前言:
如果你不認識我，只要知道我是個不務正業的浪人就好。
我沒有商科或金融背景，愛看網路資料，偶而寫點文章，歡迎各位批評指教。
近期我比較常在FB/IG出沒，如果有問題或建議，可以留言或私訊找我，我不常登入就是了。
另外很重要的一點，我誠徵書或貼紙的拍照model，如果妳/你自信開朗也願意試試，歡迎聯絡 :)
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今天談「因子」在投資中的一些概念。
如果你對因子投資有興趣，除了可以和我討論之外，我也介紹兩個台灣的朋友，Frank和Jin.
兩個都業界學界實力派，Frank是投資沙岸撿貝人，CFA 3過；Jin則是我頭號大弟子，台大財金碩+資科(他應該都可以自立宗派了但很低調)。
事實上我們很常有意見不同的地方，但討論就是這樣才好玩。
關於我對被動投資、CAPM和多因子的簡短想法，各位朋友可以參考這篇。
在我寫的被動投資書《穩定致富》中，我建議讀者可以延伸閱讀一些資產定價的書，包括實證和理論。
這裡我快速把FB文章的脈絡接起來，如果各位想要深入了解，可以依據我提到的模型去研究。
不過基於功利性，我不推薦細讀，瀏覽即可。
先來點化石級理論101，這是近代投資框架的基石。
在隨機貼現(SDF)這種大一統模型之下，如果你只用前兩階矩描述投資人要的投資策略特性，或報酬是橢球聯合分布，就會讓二次型(MV最優)能夠描述效用函數，這時決策只和她的財富兩階矩有關。
如果你再假設他所承受的風險只有市場(以及MV有效、有無風險資產、同質預期、單期、無摩擦...等一堆假設，我就不展開)，那可以簡化成一個帶Beta的訂價模型，也就是CAPM.
而Merton的ICAPM則是對於單期CAPM的跨期拓展，不但有多因子，而且關心市場和狀態的變化。
至於CCAPM，就是一種基於消費、一聽就是經濟學產物的模型。(其中Campbell Cochrane的外生習慣偏好就是個例子。)
另一方面，APT本身是基於無套利，也不關心投資人是否理性、用什麼效用函數的假設，你只要知道他用多因子來定價即可。
那你怎知上面這些模型準確?
答案是，可以用一些工具來檢測。
兩向量內積不大於兩長度的乘積，這是柯西不等式。
應用在SDF上，資產的超額報酬期望值，在貼現因子E[m]規範到1的時候，等於和SDF的負共變異數。
同時這共變異數的絕對值，也不大於資產超額報酬的標準差和SDF標準差的乘積。
或者同規範下，一個合理模型的理論最大夏普率，應該不大於SDF的標準差，這個判斷準則就是「HJ邊界」，而對應的理論SDF和可投資空間張成的SDF的最小距離就叫HJ距離。
(如果你看過John Cochrane的Asset Pricing, 可參考§5.6, 我本來要在我的書中放入此當延伸閱讀，但考量到年代太老。)
那針對多因子模型呢? 可以用GRS檢驗。
如果模型夠好，多個Alpha=0是虛無假設，而我的p告訴我排除不了，那只好當模型在現有樣本下「無法拒絕」。(這裡小樣本和p hacking又是另一回事了。)
再來，GMM可以視為用理論的矩來擬合現實矩，因為不挑剔分布模型，也接受異質性變異數和序列相關，所以很常被用在模型的檢驗上，包括SDF和CAPM.
所以，之後的FF3/5/6, q4/5, Carhart4...也都可以藉由上述的工具來深度不等的檢驗。
「狂徒我還要，有沒有特別針對因子的?」
我就知道，上進的妳會這麼問，當然妳不問我還是會跟妳說。
我們剛剛看到各種誤差檢驗法，像是Alpha, R², HJ距離等，但考量到誤差結構，因子值得更好的待遇。
像是Newey West, 針對的就是異質變異數和自相關等會造成OLS偏差的特徵，讓t, p都穩定。
或是考慮到包含時間的面板資料，雙向聚類就可以處理兩資料的相關性。
當然還有各式的修正，像是小樣本修正和Shanken修正，其中後者又和Fama Macbeth有緊密關係。
FM2可以說是傳統因子投資繞不過的工具，而隨著理論演進，後人使用截面bootstrap, 隨機檢定、樣本切分、regime shift(應該翻成市場變換，但我從來不知道中文怎說)的穩定性...來檢驗這模型的效力，也因此會有Shanken修正，讓第二步標準差能反映第一步截面回歸的估計誤差。
好，現在先換個檔，我從別的角度來輕鬆聊因子這件事。
只說大概念，不談理論推導和工程細節(當然如果我哪裡說錯也歡迎告知)。
或許你會認為，因子的知識不少，但它也只是「量化」投資的一環。
在沒有配套流程的情況下，空談因子根本沒辦法改變投資組合的收益和風險。
妳至少需要有數據、模型和策略的支持，才能讓想法落地，而這還只是非常簡陋的說法。
我沒有要細談坊間的因子投資文化，因為這很大部分是券商想要獲利、民眾想要跟風的產物，例如Smart beta和各種ETF.
不是不行，但是你要知道這中間的傳導鏈很長。
因子要有效、未來要能重現歷史、券商要能把因子曝露做到商品上、超額收益(或IR)要能蓋過內扣成本...妳才真正有得利。
問題是，你怎知道因子的效力?
據我觀察，大部分人只是買個心安，少部分人會去看原始的方法論和相關學術論文，但只有真正認真的投資人才願意且有能力「回測」復現。
(我文章是寫給投資人看的，這也是我非常早期在D卡就討論回測的原因。)
這樣說好了，回測幾百個學術因子要讓個人電腦跑一整天，時間不長，但你這樣做了嗎? 而更後面的步驟我就先不談了。
測試，是區分聽明牌心態和科學研究的分水嶺。
我就曾經幾次在酒店和朋友互相招待，一起拿著筆電跑績效回測，把我們框的陪玩小姐姐/傳播晾在旁邊喝酒唱歌，這就是投資態度。
順便八卦，我夢到某國就有機構自營因子是做錯的，而自營Alpha實力通常要強於公募，因此妳可以猜一下公募ETF的正確率。
(自營是機構自己投入，策略虧損要自己承擔；公募ETF只是賺民眾錢而已，策略虧損也能穩賺內扣。)
「狂徒，所以應該怎麼做?」
一般而言，我們會分割資料集(訓練、驗證、測試)、設定(超)參數搜索的預算、使用多重檢定控制等方式，確保嚴謹性。
其中多重檢定又有一些分支，包括錯誤發現率檢驗(BH法、Storey q值法等，隨假設效力切換)、p值逐步重採樣系列，以及之後的弱隨機對照和隨機優越檢定(這Jin做過)。
如果你聽起來有些元素很像ML，也叫做統計學習，恭喜你答對了，他們確實同源。
對於一個因子，也應該考量一些如排序法和迴歸法的一致性，Barillas Shanken的夏普型檢定(也有人用更傳統的截面Fama MacBeth和時序GRS的F檢定)，替換測試等等。
如同我之前提到的，這個因子到底是風險補償還是錯誤行為? 你該區分嗎?
還是對你來說根本沒差，你只要策略收益?
這些看看有道理、實作都是抉擇的事情，都考驗一個因子投資人的基本功。
再來看看，人們為什麼要用因子體系?
答案很簡單，算力和信心不夠做端到端。
大家都希望把原始資訊輸入系統，直接得到下單訊號，但實際上沒人能有意義的做到，所以才會以分次提取、篩選的方式，抓出因子特徵。
數據預處理、清洗、特徵工程、映射、優化...等，就是在把總工程量降低到可接受的範圍，並且讓人類可以清楚每個環節的原理，期望最小化理論和實盤的差距。
從這個角度來看，因子有重要地位，不然策略只是垃圾進出。
然而檢視整個流程，投資追求的不犯低級錯誤，比單點做到卓越還重要，所以「大因子主義」很可能會造成資源錯配。
接下來，我們來看「可解釋性」。
一方面，有些人會希望看到經濟的可解釋，這樣的因子比較可能穩健存在，且更讓投資人安心。
然而這麼做的代價，往往就是讓因子之間的「共線性」很強，導致固定因子數量下的組合特徵可解釋力縮小。
另一方面，有些人寧願不要強可解釋力，而希望用統計學的方式盡量排除純機率產物。
他們的各種因子可能沒有準確名稱，但比較好抓到報酬。
當然這兩者的偏重，隨著流派而不同，也有些方法試圖解決這種衝突，例如偏向無損的正交化、PCA/iPCA...
至於特徵縮減是提高CP值的方法，像是eigen投組、Nagel等人的SDF投射，還有嶺和收縮等技術，都是好嘗試。
可是說到底，在線性組合和模型基底的前提下，你注定會損失很多交互項的資訊。
如果要再顧及簡潔性而強制「稀疏」，那就經常(尤其在樣本充足下)進一步犧牲了模型的表現。
「狂徒，怎麼辦?」
看起來讓人類能理解，以及實際上有效果，很難兼顧。
不過好消息是，還是有一些可能的突破點。
例如「因果」性的因子，我們可以用一些因果貝氏網路圖，重現風險和報酬的傳導方向和順序(若要考慮報酬時序變化，通常用DAG/PCMCI)，篩選出真正造成影響的父節點。
這不僅能最大無損的降維，而且還能提升因子表現穩定性。
一些像是雙重差分(DiD)檢查平行趨勢或事件研究的方法，多期錯位的處理，以及試圖解決內生性問題的工具變數(IV的關聯性、外生性和弱工具穩健檢驗)，還有底層工具如雙重ML、雙重穩健、去偏誤的LASSO、後(變數)選擇推論...都在讓投資人更能掌握因果性，例如穩健推論風險價格。
另外一條路就是引入XAI(用到LIME/SHAP/GAM)，試圖用另一個較小的擬合模型、局部代理和歸因框架，解釋大因子模型的原理。
雖然這比較像是後天賦予一個機制經濟敘事的定義，包括摩擦限制、套利、資訊擴散函數，但這種proxy的因果圖或結構方程式(利用N/PC/FCI再剪枝)，效能往往好過於人類的猜測和感覺。
因此你能驗證和理論的方向是否一樣，而不是只停留在重要度排名。
你說我為何考慮這些?
大概是因為我之前和Jin討論過Pastor Stambaugh貝氏先驗(和階層先驗)和收縮、流動性因子，還有和頻率派「懲罰項」而有些心得。
至於降維這件事，又有很多非線性的作法和嘗試。
像是利用自編碼器(autoencoder)的表徵學習，提取出潛在因子(latent factor)，也可以減少壓縮損失。
或是壓到流形(manifold)上，再用距離、局部線性或NN來處理，也有人嘗試在理論上使用TDA(topology)和同胚態射(homeomorphism).
(跳脫因子區，也有人用Monte Carlo和HRP家族來解決「維度爆炸」問題，但那比較屬於處理高維資料，妥協於算力的工程手段。)
因子挖掘呢? 看你信不信工程自動化。
人力做到規模最大的應該是WorldQuant，這個有比賽過、當導師的會很清楚，但我覺得相當血汗。
(當然如果你志在進入量化機構，有WQ的經歷是加分項，只是妳最好要知道這可能僅因為妳耐操。)
至於自動生成，可以分成數據驅動和知識驅動。
數據驅動又分為rule based的算子因子生成，包括暴力因子組合、GP/GA，或是ML提取、RL改進的做派。
知識驅動可以是單純的RAG+embedding+LLM，接近使用GPT的作法，或是進一步耦合的RD agents，能自動讀論文並和策略區共優化(co optimization).
現在當妳很高興的拿到一堆單因子，好戲才要開始，因為妳要考慮怎麼組合它們。
就以混頻率、多模態的原始因子而言，你怎知Alpha decay, 動態性是什麼函數?
妳要怎麼把輿情、價量、統計、財報、影音...的資訊融合?或乾脆捨棄?
業界可以很輕易的挖到破億因子，實際入庫的從幾百到幾萬都有，怎麼辦到的?
現實就是資源有限，妳需要取捨。
你是該把時間和算力花在挖掘本身，期望大力出奇蹟?
還是透過GMM, GRS, MV spanning, 多元回歸等方式，用因子的潛能篩選?
(這裡好笑的是，三階以上風險矩像是coskew, cokurt, 理論上要被mimicking進入投資組合，但也有偷懶做法是直接被降成一階因子，走綠色通道再來。)
或是用BL和Bayesian置入一些主觀判斷(雖然這方式有點爭議)?
還是，妳想算了，在既有因子下直接優化妳的策略和最終權重?
我曾經在西濱騎車兜風，時速300左右的時候遇到條子，我後來查看GPS，也不過285公里每小時。
你知道這代表什麼嗎? 這代表我面前有個快樂度5%的時速表。
同理，當妳看著完美的因子曲線和實盤的差距，要怎樣確保損失最小? 又怎麼量化的測量誤差? 妳該花多少成本在這上面?
或者是你不考慮這些了，乾脆採取局部外包、借鑒策略，像是走Barra體系的前段因子生成、後段風險因子矩陣，但中段in house模型的「夾心」設計?
(BTW 我出席的場合上，S&P公司跟我說他們有賣六百多個因子，順便賣因子模型，但我嚴重懷疑這些有效性。)
因子是個有趣的框架，但它不是唯一。
想想看，不管你的因子功力多強，你都希望整體框架的跨資產、跨國、跨品種、跨結算規則、做空和漲停限制...之下，有最好的表現，而不是理論上冠軍，實際上吃土。
那麼你的風控能停在交易端嗎? 妳是不是應該考慮各種「壓力」，像是波動、流動性、尾部、特定環境的因子衰退?
你在算力和效力的取捨，是否也是一種優化? 包括你在思考優化這件事情的時候，是否又是一種時間-資本-優化效益的meta(元)優化?
我知道有人會把因子歸類在截面，而後套用時序模型(或是平行開發CTA)，最後再將策略混合，得出權重。
這聽起來非常合理，因為分別對應選股、擇時、配置，也確實比較省力。
不過仔細想想，這些環節實際上可以交叉使用，例如用時序特徵(通常是價量因子)選股，以及利用時序分析，包括傳統的ARCH/GARCH/ARIMA和GRU/LSTM/TFT...處理截面因子的變化。
至少在學界，已經有人利用ML處理因子的動量變化(也就是因子的因子)，還有像修大成那種做法，直接把價量轉圖、餵給CNN，抓隱藏因子。
也有一些比較從底層解決的做法嘗試，像我曾利用AWS的平台估算量子計算在不同廠家的成本，希望應用在多投資組合優化。(包括量子退火等算法，和超導、離子阱成本和持久力的取捨。)
在目睹技術提升的同時，我們也可以從更抽象宏觀的視角，俯瞰因子在投資框架中的角色。
這呼應了我開頭提到的因子存在意義，當算力和算法越來越先進時，它本身的邊界會逐漸模糊。
一些很工程導向、尤其是AI/ML類的的解決方式，像是用DRL貫穿全流程的FinRL，以及台灣就有人在研究交易用途(但有簽保密協議)的MARL，或是聽起來有點好笑的multi AI agents甚至agentic的決策流程(雖然按業界某朋友說法是startup騙VC錢的)，以及LQM(大量化模型，目前主要用於生技等)和LIM(大投資模型，主要是理論框架)的兩類端到端嘗試...隨著算力的爆發性提升，都在讓因子分析、提取、配置，逐漸融入整體。
就如同當所有人都熟悉因子加上ML的時候，ensemble, AutoML, HPO, NAS, meta learning...等工具，就從選項變成了必備。
或者換個說法，局部優化正在擴大成全局優化。
相似的，考慮到交易的點差、流動性和永久或臨時衝擊成本(包括Kyle lambda的線性，Amihud這種雙曲線，以及更時序分析導向的函數，還有像Almgren Chriss之類的優化法)，策略和因子的穩健性、泛用性和精確性權衡，疊代提升速度和生產穩定性的兼顧，模型複雜度和可解釋性，部署端的DevOps、CI/CD的pipeline和自適應、混沌性工程、版本控制和錯誤歸因樹(這也有點像因子分析)...各種深度、廣度和層級的優化，都在讓體系越來越強。
這可不僅僅是理論上的最高SR、最大四階矩優化、最大回報、最大散度或最小波動，也不是簡單的掛個RL去優化交易端...而是涉及費後、實際上的Alpha, IR、CE(確定性等價，包含報酬、風險、成本)、持倉上限、借券成本、成交率門檻、擁擠度、周轉率、研究報告成本、滑點控制、算力成本回饋到模型架構設計、預期邊際報酬和研發成本、數據壓縮和傳輸...的優化。
以上這串，我本來也沒怎麼研究。
但當我和業界避險基金、公募基金、私募基金、家族辦公室經理，投行MD、機構研究員、交易員、數據商、startup CEO/PhD朋友和工程師交流之後，就覺得問題不小。
你完全可以說這不關因子的事情，但你在測試因子的時候，根本逃不掉。
如果你高興，也可以稱此為涉及知識、資本、時間、人才、效率和膽識的「軍備競賽」。
那麼，會不會有一天，我在JPM期刊讀到的"new" Fama MacBeth，也會變成曾經的匠心獨運?
就如同第一次有人把LASSO應用到old Fama Macbeth上的時候，在模型估計表現的突破，至今變成了人類繼續前進的基石，
依然有人在努力，就像French教授的網站，還在持續更新因子簡單歸類的股票數據。
依然有人求突破，就像那位戲稱自己「被巨人踩在肩膀上」、和Fama有點恩怨情仇, 提出性能更優良的q5模型的張櫓教授等人。
依然有人在挑戰知識前緣，就像離開Fama的效率假說門下，建立AQR的Asness，後來轉向使用了ML.
學海無涯，或許一個人永遠也學不完，但市場的錢也永遠賺不完。
如果你看完我以上寫的閒聊內容，覺得有興趣了解，不妨找幾篇文獻來讀，思考自己所處的位置，以及可補強和避免的地方。
我也鼓勵各位多交流，我之前寫信給Fama, French, AQR的時候，可沒預期會收到怎樣的回覆。
我在矽谷的酒吧搭訕妹子演講者，也不知道會遇到同行；我在華爾街的聚會上，也沒想到能認識後來合作研究的朋友。
如果你對因子也有了解，歡迎討論。
我是狂徒，我們下次不知道什麼時候見。
註:
本來這篇文章是我吃完飯隨手寫的，所以也沒有什麼深度和引用(我也不可能把真的know how寫出來)。
後來想想還是找一些文獻補上，各位可以參考，當作我這些簡介的延伸閱讀。
(警告，這複製貼上自我的一些筆記和私人對話，以及我購買和參閱文獻的原始引用，僅經我肉眼確認，並用程式腳本排序編號。有些文獻後來被作者撤下，因此務必自行翻閱、驗證。)
你想真的脫離本文的淺層概念的話，還是要花時間自己提升，
保持批判和學術攻擊，當我說的都有可能出錯，這才叫「思考」。
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