DA KUƊIN BASHI, MAI CACA YANA CIN NASARA SAU ƊAYA CIKIN BIYU

Ka yi tunanin wani “kamfanin kuɗi” a mafi sauƙin siffarsa. Yana farawa da wasu kuɗi kuma yana son ninka su sau r. Yana iya yin caca ta adalci kawai — caca wadda matsakaicin ribarta sifili ne — kuma ba zai iya haɗarin fiye da abin da yake da shi a yanzu ba. Yana ci gaba da caca har ya cimma burinsa ko ya yi fatara.

Ka’idar yiwuwa (probability theory) tana ba da amsa tabbatacciya: babu yadda za a samu riba a matsakaici. Wani sanannen sakamako, ka’idar tsayawa ta zaɓi (optional stopping theorem), yana takaita damar kaiwa ninki r na kuɗin farawarka kafin fatara zuwa 1 cikin r. Kana son ninka kuɗinka sau 1,000? Damarka ita ce ɗaya cikin dubu a mafi yawa. Ana kiran wannan faɗuwar mai caca (gambler’s ruin).

Matsalar Lehman

William Kuszmaul, masanin kimiyyar kwamfuta a Jami’ar Carnegie Mellon, ya fara gajeriyar takardarsa da wani lamari na gaske. Daga 1995 zuwa 2008, Lehman Brothers Holdings ta ninka hannun jarin masu hannun jari kusan sau 8, daga kusan dala biliyan 3.4 zuwa kusan dala biliyan 26. Da caca ta adalci kuma ba bashi, wannan ya kamata ya faru da yiwuwar kusan 1 cikin 8 a mafi yawa.

Amma akwai wani abu. Lokacin da Lehman ta yi fatara a 2008, darajarta ba sifili ba ce: ta kai kusan debe dala biliyan 150. Da kusan dala biliyan 26 ɗinta, ta karɓi bashin kusan dala biliyan 640 don yin caca. A tsawon wannan lokaci, ma’aunin leverage nata — jimillar kadarori a raba da hannun jari — ya kasance tsakanin 20 da 40. Tana ta haɗarin fiye da abin da take da shi sau da yawa. Marubucin ya ƙara da cewa kwatancen nasa ya yi watsi da abubuwa da yawa, kamar haɓakar kasuwar hannun jari gaba ɗaya da sabon jari da aka tara daga masu hannun jari.

Wasan leverage

Takardar ta mayar da wannan wasa mai cikakken bayani. Wata ’yar wasa, Alice, tana farawa da $1 kuma tana nufin dala m. Kowace caca dole ta zama ta adalci, amma kafin kowace ɗaya tana iya karɓar bashi har zuwa ninki L − 1 na abin da take da shi, don haka tana iya rasa har zuwa ninki L na kuɗinta. Wasan yana ƙarewa idan ta cimma burinta ko ta faɗi zuwa sifili ko ƙasa da haka.

Matakin da ya fi bayyana, babbar caca ɗaya don dukan burin, ba ya da kyau. Ninka kuɗinta akai-akai ya fi kyau. Darasin shi ne a yi girma a hankali, don kowace nasara ta ba ta ƙarin abin da za ta yi amfani da shi a matsayin leverage a gaba. Mafi kyawun dabara tana tura wannan zuwa iyaka: a kowace caca, Alice tana nufin samun ɗan ƙaramin kaso ε na kuɗinta kawai, wanda take ci kusan kowane lokaci, yayin da take haɗarin dukan kuɗinta tare da bashin a kan asara da ba kasafai ake yi ba.

Dama ɗaya cikin biyu ta ×1,000

Marubucin ya tabbatar da cewa wannan dabara tana nasara da yiwuwar da take kusantar

m^(−1/L)

— ninkin da ake nufi a ƙarfin debe ɗaya bisa leverage — kuma cewa babu wata dabara da za ta fi wannan. Tabbacin yana amfani da wannan ka’idar tsayawa ta zaɓi, ba a kan dukiyar Alice ba amma a kan tushen L ɗinta (L-th root), wani adadi da zai iya karkata zuwa ƙasa kawai a matsakaici.

Lambobin suna da ban mamaki. Da leverage na 10, ƙasa da na Lehman, damar ninka kuɗinka sau 1,000 ba tare da taɓa yin fatara ba kusan 50% ce, tun da 1,000^(−1/10) ≈ 0.5. A gaba ɗaya, da leverage L, damar rabi da rabi tana ba ka ninki 2 a ƙarfin L. Da leverage 20, wannan kusan miliyan ɗaya ne.

Wa ke faɗuwa

Mu koma ga Lehman. Da leverage sama da 20, samfurin wasan yana cewa girma ninki takwas ba abin mamaki ba ne sosai: yiwuwarsa kusan 8^(−1/20) ≈ 90% ce, ko da da caca ta adalci kawai. Hakika bashi ba kyauta ba ne; Lehman ta ƙiyasta cewa tana biyan kusan 5% a shekara na ruwa. Saka ruwa a cikin wasan ba ya canza ƙarshen magana, muddin kuɗin ruwan ya kasance ƙasa da 1 bisa leverage.

Kuszmaul ya bayyana a fili cewa wannan samfurin wasa ne, wanda aka rubuta don darasinsa na algoridim na bazuwar (randomized algorithms), kuma ba ya da’awar bayyana harkokin kuɗi na zahiri. Darasinsa ya fi ƙunci kuma ya fi kaifi: jerin riba mai ban mamaki ba hujja ba ne na caca mai kyau. Da isasshen leverage, caca ta adalci na iya gina arziki da babban yiwuwa — kuma idan asarar da ba kasafai take zuwa ba ta zo, tana faɗuwa nesa ƙasa da sifili, a kan wani wanda ba mai cacar ba.