Scientific Context
Financial market participants have developed an extraordinary statistical expertise in risk modelling and in correlations between returns and market ratios — fuelling quantitative strategies, the most prominent family of which is factor investing. The proliferation of factors has grown so extensive that practitioners speak of a “factor zoo.”
By focusing on realised returns and their statistical behaviour, quantitative researchers turn away — explicitly or not — from financial fundamentals. A telling example: the Modigliani-Miller dividend irrelevance theorem (1961) demonstrated that dividends carry no intrinsic significance for valuation, yet Dividend Yield remains a widely used factor. In doing so, “quants” set aside a universal principle, too often overlooked:
An asset has only two sources of return: it yields, or it appreciates.
PRODIG taps into this underexplored field upstream of the data: working with financial and accounting identities to link asset performance precisely to its steady-state yield and to changes in valuation conditions.
Hence the acronym A.C.E. — Analyse en Composantes Exactes, Exact Component Analysis.
ACE and Expected Return
A preliminary note regarding the word ‘return’: unfortunately, it is often used without specifying whether it refers to a past performance or an anticipated one.
PRODIG’s tools sit upstream of expected return calculations, whether the anticipation comes from a discretionary portfolio manager or a quantitative algorithm.
Before estimating a prospective return, it is very useful to calculate what the asset yields in a stationary environment — this is precisely what the Cost of Equity of a stock measures.
This calculation simplifies the assessment of an anticipated return. For one quarter, for example:
Symmetric application: deriving the change in valuation implied by a given prospective return.
ACE and Risk Premia
Like return, “risk premia” have neither a precise definition nor a universal standard. A general definition: the risk premium is the excess return delivered by an asset above the risk-free rate.
But a fundamental distinction must be made from the outset: observed excess return or prospective risk premium? Arnott and Bernstein captured it precisely over twenty years ago — and their remark remains as relevant as ever:
“The observed excess return and the prospective risk premium [are] two fundamentally different concepts that, unfortunately, carry the same label — risk premium.”
(Arnott and Bernstein, 2002, Financial Analysts Journal)
ACE deals exclusively with prospective risk premia, calculated from current market conditions.
A second distinction: is the surplus return expected over the entire life of the asset, or over a short horizon (one day, one quarter)?
ACE defines the daily risk premium as the annualised one-day surplus return, between today and tomorrow.
(No intraday data is used.)
To compute it, one simply subtracts the risk-free rate from the Cost of Equity calculated by ACE.
By equating the risk-free rate with the security’s refinancing rate, the difference between the ACE Cost of Equity and the refinancing rate — i.e. the Risk Premium — is what practitioners call Carry.
ACE and Mathematics
The mathematics of ACE are straightforward in theory: no stochastic calculus, no regression, no statistics. For equities, it consists of exact identities linking financial and accounting data.
This apparent simplicity conceals a demanding rigour. In many contexts where stochastic calculus is routinely invoked, a careful treatment of second-order terms proves not only sufficient, but exact — keeping ACE firmly in deterministic territory, with precise results.
A further dimension is algorithmic: translating these identities into daily outputs requires reconciling financial data (available daily) with accounting data (published quarterly, and with a delay). ACE solves this to deliver a Cost of Equity that is both exact and continuously updated.
ACE and Machine Learning
Leading practitioners in the field broadly recognise two approaches to Machine Learning.
- The first: ingest large volumes of data and extract decisions from them. This is the appropriate approach for purchase recommendations on an e-commerce platform, for instance.
- The second: validate hypotheses using those same data. This is where ACE comes in.
By decomposing each return into two terms with precise financial meaning (a running steady-state yield and the effect of valuation change), ACE enriches raw data with an interpretable structure. Since its indicators are comparable across all asset classes, they are strong candidates for algorithm development. ACE is less a model than a toolkit: it provides an innovative framework precisely where Machine Learning needs financial theory to keep it grounded.
As Lopez de Prado has written:
“We need financial theories to restrict Machine Learning’s propensity to overfit.”
(Marcos Lopez de Prado, Financial Times, 11 February 2019)¹
¹“The fund industry has to use the right type of machine learning”, Marcos Lopez de Prado, Financial Times, 11 February 2019.