ࡽ࡞࠸. ෌ಖ㝤ἲࡢ㐺⏝ࡣ, ༷ࡗ࡚ၥ㢟ࢆ」㞧࡟ࡋ࡚ࡋࡲ࠺ࡔࢁ࠺. ᮏ✏࡛ࡣಖ㝤఍♫ࡢ⤌⧊ᙧែ࡟ᇶ࡙ࡃไ⣙ࢆཷࡅࡎ, ₯ᅾⓗࢹࣜࣂࢸ࢕ࣈࢆྵࡴ඲࡚ࡢ࢟ࣕࢵࢩࣗࣇ࣮ࣟࡀホ౯ྍ⬟࡛࠶ࡿ*14ᗘኚ᥮ἲࢆ᥇⏝ࡍࡿ. ࡇࡇ࡛, ␃ពࡍ࡭ࡁⅬࡣṚஸ⋡࡟ࡘ࠸࡚ᕷሙࡣ㠀᏶ഛ࡜࠸࠺ࡇ࡜࡛࠶ࡿ.
㠀᏶ഛᕷሙ࡟࠾࠸࡚ࡣ, ࣜࢫࢡ୰❧☜⋡ᗘࡣ୍ព࡟ᐃࡲࡽ࡞࠸. ࡇࡢࡓࡵ, ☜⋡ᗘࡢ㑅ᢥᡭἲࡣ」ᩘࡢࡶ

ࡢࡀᥦၐࡉࢀ࡚࠸ࡿࡀ, ᮏ✏࡛ࡣ Distortion ኚ᥮*15 ࡟ᒓࡍࡿ Wang ኚ᥮*16 ࢆ᥇⏝ࡍࡿ.

Wang ኚ᥮ࢆ᥇⏝ࡍࡿ⌮⏤ࡣ, 㛛⏣ఙ୍ [2006] ࡛ᣦ᦬ࡋࡓ࡜࠾ࡾ࡛࠶ࡿ. ➨୍࡟ Risk Measures ࡜ࡋ࡚ࡢಖ
㝤ᩱཬࡧ㈐௵‽ഛ㔠ࡀ Coherent ᛶ*17 ࢆ᭷ࡍࡿࡇ࡜, ➨஧࡟㈨⏘࡜㈇മࢆᩚྜࡋ࡚ᢅ࠼ࡿࡇ࡜, ➨୕࡟ከኚ㔞࡬ࡢᣑᙇࡀᐜ࡛᫆࠶ࡿࡇ࡜*18 , ➨ᅄ࡟ኚ᥮ᑐ㇟ࡢࢧࣥࣉ࣭ࣝࣃࢫ, ᶆ‽ṇつศᕸ㛵ᩘཬࡧ㏫㛵ᩘࡀ⏝ពࡉࢀ࡚࠸ࡿࡇ࡜ࢆ๓ᥦ࡜ࡋ࡚ᩘ್ゎᯒࡀᐜ࡛᫆࠶ࡿࡇ࡜, ➨஬࡟ኚ᥮࡟㝿ࡋ࡚ࡣ✚⋡ࡢᏑᅾࢆせồࡋ࡞࠸ࡇ࡜, ࡜࠸࠺㛗ᡤࢆ᭷ࡍࡿࡓࡵ࡛࠶ࡿ.

*12

࡞࠾, ㄗᕪ㡯ࢆṇつศᕸ N (0,  2 ) ࡜ࡍࡿሙྜࡢ࠶࡚ࡣࡲࡾࡢⰋࡉࡣ, ࣃ࣓࣮ࣛࢱ ,  ࡢḟᩘ࡟౫Ꮡࡍࡿྍ⬟ᛶࢆ➨ 27 ᅇ JAFEEኟᏘ኱఍ (2007 ᖺ 8 ᭶ 3 ᪥, ᫂἞኱Ꮫ㥴Ἑྎ࢟ࣕࣥࣃࢫ) ࡢⓎ⾲᫬࡟᳃ᖹ∝୍㑻ᩍᤵ (᪩✄⏣኱Ꮫ) ࡼࡾᣦ᦬ࡉࢀࡓ. ᥎ᐃ᫬ࡢḟᩘỴᐃ࡟ࡶ␃ពࡀᚲせ࡜࡞ࢁ࠺.

*13 ᰴᘧ఍♫ T&D ࣮࣍ࣝࢹ࢕ࣥࢢࢫཬࡧࢯࢽ࣮ࣇ࢕ࢼࣥࢩ࣮ࣕࣝ࣍ࣝࢹ࢕ࣥࢢࢫᰴᘧ఍♫ࡣᣢᰴ఍♫࡛࠶ࡾ, ᘬཷ఍♫࡛ࡣ࡞࠸.

*14 American Academy of Actuaries[2002] ࡢ♧ࡍࠕᑗ᮶࢟ࣕࢵࢩࣗࣇ࣮ࣟࡢᮇᚅ⌧ᅾ౯್ࠖࡢホ౯ཎ๎ࡢ➨୕ཎ๎࡛ࡣ, ඲࡚ࡢ࢟ࣕࢵࢩࣗࣇ࣮ࣟࢆホ౯ᑐ㇟࡜ࡍࡿࡇ࡜ࡀせồࡉࢀ࡚࠸ࡿ.

*15 Distortion ኚ᥮࡟ࡘ࠸࡚ࡣ, McLeish, D. L. and R. Mark Ressor[2003] ࡟ヲࡋ࠸.
*16 Wang ኚ᥮ࡢヲ⣽࡟ࡘ࠸࡚ࡣ Wang[2002][2003][2004] ࢆཧ↷ࡉࢀࡓ࠸.

*17 Coherent Risk Measures ࡟ࡘ࠸࡚ࡣ Artzner et al.[1999] ࢆཧ↷ࡉࢀࡓ࠸.

*18 ከኚ㔞 Wang ኚ᥮࡟ࡘ࠸࡚ࡣ Kijima[2006], ᮌᓥṇ࣭᫂⏣୰ᩗ୍ [2007] ࢆཧ↷ࡉࢀࡓ࠸.

―5―

Wang ኚ᥮࡛ࡣ, Ṛஸ⋡ qx,t ࡢศᕸ㛵ᩘ F (·) ࢆḟᘧ࡟ࡼࡾኚ᥮ࡍࡿ.
F Q (qx,t ) = [1 [F P {qx,t )} 

t]

(8)

ࡍ࡞ࢃࡕ,

[1 {F (qx,t )}  t ]dQ|F t =dP[1 {[F (qx,t )}]࡜࡞ࡿ. ࡓࡔࡋ,(9)(·) ࡣᶆ‽ṇつศᕸࡢ☜⋡ᐦᗘ㛵ᩘ, (·) ࡣศᕸ㛵ᩘ, ࣃ࣓࣮ࣛࢱ

t ࡣࣜࢫࢡࡢᕷሙ౯᱁࡛

࠶ࡾ, ࡇࡇ࡛ࡣ᫬㛫࡟౫Ꮡࡍࡿࡇ࡜ࢆ๓ᥦ࡜ࡋ࡚࠸ࡿ.
ㄗᕪ㡯࡟ṇつศᕸࢆ࠶࡚ࡣࡵࢀࡤ, Pᗘୗ࡟࠾࠸࡚ (7) ᘧࡣP
ln(qx,T) = x +࡛࠶ࡿࡢ࡛,XP
x T + WT

X
qx,T P |Ft = qx,t P exp[

(10)P
x T + WT t ](11)

WTP ࡟ᑐࡋ࡚ Wang ኚ᥮ࢆ㐺⏝ࡍࢀࡤ2dQ$ W P  =e T T 2
dPࢆᚓࡿ. ᚑࡗ࡚, Qᗘୗ࡟࠾࠸࡚ࡣ

qx,T Q |Ft = qx,t Q exp[

X

x T + ( T


T

(12)

t

t) + WTQt ]࡜࡞ࡿ.

3.2.3 ዎ⣙᮲௳ኚ᭦ᶒࡢ౯್ (ᑗ᮶࡟࠾ࡅࡿ฼┈㑏ඖࢆ᝿ᐃࡋ࡞࠸ሙྜ)ᮏ✏࡛ࡣಖ㝤఍♫ࡀಖ᭷ዎ⣙ࡢබṇ౯್ῶ౯ࢆ᭱኱໬ࡉࡏࡿࡼ࠺࡟ዎ⣙᮲௳ࡢኚ᭦ⱝࡋࡃࡣ◚⥢ฎ⌮ࢆ⾜࠺ࡶࡢ࡜᝿ᐃࡍࡿ. ᚑࡗ࡚, ኚ᭦๓බṇ౯್ࢆ⾜౑౯᱁, ኚ᭦ᚋබṇ౯್ࢆཎ㈨⏘࡜ࡍࡿ Put Otion ࡟ࡼࡾ୧⪅ࡢᶒ฼౯್ࢆᐃ⩏ࡍࡿ.
ᮏ✏࡛ࡣዎ⣙᮲௳ኚ᭦ᶒ࡟ࡼࡾಖ㝤ᩱࡣቑ㢠ࡉࢀࡎ, ಖ㝤㔠ࡢࡳࡀῶ㢠ࡉࢀࡿࡇ࡜ࢆ๓ᥦ࡜ࡋ࡚࠸ࡿ. ᚑࡗ
࡚, ➨  ᫬Ⅼ࡟ዎ⣙᮲௳ኚ᭦ࡉࢀࡓ㈐௵‽ഛ㔠බṇ౯್ࡣḟᥖࡢࡼ࠺࡟⾲⌧࡛ࡁࡿ.
f irst,M V
0 Vx+ : n

=(

start
 Vx: nf irstAx+ : n

+

start
Px:
n
f irstPx+ : nVstart M V)AM Px:
n äx+ : nx+ : n(13)ᶒ฼⾜౑ࡉࢀࡿዎ⣙᮲௳ኚ᭦ᶒࡢ฼ᚓࡣ
Vf irst,M V
P ayof f ( ) = max( Vx:start,M
0 Vx+
, 0)
n
: n

= max[(1 start Vx: nf irstAx+ : n



start
Px:
n
f irstPx+ : n

V
)AM
, 0)
x+ : n

(14)࡟ࡼࡾᐃ⩏ࡉࢀࡿࡢ࡛, ᫬Ⅼ t ࡟࠾ࡅࡿዎ⣙᮲௳ኚ᭦ᶒࡢ౯್ࡣR

V alue(RD;  | Ft ) = sup E Q [e t ru du P ayof f ( ) | Ft ]

(15)



࡜ࡋ࡚ホ౯ࡍࢀࡤࡼ࠸.
◚⥢ฎ⌮ࡢሙྜ࡟ࡣ (14) ᘧࡀḟᥖࡢࡼ࠺࡟ኚ໬ࡍࡿ.

P ayof f ( ) = max[(1 

0.9 Vx:start
n
irst
Afx+
: n

―6―



start
Px:
n
f irstPx+ : n

V
)AM
, 0)
x+ : n(16)

日本保険・年金リスク学会誌 Vol.3 No.1 2008

3.2.4 ዎ⣙᮲௳ኚ᭦ᶒࡢ౯್ (ᑗ᮶࡟࠾ࡅࡿ฼┈㑏ඖࢆ᝿ᐃࡍࡿሙྜ)ዎ⣙᮲௳ኚ᭦ᚋ࡟࠾࠸࡚, ኚ᭦ᑐ㇟ዎ⣙⪅࡟ᑐࡋ࡚฼┈㑏ඖࢆ⾜࡞࠺ሙྜࡣ Compound Option ࡟ࡼࡾၥ㢟タᐃࡍࡿࡇ࡜ࡀ࡛ࡁࡿ. ࡍ࡞ࢃࡕ, ᑗ᮶࡟࠾ࡅࡿ฼┈㑏ඖࡢᶒ฼ࢆホ౯ࡋ, ᙜヱ౯್ࢆ᥍㝖ࡋࡓዎ⣙᮲௳ኚ᭦ᶒ
ࡢ฼ᚓࢆᐃ⩏ࡍࢀࡤࡼ࠸. ฼┈㑏ඖࢆ᝿ᐃࡋ࡞࠸ሙྜ࡟ẚ㍑ࡋ, ዎ⣙᮲௳ኚ᭦ᶒ౯್ࡣᙜ↛࡟ῶᑡࡍࡿࡇ࡜࡟
࡞ࡿ.
␃ពࡍ࡭ࡁⅬࡣ, ฼┈㑏ඖ⟇ࡢ㑅ᢥ࡛࠶ࡿ. ๐ῶࡉࢀࡓಖ㝤㔠㢠ࡢᅇ᚟ᡭẁ࡟ࡼࡾ౯್ホ౯ࣔࢹࣝࡀኚ໬ࡍ
ࡿࡓࡵ࡛࠶ࡿ. ➨୍ࡢᡭἲࡣィ⟬ᇶ♏⋡ࢆኚ᭦ࡏࡎ, ๫వ㔠ศ㓄࡟ࡼࡾಖ㝤㔠ቑ㢠ࢆᅗࡿࡶࡢ࡛࠶ࡿ. ࡇࡢሙྜ࡟ࡣዎ⣙᫬ (ῶ㢠๓) ಖ㝤㔠㢠ࢆ㉸㐣ࡋࡓᚋࡶಖ㝤ᩱᨭᡶࡀ⥅⥆ࡉࢀ, ཎ㈨⏘࡜ࡋ࡚ಖ᭷㈨⏘ࡀ㏣ຍࡉࢀࡿࡇ࡜
࡜࡞ࡿ. ➨஧ࡢᡭἲࡣዎ⣙᮲௳ኚ᭦ᚋࡢබṇ౯್ཬࡧ㑏ඖ฼┈ࢆ㈈※࡜ࡋ࡚㌿᥮ࢆ⾜࠸, ዎ⣙᫬ಖ㝤㔠㢠࡟ᙜヱᡶ῭ಖ㝤㔠㢠ࡀ୙㊊ࡍࡿ㒊ศࢆ⥅⥆ಖ㝤ᩱࡢᣐฟ࡟ࡼࡾ㉎ධࡍࡿࡶࡢ࡛࠶ࡿ. ࡓࡔࡋ, ಖ㝤ዎ⣙⪅ࡀዎ⣙᫬ಖ㝤ᩱࢆ㉸㐣ࡍࡿಖ㝤ᩱ㈇ᢸࢆチᐜࡍࡿࡇ࡜ࢆ᝿ᐃࡋ࡞࠸. ᮏ✏࡛ࡣᚋ⪅ࢆ᥇⏝ࡍࡿ.
➨ 2 ᫬Ⅼ࡟ಖ㝤఍♫ࡀዎ⣙᮲௳ኚ᭦ᚋ, ➨ 1 ᫬Ⅼ (2 < 1 ) ࡟ಖ㝤ዎ⣙⪅ࡀ㌿᥮ࡍࡿሙྜࡢಖ㝤ᩱཬࡧಖ㝤㔠㢠ࡣḟᥖࡢ࡜࠾ࡾ࡜࡞ࡿ. ࡓࡔࡋ, SD ࡣኚ᭦ᑐ㇟ዎ⣙࡟ᑐࡍࡿ฼┈㑏ඖ⤥௜㢠࡛࠶ࡿ.
start
second㌿᥮ᚋ P = max[ [min[ Px:n , (1  ㌿᥮ᚋᡶ῭ S)Px+ : n ] , 0 ]1f irststart
Ax+ : nSD + 2 Vx:start
Px:
n
1
1
n
㌿᥮ᚋᡶ῭ S =+irst
f irstAsecond
Afx+
P
x+1 : n1x+ : n
: n
2

2

2

(17)

1

f irst1 2 Vx+ : n222

Asecondx+ : n
1

1

(18)

㌿᥮ᚋᖹ‽ S = max[ [min[

start
Px:
n

, 1  ㌿᥮ᚋᡶ῭ S ] , 0 ]second
Px+
: n
1

(19)

1

ᚑࡗ࡚, ➨ 1 ᫬Ⅼ࡟࠾ࡅࡿ㌿᥮ᶒ౯್ࡣ
V alue(CV ; 1 | F2 ) = sup E Q [eR

1 r duu
2

1

P ayof f (1 ) | F2 ](20)࡜ࡍࢀࡤࡼ࠸. ࡓࡔࡋ,V
start M V
P ayof f (1 ) = max[ (㌿᥮ᚋ S  ㌿᥮๓ S)AM (㌿᥮ᚋ P  Px:n )äx+ : n , 0 ]x+ : n
1

1

1

1

(21)

࡛࠶ࡿ. ࡇࡇ࡛, ㌿᥮ᚋ S ࡣ㌿᥮ᚋᡶ῭ S ࡜㌿᥮ᚋᖹ‽ S ࡢྜィ㢠࡛࠶ࡾ, ㌿᥮๓ S ࡣዎ⣙᮲௳ኚ᭦ⱝࡋࡃࡣ
◚⥢ฎ⌮ᚋࡢᡶ῭ S ཬࡧᖹ‽ S ࡢྜィ㢠࡛࠶ࡿ.
࢜ࣉࢩࣙࣥࡢୖ࡟᭩࠿ࢀࡿ࢜ࣉࢩࣙࣥ࡜ࡋ࡚ホ౯ࡍࡿࡢ࡛࠶ࡿ࠿ࡽ, ➨ 2 ᫬Ⅼ࡟࠾ࡅࡿዎ⣙᮲௳ኚ᭦ᶒࡢ฼
ᚓࡣᘧ (14) ࡼࡾ (20) ࢆ᥍㝖ࡍࢀࡤࡼ࠸. ⤖ᯝ࡜ࡋ࡚, ᫬Ⅼ t ࡟࠾ࡅࡿዎ⣙᮲௳ኚ᭦ᶒࡢ౯್ࡣ
R 2

V alue(RD; 2 | Ft ) = sup E Q [e t2࡟ࡼࡾᑟฟࡉࢀࡿ.

―7―

ru du

P ayof f (2 ) | Ft ](22)

4 ᩘ್ゎᯒ⤖ᯝ
4.1 Lee-Carter ࣔࢹࣝࡢࣃ࣓࣮ࣛࢱ᥎ᐃ
⡆᫆⏕࿨⾲ࡢ⏨ᛶṚஸ⋡ࢆධຊ᝟ሗ࡜ࡋ, ᘧ (7) ࡢࣃ࣓࣮ࣛࢱ᥎ᐃࢆ⾜࠺. ࡇࡢࡓࡵ࡟ࡣ,T
 X
X

minimize{ln(qx,t )  (x +nXx t )}

2(23)

x=0 t=1

subject L1 [0 , · · · ,  ;

X
to
x =1

0, · · · , ; 1 , · · · , T ]

x=0
T
X

t = 0t=1ࢆゎࡅࡤࡼ࠸.
ᮏ✏࡛ࡣ 1987 ᖺ࠿ࡽ 2005 ᖺࢆほᮇ㛫࡜ࡋࡓࣃ࣓࣮ࣛࢱ᥎ᐃࢆ≉␗್ศゎ࡟ࡼࡾ⾜࠸, ᑗ᮶ศࡢ t ࢆ

ARIMA ࡟ࡼࡾ᥎ᐃࡋࡓ. ㄗᕪ㡯ࡣᖺ㱋ཬࡧ᫬Ⅼ࡟౫Ꮡࡋ࡞࠸ṇつศᕸ N (0,  2 ) ࢆ㐺⏝ࡋ,  ࡢ᥎ᐃ⤖ᯝ࡜ࡋ
࡚ 0.03456 ࢆᚓࡓ.
ᅗ1

Lee-Carter Model ࡢȘ (ᖺ㱋)

ᅗ3

ᅗ2

Lee-Carter Model ࡢș (ᖺ㱋)

ᅗ4Lee-Carter Model ࡢȚ―8―Ț (1987-2005)

日本保険・年金リスク学会誌 Vol.3 No.1 2008

⾲1

Lee-Carter Model ࡟࠾ࡅࡿࣃ࣓࣮ࣛࢱ

ᖺ㱋



1

0

4 5.51010.020840.013910

4 8.9957

0.0222

40.062920

4 7.2709

0.01610.028830

4 7.1823

0.001540

4 6.5086

50

2

4

3

4

0.02970.65035

40.35400.15254

1.27890.88470.049840.18160.17370.01670.060040.1787

4

0.19410.00280.06240.02104

0.06804

0.41804 5.5243

0.00550.01150.049460

4 4.5930

0.0101

40.012370

4 3.6958

0.008140.001080

4 2.6523

0.01260.004340.015390

4 1.6839

0.01050.0043

4

0.0279⾲2

4

40.0567

0.462840.01830.08950.0312
4

0.05320.37710.220140.06090.076940.10680.4246

Lee-Carter Model ࡟࠾ࡅࡿࣃ࣓࣮ࣛࢱ (2)

ᖺᩘ

1

2

3

4

1987

11.8330

0.85250.37700.01301988

11.9430

0.83060.1436

40.01220.0287

1989

10.24900.469940.0873

4

0.02510.0181

1990

9.99790.35534

0.44641991

7.35120.15104

0.4831

1992

7.856340.2654

4

1993

5.7392

4

0.569841994

4.3411

4

1995

7.03380.01634

5
4

4

0.06560.02180.01410.03480.38180.00250.01080.30490.000840.00370.85700.06390.0131

4

0.04194

1.07650.53830.035140.01151996

40.393740.6515

40.05631997

4

2.47404

0.4675

40.07551998

40.408840.07121999

4

2.4175

2000

4

2001

4

0.03860.010840.02640.05220.2701

40.00940.14540.750740.0032

6.38780.03050.14870.026849.79750.29980.00670.01170.01272002

4

12.17800.31000.24570.00020.0027

2003

4

13.6380

0.55340.15460.021240.01032004

4

14.80400.019140.110240.016540.01932005

4

13.8480

0.0585

40.2626

40.054840.0210

4

4

―9―

4

0.01980.064840.0207ᅗ 1 ࠿ࡽᅗ 4 ࡣほ2005 ᖺࠖ࡟༊ศࡋ

ࢹ࣮ࢱࢆࠕ1962 ᖺ㸫 2005 ᖺࠖࠕ1980 ᖺ㸫 2005 ᖺࠖࠕ1987 ᖺ㸫 2005 ᖺࠖࠕ1996 ᖺ㸫

*19᥎ᐃࡋࡓ⤖ᯝ࡛࠶ࡿ. ᅗ 1 ࡼࡾ, ᖺ㱋せᅉ x ࡣほ࠼ࡽࢀࡿ. ↛ࡿ࡟, ᖺ㱋せᅉ

x ࡣほᮇ㛫࡟ࡣ౫ᏑࡏࡎᏳᐃࡍࡿࡶࡢ࡜⪃ᮇ㛫࡟ࡼࡾᙧ≧ࡣ኱ࡁࡃኚ໬ࡋ࡚࠸ࡿ. ࡲࡓ, ᫬㛫せᅉ t ࡣ᭤⥺ࡢᙧ≧ࡣఝ࡚࠸ࡿࡶࡢࡢ, Ỉ‽ࡣ␗࡞ࡗ࡚࠸ࡿ. ᙜヱ⤖ᯝ࡟ࡣ᭱⤊ᖺ㱋ࡢᕪ␗ࡀᙳ㡪ࡋ࡚࠸ࡿࡇ࡜ࡶ᝿ᐃࡉࢀࡿࡀ,Brouhns, Denuit and Vermunt[2002] ࡢࠕ་Ꮫࡢ㐍Ṍཬࡧ⎔ቃኚ໬࡟㛵ࡍࡿ㐣⛬ࢆ⪃៖ࡏࡎ, ᪤ Ṕ௨እࡢ᝟ሗࢆ཯ᫎࡋ࡞࠸ࠖ࡜ࡍࡿᣦ᦬ࢆ㚷ࡳࢀࡤ, ほᮇ㛫ࢆ㛗ᮇ໬ࡍࡿࡇ࡜ࡣᅇ㑊ࡍࡿࡇ࡜ࡀ↓㞴࡛࠶ࡿ࡜ࡶゎ㔘࡛ࡁࡼ࠺.

4.2 Wang ኚ᥮ࡢࣃ࣓࣮ࣛࢱ᥎ᐃ
ᣋ✏ [2006] ࡛ࡣࡢ≧ែኚ໬ࢆ᝿ᐃࡋ࡚࠸࡞࠸. ↛ࡿ࡟, Ṛஸ⋡పୗ࡟క࠸ᑗ᮶ࡢ⮬↛ಖ㝤ᩱࡀపୗࡍࡿࡇ

࡜ࢆ㚷ࡳࢀࡤ, ᙜ↛࡟ࣜࢫࢡࡢᕷሙ౯᱁ࡶ㐃㙐ࡋ࡚పୗࡍ࡭ࡁ࡛࠶ࢁ࠺. ࡑࡇ࡛ᮏ✏࡛ࡣ, ྛ᫬Ⅼ࡟࠾ࡅࡿணᐃ
฼⋡ཬࡧணᐃṚஸ⋡*20 ࡟ࡼࡾホ౯ࡉࢀࡿ⮬↛ಖ㝤ᩱ࡜ᩚྜࡍࡿࡼ࠺࡟ Wang ኚ᥮ࡢࣃ࣓࣮ࣛࢱ (it ; qx,t ) ࢆ
Ỵᐃࡋࡓ.
R t+1

E Q [e tPt1Қ (it ; qx,t ) =ᶆ‽⏕࿨⾲ru du Qqx,t+1 | Ft ] = vt qx,tᶆ‽⏕࿨⾲

k=0 [ln vk qx,k2Q ln B(k, k + 1)qx,k]  [ x (t  0 ) + 2 t]

 t

(24)
ࡇࡇ࡛ it ࡣ➨ t ᫬Ⅼ࡟࠾ࡅࡿணᐃ฼⋡, B(t, T ) ࡣ➨ T ᫬Ⅼ࡟‶ᮇࢆ㏄࠼ࡿಙ⏝ࣜࢫࢡࡢ࡞࠸๭ᘬമࡢ➨ t ᫬Ⅼ࡟࠾ࡅࡿ౯᱁࡛࠶ࡿ. ࡞࠾, ᘧ (24) ࡢᑟฟ㐣⛬ࡣ Appendix ࡟♧ࡍ.
⾲3⮬↛ಖ㝤ᩱィ⟬ࡢ㐺⏝ணᐃ฼⋡ᖺ㱋

0Wang ኚ᥮࡟࠾ࡅࡿࣃ࣓࣮ࣛࢱ (i1 ; qx,1 )1.5%
428.96672.0%
429.1089

2.5%
429.2503

4

⡆᫆

ᶆ‽

3.0%

⏕࿨⾲

⏕࿨⾲

29.39110.00298

0.0010810

13.4273

13.2851

13.1436

13.00280.00009

0.0001420

12.5398

12.3976

12.256112.11530.000560.0008430

3.98243.8402

3.69873.55790.000740.00086400.80000.65790.51640.37560.001440.0014850

40.312340.454540.596040.73680.00358

0.0036560

41.25614

1.3983

4

1.539741.68050.00894

0.0083470

40.134840.27704

0.4185

4

0.55930.02138

0.0219380

1.03790.89570.75430.61350.060310.0603990

3.50443.36233.22083.08000.163120.17900(ὀ 1) ⡆᫆⏕࿨⾲ࡣࠕᖹᡂ 18 ᖺ⡆᫆⏕࿨⾲ࠖࡢ⏨ᛶࡢ⋡࡛࠶ࡿ.
(ὀ 2) ᶆ‽⏕࿨⾲ࡣࠕ⏕ಖᶆ‽⏕࿨⾲ 2007(Ṛஸಖ㝤⏝)ࠖࡢ⏨ᛶࡢ⋡࡛࠶ࡿ.

*19 ⡆᫆⏕࿨⾲ࡢ᭱⤊ᖺ㱋  ࢆᇶ‽࡜ࡋࡓࢢ࣮ࣝࣆࣥࢢ࡛࠶ࡿ.

*20 ᫬Ⅼ 0 ࡟࠾࠸࡚ࡣ, ⏕ಖᶆ‽⏕࿨⾲ 2007(Ṛஸಖ㝤⏝) ࢆ㐺⏝ࡍࡿ.

―10―

日本保険・年金リスク学会誌 Vol.3 No.1 2008ᐃ⩏ᘧࡼࡾ᫂ࡽ࠿࡞ࡼ࠺࡟,↷ࡍࡿࡇ࡜࡟࡞ࡿ.

ࡣṚஸ⋡ཬࡧ฼Ꮚ⋡ࡢ≧ែ㑄⛣࡟ࡘ࠸࡚, ┤㏆ᐇ⦼ࡢࡳ࡞ࡽࡎ඲࡚ࡢᒚṔࢆཧࡢ┤ឤⓗ࡞ゎ㔘ࡣ, ಖ㝤ᩱཬࡧ㈐௵‽ഛ㔠⟬ᐃࡢᇶ♏࡜࡞ࡿ⏕࿨⾲࡟࠾ࡅࡿᏳ඲๭ቑࡢ௦

࡛᭰࠶ࡿ. ࡇࡢࡓࡵ 0 ṓཬࡧ 50  70 ṓ࡟ぢࡽࢀࡿࡼ࠺࡞ࠕ㈇್ࢆ࡜ࡿ ࠖࡣ㐪࿴ឤࢆぬ࠼ࡿ࡜ࡇࢁ࡛࠶ࢁ࠺
ࡀ, ࡇࢀࡣᙜヱ㒊ศ࡟࠾࠸࡚ᖹᡂ 18 ᖺ⡆᫆⏕࿨⾲*21 ࡢ⋡ࡀᶆ‽⏕࿨⾲ 2007(Ṛஸಖ㝤⏝) ࡢ⋡ࡼࡾࡶ㧗ࡃ࡞ࡗ࡚࠸ࡿࡓࡵ࡛࠶ࡿ.
࡞࠾, ᅗ 3 ࡟♧ࡍ Lee-Carter ࣔࢹࣝࡢ᫬㛫せᅉࡢࣃ࣓࣮ࣛࢱ t ࠿ࡽࡶ☜ㄆ࡛ࡁࡿࡼ࠺࡟, Ṛஸ⋡ࡣ㢧ⴭ࡞పῶഴྥ࡟࠶ࡿ. ࡇࢀࡣୡ⏺ⓗ࡟ほࡉࢀࡿࢺࣞࣥࢻ࡛࠶ࡿ*22 . 㐣ཤ࡟ほࡉࢀࡿࢺࣞࣥࢻࢆ཯ᫎࡍࡿࡇ࡜࡟

࡞ࡿࡓࡵ, ᑗ᮶࡟࠾ࡅࡿணᐃṚஸ⋡ࡢỈ‽ࡶ᫬㛫⤒㐣࡟క࠸పୗࡋ, 㐃㙐ࡋ࡚⮬↛ಖ㝤ᩱࡀపୗࡍࡿ. ྛ᫬Ⅼ,ྛ≧ែ࡟࠾ࡅࡿ⮬↛ಖ㝤ᩱ࡜ᩚྜࡉࡏࡿࡼ࠺࡟Ỵᐃࡉࢀࡿࡓࡵ, ⤖ᯝ࡜ࡋ࡚ࡢỈ‽ࡶపୗࡍࡿ. ࡇࢀࡣ⾲ 4

ཬࡧᅗ 5 ࡟࠾࠸࡚☜ㄆࡍࡿࡇ࡜ࡀ࡛ࡁࡿ.
⾲4

᫬ⅬWang ኚ᥮࡟࠾ࡅࡿࣃ࣓࣮ࣛࢱ (it ; qx,t )ࣃ࣮ࢭࣥࢱ࢖್ࣝ

0.5% ್

25.0% ್1

50.0% ್75.0% ್99.5% ್3.98242

2.76592.79992.82662.86142.99723

2.26112.29232.31822.35362.50194

1.95801.98742.01322.04902.196551.7515

1.78101.8064

1.84281.9997

6

1.5993

1.62831.65351.6895

1.84787

1.48081.50911.5346

1.5716

1.72868

1.3853

1.41351.43901.4743

1.633691.30581.33451.3596

1.39561.5529101.23891.26711.29251.32831.483111

1.18171.2096

1.23451.27021.420912

1.1315

1.1592

1.1841

1.2192

1.372213

1.08691.11491.1393

1.1744

1.325314

1.04771.07481.0993

1.13431.285915

1.01191.0391

1.0634

1.09811.2439(ὀ 1) ホ౯ᑐ㇟ࡢ⿕ಖ㝤⪅ࡣ 30 ṓ࡛࠶ࡿ.
(ὀ 2) ዎ⣙᫬ணᐃ฼⋡ࡣ 1.5% ࡛࠶ࡿ.

*21 2007 ᖺ 7 ᭶ 26 ᪥, ཌ⏕ປാ┬኱⮧ᐁᡣ⤫ィ᝟ሗ㒊Ⓨ⾲.

*22 ౛࠼ࡤ, ๓ᥖࠕᖹᡂ 18 ᖺᗘ⡆᫆⏕࿨⾲ࠖሗ࿌᭩࡛ࡣㅖእᅜࡢᖹᆒᑑ࿨ࡢᖺḟ᥎⛣ࡀ♧ࡉࢀ࡚࠾ࡾ, 1965 ᖺ࠿ࡽ 2006 ᖺ࡟࠾ࡅࡿ᫬

㛫⤒㐣࡟క࠺ᖹᆒᑑ࿨ࡢఙࡧ, ࡍ࡞ࢃࡕ, Ṛஸ⋡ࡢపୗࢆ☜ㄆࡍࡿࡇ࡜ࡀ࡛ࡁࡿ.

―11―

ᅗ5

(it ; qx,t ) ࡢᮇ㛫ᵓ㐀

4.3 Cox-Ingersoll-Ross Model ࡢࣃ࣓࣮ࣛࢱ᥎ᐃ
ࢫ࣏ࢵࢺ࣭࣮ࣞࢺࣔࢹࣝ࡟ࡣ

dr = [r̄  r]dt +  rWt
ࢆ᥇⏝ࡋ, 2007 ᖺ 5 ᭶ 4 ᪥ࡢ㔠฼࣭മๆࢹ࣮ࢱ࡟ࡼࡾࣃ࣓࣮ࣛࢱ᥎ᐃࢆ⾜ࡗࡓ. ⤖ᯝ࡜ࡋ࡚  ࡟ 0.07745, r̄ ࡟

0.04163,  ࡟ 0.01047 ࢆᚓࡓ.

4.4 ዎ⣙᮲௳ࡢኚ᭦ཬࡧ◚⥢ฎ⌮ࡢᶒ฼౯್ホ౯Ṛஸ⋡࡜฼Ꮚ⋡ࡢୖ࡟᭩࠿ࢀࡿࢹࣜࣂࢸ࢕ࣈ࡜ࡋ࡚ಖ㝤ዎ⣙ࡢබṇ౯್ࢆグ㏙ࡋ, ኚ᭦๓බṇ౯್ࢆ⾜౑౯᱁, ኚ᭦ᚋබṇ౯್ࢆཎ㈨⏘࡜ࡍࡿ Put Option ࡜ࡋ࡚ᶒ฼౯್ࢆホ౯ࡍࡿ. ࡍ࡞ࢃࡕ, ⾜౑౯᱁ࡣ᫬㛫ཬࡧ≧ែ࡟ࡼࡾኚ໬ࡋ, ࠿ࡘ, ௵ពࡢ᫬Ⅼ࡟ᶒ฼⾜౑ࡀྍ⬟࡞ American Moving Strike Put Option ࡛࠶ࡿ. ࡇࡇ࡛ホ౯ᑐ㇟ࡢಖ㝤ၟရࡣಖ㝤ᮇ㛫 15 ᖺࡢ㣴⪁ಖ㝤࡜ࡋࡓ. ࡞࠾, ࣜࢫࢡࡢᕷሙ౯᱁ (Wang ኚ᥮ࡢࣃ࣓࣮ࣛࢱ )ࢆ᫬㛫౫Ꮡࡉࡏࡿࡓࡵ࡟, ⮬ᅾᛶࡢ㧗࠸ᩘ್ゎᯒᡭẁ࡜ࡋ࡚ Longsta and Schwartz[2001] ࡢᥦၐࡍࡿ᭱ᑠ஧
஌ࣔࣥࢸ࢝ࣝࣟἲ (௨ୗ, LSM ἲ࡜᭩ࡃ.) ࢆ㐺⏝ࡋࡓ.
‶ᮇ௨๓࡟ᶒ฼⾜౑ྍ⬟࡞ࢹࣜࣂࢸ࢕ࣈࡣ, ᙜヱዎ⣙ࢆ⥔ᣢࡍࡿࡇ࡜࡟ࡼࡿ⥅⥆౯್࡜ᶒ฼⾜౑ࡍࡿሙྜ࡟
ᚓࡽࢀࡿ࣌࢖࢜ࣇࢆẚ㍑ࡋ, ᚋ⪅ࡀ๓⪅ࢆୖᅇࡿሙྜ࡟ᶒ฼⾜౑ࡀ⾜ࢃࢀࡿ. LSM ἲ࡜ࡣ, ➨ t ᫬Ⅼ࡟࠾ࡅࡿ࢖࣭ࣥࢨ࣭࣐ࢿ࣮ࡢࢧࣥࣉ࣭ࣝࣃࢫ࡟ࡘ࠸࡚, ᙜヱ᫬Ⅼ࡟࠾ࡅࡿ≧ែኚᩘࢆㄝ᫂ኚᩘ, ➨ t + t ᫬Ⅼ࡟࠾ࡅࡿ࣌࢖࢜ࣇࢆࢫ࣏ࢵࢺ࣭࣮ࣞࢺ࡛ t ᮇ㛫๭ࡾᡠࡋࡓ್ࢆᚑᒓኚᩘ࡜ࡍࡿᅇᖐศᯒࢆ⾜࠸, ᚓࡽࢀࡓᅇᖐᘧ࡟ࡼ
ࡾ⥅⥆౯್ࢆ᥎ᐃࡋࡼ࠺࡜ࡍࡿࡶࡢ࡛࠶ࡿ. ಖ㝤ศ㔝࡛ࡣ Andreatta and Corradin[2003] ࡢゎ⣙ᶒホ౯ࢆඛ⾜౛ࡢ୍ࡘ࡜ࡋ࡚ᥖࡆࡿࡇ࡜ࡀ࡛ࡁࡿ.

―12―

日本保険・年金リスク学会誌 Vol.3 No.1 2008Qᮏ✏࡛ࡣṚஸ⋡ࡢ୙☜ᐇᛶࢆ඲ᖺ㱋࡛ඹ᭷ࡍࡿࡓࡵ, ࢿ࢖ࣆ࢔ᩘࢆᗏ࡜ࡍࡿᣦᩘ㛵ᩘ್ exp(WT ) ࢆṚஸ⋡࡟ᑐᛂࡍࡿ≧ែኚᩘ࡜ࡋࡓ. ୍᪉, ฼Ꮚ⋡࡟ࡘ࠸࡚ࡣኚ᥮ࢆ⾜ࡗ࡚࠸࡞࠸. Cox-Ingersoll-Ross Model ࡢࣃࢫࡣṇ್࡟㝈ᐃࡉࢀࡿࡓࡵ࡛࠶ࡿ. ࡲࡓ, ᇶᗏ㛵ᩘࢆᵓᡂࡍࡿ┤஺ከ㡯ᘧ࡟ࡣ 10 ḟࡢ Laguerre ከ㡯ᘧࢆ㐺⏝ࡋ
ࡓ. ᩘ್ゎᯒࡣ Compaq Visual Fortran ࡛⾜࠸, ᅇᖐศᯒ࡟ࡣ IMSL ᩘ್ィ⟬ࣛ࢖ࣈࣛࣜࡢ DLSBRR ࢆ㐺⏝, ヨ⾜ᅇᩘࡣ 10,000 ᅇ࡜ࡋ࡚࠸ࡿ.
ಖ㝤ᩘ⌮࡟࠾࠸࡚㐺⏝ࡍࡿᶆ‽Ṛஸ⋡ࡣ⢒Ṛஸ⋡࡟ᑐࡋᏳ඲๭ቑࢆィୖࡋ, ᶒ฼ᶒ౯್ホ౯࡟࠾࠸࡚㐺⏝ࡍࡿṚஸ⋡ࡣ⢒Ṛஸ⋡ࢆᗘኚ᥮ࡍࡿ. ࡇࡇ࡛, Ᏻ඲๭ቑࡣ 2005 ᖺ⡆᫆⏕࿨⾲࡟ᑐࡍࡿ⏕ಖᶆ‽⏕࿨⾲ 2007(Ṛஸಖ㝤⏝) ࡢẚࡀ୍ᐃ࡛࠶ࡿࡶࡢ࡜௬ᐃࡋ, ᙜヱᖺ㱋ูಀᩘࢆ⢒Ṛஸ⋡࡟஌ࡎࡿࡇ࡜࡛ᶆ‽Ṛஸ⋡ࢆᐃ⩏ࡋ࡚࠸
ࡿ. ᚑࡗ࡚, ᶆ‽⏕࿨⾲࡟࠾ࡅࡿᏳ඲๭ቑࡣ⢒Ṛஸ⋡ࡢỈ‽࡟ẚ౛ࡍࡿࡇ࡜࡟࡞ࡾ, ⢒Ṛஸ⋡ࡀపୗࡍࢀࡤᏳ඲
๭ቑࡀᅽ⦰ࡉࢀࡿࡇ࡜࡟࡞ࡿ. ᙜヱᶆ‽⏕࿨⾲࡟ᩚྜࡉࡏࡿࡇ࡜࡟ࡼࡿ Wang ኚ᥮ࡢࣃ࣓࣮ࣛࢱࡢᮇ㛫ᵓ㐀ࡣඛ࡟♧ࡋࡓ㏻ࡾ࡛࠶ࡿ.
ᑗ᮶࡟࠾ࡅࡿணᐃ฼⋡ࡣྛࢢࣜࢵࢻ࣭࣏࢖ࣥࢺ࡟࠾ࡅࡿ 10 ᖺ฼௜മๆࡢࣃ࣮࣭ࢡ࣮࣏ࣥ࡜ࡋ࡚࠸ࡿ*23 .

4.4.1 ᑗ᮶࡟࠾ࡅࡿ฼┈㑏ඖࢆ᝿ᐃࡋ࡞࠸ሙྜ
ᘧ (14) ཬࡧᘧ (16) ࠿ࡽࡣ, ྛᶒ฼౯್ࡢ≉ᚩ࡜ࡋ࡚ḟᥖࡢෆᐜࢆᢕᥱ࡛ࡁࡿ.
➨୍࡟ኚ᭦๓ࡢಖ㝤ᩱ Px: n ཬࡧ㈐௵‽ഛ㔠  Vx: n ࡀᑠࡉ࠸࡯࡝, ࡍ࡞ࢃࡕ, ணᐃ฼⋡ࡀ㧗࠸࡯࡝, ࡲࡓ, ணᐃṚஸ⋡ࡀప࠸࡯࡝ዎ⣙᮲௳ኚ᭦ⱝࡋࡃࡣ◚⥢ฎ⌮࡟ࡼࡾಖ㝤⪅ࡀᚓࡽࢀࡿ฼ᚓࡀ኱ࡁࡃ࡞ࡿࡇ࡜࡛࠶ࡿ. ዎ⣙⪅ࡢᦆኻ࡜ࡋ࡚ㄞࡳ᭰࠼ࢀࡤ, ᙜ↛ࡢࡇ࡜࡜ࡋ࡚⌮ゎ࡛ࡁࡼ࠺.
➨஧࡟ྛせ⣲ࡢ฼ᚓཬࡧᶒ฼⾜౑᫬ᮇ࡬ࡢᙳ㡪࡛࠶ࡿ.
ࡁࡃ࡞ࡿࡓࡵ࡟㏴ῶࡋ,start Vx: nf irst

A

start
Px:
n
Px+ : n f irstࡣ  ࡀ‶ᮇ࡟㏆ࡃ࡞ࡿ࡯࡝ศẕࡀ኱ࡣ᫬㛫ࡢ⤒㐣࡜࡜ࡶ࡟㏴ቑࡋ, ‶ᮇ᫬࡟ࡣ 1 ࡟཰ᩡࡍࡿ. ᚑࡗ࡚, ๓⪅ࡣᶒx+ : n฼⾜౑ࢆ㐜ࡽࡏࡿ࢖ࣥࢭࣥࢸ࢕ࣈࢆ୚࠼, ᚋ⪅ࡣᶒ฼⾜౑ࢆ᪩ࡵࡿ࢖ࣥࢭࣥࢸ࢕ࣈࢆ୚࠼ࡿࡇ࡜࡟࡞ࡿ. ዎ⣙᮲௳ࡢኚ᭦࡜◚⥢ฎ⌮࡛ࡣ, ๓⪅ࡣඹ㏻ࡍࡿࡶࡢࡢ, ᚋ⪅࡟ࡣᕪ␗ࡀ⏕ࡌ࡚࠸ࡿ. ㈐௵‽ഛ㔠๐ῶࢆチᐜࡋ࡞࠸ዎ⣙᮲௳ࡢኚ᭦ࡢ᪉ࡀ, ◚⥢ฎ⌮ࡼࡾࡶᶒ฼⾜౑ࡀ᪩ࡵࡽࢀࡿࡇ࡜࡟࡞ࡿ. ࡇࢀࡣ౯್ホ౯࡜ྠ᫬࡟, ᩘ್ⓗ࡟☜ㄆࡍࡿࡇ࡜ࡀ࡛ࡁࡿ.
ᅗ6ዎ⣙᮲௳ኚ᭦ᶒཬࡧ◚⥢ฎ⌮ᶒࡢṧᏑ⋡*23 ᙜヱỴᐃᇶ‽࡟ᇶ࡙ࡃணᐃṚஸ⋡ཬࡧணᐃ฼⋡ࡣ኱ⶶ┬࿌♧➨ 48 ྕࠕᶆ‽㈐௵‽ഛ㔠ࡢ✚❧᪉ᘧཬࡧィ⟬ᇶ♏⋡ࢆᐃࡵࡿ௳ࠖ࡜ࡣ

ᚲࡎࡋࡶ୍⮴ࡋ࡞࠸.

―13―

ᅗ 6 ࡣᶓ㍈࡟᫬㛫, ⦪㍈࡟ṧᏑ⋡ࢆ࡜ࡾ, ዎ⣙᮲௳ࡢኚ᭦ཬࡧ◚⥢ฎ⌮ࡢᶒ฼ࡀ⾜౑ࡉࢀࡎ࡟ṧᏑࡍࡿࣃࢫ
ࢆ☜ㄆࡋࡓࡶࡢ࡛࠶ࡿ. ⤒㐣 1 ᖺ┠ࢆぢࡿ࡜, ఱࢀࡢணᐃ฼⋡ࡢ⤌࡟࠾࠸࡚ࡶ◚⥢ฎ⌮ࡢṧᏑ⋡ࡀዎ⣙᮲௳ࡢኚ᭦ࡢṧᏑ⋡ࡼࡾࡶ㧗ࡃ, ௨㝆ࡢᮇ㛫࡛㏫㌿ࡋ࡚࠸ࡿ. ࡲࡓ, ◚⥢ฎ⌮ࡢṧᏑ⋡ࡀዎ⣙᮲௳ࡢኚ᭦ࡢࡑࢀࡼࡾࡶ
㧗Ỉ‽࡟࠶ࡿᮇ㛫ࡣணᐃ฼⋡ࡀ㧗ࡃ࡞ࡿ࡟క࠸㛗ᮇ໬ࡋ, ணᐃ฼⋡ 2.5% ࡢ⤌࡛ࡣ⤒㐣 3 ᖺ┠ࡲ࡛, ணᐃ฼⋡3.0% ࡢ⤌࡛ࡣ⤒㐣 5 ᖺ┠ࡲ࡛࡜࡞ࡗ࡚࠸ࡿ.
ࡉ࡚, ዎ⣙᮲௳ࡢኚ᭦ࡼࡾࡶ◚⥢ฎ⌮ࡢṧᏑ⋡ࡀ㧗࠸࡜࠸࠺ࡇ࡜ࡣ, ᙜヱᮇ㛫࡟࠾ࡅࡿ๓⪅ࡢᶒ฼⾜౑ࡀᚋ⪅ࡼࡾࡶከ࠸࡜࠸࠺ࡇ࡜࡛࠶ࡿ. ࡑࡇ࡛, ḟࡢẁ㝵࡛ࡣᶒ฼⾜౑ࡢ⾜ࢃࢀࡿ≧ែࢆ☜ㄆࡍࡿ.
ࠓ⤒㐣 1 ᖺ┠࡟࠾ࡅࡿᶒ฼⾜౑㡿ᇦࡢẚ㍑ࠔ
ᅗ7ึᮇணᐃ฼⋡ 1.5% ࡢሙྜᅗ8ึᮇணᐃ฼⋡ 2.0% ࡢሙྜᅗ9ึᮇணᐃ฼⋡ 2.5% ࡢሙྜᅗ 10ึᮇணᐃ฼⋡ 3.0% ࡢሙྜ(ὀ) ⿕ಖ㝤⪅ᖺ㱋ࡣ 30 ṓ࡛࠶ࡿ.
ᅗ 7 ࠿ࡽᅗ 10 ࡛ࡣ⤒㐣 1 ᖺ┠࡟࠾ࡅࡿᶒ฼⾜౑㡿ᇦࢆ♧ࡋ࡚࠸ࡿ. ᅗࡢ⦪㍈࡜ᶓ㍈ࡣ LSM ἲࡢᅇᖐᑐ㇟࡜
࡞ࡿ≧ែኚᩘ࡛࠶ࡾ, ྛ≧ែኚᩘࡢ஺Ⅼࡀࣈࣛࣥࢡࡢ㒊ศ࡟࠶ࡿሙྜ࡟ࡣ, ᶒ฼⾜౑࡟ࡼࡾᚓࡽࢀࡿ฼ᚓࡀ⥅⥆౯್ࢆୗᅇࡿࡓࡵ◚⥢ฎ⌮ཬࡧዎ⣙᮲௳ࡢኚ᭦ࡣ⾜ࢃࢀ࡞࠸. ᅗ 6 ࡟ࣉࣟࢵࢺࡉࢀࡿࠕṧᏑࡍࡿࣃࢫࠖࡢ≧ែ
࡛࠶ࡿ. ௚᪉, ≧ែኚᩘࡢ஺Ⅼ࡟  ࡀᡴࡓࢀࡿሙྜ*24 ࡟ࡣዎ⣙᮲௳ࡢኚ᭦ཬࡧ◚⥢ฎ⌮ࡀ, × ࡀᡴࡓࢀࡿሙྜ
࡟ࡣ◚⥢ฎ⌮ࡢࡳࡀ, • ࡀᡴࡓࢀࡿሙྜ*25 ࡟ࡣዎ⣙᮲௳ࡢኚ᭦ࡢࡳࡀ⾜ࢃࢀࡿ.
ᶒ฼⾜౑㡿ᇦࡀᅗࡢୗ㒊࡟఩⨨ࡍࡿࡇ࡜ࡣ, ᶒ฼⾜౑ࡢⓎ⏕ࡍࡿ⤒῭せᅉࡢ୍ࡘࡀ, ᕷ୰㔠฼ࡢチᐜỈ‽ࢆ㉸࠼ࡿపୗ࡛࠶ࡿࡇ࡜ࢆព࿡ࡋ࡚࠸ࡿ. ࡲࡓ, Ṛஸಖ㞀ၟရ࡟࠾࠸࡚ࡣ, チᐜỈ‽ࢆ㉸࠼ࡿṚஸ⋡ࡢୖ᪼ࡀᶒ฼⾜*24 ࢢࣛࣇୗ㒊ࡢⷧ࠸࣐࣮࣮࢝㡿ᇦ.
*25 ࢢࣛࣇୖ㒊ࡢ⃰࠸࣐࣮࣮࢝㡿ᇦ.

―14―

日本保険・年金リスク学会誌 Vol.3 No.1 2008

౑ࡢⓎ⏕せᅉ࡜࡞ࡿ. ᶒ฼⾜౑㡿ᇦࡀᶓ㍈࡟ᑐࡋ࡚ᑐ⛠࡜ࡣ࡞ࡽࡎᕥഃࡀపࡃ࡞ࡿࡇ࡜ࡣ, ฼ᕪᦆࢆṚᕪ┈ࡀ⿵ሸࡍࡿࡶࡢ࡜ࡋ࡚, ྑഃࡀ㧗ࡃ࡞ࡿࡇ࡜ࡣ, Ṛᕪᦆࡀ฼ᕪᦆ࡟ຍ⟬ࡉࢀࡿࡶࡢ࡜ゎ㔘࡛ࡁࡿ. ᙜヱ⤖ᯝࡣ┤ほⓗ࡞⌮ゎ࡜ࡶᩚྜࡍࡿࡶࡢ࡛࠶ࢁ࠺.
᭦࡟, ዎ⣙᫬࡟タᐃࡉࢀࡿணᐃ฼⋡ࡢỈ‽ࡶᶒ฼⾜౑㡿ᇦ࡟ࡣᙳ㡪ࡍࡿ. 㐺⏝ࡉࢀࡿணᐃ฼⋡ࡀᙜヱ᫬Ⅼࡢ᪂ዎ⣙࡟㐺⏝ࡉࢀࡿ⋡ (1.5%) ࡛࠶ࡿሙྜ࡟ࡣ, ዎ⣙᮲௳ࡢኚ᭦࡜◚⥢ฎ⌮ࡢ㛫࡟ᶒ฼⾜౑㡿ᇦࡢ኱ࡁ࡞ᕪ␗ࡣ
⏕ࡌ࡚࠸࡞࠸ࡀ (ᅗ 7), 㐺⏝ࡉࢀࡿணᐃ฼⋡ࡀ㧗ࡃ࡞ࡿ࡯࡝ඹ㏻ࡢ㡿ᇦ () ࡔࡅ࡛ࡣ࡞ࡃ, ዎ⣙᮲௳ࡢኚ᭦࡟ಀ
ࡿ㡿ᇦ (•) ࡀᣑ኱ࡋ࡚࠸ࡿ.
ఱࢀࡢᅗ࡟࠾࠸࡚ࡶ, • ࡣ  ࡢୖ㒊࡟఩⨨ࡋ࡚࠾ࡾ, ணᐃ฼⋡ࡀ㧗ࡃ࡞ࡿ࡟ࡘࢀ࡚ᕥഃ࡟᥎⛣ࡋ࡚࠸ࡃ. ዎ⣙᮲௳ࡢኚ᭦࡟ಀࡿᶒ฼⾜౑㡿ᇦࡀ◚⥢ฎ⌮࡟ಀࡿࡑࢀࡼࡾࡶ኱ࡁ࠸࡜࠸࠺ࡇ࡜ࡣ, ࡼࡾᗈ࠸≧ែኚᩘࡢ⠊ᅖ࡛ࣃࢫࡀᾘ⁛ࡍࡿ࡜࠸࠺ࡇ࡜࡛࠶ࡿ. ࡲࡓ, 㡿ᇦࡢୖ㒊࡟ • ࡀᡴࡓࢀࡿ࡜࠸࠺ࡇ࡜ࡣ, ᕷ୰฼Ꮚ⋡ࡢపୗࡀὸ࠸ẁ㝵࡛ዎ⣙᮲௳ࡢኚ᭦ࡀ◚⥢ฎ⌮࡟ඃඛࡋ࡚⾜ࢃࢀࡿࡇ࡜ࢆព࿡ࡋ࡚࠸ࡿ. ࡍ࡞ࢃࡕ, ㈐௵‽ഛ㔠๐ῶࡢㄆྰࡣ
ᶒ฼⾜౑࡟ᕪ␗ࢆ୚࠼࡚࠸ࡿ࡜࠸࠺ࡇ࡜࡛࠶ࡿ.
᭱⤊ẁ㝵࡛⾲ 5 ࢆぢࡿ࡜, ዎ⣙᮲௳ࡢኚ᭦ࡣ◚⥢ฎ⌮࡟ẚ㍑ࡋᶒ฼౯್ࡀపࡃ࡞ࡗ࡚࠸ࡿ. ᙜヱ⤖ᯝࡣ, ಖ㝤ዎ⣙⪅ࡢ⤒῭ⓗ㈇ᢸࡀ㍍ῶࡉࢀ࡚࠸ࡿࡶࡢ࡜ㄞࡳ᭰࠼ࡿࡇ࡜ࡀ࡛ࡁࡿࡔࢁ࠺.
⾲5ᖺ㱋ዎ⣙᮲௳ࡢኚ᭦࡜◚⥢ฎ⌮ࡢᶒ฼౯್

ዎ⣙᮲௳ࡢኚ᭦

◚⥢ฎ⌮ዎ⣙᫬ணᐃ฼⋡ዎ⣙᫬ணᐃ฼⋡

1.5%

2.0%

2.5%

3.0%

1.5%

2.0%

2.5%

3.0%

0

0.00390.02070.04450.07170.01920.02970.05090.0773100.00390.02070.04450.07180.01930.02980.05100.077520

0.00390.02060.04440.07140.01920.02970.05080.077130

0.00390.02060.04430.07130.01920.02950.05080.0769400.00380.02030.04380.07050.01900.02930.05040.076150

0.00350.01940.04230.06810.01850.02840.04870.073760

0.00290.0176

0.03880.06300.01770.02640.0451

0.0686700.00180.01250.02980.04980.01670.02220.03650.05544.4.2 ᑗ᮶࡟࠾ࡅࡿ฼┈㑏ඖࢆ᝿ᐃࡍࡿሙྜ
ኚ᭦ᑐ㇟ዎ⣙⪅࡬ࡢ฼┈㑏ඖࢆ᝿ᐃࡍࡿሙྜࡢዎ⣙᮲௳ࡢኚ᭦ᶒ฼౯್ࡣ, ᘧ (22) ࡟♧ࡍ࡜࠾ࡾ࡛࠶ࡿ. ࡍ࡞ࢃࡕ, ዎ⣙᮲௳ࡢኚ᭦ࢆ᮲௳࡜ࡋ࡚฼┈㑏ඖࡀᐇ᪋ࡉࢀࡿࡢ࡛࠶ࡿ࠿ࡽ, ᑗ᮶࡟࠾ࡅࡿ㌿᥮ᶒ౯್ࢆዎ⣙᮲௳ኚ᭦ᚋࡢබṇ౯್ࡼࡾ᥍㝖, ᙜヱ౯್࡜ዎ⣙᮲௳ኚ᭦๓ࡢබṇ౯್ࢆᑐẚࡋ, ฼ᚓࢆホ౯ࡍࢀࡤࡼ࠸.
ᙜヱᶒ฼౯್ࡣ, ṇ๪ᵓ㐀ࢆᣢࡘ᱁Ꮚἲ࡟ࡼࡾホ౯ࡍࡿࡇ࡜ࡀྍ⬟࡛࠶ࡿ. ᣋ✏ [2007a] ࡛ࡣዎ⣙᮲௳ࡢኚ᭦
ཬࡧ◚⥢ฎ⌮࡟ࡘ࠸࡚, ᑗ᮶࡟࠾ࡅࡿ㌿᥮ࡢ᭷↓ࢆ⪃៖ࡋࡓᶒ฼౯್ࡢホ౯ࢆ⾜ࡗ࡚࠸ࡿ.

―15―

5 ⤖ㄽ
ᙜヱᨵṇࡣ⌧㔠࡟ࡼࡿබⓗ㈨㔠ᢞධࢆ⾜ࢃࡎ, ዎ⣙᮲௳ࡢኚ᭦ࢆㄆᐜࡍࡿࡇ࡜࡟ࡼࡾᐇ㉁ⓗ࡟ಖ㝤⪅ࡢᩆ῭ࢆ⾜ࡗࡓࡶࡢ࡛࠶ࡿ. ᶒ฼ࡢ⾜౑ࡉࢀࡿ஦ែࡣỴࡋ࡚ዲࡲࡋ࠸ࡶࡢ࡛ࡣ࡞࠸ࡀ, ᙜヱࢫ࣮࣒࢟ࡀᚑ᮶ࡢ◚⥢ฎ⌮࡟ẚ㍑ࡋ࡚ඃࢀ࡚࠸ࡿⅬ࡜ࡋ࡚ḟᥖࡢ୕Ⅼࢆᣦ᦬࡛ࡁࡿ.
➨୍࡟㈐௵‽ഛ㔠ࡢ๐ῶࢆチᐜࡋ࡞࠸ࡓࡵ, ᶒ฼⾜౑᫬ᮇࢆ᪩ࡵࡿ࢖ࣥࢭࣥࢸ࢕ࣈࢆ୚࠼ࡓ࡜࠸࠺ࡇ࡜࡛࠶
ࡿ. ᙜヱせ⣲࠿ࡽࡣၥ㢟ࢆඛ㏦ࡾࡍࡿࡇ࡜࡟ࡼࡿᦆኻࡢᣑ኱ࢆ㜵ࡄຠᯝࢆᮇᚅ࡛ࡁࡼ࠺. ᮏ✏࡛ࡣಖ㝤⪅࡟୚࠼ࡽࢀࡓᶒ฼ࢆࢹࣜࣂࢸ࢕ࣈࢫ࡜ࡳ࡞ࡋ, ᶒ฼⾜౑㡿ᇦࢆẚ㍑ࡍࡿࡇ࡜࡟ࡼࡾ, ᙜヱෆᐜࢆィ㔞ⓗ࡟☜ㄆࡋࡓ.
➨஧࡟ࡣᶒ฼⾜౑ࡢពᛮỴᐃࢆᙜ஦⪅࡟ṧࡋ, ㈐௵ࡢᡤᅾࢆ᫂☜࡟ࡋࡓࡇ࡜࡛࠶ࡿ. ហ࠿࡞⤒Ⴀ⪅ࡣ, ᴗົ೵Ṇ࿨௧ࢆⓎ௧ࡉࢀࡿࡲ࡛Ⴀᴗࢆ⥅⥆ࡍࡿ࠿ࡶࡋࢀ࡞࠸. ↛ࡿ࡟, ዎ⣙᮲௳ࡢኚ᭦ࢆ⾜ࢃࡎ࡟⤒Ⴀ◚⥢ࡍࡿࡇ࡜ࡀ࠶ࢀࡤ, ᭱ၿࡢ⤒Ⴀ⾜ືࢆᇳࡽ࡞࠿ࡗࡓࡇ࡜ࡣ᫂ⓑ࡜࡞ࡿ. ពᛮỴᐃ࡟ಀࡿ౯್ホ౯ࢆ⾜࠺ࡇ࡜࡟ࡼࡾ, ⤒Ⴀุ᩿ࡢ⤖ᯝ࡜ࡋ࡚ᣑ኱ࡋࡓᦆᐖ㢠ࡶ୍ᐃࡢ᮲௳ୗ࡛ホ౯ࡉࢀࡿ. ㈺ൾㄳồ㢠ࡢ⟬ᐃ᰿ᣐ࡟୍ᐃỈ‽ࡢᐈほᛶࢆ୚࠼ࡿࡇ࡜ࡶྍ⬟࡜࡞ࢁ࠺.
ࡲࡓ, ዎ⣙᮲௳ࡢኚ᭦࡟ࡣෆ㛶⥲⌮኱⮧ࡢᢎㄆᚋ, ᰴ୺⥲఍➼ࡢỴ㆟ࢆᚲせ࡜ࡍࡿ*26 . ࡍ࡞ࢃࡕ, ዎ⣙᮲௳ࡢኚ᭦ෆᐜ, ᇶ㔠ࡸಖ㝤ዎ⣙⪅௨እࡢ୍⯡മᶒ⪅ࡢമົࡢྲྀࡾᢅ࠸, ᑗ᮶ࡢ฼┈㑏ඖ➼࡟ࡼࡿᩆ῭᮲௳, ⤒Ⴀ㈐௵➼࡟ࡘ࠸࡚ྜពࡀᡂ❧ࡋ࡞ࡅࢀࡤ, ኚ᭦ࢆཷࡅධࢀࡎ࡟⤒Ⴀ◚⥢ࢆ㑅ᢥࡍࡿࡇ࡜ࡶ, ዎ⣙⪅ࡣ⮬ᕫ㈐௵࡟࠾࠸࡚
ྍ⬟࡛࠶ࡿ. ⮬ࡽࡢಖ᭷ࡍࡿᶒ฼౯್ࢆホ౯ࡍࡿࡇ࡜ࡀ࡛ࡁࢀࡤ, ዎ⣙⪅ࡣኚ᭦ෆᐜ࡟ࡘ࠸࡚ྜ⌮ⓗ࡞᳨ウࢆ⾜࠺ࡇ࡜ࡀྍ⬟࡜࡞ࢁ࠺.
➨୕࡟ࡣᶒ฼⦰ῶᑐ㇟ࢆ㝈ᐃࡍࡿࡇ࡜࡛ᙜヱ᫬Ⅼ࡟࠾ࡅࡿዎ⣙⪅㛫ࡢබᖹᛶࢆ☜ಖࡍࡿ୍᪉, ⤒Ⴀᨵၿ᫬࡟
࠾ࡅࡿඃඛⓗ࡞฼┈㑏ඖࢆᶒ฼⦰ῶᑐ㇟⪅࡟⣙ࡍࡿࡇ࡜࡛ᑗ᮶࡟࠾ࡅࡿዎ⣙⪅㛫ࡢබᖹᛶࢆ☜ಖࡋ࠺ࡿࡇ࡜࡛
࠶ࡿ.
࡞࠾, ዎ⣙᮲௳ࡢኚ᭦ࢆമົච㝖࡜⪃࠼ࡿ࡞ࡽࡤ, ㈚೉ᑐ↷⾲ୖࡢ㈐௵‽ഛ㔠㢠࡟ኚືࡣ࡞ࡃ࡜ࡶ, ᘧ (14)ཬࡧᘧ (21) ࡢ฼ᚓࡣ⏕ࡌ࡚࠸ࡿ. ἲே⛯ἲ➨ 22 ᮲➨ 2 㡯ࢆ㚷ࡳࢀࡤ, ᙜヱമົච㝖┈࡟ࡣᙜ↛࡟ㄢ⛯ࡍ࡭ࡁ
࡛࠶ࢁ࠺. ࡓࡔࡋ, ᙜヱ฼ᚓࡢ㢠ࡣබṇ౯್ホ౯ࣔࢹࣝ࡟౫Ꮡࡍࡿࡓࡵ, ᇶ‽࡜࡞ࡿホ౯ᡭἲࡢ☜❧ࡀᚲせ࡜⪃࠼ࡽࢀࡿ.

*26 ಖ㝤ᴗἲ➨ 240 ᮲ࡢ 5 ➨ 1 㡯 (ዎ⣙᮲௳ࡢኚ᭦ࡢỴ㆟).

―16―

日本保険・年金リスク学会誌 Vol.3 No.1 2008

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(25)

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.

1 X dx+u
lx+n+
u+1
lx u=0 (1 + i)lx (1 + i)n
n1

Sx =
=
1
ࡇࡇ࡛, v = 1+i
,

n1
X

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

(26)

lx+t
dx
x
x+1
dx ࡜ࡍࡿ࡜, ᘧ (26) ࡣ
lx =t px , lx = qx , Dx = v lx , Cx = v

Sx =
ࠉ=

࡜࡞ࡿ. ᭦࡟, Nx =


P
u=0

Dx+u , Mx =
Sx =

n1
X

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

u=0
n1
X

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


P
u=0

(27)

(28)

Cx+u , ࡜ࡍࢀࡤ,(Mx  Mx+n ) + Dx+n:= Ax:n|Dx(29)

࡜グ㏙࡛ࡁࡿ.
཰ධ⌧౯ྠᵝ࡟ࡋ࡚ಖ㝤ᩱࡢ཰ධ⌧౯ Gx ࢆồࡵࡽࢀࡿ. ಖ㝤ᩱᣐฟࡀᖹ‽ⓗ࡟⾜ࢃࢀࡿࡶࡢ࡜ࡍࢀࡤ,1 X lx+u
lx u=0 (1 + i)un1

Gx = Px:n| ·= Px:n| ·n1
Xv u u px

(30)

(31)

u=0

= Px:n| ·ಖ㝤ᩱࡢỴᐃ

Nx  Nx+n:= Px:n| · äx:n|
Dx

(32)

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

Px:n| · äx:n| = Ax:n|
Қ Px:n| =Ax:n|äx:n|(33):=(Mx  Mx+n ) + Dx+n
Nx  Nx+n

―17―

(34)

㈐௵‽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:nt|(35)(Mx+t  Mx+n ) + Dx+n
Dx+t
Gx+t = Px:n| · äx+t:nt|=

Nx+t  Nx+n
Dx+t
Қt Vx:n| = Sx+t  Gx+t
= Px:n| ·

(36)
(37)
(38)
(39)

:= Ax+t:nt|  Px:n| · äx+t:nt|

(40)

࡜࡞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

(41)

ࡓࡔࡋ, ࡇࡇ࡛ࡣᮇ୰࡛ࡢゎ⣙ࢆ᝿ᐃࡋ࡚࠸࡞࠸.
ࡉ࡚, ᘧ (41) ࡢ୧㎶࡟ v x+t+1 ࢆ஌ࡎࢀࡤ,Dx+t+1 Vt+1 = Dx+t (Vt + P )  Cx+t

(42)ࢆᚓࡿ. t = 0  (n  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤ,Dx+n Vn = Dx V0 + P

n1
X
t=0

ҚP =

Dx+t 

n1
X

Cx+t

t=0

Dx+n Vn  Dx V0 + (Mx  Mx+n )Nx  Nx+n(43)ዎ⣙᫬㈐௵‽ഛ㔠 (V0 ) ࢆ 0, ‶ᮇ᫬㈐௵‽ഛ㔠 (Vn ) ࢆᐃᮇಖ㝤࡟ࡘ࠸࡚ࡣ 0, 㣴⪁ಖ㝤࡟ࡘ࠸࡚ࡣ 1 ࡜ࡍࢀࡤಖ㝤ᩱࡀᚓࡽࢀࡿ.
ྠᵝ࡟ࡋ࡚, t = 0  (t  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤ㐣ཤἲ㈐௵‽ഛ㔠ࡀ, t = t  (n  1) ࡲ࡛㎶ࠎຍ࠼ࢀࡤ
ᑗ᮶ἲ㈐௵‽ഛ㔠ࡀᚓࡽࢀࡿ.

―18―

日本保険・年金リスク学会誌 Vol.3 No.1 2008

A.2 Wang ኚ᥮ࡢࣃ࣓࣮ࣛࢱ᥎ᐃ
R t+1

ᶆ‽⏕࿨⾲
v0 qx,t= E Q [e tru du Qqx,t+1 | Ft ]࡟࠾࠸࡚, t = 0 ࡜ࡍࡿ࡜
R1

Q
ᶆ‽⏕࿨⾲
v0 qx,0= E Q [e 0 ru du qx,1
| F0 ]Q= B(0, 1)E Q [qx,1| F0 ]Qexp[= B(0, 1)qx,0x (1  0 ) +(i0 ; qx,0 )  +Qᶆ‽⏕࿨⾲
ln v0 qx,0
 ln B(0, 1)qx,0[Қ (i0 ; qx,0 ) =


2
]
2

2
x (1  0 ) + 2 ](44)ྠᵝ࡟ࡋ࡚, t = 1 ࡜ࡍࡿ࡜
R2

Q
ᶆ‽⏕࿨⾲
v1 qx,1= E Q [e 1 ru du qx,2
| F1 ]

2
Q
exp[ x (2  1 ) + ( (i1 ; qx,1 ) 2  (i0 ; qx,0 )) += B(1, 2)qx,1
]
2
2

Q
ᶆ‽⏕࿨⾲
ln v1 qx,1
 ln B(1, 2)qx,1
 [ x (2  1 ) + 2 ] + (i0 ; qx,0 )

 2
P1
2
Q
ᶆ‽⏕࿨⾲
 ln B(k, k + 1)qx,k
 x (k  0 )  2 ]k=0 [ln vk qx,k

=
 2
P1
Q
ᶆ‽⏕࿨⾲
 ln B(k, k + 1)qx,k]  x (2  0 )   2
k=0 [ln vk qx,k= 2Қ (i1 ; qx,1 ) =(45)
௨ୗྠᵝ࡟ࡋ࡚, ᘧ (24) ࢆᚓࡿ.
᭦࡟௵ពࡢ t ᫬Ⅼ࡛ᘧ (24) ࡀᡂ❧ࡍࡿ࡜ࡁ, t + 1 ᫬Ⅼ࡛ࡣ


2
Q
ᶆ‽⏕࿨⾲
 ln B(t, t + 1)qx,t
 [ x (t+1  t ) + 2 ] + (it ; qx,t ) t
ln vt qx,t

 t+1
Pt
2
Q
ᶆ‽⏕࿨⾲
 ln B(k, k + 1)qx,k]  [ x (t+1  0 ) + 2 t]
k=0 [ln vk qx,k

=
 t+1

Қ (it+1 ; qx,t+1 ) =࡛࠶ࡿ. ᚑࡗ࡚, ᩘᏛⓗᖐ⣡ἲ࡟ࡼࡾᘧ (24) ࡀド࡛᫂ࡁࡓ.

―19―

(46)

ཧ⪃ᩥ⊩
[1] ᮌᓥṇ࣭᫂⏣୰ᩗ୍ [2007], ࠗ㈨⏘ࡢ౯᱁௜ࡅ࡜ ᗘኚ᥮࠘, ᮅ಴᭩ᗑ, 2007 ᖺ 6 ᭶ 20 ᪥.
[2] ᅜ཭┤ே࣭ᐊ஭ⰾྐ [2006], ࠕಖド௜ࡁᖺ㔠࢜ࣉࢩࣙࣥࡢホ౯ἲ࡟ࡘ࠸࡚ࠖ, ࠗ᪥ᮏಖ㝤࣭ᖺ㔠ࣜࢫࢡᏛ఍
➨ 4 ᅇ ◊✲Ⓨ⾲኱఍ ண✏㞟࠘, 2006 ᖺ 10 ᭶ 14 ᪥, 15-23 㡫.

[3] ᑠᬽཌஅ࣭㛗㇂ᕝ▱ᘯ [2005], ࠕᑗ᮶⏕࿨⾲ࡢ⤫ィࣔࢹࣜࣥࢢ:Lee-Carter ἲ࡜ࡑࡢᣑᙇ -ࣄ࣮࣐࣭ࣗࣥࢭ
࢟ࣗࣜࢸ࢕࡬ࡢᇶ┙◊✲-ࠖ,ࠗ⥲ྜᨻ⟇Ꮫ࣮࣡࢟ࣥࢢ࣮࣌ࣃ࣮ࢩ࣮ࣜࢬ θ 71࠘, 2005 ᖺ 4 ᭶.

[4] ᑠᬽཌஅ [2005], ࠕṚஸ⋡ࡢࣔࢹࣜࣥࢢ࡜ண ࠖ, ࠗ⤫ィ࠘, ➨ 56 ᕳ, ➨ 4 ྕ, 2005 ᖺ 4 ᭶ 1 ᪥, 19-24 㡫.
[5] ᅜ❧♫఍ಖ㞀࣭ேཱྀၥ㢟◊✲ᡤ [2006], ࠗ᪥ᮏࡢᑗ᮶᥎ィேཱྀ (ᖹᡂ 18 ᖺ 12 ᭶᥎ィ)࠘, 2006 ᖺ 12 ᭶.
[6] ῰㇂ᨻ᫛࣭⳹ᒣᐉ⬍ [2004], ࠕᖺ㱋᫬௦༊ศࢹ࣮ࢱ࡟ࡼࡿ㉸㧗㱋⪅ᑑ࿨ศᕸࡢ᥎ ࠖ, ࠗ⤫ィᩘ⌮࠘, ➨ 52
ᕳ➨ 1 ྕ, 117-134 㡫.

[7] ஧ぢ㝯 [1988], ࠗ⏕࿨ಖ㝤ᩘᏛ࠘, ㈈ᅋἲே⏕࿨ಖ㝤ᩥ໬◊✲ᡤ, ᫛࿴ 63 ᖺ 1 ᭶ 24 ᪥.
[8] ᳃ᖹ∝୍㑻 [2006], ࠕᑑ࿨ࣜࢫࢡ࡜ࡑࡢドๆ໬:ࣔࢹࣜࣥࢢࡢᒎᮃࠖ, ࠗ᪥ᮏಖ㝤࣭ᖺ㔠ࣜࢫࢡᏛ఍ ➨ 4 ᅇ◊✲Ⓨ⾲኱఍ ண✏㞟࠘, 2006 ᖺ 10 ᭶ 14 ᪥, 103-122 㡫.

[9] 㛛⏣ఙ୍ [2006], ࠕ⏕࿨ಖ㝤ዎ⣙࡟࠾ࡅࡿ㌿᥮ᶒࡢ࢜ࣉࢩࣙࣥ౯್ホ౯ࠖ, ࠗ᪥ᮏಖ㝤࣭ᖺ㔠ࣜࢫࢡᏛ఍
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[10] 㛛⏣ఙ୍ [2007a], ࠕ⏕࿨ಖ㝤ዎ⣙ࡢ₯ᅾⓗࢹࣜࣂࢸ࢕ࣈ㸸㌿᥮ᶒ࡜ൾ㑏ᶒࠖ, ࠗJAFEE2006 ෤Ꮨ኱఍࠘,2007 ᖺ 1 ᭶ 23 ᪥-24 ᪥, 235-263 㡫.
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