How the Cross-Section Makes the Curve Visible
This chapter examines how the 2026 New York Fed study turns the equity-premium term structure from an economic idea into an estimable object. The key question is why a panel of firms can reveal several maturity-specific discount rates when one firm’s price normally gives only one overall valuation condition. Understanding that identification chain also makes the result easier to judge: the curve is inferred from expected cash flows, a chosen shape, and observed prices, so those ingredients are part of the evidence rather than invisible technical details.
A price, a cash-flow path, and a constrained curve
Start with the valuation condition. For firm i at time t, the market value of equity is written as the present value of expected shareholder free cash flows arriving at dates m = 1, 2, and onward. Each cash flow is divided by a discount rate that can depend on its maturity. The equation is conceptually straightforward: forecast what shareholders may receive, discount each dated payment, and make the total line up with the firm’s observed market value.
An infinite stream cannot be estimated directly, so the study introduces a terminal horizon, T. Up to T, cash flows and discount rates can vary by maturity. After T, expected cash flows grow at a common steady-state rate and the discount rate remains at its terminal level. The distant stream then becomes a terminal-value term. This is a practical boundary for the calculation, not a claim that firms suddenly become identical in year 15 or any other chosen year.
The next step solves the problem that makes a term structure difficult. A conventional implied-cost-of-capital calculation would use the observed price and expected cash flows to find one firm-specific rate. That rate may fit the whole stream, but there would be no remaining information inside that one equation to estimate separate rates for years one, five, or fifteen. The New York Fed approach instead assumes that the discount-rate curve has a common structure across observations while allowing it to vary with observable characteristics.
In broad terms, each maturity-specific total discount rate is the risk-free rate at that maturity plus an equity risk-premium function. The premium function depends on maturity and can also depend on characteristics such as size, book-to-market, and credit risk. The shared structure supplies discipline; differences in firms’ expected cash-flow patterns supply the identifying variation. A firm whose expected value is concentrated in near-term cash flows loads the price condition differently from a firm whose value depends on distant cash flows. Across many firms, those different loads help distinguish a change in the curve from a change in the cash-flow path.
The authors estimate the unknown curve parameters jointly across firm-year observations. For each observation, they calculate a relative pricing error: the model value divided into the expected discounted cash flows, minus one. They then choose the parameters that minimize the weighted sum of squared errors. This is not a procedure that observes a separate premium at each maturity. It is a calibration of one structured schedule so that it explains the cross-section of observed equity values as well as possible.
The direction of the identification can be understood without solving the optimization. Imagine two firms with the same risk-free curve but different expected cash-flow timing. If the first firm receives most of its expected value soon and the second receives most of it far in the future, changing the long-maturity premium changes the second model value much more than the first. If the model systematically overvalues the long-duration firms while fitting the short-duration firms, a higher long-horizon discount rate is one possible way for the shared curve to reduce that error. The panel contains many such contrasts, allowing the estimation to use relative sensitivity to dates rather than pretending that each firm directly reports a rate.
This logic also explains why observable characteristics are included. Size, book-to-market, and credit risk may be associated with discount-rate differences that are not simply consequences of payment timing. Letting the level and slope depend on those characteristics gives the model a way to represent systematic cross-firm differences instead of forcing all pricing variation into cash-flow duration. It does not guarantee that every relevant risk is controlled for. It is a structured compromise between an unconstrained firm-by-firm calculation, which cannot identify a curve, and a single curve for every firm, which may hide real heterogeneity.
The structure used for that schedule is a modified Nelson–Siegel curve, a functional form familiar from bond-yield modeling. Its two main components have an intuitive reading. The level sets the general height of the premium curve. The slope controls how much the short end differs from the long end. A decay parameter determines how quickly the curve approaches its long-run level. In the 2026 specification, the level and slope can interact with firm characteristics, so two firms need not have the same shape.
The reported version omits the Nelson–Siegel hump term. As a result, it is designed to capture smooth monotonic increases or decreases rather than every possible pattern. A curve that falls at the very short end and then rises later—a U-shaped pattern—cannot be freely represented by this baseline form. That limitation does not make the curve useless; it tells us what kind of evidence the model is able to express and what kind it may smooth away.
The benchmark choices make the reported slope precise in meaning. The terminal horizon is set to 15 years, and the slope is normalized as the 15-year premium minus the 5-year premium. The five-year anchor reflects the authors’ concern that one- to four-year estimates are the least precise because many firms’ cash flows have longer duration. Long-run nominal cash-flow growth is set to 6%, value weights are used within years that receive equal overall weight, and the decay parameter is fixed at 4. These are transparent conventions for estimation, not properties that the market is known to obey.
The distinction between level and slope is especially important. The paper reports that changing the long-run growth assumption can move the long-term premium nearly one-for-one—for example, a 3% growth assumption produces a long-term premium near 5%, rather than the benchmark’s roughly 8.3%. The slope changes much less. Thus, the evidence for an upward shape should not be summarized as evidence that the quoted long-term level is a universal cost of equity.
The terminal value makes this distinction especially visible. Beyond the chosen horizon, a growing perpetuity is valued using the terminal discount rate relative to the steady-state growth rate. A small change in either assumption affects a large block of distant cash flows at once. The method can still learn about the curve before the terminal date, but the long end is not independent of what the model assumes about the firm after the explicit forecast period. This is why the authors vary the forecasting horizon and long-run growth rate rather than presenting the terminal level as a directly measured market quantity.
There is also a difference between estimating a curve and proving that its shape is economically fundamental. The Nelson–Siegel form reduces the number of free parameters and keeps nearby maturities from moving independently, which is valuable when the optimization is nonlinear and the cash flows are imperfectly forecast. The price of that stability is that some shapes are ruled out before the data are fitted. Statistical precision around a smooth curve tells us how uncertain the fitted parameters are within that representation; it does not test every alternative representation of the term structure.
The study also reports uncertainty rather than treating the fitted line as exact. Because the estimator is a weighted nonlinear least-squares procedure, the authors adjust standard errors for heteroscedasticity and for time-series and cross-sectional correlation in pricing errors. They then use the delta method to obtain uncertainty at each maturity. The result is a curve with a statistical envelope, not a row of equally reliable point estimates.
Forecasting the cash-flow clocks
The cross-section can identify a slope only if the cash flows assigned to different firms really differ across time. The forecast system is therefore central. The study starts from a clean-surplus accounting identity: shareholder free cash flow equals earnings minus the change in book equity. It proxies book-equity growth with sales growth, following evidence that sales growth predicts book-equity growth better than many other accounting variables. The forecast is not just a technical input; it determines how much value the model places at each maturity.
For the first one to five years, the authors use analyst information when it is available. Revenue forecasts provide a path for sales growth. Direct forecasts of return on equity are used where coverage exists. Because ROE forecasts are relatively scarce, the authors also construct implied ROE forecasts from median earnings-per-share forecasts divided by projected book equity per share, assuming the number of shares stays fixed. This expands coverage, but it also introduces a chain of assumptions: an EPS forecast becomes a profitability forecast only after book equity has itself been projected.
This construction matters because the model needs a sequence, not just a forecast for next year. A one-year earnings estimate says something about the first payment interval, but it cannot by itself determine how much of today’s price depends on years four, ten, or fifteen. Revenue forecasts help extend the growth path; projected book equity provides the denominator for profitability; and the accounting identity converts those operating forecasts into cash available to shareholders. Each link affects the location of value along the maturity axis.
The authors compare the implied forecasts with actual ROE forecasts and report correlations of roughly 70% to 80% at shorter horizons and about 55% at five years. Those figures support the construction as a useful approximation, not as a direct observation of investors’ expectations. Analyst forecasts can be optimistic or noisy, and the paper later tests corrections for documented forecast bias. The fact that the forecasts are informative and imperfect is precisely why their role must remain visible when interpreting the curve.
After the analyst horizon, the model uses mean reversion. ROE moves toward a long-run level of 12% with a baseline autoregressive coefficient of 0.81. Sales growth, used as the book-equity-growth proxy, moves toward 6% nominal growth with a coefficient of 0.40. These coefficients imply that initial differences in profitability and growth fade gradually rather than disappearing immediately. The authors prefer estimates based on more distant lags because simple short-lag pooled regressions can make the initial reversion look too fast while missing the slower decay that matters for valuation.
The choice is economically consequential because a forecast that reverts quickly puts less weight on a firm’s distinctive current condition in later years. A forecast that reverts slowly allows present profitability or growth to affect the terminal approach for longer. The first choice can make firms’ cash-flow paths look too alike; the second preserves more cross-sectional variation but risks carrying temporary conditions too far into the future. The study’s reported coefficients are an attempt to describe long-term behavior, not a claim that every firm follows the same transition at the same speed.
This persistence affects identification in two ways. If every firm is forced into the same steady state almost immediately, their future cash-flow patterns become too similar. Cross-sectional pricing differences then have to be explained mainly by initial valuation ratios or by large differences in discount rates, making the separation between short- and long-run premia weaker and noisier. If persistence is plausible, firms retain distinct cash-flow clocks long enough for the panel to contain information about several maturities. But if persistence is set to an extreme value, the estimated slope can change sign. The modeling choice is therefore a boundary of the evidence, not a footnote.
The sample is broad enough to make that cross-sectional strategy possible but not neutral. The analysis uses U.S. data from 1980 through 2024 and an annual panel with June as the valuation month. Starting in 1980 ensures more than 500 firms per year in the relevant cross-section. After standard filters, the final Compustat–CRSP–I/B/E/S sample contains 56,126 firm-year observations. The analyst-covered firms are, on average, larger than the broader accounting sample, so the panel gains forecast information partly by selecting firms that analysts follow more closely.
The authors exclude financial and utility firms, firms outside the major exchanges, observations with large market-cap mismatches between databases, and extreme implied internal rates of return. They also winsorize some variables and reduce the influence of outliers. These choices make the valuation exercise less dominated by observations that standard fundamental models find difficult to fit. They also mean that the resulting curve describes a selected U.S. equity sample, not every listed company and not every economy.
Weighting determines whose pricing errors matter most. Value weighting puts more emphasis on economically large firms, while equal weighting across years prevents later observations with larger dollar market values from dominating the long sample. This is a defensible way to target an economically meaningful aggregate curve, but it is still a choice about the population being summarized. A different weighting scheme could emphasize a different mix of cash-flow durations and characteristics.
The authors test several ways that the forecast system might be distorting the result. They correct analyst earnings forecasts for documented optimism, allow the steady-state growth rate to vary with historical or professional-forecaster information, average the analyst-based path with a simpler mean-reversion path, cap extreme cash flows, alter winsorization, and vary the terminal horizon. The aggregate curve remains upward sloping across these exercises, often with a stronger slope. At the same time, the persistence experiments show why “robust” must be read carefully: plausible alternatives preserve the qualitative pattern, while extreme reversion can weaken or reverse it. Robustness narrows the range of concern; it does not remove the dependence on how future cash flows are modeled.
The identification chain now has a clear interpretation. Prices provide the valuation target. Forecasts place expected cash flows along the time axis. Cross-firm differences make that time axis informative. The Nelson–Siegel form prevents the estimates from becoming too flexible, and the panel objective converts the information into common level and slope parameters. The output is a disciplined ex-ante inference, but it remains conditional on the forecast system, the terminal horizon, the selected sample, and the allowed curve shape.
That conditional character is a strength as well as a limitation. It lets the study examine a long period without relying on a short and noisy market for dividend derivatives, and it can let the curve vary with firm characteristics. But the same design raises a second question: what if expected compensation is not primarily revealed by differences among firms, but by the changing state of the whole financial system? A different framework answers that question by making one pricing kernel price both bonds and stocks.