From a Stream of Cash Flows to a Risk-Premium Curve
This chapter focuses on the valuation mechanism behind an equity-premium term structure: how the timing of expected cash flows changes the discounting problem. The practical judgment it supports is simple but important—before comparing two premiums, determine whether they are one average rate for an entire stream or rates attached to particular horizons. That distinction explains why a familiar dividend-discount calculation can be useful without telling the whole story.
When one discount rate compresses a whole cash-flow stream
The value of a share is the present value of what an investor expects to receive from it. In a simplified dividend-discount model, the calculation looks like this: expected dividends are projected into the future, each is discounted, and the discounted amounts are added together. The conventional version applies one required return to every future payment. If the stock price and the dividend forecasts are known, the analyst solves for the single return that makes the sum equal to the observed price.
That return is not meaningless. It is a compact answer to a real valuation question: what constant rate would price this entire projected stream? It behaves much like a bond’s yield to maturity. A bond’s yield to maturity is one rate that summarizes the price of a bundle of coupons and principal, even though the bond’s cash flows arrive on different dates. The dividend-discount rate performs a similar compression for equity.
In symbols, the constant-rate calculation asks for the value of expected cash flow at date m after discounting it by the same rate for every date: each payment is divided by a corresponding power of one plus that rate, and the results are summed. The unknown rate is chosen so that the sum equals the observed price. This makes the rate a property of three things taken together—the price, the expected cash-flow path, and the assumption that one rate can stand in for all dates. Change any of those inputs and the implied rate can change.
Consider a deliberately simple example. Suppose one claim is expected to pay 60 units in the first two years and 40 units later, while another is expected to pay only 10 units early and 90 units later. They have the same total projected cash, but they do not have the same exposure to discounting at each date. If distant compensation rises relative to near-term compensation, the second claim loses more present value for the same change in the long end of the curve. A single rate can summarize each price, but the two summaries will not reveal that common change in the same way. The payment schedule is part of the measurement.
The three-stage version of the dividend-discount model makes the expected-growth path less crude, not the return horizon-specific. In the ECB implementation, analyst earnings-growth expectations describe a medium-term stage, a long-run dividend-growth assumption describes the distant stage, and the intervening years are filled by a gradual transition. The cited calculation uses a 3.5% long-run real growth assumption and subtracts a ten-year real yield when it converts the solved return into a premium. Those choices can be sensible for the intended application, but they show why the resulting premium remains a whole-stream summary: one return is still applied to every future dividend.
But compression is not the same as resolution. Suppose two firms are each expected to deliver the same total cash over ten years. The first pays most of it during the next two years. The second pays little at first and is valued mainly on cash flows expected late in the decade. If the required compensation for near-term and distant cash flows differs, the two streams should not be discounted in the same way. A single rate can still be calculated for each firm, but it will be an average shaped by that firm’s payment schedule. It will not say which horizon carries the highest compensation.
This is the point of a term structure. Instead of asking only for the one rate that prices the whole stream, we ask for a schedule of rates: one attached to a cash flow arriving after a short interval, another attached to an intermediate date, and another attached to a distant date. In a discounted-cash-flow representation, the schedule determines how much each expected payment contributes to today’s value. The curve’s level describes the general compensation being required; its slope describes whether later cash flows carry more or less compensation than earlier ones.
The three-stage dividend-discount model makes the cash-flow side more realistic without removing this compression. In the implementation described by the ECB paper, medium-term earnings-growth expectations from the I/B/E/S analyst survey inform the first stage, and a long-run growth assumption governs the distant stage, with growth interpolated between them. The model still solves for one return on equity and applies it across all future dividends. Subtracting a long-term government yield produces an equity-premium estimate, but that estimate is best interpreted as an average over the horizons represented in the dividend stream.
A term structure asks a finer question because it keeps the discount factor tied to the date of the payment. The difference matters most when the stream is uneven. For a mature company with substantial current payouts, the observed price may depend heavily on short and intermediate cash flows. For a young or rapidly changing company, more of the price may depend on whether distant growth actually arrives. A single average rate can conceal that contrast, while a curve makes it part of the valuation problem.
The same logic applies to projects and corporate decisions. A project whose benefits arrive next year is not exposed to the same discounting schedule as one whose benefits arrive after a decade. Using one cost of capital for both may be a reasonable approximation in some settings, but it silently assumes that the horizon profile of compensation is the same. If the approximation is poor, the calculated net present value can give too much weight to distant benefits or too little weight to near-term ones.
This does not mean that every valuation model must use a separate rate for every date. It means that a single rate should be recognized for what it is: a summary that depends on the cash-flow path used to construct it. The more a decision depends on distant cash flows, the more important it becomes to know whether the rate is genuinely horizon-specific or merely an average that happened to fit a particular stream.
Cash-flow duration is useful language for this dependence. It does not mean that a company has a fixed expiration date. It describes how much of the present value is tied to cash flows arriving soon versus cash flows that require the firm to keep operating and growing. Two businesses can have the same market value today while having different duration because investors expect their value to be realized through different portions of the future stream.
That is also why an average market premium can conceal important differences. A broad index combines firms with mature payouts, long growth expectations, and varying financial strength. Its single implied rate may price the aggregate stream reasonably while saying little about the premium attached to one component’s near-term cash flow or another component’s distant cash flow. A term structure does not eliminate aggregation, but it gives aggregation another dimension: the weights can differ by date as well as by firm.
Three different meanings of a horizon return
Once payments have dates, “the return over five years” can still mean several things. The first is a maturity-specific discount rate. In the cross-sectional discounted-cash-flow approach, it is the rate used to price a cash flow at a particular maturity, after accounting for the risk-free rate at that maturity. This is a pricing object: it tells us what rate makes the expected payment consistent with the observed equity value.
The second is a hold-to-maturity return. Imagine a claim that pays one cash flow at a specified date, such as a dividend strip. The relevant return is the expected annualized compensation from buying that claim and holding it until the payment arrives. The New York Fed study uses this kind of maturity-specific discounting as its central object. It is not the same as looking at a stock’s realized return over the next year and then labeling that outcome a five-year premium.
The third is a one-period or forward return on a claim that itself matures later. An investor may buy a long-dated claim today, hold it for one year, and sell it before its final payment. The expected one-year return then depends on how the claim’s price changes as it moves closer to maturity. Under time-invariant discount rates, the one-period return on a claim maturing after m periods is closely related to a forward rate between nearby maturities. With time-varying discount rates, the relationship also depends on how discount rates evolve and how they covary with the cash flows.
These definitions can point in similar directions over long samples, but they should not be mixed casually. A hold-to-maturity premium describes compensation accumulated over the life of a dated claim. A one-period premium describes the next holding interval. A dividend-discount premium describes the single return that prices an entire expected stream. They answer related questions at different levels of aggregation.
A useful way to see the difference is to track a claim that pays only at the end of a stated horizon. If its price falls during the next year, the investor’s one-year return can be poor even when the expected return from buying it today and holding it to the final payment is positive. The change in price may reflect a revision in the discount rates used for the remaining years. Conversely, a claim can have a reasonable one-year expected return while carrying a lower average compensation when held all the way to maturity. The holding period determines which part of the curve is being tested.
The reinvestment convention matters for an equity index because dividends arrive before the final horizon. In the affine framework, the multi-period return assumes that those receipts are reinvested in the associated stock or index. That convention turns the sequence of interim payouts and the terminal price into one compounded investment outcome. A maturity-specific equity discount rate for a single cash flow is conceptually cleaner than an index return with reinvested payouts, but the two are connected through the way each treats interim cash. A reader should ask what happens to dividends before comparing the numbers.
The curve’s slope should therefore be read as a statement about relative compensation, not as a forecast that one maturity will necessarily earn more in every future realization. An upward-sloping expected-premium curve means that the model assigns higher required compensation to later horizons than to earlier ones, holding the stated conditioning information and benchmark fixed. It does not guarantee that a long-horizon realized return will exceed a short-horizon realized return. Outcomes can depart from expectations, and the curve can change when the state of the market changes.
The benchmark matters just as much as the equity return. An equity premium is an excess return, so the risk-free comparison must have a compatible horizon and currency basis. A short-horizon equity return minus a long-term government yield is not a clean measure of compensation for either horizon. The ECB model makes this matching explicit by defining the n-period premium as expected n-period stock return minus the model-implied real bond yield for that period. The New York Fed approach similarly separates the term structure of risk-free rates from the firm’s risk-premium curve.
The distinction between real and nominal values also matters. A nominal equity return includes compensation measured in the currency’s changing purchasing power; a real return removes inflation. A risk-free rate used as the benchmark must match that choice. The ECB model is built to connect nominal bond yields to a real pricing kernel through inflation, while the New York Fed model works with a risk-free term structure and a long-run nominal cash-flow-growth assumption. The two conventions can both be coherent, but their levels should not be compared as though they used identical units.
Horizon changes the interpretation of firm characteristics as well. Consider a value premium—the difference in expected returns between high and low book-to-market firms. If high book-to-market firms have cash flows concentrated earlier, a premium associated with those firms may be especially sensitive to short and intermediate discount rates. The observed difference need not mean that “value” is rewarded by one timeless risk factor whose effect is identical at every date. It may partly reflect where each group’s expected cash flows sit on the curve. The same caution applies to size and credit quality.
That is a mechanism, not a verdict. Different cash-flow duration can help explain a characteristic premium, but it does not establish that duration is the only cause. Firms with short or long cash-flow profiles may also differ in profitability, distress, liquidity, growth opportunities, or exposure to macroeconomic shocks. A term structure makes the timing channel visible; it does not automatically identify every other channel.
The horizon distinction also changes how a curve should be read during market stress. A longer investment horizon does not guarantee a smaller premium or protection from a recession. It only refers to when the cash flow or return is measured. In the affine evidence, short-horizon premia are more volatile than long-horizon premia, but all horizons respond to major market episodes. In the cross-sectional evidence, the aggregate curve remains upward sloping in the benchmark while becoming flatter in bad times. The short end can reprice sharply without the long end moving in the same direction or by the same amount.
A useful interpretation therefore has three parts. First, identify the cash-flow or holding horizon. Second, subtract a risk-free rate that matches that horizon and real or nominal basis. Third, ask whether the reported rate is a price-implied discount rate, a hold-to-maturity expectation, a forward one-period return, or a whole-stream average. Only after those choices are clear can the slope of an equity-premium curve be given economic meaning.
The remaining difficulty is identification. One firm’s observed price still gives only one aggregate valuation condition. To recover several maturity-specific rates, researchers need either systematic differences in cash-flow timing across many firms or a structural system that links stocks and bonds through common state variables. The next problem is therefore not how to define the curve, but how the data can make its separate horizons visible.