Implementing Fischer Black's Simple Discounting Rule

Corporate managers typically estimate the value of capital projects by discounting the project's expected future net cash flows at the cost of capital. The capital asset pricing model (CAPM) is generally used to estimate that cost. But, as anyone who has worked on the finance or business development staff of a public company can attest, there are major challenges in applying the CAPM, including largely unresolved questions about what constitutes the “market portfolio,” how to estimate market risk premiums, and how to estimate the betas of projects. In a short article published in Financial Management in 1988, Fischer Black proposed a valuation “discounting rule” that avoids all these problems—one that involves discounting a relatively certain (as opposed to an expected or average) level of operating cash flows at the risk-free rate. But Black's article does not address the question of how to calculate these “certainty equivalent” or “conditional” cash flows. In this article, the authors propose a way of implementing Black's rule that involves estimating the “conditional” cash flows in a three-step procedure:

• • Find a benchmark security that correlates with the project's cash flows;
• • Estimate the percentiles of the distribution in which the benchmark return equals the risk-free rate over different investment horizons;
• • Use information from corporate managers to assess the cash flows that define the same percentiles in the cash flow distributions.

As the authors point out, the virtue of Black's rule is that it shifts the focus of the analyst away from the assessment of discount factors and puts it squarely on the more challenging, and arguably more relevant, problem of estimating the project's cash flows.     

The Role of Real Options in Capital Budgeting: Theory and Practice1

Most discussions of capital budgeting take for granted that discounted cash flow (DCF) and real options valuation (ROV) are very different methods that are meant to be applied in different circumstances. Such discussions also typically assume that DCF is “easy” and ROV is “hard”—or at least dauntingly unfamiliar—and that, mainly for this reason, managers often use DCF and rarely ROV. This paper argues that all three assumptions are wrong or at least seriously misleading. DCF and ROV both assign a present value to risky future cash flows. DCF entails discounting expected future cash flows at the expected return on an asset of comparable risk. ROV uses “risk‐neutral” valuation, which means computing expected cash flows based on “risk‐neutral” probabilities and discounting these flows at the risk‐free rate. Using a series of single‐period examples, the author demonstrates that both methods, when done correctly, should provide the same answer. Moreover, in most ROV applications—those where there is no forward price or “replicating portfolio” of traded assets—a “preliminary” DCF valuation is required to perform the risk‐neutral valuation. So why use ROV at all? In cases where project risk and the discount rates are expected to change over time, the risk‐neutral ROV approach will be easier to implement than DCF (since adjusting cash flow probabilities is more straightforward than adjusting discount rates). The author uses multi‐period examples to illustrate further both the simplicity of ROV and the strong assumptions required for a typical DCF valuation. But the simplicity that results from discounting with risk‐free rates is not the only benefit of using ROV instead of—or together with—traditional DCF. The use of formal ROV techniques may also encourage managers to think more broadly about the flexibility that is (or can be) built into future business decisions, and thus to choose from a different set of possible investments. To the extent that managers who use ROV have effectively adopted a different business model, there is a real and important difference between the two valuation techniques. Consistent with this possibility, much of the evidence from both surveys and academic studies of managerial behavior and market pricing suggests that managers and investors implicitly take account of real options when making investment decisions.     

The Beta Dilemma in Emerging Markets

In principle, emerging markets analysts employ the same analytical framework when estimating the value of businesses as their counterparts in developed economies: they forecast future cash flows and discount those to the present with appropriate costs of capital that are estimated using the Capital Asset Pricing Model (CAPM) framework. But in practice, emerging market analysts have a more complicated job because the task of estimating costs of equity in emerging markets is more difficult. Whereas developed economies have an abundance of historical data on overall stock market movements, industry share price behavior, and many individual share price histories, emerging market economies often do not. There may be no comparable local firms that are publicly traded—or if there are, their CAPM betas may be unreliable. And if analysts instead use the beta of a U.S. competitor as a surrogate for the emerging market beta, they face the question of whether domestic betas are equivalent across borders. As a consequence, appraisers of emerging market companies confront a “beta dilemma.” Part of this is a data problem stemming from shorter share price histories in emerging markets and the absence of publicly traded companies in some industries. In such cases, analysts may be inclined to use industry betas calculated with U.S. share prices as a substitute. But this creates an equivalence problem—the possibility, as confirmed by the author's research, that domestic U.S. and emerging market betas are not statistically equivalent for most industries. The author proposes a solution to this problem that involves grouping emerging markets into a single, distinctive asset class that allows for reliable calculations of industry betas. He also suggests ways of testing emerging market industry betas to determine whether they are statistically comparable.     

INCORPORATING COUNTRY RISK IN THE VALUATION OF OFFSHORE PROJECTS

Offshore projects, especially those in emerging economies, are generally viewed as more risky, and thus as contributing less to shareholder value, than otherwise comparable domestic investments. Emerging economies are typically more volatile than the economies of industrialized countries. They also present a greater array of risks that are (perceived as being) primarily of a downside nature, such as currency inconvertibility, expropriation, civil unrest, and general institutional instability. Further, because such risks are relatively unfamiliar to the investing companies, the companies are likely to make costly errors in early years and to require more time to bring cash flows and rates of return to acceptable steady-state levels. To reflect these higher risks and greater unfamiliarity, many companies include an extra premium in the discount rate they apply to offshore and, particularly, emerging-market projects. However, the basis for these discount rate adjustments is often arbitrary. Such adjustments do not properly reflect objective information available about either the nature of these risks, or about the ability of management to manage them. Nor do they take into account the reality that the risks stemming from unfamiliarity fall over time as the firm progresses along the learning curve. As a result, companies often “over discount” project cash flows in compensating for these risks, and thus unduly penalize offshore projects. More important, adjusting for country risk using arbitrary adjustments to the discount rate fails to focus management's attention on strategic and financial actions can be taken to reduce risk—notably, actions capable of transferring some of the company's exposures to specific risks to different parties with comparative advantages in bearing those risks. This paper outlines a four-step procedure for assessing overseas risks that integrates these various aspects:

• ? Classify risks in terms of various stakeholders' comparative advantage in risk-taking based on their: (a) existing portfolio of assets; (b) access to information; and (c) capabilities for reducing risk.
• ? Allocate risk through project structuring and financial engineering to exploitthese comparative advantages.
• ? Adjust resulting cash flows (relative to their most-likely levels) for (a) the impact of “asymmetric” risks; (b) learning effects; and (c) potential competitive options and/or barriers to entry resulting from comparative advantage in dealing with risks.
• ? Discount resulting expected cash flows at a risk-adjusted discount rate that reflects the covariance of the cash flows with the benchmark portfolio.

     

Risk-adjusted discount rates and capital budgeting under uncertainty

This paper is concerned with the valuation of multiperiod cash flows in a world where prices are determined according to the Sharpe-Lintner-Black model of capital market equilibrium. We find that the current market value of any future net cash flow is the current expected value of the flow discounted at risk-adjusted discount rates for each of the periods until the flow is realized. The discount rates are known and non-stochastic, but the rates for the different periods preceding the realization of a cash flow need not to be the same, and the rates relevant for a given period can differ across cash flows. The risk adjustments in the discount rates arise because of uncertainties about reassessments through time of the expected value of a flow and the relationships between these reassessments and the corresponding reassessments of the expected cash flows of all firms.     

Quantifying time value errors

Time valuation of cash flows is an essential part of personal financial planning and management. Many financial arrangements are priced according to a cash-flow valuation model. Expected cash flows associated with a stock or bond are discounted at an appropriate risk-adjusted rate in order to determine the fair value of the financial asset. Home mortgage loans are priced according to the discounted value of the future principal and interest cash flows. Yet, despite the importance of the discounted cash flow methodology in pricing assets, computational errors are often made when discount factors are not calculated precisely. This article attempts to quantify the magnitude of the error when the mathematical function for present value is ignored and interpolation is used instead to determine the discount factor.     

Real Asset Valuation: A Back‐to‐basics Approach

Different valuation methods can lead to different corporate investment decisions, and the conventional “static, single discount rate” DCF approach in particular is biased against many of the kinds of decisions that corporate managers tend to view as “strategic.” Reducing the bias from valuations involves two main tasks: treating risk in a way that is consistent with observed market pricing, and accounting for the ability of companies to make decisions “dynamically” over time. The authors propose two separate tools, market‐based valuation and complete decision tree analysis, for accomplishing these two improvements in valuation. The authors also suggest working with the full distribution of future cash flows, one possible realization at a time, rather than working with the aggregate measure of expected cash flow. From a technical perspective, it is necessary to work with the full distribution to value real options properly. Valuing the cash flows one realization at a time also leads to a much better understanding of the interaction between economy‐level, systematic risks and local asset‐level, technical risks. Just as important, the proposed approaches support an effective division of labor between local asset managers, who are better positioned to model technical considerations and other asset specifics, and the central finance staff, who can ensure the consistent treatment of economy‐wide risk and to create the rules of engagement for evaluating opportunities. After presenting an overview of both the valuation and the organizational issues, the authors present a case involving a corporate investment in carbon capture and storage that illustrates both the application of the proposed methods and the various sources of bias in the typical DCF analysis.     

Estimating cost of capital in firm valuations with arithmetic or geometric mean – or better use the Cooper estimator?

We test the extent and determinants of bias effects of the arithmetic as well as the geometric mean estimator and the estimator of Cooper [1996. Arithmetic versus geometric mean estimators: Setting discount rates for capital budgeting. European Financial Management 2 (July): 157–67] regarding discount rate estimation for firm valuation by way of a bootstrap approach for 13 different countries. The Cooper estimator is superior to both the geometric and the (conventional) arithmetic mean estimator. However, a ‘truncated’ version of the arithmetic mean estimator leads generally to better estimation outcomes than the Cooper estimator. This means that, in order to reduce problems of upward-biased firm value estimates, expected cash flows beyond a certain time horizon are completely neglected in terminal value estimation. Such an approach seems particularly reasonable for the valuation of young growth companies as well as for companies from quickly developing countries such as Brazil, China, or Thailand, because the bias in terminal value estimation is increasing in the growth rate of future expected cash flows.     

AN EMPIRICAL TEST OF AN EX-ANTE MODEL OF THE DETERMINATION OF STOCK RETURN VOLATILITY

The assumption that changing expected cash flows and discount factors affect a security's return is at the foundation of many financial models. This study examines empirically the hypothesis that expected stock return variability is a function of cash flow and discount rate uncertainty. Maximum likelihood estimation techniques and expectational data are employed. Strong, positive relationships are found, verifying the foundations of the ex-ante models with ex-ante data and providing a better understanding of security markets by explaining, in part, the causes of expected stock price variability.     

Do Analysts Disclose Cash Flow Forecasts with Earnings Estimates when Earnings Quality is Low?

Cash flows are incrementally useful to earnings in security valuation mainly when earnings quality is low. This suggests that when earnings quality decreases, analysts will be more likely to supplement their earnings forecasts with cash flow estimates. Contrary to this prediction, we find that analysts do not disclose cash flow forecasts when the quality of earnings is low. This is because cash flow forecast accuracy depends on the accuracy of the accrual estimates and the precision of accrual forecasts decreases for firms with low quality earnings. Consequently, as earnings quality decreases, cash flow forecasts become increasingly inaccurate compared to earnings estimates. Cash flow estimates that lack reliability are not useful to investors and, consequently, unlikely to be reported by analysts. This result provides an explanation for why analysts are less likely to report cash flow estimates when earnings quality is low.     

Discount-Rate Risk in Private Equity: Evidence from Secondary Market Transactions

Measures of private equity (PE) performance based on cash flows do not account for a discount-rate risk premium that is a component of the capital asset pricing model (CAPM) alpha. We create secondary market PE indices and find that PE discount rates vary considerably. Net asset values are too smooth because they fail to reflect variation in discount rates. Although the CAPM alpha for our index is zero, the generalized public market equivalent based on cash flows is large and positive. We obtain similar results for a set of synthetic funds that invest in small cap stocks. Ignoring variation in PE discount rates can lead to a misallocation of capital.     

Estimating the Cost of Capital Using Stock Prices and Near‐term Earnings Forecasts

Earnings‐based valuation models, although long used by finance practitioners, have become increasingly popular among finance academics as well. Among the most important reasons for academics' increased acceptance of earnings‐based valuation is the well‐documented claim that earnings over a short (three‐ to four‐year) forecast horizon tend to capture a large fraction—as much as 80%—of today's value, much more than is captured by near‐term forecasts of free cash flow, the measure long advocated by finance theorists as the basis for DCF valuation. But most important for the purposes of this article, the recognition that such a large percentage of the current values of many public companies is captured within a short forecast horizon has led to a large academic literature that uses earnings‐based valuation models together with current stock prices to “back out” estimates of the companies' implied expected rates of return and costs of equity capital. The effectiveness and precision of such reverse engineering depend on the reliability of the forecasts both within a finite forecast horizon and beyond. And although the models tested in academic work, which are based on large samples of forecasts and hard‐to‐verify assumptions about earnings beyond the forecast horizon, often do not appear to provide useful estimates, the author argues that such reverse engineering of the valuation models should become straightforward and workable once reliable forecasts of earnings are obtained—say, from the corporate (or investment) analysts who are familiar with the operations of the companies they work for (or cover).     

ON THE RATIONALITY OF COMMON STOCK RETURN VOLATILITY

This paper derives the relationship between the population unconditional variance of common stock returns and the variance of expected returns conditional on a well-specified information set. As a consequence, a lower bound is obtained for the variance of common stock returns. The sample counterpart of this bound is then empirically tested against the sample variance of returns. The paper's main conclusion can be stated as follows: the observed volatility of real (inflation-adjusted) common stock returns is not “irrationally” large. The paper admits of this conclusion because the point estimate of the lower-bound variance derived in this model is actually larger than the point estimate of common stock return volatility. However, since these point estimates are found to have a statistically insignificant difference, equality of the two variances cannot be ruled out. Hence, “rationality” of common stock returns—as implied by a utility-based valuation conditional on a specified information set—cannot be rejected.     

What Do FAS 157 “Fair Values” Really Measure: Value Or Risk?

FAS 157, the U.S. accounting standard that prescribes how fair values of assets and liabilities are to be measured when other U.S. GAAP standards require fair valuation, stipulates that fair values be measured as the exit values of assets and liabilities—the proceeds for assets hypothetically sold on the date of the financial report, and, correspondingly, the amount required to settle liabilities on the date of the financial report. This conceptual article argues that exit values do not reflect the value of the net assets of the firm to shareholders, which is best reflected by discounted cash flows to maturity. Moreover, exit values—biasing fair values downward when markets are illiquid—have a pernicious, systemic risk effect; specifically, they give rise to write‐downs that in turn cause contagion: prices of equities and other financial instruments of peers react negatively, leading to further write‐downs by those peers. This may have aggravated the recent financial crisis. However, while exit values are not proper measures of value to shareholders, they are useful measures of downside risk when prospects turn sour for a firm. Thus, both exit values and discounted cash flows should be presented in financial statements.     

What's your real cost of capital?

In valuing any investment project or corporate acquisition, executives must decide what discount rate to use in their estimates of future cash flows. The traditional approach is to apply the capital asset pricing model (CAPM), which has remained fundamentally unchanged for 40 years. But the formula--in particular, its beta element--has long been a source of frustration. In fact, corporate executives and investment bankers routinely fudge their CAPM estimates, say the authors, because experience and intuition tell them the model produces inappropriate discount rates. CAPM has three main problems: First, beta is a measure of both a stock's correlation and its volatility; second, beta is based on historical data; and third, CAPM rates don't take into account the term of the investment. These factors together result in discount rates that defy common sense. As an alternative to CAPM and its beta element, the authors developed a forward-looking approach to calculating a company's cost of capital, the market-derived capital pricing model (MCPM). It does not incorporate any measure of historical stock-to-market correlation, relying instead on estimates of future volatility derived from the options market. This is helpful since investor expectations from the options market are built into a company's current stock price. Using GE as an example, the authors give step-by-step instructions for how to calculate discount rates with MCPM. They also offer evidence from a range of industries to show that MCPM's discount rates are more realistic--especially from the corporate investor's perspective--than are CAPM's.     

USING DISCOUNTED CASH FLOW ANALYSIS IN AN INTERNATIONAL SETTING: A SURVEY OF ISSUES IN MODELING THE COST OF CAPITAL

Based on the results of a recent survey of University of Chicago Graduate School of Business alumni, the authors of this article suggest that prevailing corporate practice in valuing overseas investments reflects a flawed understanding of finance theory. Although the survey finds that almost all respondents use the discounted cash flow (DCF) method in some fashion or another, there is significant variation both in the application of DCF and in the weighting that different groups assign to DCF in dealing with segmented markets. Of greater interest, the survey also shows that, as the complexity and uncertainty involved in valuation tasks increases, practitioners appear to place greater reliance on heuristics, or conventional rules of thumb. And in relying on heuristics as perceived risk increases, the authors warn, “people tend to become less consistent, less systematic, and less rigorous in the methods they use to measure and evaluate risk.” Also of interest to the authors, many practitioners doing international valuations appear to be unwitting adherents to a “multi-factor” asset pricing model. For, in addition to traditional market factor proxies, they tend to incorporate country-specific risks, such as political and sovereign risk, into the discount rate. The authors attribute this practice to the implicit (and generally mistaken) assumption that there is a significant relationship between systematic risk and the degree of foreign market segmentation. Following presentation of their survey results, the authors explore several important issues surrounding international cost of capital. Perhaps most important is the degree of market “segmentation” and how it affects the pricing model (whether a global or a local version of the CAPM, for example) used to calculate the cost of capital. The article provides a framework to help practitioners decide which pricing model is appropriate for valuing a given investment. Moreover, since the cost of capital affects EVA-type measures of operating performance that are often used in performance evaluation schemes, the framework also can be used to guide senior management in thinking about the proper hurdle rates for their overseas business units.     

Brealey,Myers, and Allen on Real Options

Real options are valuable sources of flexibility that are either inherent in, or can be built into, corporate assets. The value of such options are generally not captured by the standard discounted cash flow (DCF) approach, but can be estimated using a variant of financial option pricing techniques. This article provides an overview of the basics of real option valuation by examining four important kinds of real options:

• 1 The option to make follow‐on investments. Companies often cite “strategic” value when taking on negative‐NPV projects. A close look at the payoffs from such projects reveals call options on follow‐on projects in addition to the immediate cash flows from the projects. Today's investments can generate tomorrow's opportunities.
• 2 The option to wait (and learn) before investing. This is equivalent to owning a call option on the investment project. The call is exercised when the firm commits to the project. But often it's better to defer a positive‐NPV project in order to keep the call alive. Deferral is most attractive when uncertainty is great and immediate project cash flows—which are lost or postponed by waiting—are small.
• 3 The option to abandon. The option to abandon a project provides partial insurance against failure. This is a put option; the put's exercise price is the value of the project's assets if sold or shifted to a more valuable use.
• 4 The option to vary the firm's output or its production methods. Companies often build flexibility into their production facilities so that they can use the cheapest raw materials or produce the most valuable set of outputs. In this case they effectively acquire the option to exchange one asset for another.

The authors also make the point that, in most applications, real‐option valuation methods are a complement to, not a substitute for, the DCF method. Indeed DCF, which is best suited to and usually sufficient for safe investments and “cash cow” assets, is typically the starting point for real‐option analyses. In such cases, DCF is used to generate the values of the “underlying assets”—that is, the projects when viewed without their options or sources of flexibility.     

Is Country Diversification better than Industry Diversification?

In this paper, we develop a framework in which one can examine the source of industry and country diversification by examining their underlying return components. We find that the global cash flow factor explains on average 39% of the variation of country cash flows and global discount rates explain 55% of the variation of country discount rates. These are much less than the explanatory power of the two factors over industry cash flow and discount rate variations, which are 72% and 78% respectively. This suggests that global factors are much less important for return components at country level than at the industry level. As a result, both better diversification of expected returns and cash flows across countries determine the larger benefits of country diversification versus industry diversification. Moreover, emerging markets tend to have much smaller co‐movements of both dividends and expected returns with those of the world, suggesting a lower degree of integration with the world goods and financial markets. Our results cast doubt on the prevailing wisdom that country diversification should be replaced by industry diversification.