Drilling For Value, Appendix C: Discounting Theory and Practice


  • Discount rates are a cornerstone of modern valuation methods for discounting the value of expected future cash flows.
  • Upstream valuation professional systemically utilize elevated discount rates well in excess of rational expectations for long-run capital growth.
  • The use of elevated discount rates may have roots in Modern Portfolio Theory, heuristics regarding the aggregation of well-level economics, and as proxies for high expected rates of depletion.
  • Re-calibration of investors’ rational expectations indicates that lower discount rates may be more appropriate for evaluating long-run returns.
  • Discount rates are simply a means by which to equate dollars in different time-periods — any further deliberation is likely to suffer from diminishing returns.

Figure 1: Sunburst – Pumping UnitSource: Greg Evans. Sunburst – Pumping Unit. Art Gallery of Greg Evans

Discount rates are at core of valuation; they represent investor preferences on the timing of money flows. Discounting is simply a means of equating dollars to dollars in different time periods under the time value of money (TVM) principle (i.e., a dollar today is worth more than a dollar tomorrow). Nominally, the discount rate represents the rate at which cash flows can be reinvested. A rate can also be thought of a hurdle or benchmark rate; returns above which add value added and below which destroy value. That’s about where general consensus stops. In practice, investors select discount rates any number of ways: corporate and/or aggregated weighted average costs of capital (WACC); the Capital Asset Pricing Model (CAPM); multi-factor beta models; and more. It has also become common financial practice to elevate discount rates to compensate for risk — this is a weak assumption.

TVM has an illustrious history, presumably dating back to ancient Babylonian civilization1. The modern notion for interest further can be traced back to the 16th century2. It was not until the 20th century, however, that modern methods of discounted cash flows (DCF) were conceived. Irving Fisher formalized the cornerstones of modern TVM — the exponential utility function and the definition of real (i.e., inflated) interest — in his 1930 book, The Theory of Interest. Although practitioners and lenders had long ascribed value to time (thereby anticipating utility), Fisher formalized the distinctions between and integration of nominal interest rates, inflation, and real growth3. In his 1938 paper, The Theory of Investment Value, John Burr Williams was first to formally express the value of equity in terms of discounted cash flows, terms previously ascribed as relevant only to interest bearing liabilities. Williams’ work anticipated the Modigliani-Miller (MM) theorem on capital structure. Although recent findings in behavioral economics suggest that humans act inconsistently with classical utility, these findings do not refute the optimality of exponential utility: that — assuming profits can be reinvested a constant rate of return — continuous reinvestment maximizes the long-run growth rate of the bankroll.

Traditionally, discount rates simply represented the rate at which excess cash flows could be reinvested. In this conventional stream of thought, the mathematics of interest consist of a risk-free rate of return, inflation expectations, and expectations for future excess growth. Then the CAPM happened: by adding a risk premia component to discount rates, investors can now magically-mystically compensate themselves for risk by elevating the discount rate. This implication of CAPM is a redirection of earlier work by Merton Miller and Harry Markowitz on Modern Portfolio Theory (MPT) which elegantly demonstrated the free lunch of diversification.

As with many solutions in life, the optimal discount rate is often the simplest. When selecting a discount rate, investors should consider opportunity cost, meaning the average expected return on invested capital for competing dollars. Moreover, adjusting discount rates for risk has no basis in the time value of money principle; such adjustments are more appropriate for the numerator (i.e., cash flows themselves). As Warren Buffett points out, “You can’t compensate for risk by using a high discount rate.” But most importantly, investors need not second guess themselves — it is better to be vaguely right than exactly wrong.

Industry Norms
Upstream valuation professionals typical value cash flows at discount rates which represent the weighted average cost of capital (WACC). Utilizing the WACC is consistent with the theoretical framework underlying the TVM, since rational investors should expect a return on their capital above its opportunity cost. However, a comparison of financing data from publicly traded companies and industry surveys indicates that valuation professionals use elevated discounted rates well in excess of rational expectations for return. In part, professionals elevate discount rates in order to compensate for perceived risk, but there may be other reasons as well.

The industry standard discount rate of 10% mirrors guidance from the Securities Exchange Commission (SEC) and Federal Accounting Standards Board (FASB) that requires public companies to disclose fair value estimates of petroleum resources within annual 10-K or 20-F reports. Present value calculations prepared using a 10% discount rate are informally known as “PV-10”. Banks typically discount at 9%. The June 2014 SPEE survey indicates that applied discount rates range from 4.96% to 29.24%. The median, however, tends to be tightly clustered; a slightly older SPEE survey shows that 71% per cent of respondents used discount rates in the range 9 per cent to 11 per cent, presumably mirroring SEC guidance4.

Financing data gathered from publicly traded oil and gas companies indicates that industry standard discount rates are unusually high in relation to the expected return on equity. A capitalization weighted index of US exchange traded oil and gas companies, constructed to S&P indexing standards, had an annual growth rate of approximately 7.5% since 1999; the growth rate is negative since 2007. If anything, discount rates should not exceed long-term expectations, especially since expectations have come down significantly as a result of lower interest rates.

Figure 2: Upstream Total Return Index versus Crude Oil PricesNote to Figure 2: Custom indices are constructed according to a modified capitalization weighted indexing methodology. Upstream Total Return Index includes all U.S. publicly traded companies in the Portfolio123 database which are assigned primary GICS Codes 10102010 (Integrated Oil & Gas) and 10102020 (Oil & Gas Exploration & Production). Dividends explain nearly half of the returns of holding a capitalization weighted index of oil and gas producers. A price return index which excludes dividends would have increased about 58% of the rate of the total return index.

Furthermore, empirically derived industry WACCs have been in secular decline since this time; in 2016, they ranged between 3 to 4%. This secular decline in cash returns on investment is definitely related to lower interest rates and bond yields (i.e., products of monetary policy and central bank machinations), but I can’t help but to consider the possibility that, similarly to declining cash flow margins, there is also a causal linkage between declining returns on equity and declining energy returns on energy invested (EROEIs). EROEI is more fully explored in Appendix A.

Figure 3: Weighted Average Cost of Capital (WACC) for Publicly Traded Oil and Gas Companies, 1999-2016

Source: Portfolio123; author’s calculations

Note to Figure 3: Industry aggregates consider all U.S. publicly traded companies assigned to primary GICS Codes 10102010 (Integrated Oil & Gas) and 10102020 (Oil & Gas Exploration & Production). The weighted average cost of capital is the sum of all quarterly cash dividends and interest payments divided by the sum of all invested capital (i.e., book values of equity plus debt and non-controlling interests).  

The disconnect between industry norms and rational expectations is striking. One contributing factor is that analysts modify the empirically derived WACC to compensate for risk. Another common practice is to elevate discount rates when aggregating individual well economics as a proxy for corporate overhead and “non-core” operating costs. Both of these rely on weak assumptions. However, there may be other perfectly valid reasons for the systemic use of elevated discounted rates.

According to a 2008 SPEE survey, 48% of respondents used a higher discount rate to account for “a profit or expected rate of return for the buyer, and any risk/uncertainty that the evaluator may choose to impute to the asset”5. The common practice of modifying discount rates to compensate for risk comes from Modern Portfolio Theory (MPT), namely the CAPM. Confusingly, in the world of finance, there are two types of WACCs: a) the empirical cost of capital (which, when expressed as a percentage, is financial costs divided by the invested capital); and b) an implied cost derived from the CAPM consisting of a risk-free yield plus an equity risk premium (ERP). The most common implementations of the ERP presume that returns are commensurate with risk defined by asset price standard deviation (i.e., financial volatility).

Henceforth, WACC refers to the empirical cost of capital.

This intuition that expected returns should be commensurate with risk is perfectly sound. Furthermore, the presence of an equity risk premium is not controversial. However, the expectation that returns should exceed risk is fundamentally different than the claim that returns are the reflexive assumption of risk. Moreover, the implementation of a standard deviation-based proxy for expected return is contradicted by empirical equity returns in which low volatility and low beta portfolios outperform. In response, Post-Modern (Post-Mortem???) Portfolio Theory revises the original CAPM to account for higher-order moments of the presumed underlying probability distribution without solving any core problems. Realizing that quantification of expected risk and reward in an efficient market could not be nearly as straightforward as a single linear regression, Eugene Fama and Kenneth French (FF) devised additional beta factors (i.e., Fama-French factors). Currently, FF has five factors while Cliff Asness’ firm, AQR, adds a sixth factor for momentum. These multi-factor approaches still suffer from fallacies of hindsight and assumed linearity.

Moreover, for the typified CAPM approach to have merit, the MM theorem on the irrelevance of capital structure must be wrong since it posits that the value of a firm is unaffected by capital structure. MM was originally premised on a tax-free world, but in the presence of debt interest tax deductions, it suggest that optimal capital structure incorporates at least some debt. This is in contradiction to most implementations of CAPM which elevate cost of capital discount rates for firms with volatile securities prices.

Another common practice among upstream valuation professionals is to elevate discount rates as proxies for corporate overhead and “non-core” operating costs when aggregating individual well values. The justification is that since their models do not capture “all the costs”, raising the discount rate should offset the fact that the models are only capturing about half of the full-cycle economic costs6. These practices echo specifications for reservoir valuation as defined by the SPE7, FASB8 9, and SEC10 11 12. A more thorough discussion on reserve-based valuation metrics can be found in Part 2.

The heuristic practice of elevating discount rates as a rough substitute for capturing economic costs is a practice which I presume has roots in the difficulty of evaluating companies with a negative cash flows and earnings. It also mirrors approaches commonly used elsewhere in private equity and investment banking in which discount rates applied to EBITDA are elevated in relation to those applied to actual earnings. However, modifying discount rates in order to offset economic costs has zero basis in any financial theory with which I am familiar. There are better ways to price “out of the money” assets. Since at least the 1960s, culimating with Black-Scholes in 197313, financial theory has provided adequate ways to deal with precisely this issue.

To some degree, industry standard practices of using elevated discount rates may be attributed to their roles as proxies for high rates of asset depreciation and depletion (vis-à-vis a negatively growing annuity). The math behind this is actually valid, but when used superficially only serves as a rough substitute for actual production decline analysisInterestingly, substituting the depletion rate — which might be inferred from either accounting data (if the firm uses the units of production depletion method)14 and/or production data — in place of the growth constant in the growing annuity formula is consistent with the analytical solution governing exponential reservoir decline (i.e., the expected production decline rate within an idealized, constant flowing pressure, closed reservoirs is equal to the depletion rate). Even though hyperbolic (i.e., declining decline rate) curves provide better fits to empirical decline curves, the mathematical convenience by which exponential decline rates are mathematically equivalent to exponential discounted utility functions ubiquitous in TVM calculations is astoundingly convenient15. This particular convergence between engineering and finance may be one of the more concrete examples of Albert Einstein’s claim that compound interest is the most significant scientific discovery ever made.

Perhaps the most notable unification between reservoir engineering and economic theory can be found through Arps’ equations. In 1945 and 1956, J.J. Arps published a class of functions based on observed hyperbolic declines found in many oilfields globally. Exponential decline and harmonic declines are special cases of Arps’ general hyperbolic function dependent on a scalar parameter, \beta. Notice the following convergences in Table (1): production quantity, q(t), and periodic discounted payoff, e^{\int_0^T r_t dt}C_t; cumulative production, Q(t), and the present value of an annuity of C_t; and, ultimately recoverable resource (URR) and undiscounted future cash flows. Also notice how exponential discounting is a special case of the hyperbolic function as the limit of \beta \to 0. In a fiscal setting, \beta may be interpreted as the fraction of profits which are notreinvested at a given rate of return. In the special case of \beta = 1, the equation becomes a linear growth function, representing a fixed amount which is reinvested. When \beta =0, the entire bankroll is continuously reinvested, implying exponential growth. To the extent that maximal reinvestment is possible, the exponential function is an optimal discounting solution which maximizes the long-run growth rate. It should also be noted that the lower the rate of discount/decline, the less divergence there will be between any two curves over both the short and long run.

Table (1): Arps’ Equations

Source: Linnea Lund. Decline Curve Analysis of Shale Oil Production: The Case of Eagle FordUppsala University. Oct 2014

Estimating an Appropriate Discount Rate
Rational investors expect a return on capital in excess of financing and/or opportunity costs. It is sensible therefore to derive an appropriate discount rate from the weighted average costs of capital (WACC). This number represents the required rate of return by which sums of money must grow in order to realize a real return on investment — it can be thought of a yard stick or hurdle. Returns above this hurdle add value, while returns below which destroy investment value. In practice, it may be appropriate to consider an individual project’s, firm’s, or peer group’s WACC.

When valuing gross cash flows (before capital costs), the discount rate may be set to a company’s WACC, considering all capital costs (i.e., dividends, interest payments, and other forms of cash and non-cash compensation paid to capital investors) and invested capital. This method is functionally equivalent to netting the difference between returns on invested capital (ROIC) and the WACC in order to derive an estimate for economic profit (à la Sterns Value Management implementation of Economic Value Added (EVA) analysis).

If valuing cash flows net of capital costs, it is more appropriate to utilize a single discount rate for all comparable analyses, estimated by aggregating an appropriate peer group’s cost of capital. This approach is most consistent with the MM theorem on the irrelevance of capital structure.

Rational Expectations and Lower Discount Rates
Over time, I’ve used progressively lower discount rates for valuing E&P assets. Compellingly, Figure (3) shows that the aggregate WACC for upstream oil and gas companies has been in secular decline for the last few decades. Whereas high discount rates lend greater weight to the near-term, lower rates emphasize the long-term. If we lend any credence to Howard Marks’ advice in his recent interviews on Master’s in Business with Barry Ritholz and The Investor’s Podcast, most investors tend to be overly focused on the short-term. In his view, focusing on the long-run — i.e., counter-cyclical investments with a long-tail of cash flows — is more conducive to excess returns.

Moreover, nowadays one can rarely identity equity of any oil and gas producer which trades below its economic net asset value when a 10% discount rate and full-cycle costs are used. Necessity dictates that rational investors in upstream energy re-calibrate their expectations in order to justify any capital allocation.

From this perspective, industry standard use of higher discount rates is a form of cognitive dissonance. The ability to avoid this sort of groupthink should be viewed as a distinct advantage. If other investors systematically undervalue long-tail and counter-cyclical investments, there may be some opportunity left for long-term investors to identify dislocations between price and value.

Academic criticism indicates that exponential utility functions fail to explain empirical human behavior whereby a) perceived value (i.e., utility) falls rapidly during earlier periods and then falls more slowly over longer periods (vis-a-vis, hyperbolic discounting); and, b) investors avoid negative outcomes more strongly than they seek equal but inverse positive outcomes (i.e., bad is asymmetrically stronger than good). Furthermore, it should be apparent there is no such thing as a single empirical discount rate. Rather, there are multiple discount rates at multiple given maturities on any continuous yield curve to choose from at any given point in time. Moreover, interest rates are stochastic. Dealing with behavioral asymmetry is difficult; this discussion belongs to realm of behavioral economics, namely prospect theory16. Dealing with multiple interest rates and/or the probabilistic expectancy for stochastic discount rates is also beyond simple resolution. However, re-calibrating for lower discount rates at least partly reconciles discrepancies between observed and optimal investor preferences — the difference between hyperbolic and exponential utility is far less pronounced for low discount rates17.

In any case, lower discount rates for evaluating capital investments a) mitigates the differences between inter-temporal choices based on exponential, hyperbolic, and/or harmonic utility preferences; and, b) places greater significance on far afield future economic ramifications. A willingness to defer short-term gains in favor of long-run optimal outcomes, in my opinion, can confer a distinct advantage to long-term planners and investors.

Risk and Margin of Safety
In order to compensate for risk, base case estimates should reflect a scenario which is no more likely than not, erring on the side of conservatism when there is a high degree of uncertainty. In other words, spending more energy on reflecting “all the costs” and decreasing uncertainty in the numerator (i.e., the amounts, timings, and likelihoods of discrete cash flows) greatly decreases the need to adjust for risk in the denominator (i.e., discount rate). Warren Buffett understands this approach through the margin of safety principle.

Furthermore, the fields of financial theory and practice have developed concrete ways of dealing with uncertainty that are not premised on modifying discount rates. Part 4 contains a detailed discussion on the use of options analysis as a means of dealing with underlying commodity price stochasticity as well as managerial flexibility.

If all this focus on the discount rate seems superfluous, it is because it is. But that’s the point: it is the author’s opinion that rational investors must pick one number and method to use with all comparable discounting analyses which represents the opportunity cost (i.e., the average expected return on invested capital), and importantly, not overthink this decision. There is already enough complexity and uncertainty concerning the numerator; it is superfluous to add more. Dollars are dollars no matter where they come from and discount rates are simply tools to compensate for the time value of those dollars; nothing more. Any further deliberation on this topic is likely to suffer from its own sort of diminishing returns.

A more eloquent man explained this advice in simpler terms: “it is better to be vaguely right than exactly wrong”.

Footnotes   [ + ]

1. William N. Goetzmann. Financing Civilization
2. Martín de Azpilcueta. Wikipedia.
3. Refer to the Fisher Equation for a simple derivation of the real interest rate
4, 5. C.R.K. Moore. Perspectives on the valuation of upstream oil and gas interests: An overview. Journal of World Energy Law & Business, 2009, Vol. 2, No. 1.
6. A more detailed discussion on the practice of aggregating well economics using half-cycle measures is contained in Part 4.
7. Petroleum Resources Management System. Society of Petroleum Engineers. 2007.
8. Financial Accounting Standards Board. Extractive Activities—Oil and Gas (Topic 932): Oil and Gas Reserve Estimation and Disclosures. January 2010.
9. FASB. Accounting for Extractive Activities–Oil & Gas: Amendments to Paragraph 932-10-S99-1. April 2010.
10. SEC. 17 CFR Parts 210, 211, 229, and 249 [Release Nos. 33-8995; 34-59192; FR-78; File No. S7-15-08]: Modernization of Oil and Gas Reporting. January 2010.
11. Electronic Code of Federal Regulations. Regulation S-K [17 CFR Part 229] >> Subpart 229.1200—Disclosure by Registrants Engaged in Oil and Gas Producing Activities
12. Electronic Code of Federal Regulations. Regulation S-X [17 CFR Part 210] >> §210.4-10 Financial accounting and reporting for oil and gas producing activities pursuant to the Federal securities laws and the Energy Policy and Conservation Act of 1975.
13. Fischer Black, Myron Scholes. The Pricing of Options and Corporate Liabilities. The Journal of Political Econony, Volume 81 Issue 3 (May-Jun 1973), 637-654.
14. US Internal Revenue Service. IRS Examining Process. Oil and Gas Handbook, Section Reserves of Oil and Gas.
15. Although empirical decline curves tend to fit hyperbolic (i.e., declining decline rate) curves, the math to calculate the present values of hyperbolically declining cash flows is also astoundingly convenient (even though it is less studied). However, the decline rates of a hyperbolically declining reservoir do not converge as neatly with periodic depletion rates, making the initial parameters more difficult to estimate from both production and financial data. The added utility of hyperbolic discounting of expected future cash flows is also suspect.
16. Michael Lewis’ The Undoing Project: A Friendship That Changed Our Minds tells the story on how the works of Daniel Kahneman and Amos Tversky invented the field of behavioral economics and changed our minds on human decision making
17. In the grander scheme of things, the present valuation of far-dated potential economic risks (due to deleterious political, social, and environmental impacts) are not as insignificant as implied by the traditional interpretation of the time value of money and normative (i.e., geometric and/or exponential) discounting methods. Normative discounting methods imply that utility decreases at a constant rate with respect to time. E.g., one unit of impact 20 years from today is worth .15 units of impact today under a 10% discount rate, but .55 unit of impact under a 3% discount rate. However, research in behavioral economics indicates that perceived value of a unit falls rapidly during earlier periods and then falls more slowly over longer periods. Hyperbolic discounting methods, in which the rate of utility decay decrease with respect to time, are more consistent with empirical human preferences regarding consumption and time value. They are also less prone to overly discounting long-tail risks and rewards. Still, normative discounting is more entrenched than hyperbolic discounting, even though both methods are mathematically simple to employ. In addition, normative methods provide solutions that are consistent with rational behavior whereby excess returns are reinvested at an invariant discount rate. Arguably, those in the business of making money should be more concerned with making optimal decisions than accurately modeling the behavior of others. A reconciliation between the seemingly disparate normative and behavioral models can be simplified, however: the difference between hyperbolic and exponential utility is far less pronounced for low discount rates. The net result is that lowering the hurdle rate decreases the likelihood that present-day economic decisions overly discount far afield economic costs and it more accurately reflects observed human preferences.