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A market-to-book formula for equity strategies

A new proxy formula for equity market-to-book ratios suggests that (the logarithm of) such a ratio is equal to the discounted expected value of (i) differences between return on equity and market returns and (ii) the net value added from share issuance or repurchases. A firm with a higher market-to-book ratio must have lower future returns, higher return on equity, or more valuable growth or repurchase opportunities. One can cleanly decompose return forecasts into forecasts of future profitability, investment, and the market-to-book ratio at any horizon. Empirical evidence confirms that market-to-book ratios predict returns in the long run, but only to the extent that the ratio itself is not affected by profitability and investment. Indeed, profitability and investment value-added jointly explain about 60% of variations in market-to-book ratio and – hence – should be taken into consideration for investment strategies based on valuation ratios. This broader view helps to forecast returns on value stocks versus growth stocks.

Cho, Thummim, Lukas Kremens, Dongryeol Lee and Christopher Polk, “Scale or Yield? A Present-Value Identity”.

The below are quotes from the paper and some other sources which are linked next to the quote. Emphasis, headings, and text in brackets have been added for clarity.

This post ties in with this site’s summary on fundamental value estimates.

Basics of the market-to-book ratio

“The market-to-book ratio (also called price-to-book ratio) is…a company’s current market value relative to its book value. The market value is the current stock price of all outstanding shares. The book value…equals the net assets of the company and comes from the balance sheet…The ratio is used to compare a business’s net assets that are available in relation to the sales price of its stock.” [CFI]

“Two frameworks…heavily influencing asset-pricing research are the market-to-book equity ratio decompositions of Vuolteenaho (2002) and of Fama and French (2006).

  • Vuolteenaho’s log-linear decomposition measures a stock’s cash-flow fundamentals…approximates the log of the market-to-book ratio as a spread between future log returns on equity and future log [market] returns. [Specifically, the ratio is roughly equal to the discounted expected value of future differences between returns on equity and market returns]… missing the important role of future book equity investment.
  • Fama and French use [a] restatement of the dividend discount model to relate expected [market] returns to expected future profitability and investment as well as the current market-to-book ratio…[This] motivates ‘profitability’ and ‘investment’ as natural predictors… What limits the applicability of Fama and French (2006)’s identity is that it is nonlinear and includes nonstationary variables.”

“Book equity this year equals book equity last year plus earnings less dividends…[On this basis] one derives a model for the log book-to-market ratio [approximating it as the discounted future excess log stock returns minus the discounted future excess return on equity]…The book-to-market ratio can be temporarily low if future cash flows [or returns on equity] are high and/or future excess stock returns are low.” [Vuolteenaho]

Return on equity is a measure of financial performance calculated by dividing net income by shareholders’ equity…Net income over the last full fiscal year, or trailing 12 months, is found on the income statement—a sum of financial activity over that period. Shareholders’ equity comes from the balance sheet—a running balance of a company’s entire history of changes in assets and liabilities.”[Fernando and Mansa]

“Valuation theory says that expected stock returns are related to three variables: the [market-to-book] equity ratio, expected profitability, and expected investment. Given a market-to-book ratio and expected profitability, higher expected rates of investment imply lower expected returns. But controlling for the other two variables, more profitable firms have higher expected returns, as do firms with [lower market-to-book ratios]. These predictions are confirmed in our tests.” [Fama and French]

A new present value identity

“A present-value identity, such as…log-linear price-dividend ratio decomposition, is a relation among a valuation ratio, stock returns, and cash-flow fundamentals that holds in almost all situations, without assumptions on investor rationality or market structure…We provide a log-linear present-value identity in which investment, profitability, and discount rates together determine a firm’s market-to-book ratio…We decompose the log market-to-book ratio into contributions from future discount rates, returns on equity, and the value added from future value-enhancing investment.”

“We provide an identity that is linear and features stationary variables for profitability and investment… Our identity states [that the log market-to-book ratio is approximately equal to the discounted expected values of (i) future differences between returns on equity and market returns, and (ii) the log ‘investment value added’]…This refinement provides the important advantage that we can cleanly decompose return forecasts of any variable into forecasts of future profitability, investment, and the market-to-book ratio at any horizon.”

“The investment value-added [is] the value-added or lost from the firm’s net investment in equity capital through share issuances or repurchases… Issuance at market-to-book above one increases book-equity per share. Issuance at market-to-book above below one decreases book-equity per share…The opposite holds for repurchases…investment can take place through the payout-versus-plowback decision. Plowback is valuable if a dollar inside the firm is more valuable than a dollar outside.”

“Ceteris paribus, a firm with a higher market-to-book ratio must have lower future returns, higher return on equity, or more valuable growth or repurchase opportunities.”

Empirical analysis

“We combine monthly stock price data from the Center for Research in Security Prices and annual accounting data from CRSP/Compustat Merged to create a merged annual dataset for the period 1963–2020.”

“[We] show that (i) market-to-book ratios vary cross-sectionally due to cashflow growth differentials driven by both profitability (‘yield’) and investment (‘scale’), and thus firm-level return news reflects shocks to both cash-flow components, (ii) the asset growth anomaly involves short-run return predictability associated with a long-horizon decline in profitability, and (iii) variables that predict the two components of cash-flow growth improve time-series forecasts of portfolio returns beyond what the market-to-book ratio predicts.”

“We document the relative importance of profitability and investment [value added]for valuations and returns…Profitability and investment account for approximately 39% and 18% of the cross-sectional variance in market-to-book ratios respectively. Interestingly, while growth (high market-to-book) firms have always had faster-growing cash flows, this has increasingly been driven by the expectation of growth firms making more value-adding scale decisions [rather] than the expectation of higher profitability.”

“Profitability news and investment news—the sum of which is cash-flow news—account for around 67% and 18% of the total return news variance of a typical stock, with discount-rate news accounting for the rest.”

Important clues for investors

“[We] decompose cross-sectional return predictability (and its term structure)…into the predictability of profitability (return on equity), investment value added, and covariance with the market-to-book ratio….In particular, we examine the cross-sectional return predictability of asset growth, the primary investment-based return predictor…(i) Cross-sectional differences in asset growth are transitory, with little serial correlation beyond three to four years. (ii) The negative return prediction is short-lived and dies out at a similar horizon. (iii) The negative short-run return is the pricing-reflection of a steady long-run decline in shareholder cash flows from a fall in return on equity.”

“Our evidence that a variety of investment variables predict the yield and scale components of future cash flows should also help improve the ability of the value spread to forecast value-minus-growth returns. Indeed, we find that the investment value-added spread significantly improves…such forecasts, even in [recent times], when the link between the value spread and time-variation in the value premium [was] weaker.”

“In recent years, a low realized value premium has led to a lively discussion among academics and practitioners about the ‘death of value’. Just like many of the theoretical explanations for a positive value premium, the negative realization suggests a particularly large cash-flow growth differential between high and low market-to-book ratio firms…Growth firms do have higher realized cash-flow growth, justifying their higher valuations…Over the whole sample, we confirm the well-established result that value firms have higher returns, although with some important time-variation in the magnitude of this result, e.g., around the build-up of the dotcom bubble and its subsequent crash.”


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