5 The social discount rate
5.1 Introduction
From the terms of reference of the Committee:
The discount rate level has a significant impact on the profitability of long-term measures. The guidelines for determining the discount rate are based on exponential discounting and the so-called Capital Asset Pricing Model. However, financial markets provide limited information about risk premiums for projects with a long economic life, such as for example transportation investments. The Stern Review has recommended a discount rate of 1.4 percent for climate calculations, whilst other economists have argued that such estimate is too low. The Committee shall assess, against this background, which discount rate should be applied for long-term measures, and whether the required rate of return should be differentiated on the basis of the duration of such measures. The Committee shall review existing literature within this area, and assess various methods for determining the discount rate. The Committee shall consider, in this context, whether the theoretical framework for determining the discount rate should be based on the opportunity cost of capital or on consumer behaviour. Moreover, both the cost side and the benefit side are subject to systematic uncertainty from a portfolio perspective. The Committee shall make a recommendation as to how systematic uncertainty should be handled in public investment analysis.
Most public measures have both cost and benefit effects that materialise over a number of years. In order to evaluate measures we therefore need to be able to weigh economic cost and benefit effects that occur at different points in time. At the same time, effects arising in the distant future are subject to considerable uncertainty. A cost-benefit analysis should also take such uncertainty into consideration.
The NOU 1997: 27 Green Paper recommends that these considerations be taken into account through the application of a discount rate that both captures the trade-off between different periods and the consequences of an uncertain future for a decision maker who would prefer to avoid uncertainty. Alternatively, one may use a risk-free rate, whilst correcting for risk in the uncertain net project benefits, thus implying that the discount rate only captures the trade-off between different periods.
Some public measures may have effects into the very distant future. These may include, for example, climate measures or measures targeting biodiversity. The NOU 2009: 16 Green Paper discusses uncertainty and the discounting of such measures in more detail.
The present Chapter starts out by examining various theoretical approaches to the discount rate and discusses the handling of systematic risk (5.2). Thereafter, we explain the discount rate guidelines in Norway (5.3) and address theories as to how the discount rate may develop in the very long run (5.4). We will then discuss the discount rate level (5.5), before reviewing empirical estimates as to the social discount rates applied in some other countries (5.6). The assessments and recommendations of the Committee are presented in Chapters 5.7 and 5.8, respectively.
5.2 Discount rate
Discounting future values by applying a discount rate converts these to cash-equivalent values as per a specific reference date. The reference date normally chosen for an investment is the start date of the future costs and revenues of the project. The cash-equivalent value is in such case referred to as the net present value. Hence, discounting facilitates comparison between, and ranking of, measures with economic effects that occur at different dates.
The discount rate concept may be approached in two ways. It may be interpreted as a required rate of return in the form of the minimum economic compensation per krone invested that would be required for one to be willing to forego consumption at present in return for higher consumption one period later. Alternatively, it may be interpreted as a market-determined opportunity cost, inasmuch as it represents the additional consumption one would have achieved after a period by depositing one krone in the bank or investing it in another interest-bearing investment instead of consuming it now. The consumption and savings profile of an individual is optimal when the subjective required rate of return of the individual is equal to the opportunity cost as defined by the market interest rate.
Uncertainty that influences discount rate estimates may be separated into two categories. Firstly, there is uncertainty concerning developments in the economy as such, which were not discussed in this context in the NOU 1997: 27 Green Paper. This encompasses uncertainty relating to both future consumption developments that affect the required rate of return of the consumer, as well as future returns forgone in the capital market. This may influence the choice of discount rate for use in the evaluation of public projects and, if applicable, the time structure of such rate in the very long run. This is discussed in more detail in Chapter 5.4.
Secondly, there is uncertainty about the economic outcome of the projects and measures to which the capital is committed. This uncertainty is reflected in the discount rate in the form of a project-specific risk premium. This is discussed thoroughly in the NOU 1997: 27 Green Paper from the perspective of the Capital Asset Pricing Model. Financial theory has developed well-established models for the pricing of the risk associated with financial assets, including the Capital Asset Pricing Model. However, these are based on assumptions that may be problematic for the evaluation of risk premiums for long-term public projects. This is discussed in more detail in Chapter 5.4.2.
Different theoretical approaches to estimating discount rates shed light on different aspects of the discount rate problem. In Chapter 5.2.1, we start out by looking at a key model for the consumer’s required rate of return.
5.2.1 The consumer’s required rate of return
An investment is profitable if the future return, as evaluated at present, is deemed to be worth more than the utility loss from foregoing consumption today. A simple approach to such profitability assessments is that they are influenced by the return on the investment, the impatience of the person making the investment and the extent to which the consumer, when faced with an uneven time profile for lifetime income, prefers consumption smoothing over time.
If we assume that this forms the basis for the decisions of a representative consumer, we get an expression for the marginal condition for optimal saving in a situation without any uncertainty with regard to either the return on the project or developments in the economy in general. Impatience may be expressed as a rate of time preference stating by how much a unit of utility shall be adjusted based on how far into the future it comes. The more impatient is the consumer, the higher is the rate of time preference. The preference for consumption smoothing may be expressed through the elasticity of marginal utility of consumption, which shows the percentage change in marginal utility when consumption is changed by one percent. The higher the numerical value of the elasticity of marginal utility of consumption, the stronger is the preference for consumption equalisation over time. Furthermore, we may look at the change in consumption we experience from one period to the next as the result of economic growth. On this basis, we can derive the so-called Ramsey condition for optimal saving:
(1)r = p + µg
where r denotes the return on investment, r is the consumer’s rate of time preference, g is relative consumption growth per capita, and µ is the numerical value of the elasticity of marginal utility of consumption.1 The product µg shows the percentage change in marginal utility when consumption is changed by g percent. When also including the rate of time preference, the right-hand side of (1) thereby expresses the consumption-based required rate of return. If, for example, µ = 2 and future consumption is increased (reduced) by 1.5 percent, the required rate of return on savings will increase (decrease) by 3 percentage points.
Under economic growth, the population will be better off in the future than at present in material terms, which will manifest itself by an increase in consumption per capita. According to the optimum condition (1), this results in a higher required rate of return. A higher rate of return requirement means that one attaches relatively more weight to immediate consumption, as opposed to realising more consumption later. With a high discount rate, a project will only be profitable in those cases where it yields a relatively high return in subsequent periods. Declining per capita consumption – for example as the result of strong population growth – will correspondingly result in a lower required rate of return and a stronger incentive to save. The last element of the required rate of return defined by (1) will result in consumption smoothing over time.2 It is reasonable to refer to this element as a wealth effect. A presumption of continued positive (negative) wealth developments in a generational perspective should therefore be reflected in an increasing (declining) discount rate over time for projects with correspondingly long-term effects.
Table 5.1 Different social discount rates derived from the Ramsey condition
Source | Pure rate of social time preference, θ | Elasticity of marginal utility of consumption, η | Consumption growth rate, g) | Discount rate = θ + ηg |
---|---|---|---|---|
per cent | per cent | per cent | ||
Stern (2007) | 0.1 | 1 | 1.3 | 1.4 |
Quiggin (2006) | 0 | 1 | 1.5 | 1.5 |
Cline (1993) | 0 | 1.5 | 1 | 1.5 |
Garnaut (2008) | 0 | 1-2 | 1.3 | 1.3-2.6 |
HM Treasury (2003) | 1.5 | 1 | 2 | 3.5 |
Nordhaus (2007) | 1.5 | 2 | 2 | 5.5 |
Weitzman (2007) | 2 | 2 | 2 | 6 |
Arrow (2007) | 0 | 2-3 | lf 1-2 | 2-6 |
Dasgupta (2006) | 0 | 2-4 | lf 1-2 | 2-8 |
Gollier (2006) | 0 | 2-4 | lf 1.3 | 2.6-5.2 |
Empirical evidence | 0-3 | 0.2-4 | 1.2-2.1 (for Australia) | 0.24-11 (given range) |
The table provides an overview of different social discount rates from different studies. The table contains an overview of discount rates for use on both marginal measures and the global climate issue (for example Stern, 2007, and Cline, 1993).
Source Harrison (2010), p. 36.
It is reasonable for there to be a connection between how consumption is distributed between the rich and the poor at a given point in time and between the poor and the rich at different points in time. However, the elasticity of marginal utility of consumption is not necessarily the same with respect to the two issues, since distribution at a given point in time will by definition be distribution between different individuals, whilst distribution over time may be distribution between “yourself today” and “yourself in future” (see Atkinson et al., 2009, for en empirical study of how these may vary).3 To estimate a discount rate for use in cost-benefit analysis, one may assume continuous positive economic growth and identify probable values for the various parameters in the Ramsey condition (equation 1). However, different estimates and assumptions as to the parameter values included in the Ramsey condition may produce very different estimates as to the required rate of return. Harrison (2010) shows that different sources arrive at required rates of return that range from 1.4 percent to 8 percent, cf. Table 5.1, which provides an overview of discount rates for use on both marginal measures and the global climate issue (for example Stern, 2007, and Cline, 1993). This illustrates that there is no straightforward answer to what is an appropriate discount rate when using this simple approach, even before introducing the issues of uncertainty as to the return on the project, as discussed in 5.2.2, and uncertainty as to the long run opportunity cost of capital, as discussed in Chapter 5.3, respectively. Box 5.1 illustrates how the choice of discount rate has been a key issue in the climate debate.
Textbox 5.1 The discount rate and the climate issue – an example
Measures aimed at influencing the global climate need to be analysed over a very long time horizon. Profitability assessments are therefore critically dependent on the chosen discount rate. This has, in particular, been discussed in view of the consumption-based approach (cf. the Ramsey condition, Chapter 5.2.1):
r = p + µg
The debate evidenced by the Stern Review on the climate problem (Stern, 2007) and a response to that report (Nordhaus, 2007a), illustrates key issues in choosing a long-term discount rate. Stern (2007) puts the pure rate of time preference, p, close to zero, based on an ethical assessment to the effect that the utility of future generations shall carry as much weight as that of current generations when making intertemporal trade-offs.
Stern (2007) also attributes a low value to µ. This implies that one assumes a low preference for consumption smoothing over time, since little weight is accorded to the expectation that future generations may be richer than current generations. A higher µ value would, when taken in isolation, have implied that the current generation should save less for future generations.
Nordhaus (2007a) emphasises that the return on alternative investments in the market should guide climate investments. This in order to ensure efficient capital allocation in the economy, across sectors. Hence, Nordhaus (2007a) argues that the parameter values in the Ramsey condition should be such as to bring the discount rate on a par with observable market interest rates. Nordhaus (2007a) notes that the discount rate chosen by Stern implies that investment decisions conditioned on such a rate will result in too much being invested too early in low-return climate measures, when compared to the decisions one would have made if applying a discount rate that matches observable market interest rates. According to Nordhaus (2007a), a more effective strategy would be to invest more in conventional capital in an early phase and thereafter use the return on these investments to make large subsequent investments in climate measures.This is consistent with applying the market interest rate as the discount rate for climate measures as well. Moreover, if one is concerned about exceeding potential natural threshold values, such as the melting of the Greenland ice sheet or the disintegration of the West Antarctic ice sheet, Nordhaus (2007b) shows that economic analyses of the climate issue may apply caps that define the maximum permissible temperature increases/carbon dioxide concentrations. Such caps may be introduced into the analyses without changing the chosen discount rate.
5.2.2 Market-based opportunity cost of capital
The use of scarce resources for a specific purpose implies an economic cost inasmuch as it supplants the potential value added from the best alternative use. This is often termed the opportunity cost. It applies to the use of capital as well. When capital is tied up in a specific project, alternative profitable use of such capital is ruled out. The cost of this is the value added lost by deselecting the best alternative. In an economy without capital rationing, all projects delivering a return in excess of the alternative return on the capital in the financial market will normally be realised. Hence, the return in the financial market is the return on the marginal project, and the market interest rate determines the demand for capital.
Society will normally be faced with alternative uses for scarce resources, and the discount rate should, as an opportunity cost, reflect the best alternative return on the capital tied up in a proposed measure. In a closed economy without market failure, the market interest rate realised in an equilibrium between the supply of, and the demand for, capital will therefore express the return on the best alternative investment, which will be concurrent with the required rate of return of the consumers who contemplate whether to save or consume today. In an open economy with a given international interest rate level and free capital movement, both capital supply and capital demand will reflect the world market interest rate, and this interest rate will represent the opportunity cost of capital in the financial market and hence the required rate of return of those who supply capital.4 This will therefore be the relevant calculation price for trade-offs between consumption in different periods. The scope for deviations between the required rate of return of consumers and the real marginal return on capital as well as the interest rate must be related to imperfections in the economy, for example in the form of distorting taxes. The NOU 1997: 27 Green Paper discussed such capital market imperfections in more detail.
The practical application of the opportunity cost principle in the Norwegian context has been that the discount rate of a project comprises two elements. Firstly, all projects must have a return that is at least as high as a presumed risk-free return in international capital markets. Secondly, a premium is added to the discount rate, reflecting the presumed risk associated with the project. This has been used to express the risk-adjusted opportunity cost of capital of a public project, as it is the return forgone in the financial market on a financial investment with the same presumed risk profile. Models for such pricing of risk are presented in Chapter 5.2.3 below. In Chapter 5.4, we will take a closer look at the relationship between uncertainty in the very long run and the relevant discount rate.
5.2.3 Models for pricing uncertain financial claims
Financial theory is concerned with how investors should balance their portfolios when the return on future claims is uncertain. The models are based on the general premise that investors maximise the expected utility of consumption. However, well-established models, like for example the Capital Asset Pricing Model, are directly linked to prices and returns in the market. This makes them practical to use, but also requires several simplifying assumptions that may be problematic for public projects.
If we adopt the standard assumptions that the economy comprises rational investors who are able to make marginal investments, we can derive a basic pricing model for all types of uncertain financial claims. Box 5.2 presents the details of such model.5
The Capital Asset Pricing Model is such a price model, and is an equilibrium model in which the consumption of investors is linked to one single portfolio comprising all assets in the economy (the market portfolio). The model is normally expressed in return format:
E(Ri) = r + ß(E(RM) – r)
where E(Ri) is the expected return on a security i, r is the risk-free interest rate, E(RM) is the expected return on the market portfolio, and ß specifies the degree to which the return on the investment is correlated with the return on the market portfolio. The model says that investors will demand higher risk compensation the more the return on an asset is correlated with the return on the market portfolio.
The key message of the model is that only the risk priced in the market is risk that cannot be diversified away by holding different securities. This is called systematic risk. One cannot achieve an expected return in excess of the risk-free interest rate without assuming systematic risk. Box 5.3 contains a more detailed discussion of the distinction between systematic and unsystematic risk.
Textbox 5.2 Systematic and unsystematic risk
Uncertainty in a project may be divided into systematic and unsystematic risk, respectively. Risk that depends on project-specific circumstances is referred to as unsystematic risk. This may, for example, be the uncertainty relating to the geology of a mountain one intends to drill into to construct a tunnel. In other projects we will also encounter uncertain costs that depend on project-specific circumstances only. In some of these projects the actual costs will be lower than planned, whilst they will be higher in others. Since there is, generally speaking, no correlation between the costs of the various projects, the outcomes of this type of uncertainty will cancel each other out when we look at the portfolio of projects for society as a whole. We say that the unsystematic risk can be diversified away, which implies that we shall not increase the required rate of return of a measure in the face of this type of risk.
By systematic risk, on the other hand, is meant the degree to which the gains from the measure are sensitive to volatility in the marginal utility of consumption. Furthermore, it can be assumed that the marginal utility of consumption is lower when one gets richer. One may, for example, envisage that the return on a transportation investment varies at different stages of the business cycle. The predominant benefits from transportation measures are reduced travel time and increased safety. The Committee recommends that both the value of time and the value of a statistical life be real-income adjusted, and that estimated GDP growth per capita be used as an estimate of such growth, cf. the discussion in Chapter 4 on real-price adjustment. One consequence of this is that the benefit side of transportation projects is positively correlated, to a higher extent than before, with the return on the national wealth, and consequently being perceived as measures with higher systematic risk than would be the case without such real-income adjustment. The value of other measures in society may also be sensitive to business cycle developments. Meanwhile, other measures, for example within welfare services for the elderly or prison services, will have an economic return that cannot be assumed to be sensitive to the business cycle. In practice, one will of course encounter large grey areas when seeking to classify specific uncertainty elements as either systematic or unsystematic.
In the presence of systematic risk, the economic profitability calculation needs to be corrected for this type of uncertainty. The present Chapter notes how to do so by including a risk premium in the discount rate. In theory, the magnitude of the risk premium shall depend on the degree of correlation between the project return and the marginal utility of consumption. The economic return on priced elements is comprised of benefit elements less cost elements. A positive correlation between the net benefits of the project and the remainder of the economy increases the risk, whilst a positive correlation between net cost and the remainder of the economy reduces the risk. Both aspects merit the inclusion of a premium in the discount rate, which reduces the net present value of net benefits (lower profitability) and reduces the net present value of net costs (higher profitability).
Another and more direct approach for taking risk aversion into account in the calculations, is to calculate the so-called certainty-equivalent values of the various cost and benefit components at different points in time, and use the risk-free interest rate for discounting. The resulting figure expresses a certainty-equivalent project profit that will, in the face of risk aversion, be lower than the expected profit, assuming that the project profit is positively correlated with the return on the national wealth. In theory, these two approaches should be equivalent. In practice, the calculation of certainty-equivalent cost and benefit values require detailed knowledge of the various components to be valued, and such calculation also needs to rely on a number of assumptions. All in all, this may imply that the approach involving the use of certainty-equivalent cost and benefit values is impracticable and characterised by low transparency.
The Capital Asset Pricing Model has made a major impact since it was developed in the mid-1960s by Sharpe (1964), Lintner (1965) and Mossin (1966). One of the main reasons for this is that the model can be used to estimate the average risk premium in the stock market on the basis of market return data. The model is therefore well-suited for opportunity cost assessments. If one has a project that needs funding, and one has an estimate as to the magnitude of the project risk relative to that of an average project funded via the stock exchange (β), one can obtain an estimate of the project’s risky returns by identifying objects in the financial market with the same risk profile. The risk premium on public measures can then be derived from the market rate of return on such financial assets with the same risk characteristics. The Ministry of Finance (2005) provides a more technical discussion of such an approach.
The Capital Asset Pricing Model offers a simple and pedagogic presentation of important asset management principles, and is a well-established model for the pricing of projects that involve risk. However, the model is based on a number of simplified assumptions. The two most important ones are that investors live for one period only and that they have no form of labour income during that period. Moreover, the assumption of a linear relationship between marginal utility and the return on the market portfolio implies either that the preferences of investors are based on expectation and variance only, or that the return on all assets follows a normal distribution.6 The model also assumes the existence of an investable market portfolio comprising all assets in the economy. As noted in Report No. 17 (2011-2012) to the Storting, Box 2.10, empirical research suggests that the model fails to provide a fully adequate description of how investors act and of how financial markets work. For example, research clearly indicates that systematic parts of the return on equities are determined by other factors in addition to beta (Banz, 1981, Fama and French, 1993, Jagadeesh and Titman, 1993).
A project’s contribution to economic value added will normally be subject to uncertainty. The systematic uncertainty of a public project may be made operational by taking into account the extent to which the return on the project co-varies with the return on the national wealth, which is typically measured by the national income. The term “national income” must in this context be interpreted in the widest sense, and shall in principle include everything that contributes to the welfare of the country. Using, for example, the national income figures of Statistics Norway will thus only give an approximation of this theoretical variable. By estimating how uncertain the return is relative to the return on an average private project, the pricing of project risk based on the Capital Asset Pricing Model can express the economic risk-adjusted alternative return against which the project can be evaluated.
However, two considerations, in particular, make it problematic to use the Capital Asset Pricing Model to determine discount rates for public projects. One of these is that the model, in its simple form, is valid for one period only. The other is that the model assumes that all assets are tradable and have a market price, whilst large parts of national wealth are not tradable.
In practice, it is common to assume a constant risk-free interest rate and a constant risk premium in the valuation of projects involving recurring cash flows, when applying a required rate of return calculated on the basis of the Capital Asset Pricing Model using market data. Both of these assumptions may be questionable for public projects with a long investment horizon.
With a long time horizon, it is reasonable to assume that interest rates, risk premiums and volatilities may all change. This makes the modelling of dynamic consumption and investment projects complex. The problem with uncertainty over time with regard to the so-called risk-free interest rate will be addressed later in the present Chapter. The classical solution to a portfolio choice problem over several periods, of which Mossin (1968), Samuelson (1969) and Merton (1969) have contributed one version each, is that the investment horizon is, under certain assumptions, irrelevant to optimal portfolio choice. One important assumption underpinning the irrelevancy result is that the return on risky assets is independent and identically distributed over time7. How the return on risky assets develops over time is a matter of debate. However, several empirical works suggest that equity returns are mean reverting, i.e. that returns tend to revert to a mean over time if they deviate, for various reasons, from such mean at a given point in time. A rational explanation for mean reversion is that risk premiums vary over time in tandem with business cycle fluctuations in the economy.
If the Capital Asset Pricing Model shall, nevertheless, be used as a framework for establishing a discount rate for public measures, the risk premium and beta estimates should, ideally speaking, both be based on national income. In practice, however, all price data are from the listed equity markets. National income figures are obtained from the national accounts, and include no information about risk considerations. This means that we need to believe that stock premium estimates express a correct price for the risk associated with public projects, in order for these to be used in determining a discount rate for public measures. This is probably justifiable from an opportunity cost perspective, provided that we accept that the risk aversion of stock market investors is reasonably representative of the risk aversion of those carrying the risk associated with public projects, cf. the discussion in the NOU 1997: 27 Green Paper. In addition, we need a method for estimating the beta of public projects. This is difficult because figures on the net benefits of a public project – as opposed to on the return of an equity – are not readily available and based on market prices.
There also exist models for estimating the beta of non-tradable objects, see Minken (2005). However, such calculations are fairly complex, whilst data availability is poor. The analysis in Minken (2005) shows that the output from such calculations is highly sensitive to changes in the data.
Textbox 5.3 Basic pricing model for all types of uncertain financial claims
Let P_{t} be the price of an uncertain claim at time t, let C be consumption, let X be cash flow and let r express the pure time preference of investors. Maximising expected utility, subject to a budget restriction, gives the first-order condition:
P_{t}U’(C_{t}) = E_{t}[pU’(C_{t+1})X_{t+1}]
The first-order condition says that the marginal cost of purchasing a unit of the claim must be equal to the expected marginal utility of owning a unit of the claim in the next period. If we solve the equation for P, we get the basic pricing model:
M is often referred to as the stochastic discount factor. The model says that the price of an uncertain future claim is equal to the expected discounted cash flow of the claim, with the discounting reflecting intertemporal substitution, risk preferences and time preferences (expressed by marginal utility and ρ). Investors prefer claims that provide high returns in those states and at those times when they have relatively low consumption, and thus relatively high marginal utility from an increase in consumption (difficult times). This will put upward pressure on the prices of such claims. Conversely, the prices of claims that do well in good states or at good times (and deliver poor performance in difficult states or at difficult times) will be subject to downward pressure.
In order to get from the general model to a model of practical use, we need to specify the characteristics of the stochastic discount factor. We also need to get around the problem that it is difficult to obtain good consumption data. Various assumptions and simplifications of the model give rise to many of the price models we know from economic theory, including the Capital Asset Pricing Model and other factor models like ICAPM (the Intertemporal Capital Asset Pricing Model) and the Fama-French model.
The factor models solve the problem of poor consumption data by replacing consumption by one or more other factors that are assumed to be good proxies for consumption, and for which better data exist. In addition, one assumes that the relationship between the marginal utility of investors and the said factors is linear.
5.3 Norwegian guidelines on choosing the discount rate
5.3.1 Historical guidelines
From 1967 to 1999, Norway used an approach where the discount rate was based on the Ramsey model. Based on a report by Leif Johansen in 1967, in which it was assumed that ρ = 1 percent, µ = 3 and g = 3 percent, the discount rate was put at 10 percent in circular R-3/1975. In circular R-25/78, the discount rate was changed to 7 percent.
When revising the cost-benefit analysis framework in 1998, the assumption of a small, open economy was held to be reasonable for Norway. The risk-free interest rate and risk premiums were deemed to be increasingly determined by international markets. Hence, estimates of the relevant discount rate were based on the so-called Capital Asset Pricing Model (see Chapter 5.2.2).
The recommendations in the NOU 1997: 27 Green Paper were based on the Capital Asset Pricing Model. The Ministry of Finance guides from 2000 and 2005 discuss, in more detail, how the model may be adapted to provide an expression for a reasonable discount rate for use in cost-benefit analysis. Circular R-14/99 stipulates that a risk-free real rate of 3.5 percent should be assumed for cost-benefit analysis purposes. Three different risk classes were defined, with a risk-adjusted rate of 4, 6 and 8 percent, respectively. Specific calculation of the risk-adjusted required rate of return was recommended for large projects or groups of projects.
5.3.2 Current guidelines
A new circular R-109/2005, replacing R-14/99, was issued following revision of the Ministry of Finance cost-benefit analysis guide in 2005. The risk-free rate for use in the cost-benefit analysis of central government measures was here put at 2 percent, and it was noted that a normal project will have a risk premium of 2 percentage points, and thus a risk-adjusted required rate of return of 4 percent. For measures where considerable systematic risk may reasonably be assumed, it is stated that a risk premium of 4 percentage points, and thus a discount rate of 6 percent, may be appropriate. Specific calculation of the risk-adjusted required rate of return continued to be recommended for large projects or groups of projects. This applies, in particular, to projects falling within the scope of the system for quality assurance of major public projects (QA1), cf. the discussion in Chapter 5.3.3 below.
The Ministry of Transport and Communications has initiated a project to examine the discount rate for projects within its areas of responsibility. Minken (2005) presents an analysis of transportation projects in Norway in which he recommends a discount rate of 4.5 percent for road and railway projects and 5 percent for harbours and airport infrastructure. The said rates comprise an estimated risk-free rate of 2 percent and a systematic risk premium of 2.5 percent and 3 percent, respectively. However, Minken (2005) notes that the analysis is highly sensitive to the data used in the analysis. Moreover, he notes that the absence of real price adjustment of those unit prices that depend on income (the value of time and life, in particular) may, when taken in isolation, suggest a 0.5 – 1 percent reduction in the resulting discount rate. The Ministry of Transport and Communications concluded by applying a discount rate of 4.5 percent to all projects within its area of responsibility, whilst for practical reasons not distinguishing between road, railway and aviation.
5.3.3 Current guidelines for projects falling within the scope of the quality assurance regime for major central government investments
In the autumn of 1997, the Government initiated a project to review and propose improvements to the systems for planning, implementing and following up on major central government investment projects. The quality assurance regime now encompasses choice of concept (QA1) and the basis for control, as well as the cost estimate, including uncertainty analysis of the chosen project alternative (QA2). The system for quality assurance of major projects applies to projects with an estimated overall investment cost in excess of NOK 750 million (Ministry of Finance, 2011).
QA1 concerns quality assurance of the basis for decision making with regard to the commencement of the planning phase. Specifically, six aspects are subject to quality assurance: a needs assessment, a general strategy document, a general requirement document, a feasibility study, an analysis of alternatives, as well as guidelines for the planning phase. The analysis of alternatives includes cost-benefit analysis of the “do-nothing option” and at least two alternative main concepts. Input data from dedicated uncertainty analyses are used, with expectation values and dispersion measures being calculated for the various uncertainty elements. The uncertainty analysis seeks to directly calculate the systematic uncertainty of relevance to society. It encompasses, inter alia, differences in systematic uncertainty relating to the investment expenditure, variations in the level of the systematic uncertainty relating to the benefits and, not least, variations in how the uncertainty relating to benefits is eliminated over the lifespan of the project.
The current guidelines note that this provides a more reliable indication of systematic uncertainty than does adding a standard risk premium to the discount rate. The general risk premium in the discount rate is therefore not made applicable to projects falling within the scope of the central government rules on external quality assurance. The analyses have consequently been based on the risk-free discount rate of 2 percent, adjusted for estimated systematic risk if applicable. It has proven difficult to thus adjust for systematic risk, and specific risk-adjustment for individual projects has only been estimated in a small number of cases. The outcome has therefore been that many major projects are only discounted at the risk-free rate of 2 percent, with no real adjustment for systematic uncertainty through a risk premium or certainty-equivalent values, whilst minor projects (outside the QA scheme) have relied on the general recommendation of the Ministry of Finances (2005) for a discount rate of 4 percent for projects with normal systematic risk (Vennemo, 2011).
5.4 Very long-run considerations in the face of uncertainty
The challenge in using market data to determine the discount rate is that the most long-term interest-bearing financial instruments being traded do not have maturities that match the lifespans of the very longest-term public projects. How uncertainty develops and influences the rate must therefore be considered from the perspective of economic theory. In Chapter 5.4.1, we take a closer look at uncertainty about general economic developments, and which consequences such uncertainty has for the discount rate and its time structure. The fact that the project return may also include systematic risk is disregarded at this stage. In Chapter 5.4.2, on the other hand, we discuss the relationship between the systematic uncertainty associated with the return on the project and discount rate developments over time. In Chapter 5.4.3, this is discussed from the perspective of the global environmental challenges, as discussed in the NOU 2009: 16 Green Paper.
5.4.1 Uncertainty about general economic developments
Cost-benefit analysis of global climate policies and climate measures necessitates analysis of effects in the very long run. This has initiated new research on which discount rate it is appropriate to use in the very long run.
A key aspect that is highlighted is the basic uncertainty about future global economic developments. A joint feature of analyses addressing this is the assumption that consumers are risk averse. One way of approaching uncertainty about future wealth developments is to start out from the Ramsey condition, presented in Chapter 5.2.1, and let economic growth be uncertain. Gollier (2008), for example, presents such an expanded Ramsey condition in which uncertainty about future economic developments gives rise to an insurance motive for savings, which is indicative of a lower discount rate. In the face of uncertainty, one will, all else being equal, invest somewhat more for the future to guard against unfavourable future outcomes with regard to one’s economic situation. This approach implies, under certain assumptions relating to preferences and consumption developments over time, a stable rate for all years.8 Given the uncertainty assumption that suggests a stable discount rate over time, Gollier (2011) takes a closer look at the magnitude of the precautionary element in the required rate of return. It is noted that the insurance element is moderate for rich countries as the result of relatively stable economic growth in these countries. Assuming an elasticity of marginal utility of 2, which is stated as being based on a discretionary assessment, he finds, for example, a rate for the United States that is reduced from about 3.5 percent (if not taking the uncertainty into account) to 3.35 percent (if taking it into account). If, on the other hand, one takes into account the large variation in income globally, he finds that the precautionary element is much larger, with the global rate being reduced from about 4.5 percent to 0.7 percent. This indicates that in examining issues that affect the entire planet, for example the formulation of a global climate policy, it is important to consider such precautionary element. Under other assumptions – especially that the uncertain consumption is positively correlated over time − Gollier (2008) shows that the required rate of return declines over time.9
If examining measures in the very long run, it may also be argued that economic growth can be expected to decline over time. This will also, when taken in isolation, indicate a declining rate over time, cf. the Ramsey condition with a declining growth rate in consumption per capita. Such arguments may, for example, be linked to limited aggregate global resources and to economic growth in the history of humanity only having been the norm for the last 250 years.
Another approach to uncertainty about future economic developments is modelled in Weitzman (1998) as uncertainty with regard to the opportunity cost of capital, independently of uncertainty as to the actual return on the project. This is based on the premise of the discount rate as a price that reflects the forgone market-based alternative return on the capital invested in the project, cf. Chapter 5.2.2. This raises two issues. One of these is that when the alternative return is uncertain, the discount rate needs to take the form of a certainty-equivalent opportunity cost of capital. This raises the issue of the relationship between the certainty-equivalent rate and the expected rate.
Geometric discounting makes the net present value a convex function of the discount rate. This implies that uncertainty about the discount rate results in the expected net present value of future project profits being higher than the net present value calculated on the basis of the expected rate (cf. the so-called Jensen’s inequality). Consequently, the certainty-equivalent discount rate is lower than the expected rate. The certainty-equivalent discount rate is implicitly determined by the expected value of the discount factors corresponding to the uncertain rates, and not by the expected value of the discount rates (cf. Weitzman, 1998, and Weitzman, 2012, Box 5.4).
Secondly, there is the issue of developments in the certainty-equivalent discount rate over time. Such time structure depends on how the uncertainty associated with the alternative return changes over time. Newell and Pizer (2003) undertake a more general analysis of discounting under uncertainty based on capital market returns, in which the assumptions concerning rate developments over time are discussed. It is there noted that developments in the certainty-equivalent rate over time depend on whether one assumes that the uncertain future rate tends to revert to a long-term mean (so-called mean reversion) or whether one has positively serially correlated market rates as the result of random macroeconomic shocks. In the former case, the presence of uncertainty has relatively minor implications for the discount rate. In other words, the certainty-equivalent rate is in such case marginally lower than the expected rate. In the latter case, the certainty-equivalent rate will be considerably lower than the expected rate, especially in the long run. Newell and Pizer (2003) look at historical interest rate developments for long-term government bonds in the United States over the last 200 years, and estimate developments in the certainty-equivalent rate on that basis. They find, under the assumption of random walk-based rate developments, that stochastic, macroeconomic shocks with persistent effects will imply positively serially correlated rates. This results in the certainty-equivalent rate declining from 4 percent to 2 percent after 100 years, to 1 percent after 200 years and to 0.5 percent after 300 years. This is in line with the approach in Weitzman (1998). However, a model assuming mean reversion indicates that the same historical data imply a certainty-equivalent rate that is in practice identical to the expected rate of 4 percent for the first 30-40 years, and that remains above 3 percent for the next 200 years. It only declines to 1 percent after 400 years. Newell and Pizer (2003) state that evidence based on standard statistical tests provides no clear answer as to which model best fits the data. The authors nevertheless note that random walk is the model that best fits the data in most attempts at splitting the historical data into two periods (so-called split samples), which is indicative of a declining rate over time.
Like Newell and Pizer (2003), Hepburn et al. (2009) start out with an analysis of variations in the historical risk-free rate, looking at Australia, Canada, Germany and the United Kingdom. Different specifications of the econometric model provide somewhat different outcomes with regard to the time structure of the long-term certainty-equivalent rate. Based on a risk-free rate in year 1 of 3.5 percent for the United Kingdom, they find, for example, a rate that varies between 3.54 percent and 3.31 percent after 40 years and between 3.42 percent and 3.22 percent after 100 years. The article notes that the model with the steepest decline in the rate enjoys the best empirical support.
Instead of seeking to estimate a discount rate from historical data, Weitzman (2001) shows the findings from a survey amongst a large sample of economists as to which real discount rate they believe to be reasonable, “everything taken into consideration”, for assessing a long-term climate issue. The mean response is 4 percent with a standard deviation of about 3 percent. Weitzman (2001) uses this sample as a distribution of potential future discount rates and shows, in line with the more general observation in Weitzman (1998), how such a distribution suggests a declining certainty-equivalent rate over time. By using the findings to estimate a specific distribution for this certainty-equivalent discount rate, he notes that this suggests, given his approach, a discount rate of 4 percent for the first five years, thereafter 3 percent for the next 20 years, thereafter 2 percent for the next 50 years, thereafter 1 percent until year 300 and thereafter 0 percent. However, Freeman and Groom (2012) note that these findings follow from a specific assumption to the effect that the respondents in the survey gave their responses as a normative assessment as to what they believed the rate should be, and not as an estimate as to what will in fact be the average rate. If respondents did actually give their best estimates of an average future rate, it is the distribution of individual estimated averages that is relevant, whilst it is the distribution of the respondent’s normative estimates that is relevant under the normative interpretation. In the latter case, the variance will be much higher than the variance of the future rate estimates, which will be given by the variation in the distribution of averages. If the responses are interpreted as each respondent’s best estimate of the future rate, it will result in a much flatter term structure for the discount rate than follows from Weitzman’s approach. Since there will always be an element of normative replies, Freeman and Groom (2012) can be taken to indicate that Weitzman’s findings are toned down, but without the declining tendency being eliminated altogether.
Freeman and Groom (2012) note that it is not possible to know on what basis the respondents in Weitzman (2001) submitted their responses. The authors note that different interpretations of the responses reflect, to a large extent, approaches to discount rate recommendations in the United Kingdom, France and the United States. The United Kingdom and France have public recommendations for a time structure of the discount rate that resembles what would result from a normative interpretation of the responses from the respondents. Recommendations in the United States, on the other hand, with less dramatic rate declines, resembles what one would get by interpreting the responses of the respondents as estimates of expected averages.
The literature reviewed above suggests, all in all, that growing uncertainty about the market return is indicative of a declining discount rate as given by the opportunity cost of capital. The reason for this is that it does, when taken in isolation, become more attractive to realise the project as the uncertainty associated with the alternative return that can be obtained in the market increases. This results in a lower required rate of return, and thus a lower discount rate. The literature reviewed in the present Chapter has found support for the contention that uncertainty with regard to macroeconomic developments - and hence the alternative return as given by the market rate of interest – increases in the very long run, i.e., beyond the period in which market returns can reasonably be hedged in financial markets. Consequently, this literature suggests that the discount rate, as assessed on the date of analysis, will eventually decline over time.
So-called hyperbolic discounting, or bias towards the present, has also been invoked as an argument for a declining discount rate over time (Harstad, 2012). Hyperbolic discounting implies that people’s degree of impatience is not constant, and that the present is accorded relatively more weight than would be suggested by exponential discounting at a constant rate. This means that the trade-off between utility in subsequent periods changes over time. This gives rise to dynamic time inconsistency, which implies that the profitability of a decision is dependent on when such decision is made, although the informational basis for the decision remains unchanged. However, a number of empirical studies show that a bias towards the present is commonly found in individuals, and therefore must be assumed to influence the population’s willingness to pay for future consequences. See for example Laibson (1997), Frederick et al. (2002), Dasgupta and Maskin (2005), for a more detailed discussion.
5.4.2 Uncertainty as to the return on the measure
Chapter 5.4.1 addresses how uncertainty with regard to general economic developments influences which risk-free rate should be adopted. This is examined quite extensively in economic theory. Another issue is how uncertainty concerning the actual return on the project will influence the discount rate in the long run. Such uncertainty will influence the risk premium element of the discount rate, cf. Chapter 5.2.3, Models for pricing uncertain financial claims, and related fact boxes. The literature examining the very long-term time structure of such risk premiums is more limited.
Weitzman (2012) presents a theoretical approach to the problem. Based on a consumption-based, multi-period version of the Capital Asset Pricing Model, in which net consumption from a project is comprised of an unsystematic component plus a systematic component that varies proportionally with the uncertain aggregate consumption in the economy at a factor of 0 ≤ß ≤1, an optimal discount rate is derived. This is implicitly determined by the risk-adjusted rate corresponding to a discount factor that is a ß-weighted average of the discount factor calculated with the risk-free rate and the discount factor calculated with the expected return on the capital portfolio generating the value added corresponding to the uncertain consumption component. The one-period version of this model results in a risk-adjusted required rate of return that is concurrent with the required rate of return from the Capital Asset Pricing Model. Moreover, for any ß between 0 and 1, the risk-adjusted required rate of return will be declining with the time horizon and will approach the risk-free rate when the horizon becomes sufficiently long.
The world considered here is genuinely uncertain. Over time, any project with a return that is less uncertain than is the value added in the risk-exposed part of the economy, will represent a form of insurance in a long-term perspective. For such projects the unsystematic part of the value added will therefore become increasingly important over time, whilst the component exposed to systematic risk “discounts itself out”. The discount rate for projects with a certain systematic risk shall therefore decline over time, according to Weitzman (2012). Projects without systematic risk will, in line with the theory, have a constant rate equal to the risk-free rate. The model in Weitzman (2012) is presented in more detail in Box 5.4.
The discount rate used in the evaluation of public projects must be assessed on the basis of the sum total of the risk-free rate and the relevant risk premium. However, different model approaches share the feature that increasing uncertainty over time with regard to the alternative market return implies a declining risk-adjusted required rate of return.
It has not previously been common practice to use a rate that varies over time for cost-benefit analysis purposes. It is therefore worthwhile to make two observations. Firstly, the basis for a declining rate, as presented above, is increasing uncertainty over time. In an assessment situation in which such a declining rate is to be used, it will therefore be appropriate to assume that the rate structure will apply from the date of analysis. Secondly, values in the same period shall be discounted at the same discount rate, since they are exposed to the same macroeconomic uncertainty. As an example, assume that a declining discount rate is made operational by applying one rate for the first 40 years and a lower rate for subsequent years. This means than an economic effect in year 50 shall first be discounted to year 40 by applying the low rate, and then discounted from there at the shorter-term rate. In other words, the value of an effect cannot be changed significantly by moving it from year 40 to year 41, etc.
Textbox 5.4 Summary of Martin Weitzman (2012): Rare Disasters, Tail-Hedged Investment, and Risk-Adjusted Discount Rates
From the welfare effects resulting from a marginal investment project in a simple dynamic stochastic general equilibrium model, one can derive the optimal risk-adjusted discount rate schedule to be applied to the project’s time-dependent pay-offs . That is, the investment will improve welfare if and only if the present value of its cost is less than the expected present discounted benefits calculated at these discount rates.
Assuming constant relative risk aversion, the investment opportunities of the economy may be depicted as consisting of two assets; one risk-free asset which gives a fixed unit of consumption in each period, and one risky asset which may be thought of as equity yielding some fraction of all future risky consumption pay-offs.
Net benefits from the investment are random variables that are assumed to be made up by two components, and given by
In (1), ß_{t} is the fraction of the pay-offs at time t that replicates the risk profile of the aggregate economy at that time, while the fraction (1 - ß_{t}) is stochastically independent of the aggregate economy. Hence, without loss of generality, the latter component can be normalised by setting E[I_{t}] =1 for all t. That implies that expected net benefits at time t are given by b_{t}.
The role of the fractions ß_{t} resembles that of the investment beta in the Capital Asset Pricing Model. However, in the present context the beta at time t reflects the correlation between the project’s pay-offs in terms of net benefits and the aggregate consumption at t, which is here called the real project beta.
For simplicity, it is assumed that ß_{t}= ß for all t.
The relationship between the instantaneous risk-free rate, rf, and the gross return on the risk-free asset is given by r_{f}= ln(R_{f}) ≡ ln(e^{rf}) and similarly for the instantaneous risky rate of return on equity, i.e. r_{e} = lnE[R_{e}]
Assuming an investment at time 0 yielding a single pay-off at time t given by the expression for B_{t} as in equation (1), one can define X as the loss of consumption at time zero that makes the consumer precisely indifferent between accepting or rejecting this hypothetical project. The consumption loss X must then satisfy
where a is the constant discount factor representing the pure time preference for utility. The corresponding discount rate for a project with a beta value = b must satisfy the equation
Relying on the simplifying properties of the iso-elastic utility function, one can derive the critical value of X as given by
Using (3), this can be expressed in terms of discount factors as
Taking logarithms, (5) can be rewritten as
Hence, a marginal investment project will increase welfare if and only if the expected present values of its net benefits discounted by the discount rate given by (6), is positive.
Compared with the Capital Asset Pricing Model where the risk-adjusted discount rate is given by a beta-weighted average of the risk-free rate and the expected return on the market portfolio, the optimal risk-adjusted discount rate in the consumption-based investment model is implicitly given by the optimal discount factor being a beta-weighted average of the discount factors associated with the risk-free rate and the expected return on equity, respectively. Hence, the discount rate given by (6) represents a multi-period generalisation of the stationary Capital Asset Pricing Model.
The static Capital Asset Pricing Model relies on mean-variance preferences, whilst its multi-period investment counterpart relies on constant relative risk aversion preference, both of which yield portfolio separation.
For t = 1, the discount rate equals the beta-weighted discount rate in the Capital Asset Pricing Model and approaches the risk-free rate as t becomes very large. Thus, for any risk profile as represented by the fraction 0 < ß < 1 of the project’s pay offs that correlates perfectly with total consumption, the appropriate discount rate will be declining over time and approach the risk-free rate as a limiting value.
5.4.3 Discounting and global environmental challenges
The NOU 2009: 16 Green Paper, Global Environmental Challenges – Norwegian Policy, includes an analysis of uncertainty and discounting in view of, inter alia, the above literature, based on the work published at that time.
It was noted that there are arguments in favour of long-term climate measures being evaluated at a lower discount rate. According to the report, these arguments can be divided into two groups. Firstly, it is uncertain whether we attach sufficient weight to later generations if we also apply a relatively high discount rate for long-term projects. Secondly, there may be doubt as to whether the information in observable market interest rates is also of relevance to subsequent periods.
The report illustrated the issue by assuming that both risk-free rates and risk premiums will continue to develop more or less in line with what we can estimate on the basis of available market data, and that the project we are examining is not sufficiently large to influence market prices. Effects in the distant future will in such case have a very low net present value. With a discount rate of 5 percent, the net present value of 1 krone in, for example, 50 years will be 8.7 øre, and in 100 years only 0.76 øre. It was noted that it is often argued, on this basis, that projects with effects in the distant future should be evaluated on the basis of a low discount rate out of concern for future generations. However, the NOU 2009: 16 Green Paper noted that it is not obvious that a lower discount rate addresses the problem we want to solve.
It was noted, assuming that it is not a problem in itself to value all income and costs in kroner, that the analysis should, in the usual way, identify which groups profit from the project, and which groups incur a loss from it. If the project has a negative net present value when not taking distribution effects into account, whilst those who would profit from the project are future generations, we are, generally speaking, dealing with a distribution problem, and not a discounting problem. Those conducting the project analysis must in such case identify the distribution effects, and the decision maker should consider whether the project should be implemented due to the positive effects for future generations. The present Green Paper discusses the distribution issue in more detail in Chapter 3.
Furthermore, the NOU 2009: 16 Green Paper noted that it is also necessary to properly address other prices than the discount rate when dealing with long-term projects. If, for example, environmental goods become scarcer over time, there is reason to believe that the value (the calculation price) of environmental goods will increase relative to the value of other goods. The implications of such changes in relative prices may, according to the NOU 2009: 16 Green Paper, outweigh the implications of discounting, and make projects economically profitable even though the income is realised in the distant future. The Committee discusses real price adjustment in more detail in Chapter 4 and specifically addresses real price developments with regard to the shadow price of greenhouse gas emissions in Chapter 9. Reference is also made to the committee on the valuation of biodiversity appointed by the Ministry of the Environment.
The NOU 2009: 16 Green Paper then addresses the issue of determining a discount rate for periods for which we have no observable market prices to rely on. It is noted in this context that we need to examine what determines the risk-free rate and the risk premium in the long run. A number of the models outlined above are presented, and some factors, in particular, are highlighted: changes in economic growth, changes in uncertainty, weakness in the available data and market failure.
The NOU 2009: 16 Green Paper concluded, inter alia, that the discount rate applied for cost-benefit analysis purposes should be a real rate calculated before tax, and with a maturity tailored to the project duration. One should, as a main rule, use market-based estimates for the risk-free rate and the risk premium. However, it was noted that there is considerable uncertainty associated with estimates of long-term risk premiums and, in part, also long-term, risk-free rates. The NOU 2009: 16 Green Paper proposed no changes in current practice for determining the discount rate for public projects in Norway, whilst at the same time noting that the risk associated with many public projects is low.
5.5 The level of the discount rate
By comparing the interest rates charged for short-term and long-term loans, one can get an impression of how market players consider trade-offs in the long run. However, this only applies to periods in which a certain volume of securities is actually traded. Developments in the wake of the financial crisis in 2008 and, in particular, since the last year’s Euro crisis, have illustrated how unclear it is what can be considered a risk-free interest rate and what constitutes a risk premium. Generally speaking, all yield curves are upward sloping in the current market, i.e. the interest rate is higher the longer the maturity, as far as those periods for which liquid markets exist are concerned.
The current interest rate level in the market for securities that are presumed to be secure is, generally speaking, very low from a historical perspective. It is uncertain whether this can be considered a long-term tendency, or must be considered a special phenomenon. This may be due to the specific circumstances in the current international currency market, where the ability of banks to manage without central government support in Europe and the United States is associated with significant risk, in combination with weak confidence in the ability of some states to handle their explicit and implicit liabilities.
Assessments of returns in international financial markets in the very long run are of special interest to the Norwegian State, because the return on the Government Pension Fund Global comes from these markets. Report No. 17 (2011-2012) to the Storting provides a detailed discussion of both historical and future returns in the very long run. It is shown that the average annual real return on the Government Pension Fund Global of 2.7 percent is well within normal fluctuations around an expected return of 4 percent.10 It is noted that current real interest rates are very low, also from a historical perspective. According to the said Report to the Storting, this is partly caused by the steep slump experienced by the world economy since 2007, and partly by central banks wanting to stimulate economic growth. The Ministry of Finance is of the view that the extraordinary nature of the current situation suggests that one should be cautious about changing the estimates for the expected real return on the Government Pension Fund Global exclusively on the basis of today’s low real interest rate levels. The expected real return on the Fund is based on an estimate of 2.5 percent for the mean return on the Fund’s portfolio of government bonds and an estimate of 2.5 percent for the stock premium. Other investments are assumed to have a risk profile between those of government bonds and equities, respectively.
The required rate of return for public projects should be determined as an arithmetic mean11 before tax. The expected long-term return on equities in the Government Pension Fund Global is determined net of corporation tax and represents a geometric mean. It therefore needs to be adjusted to allow comparison with the required rate of return for public projects. If one uses the 2.5 percent expected return on the government bond portfolio as the risk-free benchmark return, one arrives, through the numerical example in the NOU 2009: 16 Green Paper, Box 8.2, at a real risk premium (before tax) of 3.5 percent for an average project funded in the stock market, given the assumptions made in Box 8.2 in the NOU 2009: 16 Green Paper.12 Alternatively, we could have disregarded an estimated maturity premium in the government bond portfolio and thus applied 2 percent as a risk-free benchmark interest rate, which would have put the relevant risk premium at 4 percent. Both of these approaches suggests that a real required rate of return of 6 percent for projects with about the same systematic risk as an average project funded in the global stock market is consistent with expectations as to the long-term return on the equity portfolio of the Government Pension Fund Global.
If we assume that an ordinary public measure, being a transportation measure, has a risk profile that is somewhat closer to a government bond than to an average project funded via the stock exchange, the calculations above suggest that a risk-adjusted real required rate of return (before tax) of about 4 percent is reasonable. This is indicative of a risk premium of 1.5 percentage points. Using the current recommendations on the direct calculation of risk premiums, as set out in the Ministry of Finance guide from 2005, has turned out to result in a lower risk-adjusted rate than would be suggested by such a reasonableness test.13
5.6 Social discount rates in other countries
Discount rate recommendations vary considerably between countries. One review identifies recommendations ranging from 1 to 15 percent (Harrison, 2010). We will present some selected recommendations below. Different countries attach weight to different considerations in their discussion and determination of discount rates. We have chosen to present the recommendations as set out in the documents that have been reviewed.
5.6.1 Recommendations for the EU area
A consortium lead by the German institute IER (University of Stuttgart, Institute of Energy Economics and the Rational Use of Energy) has over the period 2004-2006 prepared, inter alia, a proposal for harmonised guidelines for transportation infrastructure investments in Europe through an EU-funded research project called HEATCO (Harmonised European Approaches for Transport Costing and Project Assessment). It does not provide one uniform recommendation on the discount rate for infrastructure projects. It is instead noted that some countries start out from the “social rate of time preference” as expressed through the Ramsey condition, whilst others base themselves on estimates as to the opportunity cost of capital. The recommendation for analyses of cross-border transportation infrastructure investments is to use a risk-free rate or a weighted average of the required rates of returns in the various countries, weighted by the funding contribution of each country. It is not stipulated how the risk-free rate shall be calculated. The use of sensitivity analyses is recommended, applying a rate of 3 percent, which in the guidelines is explained by reference to the Ramsey condition, with p=1.5, µ=1 and g=1.5. For projects in countries that themselves recommend the use of a declining rate path in the long run, the recommendation calls for sensitivity analysis with a declining rate path for effects that occur from year 40 and beyond.
No special guidance is provided as to whether costs or discount rates should be corrected for risk aversion. It is, on the other hand, recommended that risk and uncertainty be highlighted through sensitivity analyses, scenario analyses and/or Monte Carlo simulations, depending on the available resources and data. The treatment of systematic and unsystematic risk is not specifically addressed.
5.6.2 Sweden
There exists no intersectoral national guidance on discount rate choice in Sweden. For the transportation sector, a committee appointed by the Swedish Transport Administration recommends using HEATCO and the recommendations in the United Kingdom (Swedish Transport Administration, 2012) in determining the discount rate. The Ramsey model is used as a starting point (see the above discussion). The pure rate of time preference and the elasticity of marginal utility of consumption are put at the same levels as in the United Kingdom (HM Treasury, 2003) and as referred to in the HEATCO guidelines. Annual economic growth in Sweden is estimated at 1.78 percent. Taken together, this suggests a discount rate of 3.28 percent. Reference is made to arguments in favour of a declining rate over time, but it is noted that the methodological tools used in the cost-benefit analysis of transportation measures in Sweden are not adapted to such a declining rate path. A somewhat lower discount rate than the 4 percent previously recommended is assumed to partially compensate for this. The discount rate was 4 percent until September 2012, but the new guidelines applicable from September 2012 recommend 3.5 percent.
5.6.3 Denmark
The Danish Ministry of Finance (Danish Ministry of Finance, 1999) assumes, on a discretionary basis, that the social return on capital in alternative use will fall within the 6-11 percent range. 6 percent is chosen as the discount rate on the assumption that public investments are likely to involve lower systematic risk than the average project on the stock exchange. It is noted in the Danish guidelines that the chosen level is also held to fall within the likely range of the social rate of time preference.
5.6.4 United Kingdom
In the United Kingdom the discount rate is based on the social rate of time preference and quantified at 3.5 percent in real terms (HM Treasury 2003). The British economic analysis guide, the “Green Book”, was revised in 2003. The former discount rate of 6 percent included a risk premium for various risk elements. However, it was concluded during the revision that it was a better solution to address relevant risk specifically for each project through various risk analysis methods. Moreover, the British have prepared separate guidelines on how the authorities may correct, in their calculation of the expected cost and benefit flows, for a tendency to be overly optimistic in assessing projects, the so-called optimism bias, with a view to obtaining more unbiased estimates. The “Green Book” assumes that systematic risk is usually negligible for most individual projects, given their magnitude relative to national income.14
Due to uncertainty about the future, the British guide recommends a declining discount rate over time for projects with effects beyond 30 years. Reference is made to Weitzman (1998, 2001) and to an unpublished paper by Christian Gollier from 2002. The guidelines include a table specifying which discount rates shall be applied for effects at various time intervals, with the rate being 3.5 percent until year 30, thereafter 3.0 percent until year 75, and with successive reductions down to 1 percent for effects that occur in year 300 and beyond. It is not specified how one has arrived at the specific recommended rate path.
5.7 The assessment of the Committee
In cost-benefit analysis, it is necessary to compare effects that occur at different points in time. A systematic and transparent way of doing so is discounting by way of a discount rate, with all monetised effects being converted to the value they will have in a given reference year, calculated at such rate, and therefore being comparable. When the reference year comes at the beginning of the lifespan of the measure, this is called a net present value calculation.
It is difficult to arrive at a universally valid answer to the question of what is the “correct” discount rate or how such a rate shall be estimated. However, it is reasonable to start out from the premise that a lower value is attributed to future values than to current values, from a current perspective.
A simple approach is to say that the required rate of return depends on a pure rate of time preference, the elasticity of marginal utility of consumption, which shows relative change in marginal utility divided by relative change in consumption, as well as estimated consumption growth. However, all of these variables are highly uncertain in the long run, and Chapter 5.2.1 notes that different estimates for these key variables have resulted in estimated discount rates that range from 1.4 to 8 percent. The United Kingdom and Sweden recommend a short-term rate of 3.5 percent based on a set of assumptions that are held to be reasonable there.
Likewise, the required rate of return in the financial market reflects the return on private investments, assuming well-functioning markets. Theoretically, the required rate of return for corresponding investments abroad represents the opportunity cost of a public measure, and is consequently the discount rate for a small, open economy. The NOU 1997: 27 Green Paper notes that Norway is a small, open economy, and especially so since the liberalisation of the capital market. Such developments have been reinforced since 1997, which suggests, when taken in isolation, that the arguments from the NOU 1997: 27 Green Paper about using a rate model based on an alternative assessment of the price of capital remain equally valid.
The real risk-adjusted discount rate (before tax) should reflect the risk-free rate and the risk associated with the project, and thus reflect the opportunity cost of the project. From an opportunity cost perspective, it will generally speaking make sense to adjust the discount rate to be applied to each measure on the basis of the systematic risk of such measure. The systematic risk depends on whether the country is performing well or not when the return from the project is expected to materialise. If the economy is performing well, one can expect a lower value to be attributed to the project’s contribution to the country’s value added. This may be reflected in the analysis by applying a higher discount rate.
However, it is not obvious how such a project-specific risk premium shall be estimated. A correct calculation of the risk associated with individual measures is complex, whilst the availability of data is poor. Experience from such project-specific calculation of the risk premium has shown that calculations like these have only been carried out to a limited extent, and in those cases where they have been carried out the findings are sensitive to modified assumptions and minor changes in the data used. As noted in Chapter 5.5, direct use of the approach in the Ministry of Finance guide from 2005 has also resulted in a lower risk premium than would be indicated by a reasonableness test. Chapter 5.2.4 discusses theoretical weaknesses associated with applying the Capital Asset Pricing Model to estimate the risk premium for public measures. The present Chapter has noted weaknesses in the Capital Asset Pricing Model, both with regard to the identification of the systematic risk in the stock market, and as a basis for estimating the relevant risk premium for a public project. The Committee therefore holds the view that this model does not constitute a good basis for determining the discount rate for public projects.
Theoretical developments, especially in view of the climate debate, have generated much new and interesting analysis of the valuation of effects in the very long run. In summary, we may say that the theory reviewed suggests that the risk-free rate shall be declining over time if the uncertainty pertaining to growth rates in the economy accumulates over time. If such uncertainty does not accumulate, one can nevertheless argue in favour of declining rates over time on the basis of an expected decline in growth rates over time. Hyperbolic discounting is an alternative argument in favour of declining rates. However, there is no corresponding consensus as to how the rate path should specifically develop over time, and the discussion in Chapter 5.4.1 shows that different assumptions give rise to different time profiles. There is, at the same time, an ongoing debate about how project-specific systematic risk in the long run affects the economically relevant risk premium. The contribution from Weitzman (2012) notes that one will, for projects with lower systematic risk than the market average, get a rate that declines towards the risk-free rate in the long run.
All in all, the literature reviewed above indicates that increasing uncertainty about the opportunity cost of capital suggests a declining discount rate. The economic reasoning behind this is that it becomes more attractive to realise the project as the uncertainty associated with the alternative market return increases. This results in a lower certainty-equivalent required rate of return, which means a lower discount rate. The literature review in the present Chapter has demonstrated support for the contention that the uncertainty associated with macroeconomic developments - and hence with the alternative return – is increasing in the very long run, beyond the period that can reasonably be hedged in the financial markets. Consequently, this literature suggests that the discount rate, as assessed per the date of analysis, should decline over time. The Committee is of the view, based on an overall assessment, that the discount rate for use in the cost-benefit analysis of public measures should reflect these arguments in favour of a declining discount rate, based on the theory presented in this Chapter, especially for very long-term projects.
When the Committee is to make recommendations on the discount rate for use in the evaluation of public measures, we also need to take into consideration the decision structure within which this will be applied. Considerable room for discretionary assessments with regard to estimates as to project-specific risk, the time structure of the discount rate and the lifespan of the project may offer incentives to choose assumptions that may influence the outcome of the analysis in the direction favoured by various interested parties. In addition, experience from previous practice with several risk classes suggests that many project analysts have been uncertain about the technical criteria for choosing the risk class, and that such choices may therefore at times seem somewhat arbitrary. These circumstances suggest that it may be preferable to recommend simple and transparent rules that capture the most important aspects of the matter, without being too complex to understand or to apply. Simple, but rigid, rules will, on the other hand, necessarily limit the scope of the analyst for taking more specific knowledge about each project into account, for example with regard to systematic risk.
There is no one correct way of providing specific estimates for the risk-free market interest rate, the risk premium and the time profile of interest rate developments. However, a reasonable approach may be to assume that it will, under normal market conditions, be possible to secure a risk-free real interest rate of 2.5 percent within a time span of 40 years through investments in the international financial market. This is on a par with the unconditional expected return on government bonds in the Government Pension Fund Global. Beyond 40 years, it is reasonable to assume that one cannot secure a long-term interest rate in the market, and the discount rate should be determined on the basis of an assessment of the certainty-equivalent rate. A certainty-equivalent discount rate of 2 percent may be in line with reasonable assumptions as to developments in the uncertainty associated with future economic developments.
Furthermore, the discount rate needs to reflect the fact that ordinary public measures, like a transportation measure, will to some extent be sensitive to changes in the general economic situation (systematic risk). The Committee is of the view that the Capital Asset Pricing Model is not a good starting point for determining which risk premium will reflect this in a good manner. In line with the principle that the discount rate shall reflect the opportunity cost of capital, and moreover being a simple and transparent rule, the Committee believes that it is reasonable to look at the expected return on the Government Pension Fund Global, which owns a small portion of the world’s production capacity and debt. The estimated expected mean return on government bonds for the Fund’s portfolio is 2.5 percent, with a risk premium for equities of 2.5 percent. Chapter 5.5 explains that a required rate of return for an ordinary public measure, like a transportation project, of 4 percent is consistent with these expectations. Table 5.2 outlines a discount rate structure for such an ordinary project on this basis.
It is the recommendation of the Committee that the structure outlined in Table 5.2 normally be used for the discounting of all types of public measures. Consequently, the Committee has decided, based on an overall assessment, not to recommend the establishment of several risk classes with different risk-adjusted discount rates. For measures that quite obviously involve low or negative systematic risk, like for example labour market measures, it will be appropriate to apply a lower discount rate. For measures that quite obviously involve higher systematic risk it will, correspondingly, be appropriate to apply a higher discount rate. If one would otherwise like to conduct a sensitivity analysis to examine how the net benefits from the project will be influenced by other assumptions as to systematic risk, it will be appropriate to do so as a supplement, thus retaining a basic analysis that ensures comparability in the decision-making process. For projects where it is primarily the costs that are quantified, the evaluation of sensitivity to business cycle fluctuations must focus on the cost aspect. For commercial public sector operations in direct competition with the private sector it will, however, be appropriate to use a risk premium faced by corresponding private enterprises.
Table 5.2 Discount rate structure for an ordinary project
Discount rate | |||
---|---|---|---|
Years 0-40 | Years 40-75 | From year 75 (i.e. largely environmental effects) | |
-risk-free rate | 2.5 percent | 2 percent | 2 percent |
-premium | 1.5 percent | 1 percent | 0 percent |
Risk-adjusted rate | 4 percent | 3 percent | 2 percent |
It has not previously been common practice to use a rate that varies over time for cost-benefit analysis purposes. It is therefore worthwhile to make two observations. Firstly, the basis for a declining rate, as presented above, is increasing uncertainty over time. In an assessment situation in which such a declining rate is to be used, it will therefore be appropriate to assume that the rate structure will apply from the date of analysis. Secondly, values in the same period shall be discounted at the same discount rate. Assume that a declining discount rate is made operational by applying one rate for the first 40 years and a lower rate for subsequent years. This means than an effect in year 50 shall first be discounted to year 40 by applying the lower rate, and then discounted from there at the more short-term and higher rate. In other words, the value of an effect cannot be changed significantly by moving it from year 40 to year 41, etc.
For the time being, there is an ambition for the pricing of systematic risk to be handled individually for projects that fall within the scope of the central government scheme for quality assurance of concept choice (QA1). Under the current guidelines, this applies to projects with an expected cost in excess of NOK 750 million. The Committee notes, however, that it appears to be more difficult than previously assumed to identify the risk premium that reflects, in a correct manner, the systematic risk associated with individual projects, and, more specifically, that the method recommended thus far, based on the Capital Asset Pricing Model, has weaknesses. This suggests, when taken in isolation, that no requirement for a project-specific risk premium should be applied. Moreover, the Committee holds the view that it would produce a more predictable system if the discount rate is determined in the same way for all public measures, irrespective of whether these fall within the scope of the scheme for external quality assurance of the choice of concept (QA1).
Finally, the Committee highlights the importance of correct valuation of the effects included in the analysis, before discounting. The recommendations made by the Committee in the present Report will to a large extent contribute to more correct valuation of the effects in the analysis before discounting, cf. Chapter 4 on real price adjustment and Chapter 9 on carbon pricing. Where the effects of the measure are subject to major uncertainty, considerable weight should be attached to ensuring unbiased estimates and the best possible examination of the uncertainty, preferably through separate sensitivity analyses of such effects. The uncertainty analyses performed in preparing choice of concept reports and QA1 are valuable, inter alia, through their contribution to obtaining more unbiased estimates for cost and benefit effects.
5.8 Summary recommendations
In principle, the real risk-adjusted discount rate should reflect the risk-free interest rate and the risk associated with the project and, consequently, reflect the project’s risk adjusted opportunity cost of capital. The discount rate applied in the assessment of public measures should, however, nonetheless be based on simple rules that address the main aspects of the matter.
For commercial public sector operations in direct competition with the private sector, it will be appropriate to use a discount rate faced by corresponding private enterprises.
A real risk-adjusted discount rate of 4 percent will be reasonable for use in the cost-benefit analysis of an ordinary public measure, such as a transportation measure, for effects in the first 40 years from the date of analysis.
Beyond 40 years, it is reasonable to assume that one will be unable to secure a long-term rate in the market, and the discount rate should accordingly be determined on the basis of a declining certainty-equivalent rate as the interest rate risk is supposed to increase with the time horizon.. A rate of 3 percent is recommended for the years from 40 to 75 years into the future. A discount rate of 2 percent is recommended for subsequent years.
5.9 Bibliography
Atkinson, G., S. Dietz, J. Helgeson, C. Hepburn and Håkon Sælen (2009). Siblings, Not Triplets: Social Preferences for Risk, Inequality and Time in Discounting Climate Change. Economics e-journal, Vol. 3 (26).
Banz, R. (1981). The relationship between returns and market value of common stock. Journal of Financial Economics, 9, pp. 3-18.
Campbell, J, Y. and L. M. Viceira (2003). A multivariate model of strategic assets allocation, Journal of Financial Economics 67, pp. 41–80.
Cline, W.R. (1993). Give Greenhouse Abatements a Fair Chance, Finance and Development (3): pp. 3-5.
Cochrane, J. H. (2005). Asset Pricing: Revised Edition. Princeton, NJ: Princeton University Press.
Dasgupta, P. and E. Maskin (2005). Uncertainty and Hyperbolic Discounting, American Economic Review, 95(4), pp. 1290-1299.
Dovre Group (2010). Guide No. X. Systematic uncertainty – conversion to a risk-adjusted rate.
Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, pp. 3-56.
Norwegian Ministry of Finance (2005). Cost-benefit analysis guide.
Norwegian Ministry of Finance (2005b). Circular R-109/2005.
Norwegian Ministry of Finance (2011). Proposition 1 (2011-2012) to the Storting for the Ministry of Finance.
Danish Ministry of Finance (1999). Guide on the preparation of economic impact studies.
Frederick, S., G. Loewenstein and T. O'Donoghue (2002). Time Discounting and Time Preference: A Critical Review, Journal of Economic Literature, 40, pp. 351-401.
Freeman, M. and B. Groom (2012). Positively Gamma Discounting! Mimeo. Submitted to the Economic Journal, April 2012.
Gollier, C. (2002). Time horizon and the discount rate, Journal of Economic Theory, 107, pp. 463-473.
Gollier, C. (2008). Discounting with fat-tailed economic growth, Journal of Risk and Uncertainty, 37, pp. 171-186.
Gollier, C. (2011). On the under-estimation of the precautionary effect in discounting. CESifo working paper No. 3536, July 2011.
Harrison (2010). Valuing the Future: the social discount rate in cost-benefit analysis. Visiting Research Paper, April 2010. Australian Government. Productivity Commission.
Harstad, B. (2012). Investment Policy for Time-Inconsistent Discounters. Mimeo., presented to the Committee’s conference on the social discount rate in Bergen, 24-25 May 2012. Available at http://www.sv.uio.no/econ/personer/vit/baardnh/dokumenter/discounting.pdf.
HEATCO (2006). Deliverable 5. Proposal for Harmonised Guidelines. Report, Developing Harmonised European Approaches for Transport Costing and Project Assessment.
Hepburn, C., Koundouri, P., Panopoulou, E. and Pantelidis, T. (2009). Social discounting under uncertainty: a cross country comparison. Journal of Environmental Economics and Management 57, pp. 140-150.
HM Treasury (2003). Appraisal and Evaluation in Central Government (The Green Book), HMSO, London.
Jegadeesh, N. and S. Titman (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48, pp. 65-91.
Johnsen, T. (1996). Required rate of return in the evaluation of the profitability of central government-owned commercial enterprises. Institute for Research in Economics and Business Administration (“SNF”) report 90/96.
Laibson, D. (1997). Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, May 1997, pp. 443-477.
Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics. 47:1, pp. 13–37.
Report No. 17 (2011-2012) to the Storting. The Management of the Government Pension Fund in 2011.
Merton, R.C. (1969), Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics (51), pp. 247-257.
Minken, H. (2005): Cost-benefit analysis in the transportation sector: The risk premium in the discount rate. Institute of Transport Economics (“TØI”) report 796/2005.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica 34: pp. 768-783.
Mossin, J. (1968). Optimal multiperiod portfolio policies, Journal of Business, pp. 215-229.
Newell, R. G. and W. A. Pizer (2003). Discounting the distant future: How much do uncertain rates increase valuations? Journal of Environmental Economics and Management. (46) pp. 52-71.
Nordhaus, W. (2007a). The Stern Review on the Economics of Climate Change.
Nordhaus, W. (2007b). The Challenge of Global Warming. Economic Models and Environmental Policy.
NOU 1997: 27 Green Paper. Cost-benefit analysis.
NOU 2009: 16 Green Paper. Global environmental challenges – Norwegian policy.
Samuelson, P.A. (1969). Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics (51), pp. 239-246.
Sharpe, W, F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance. 19:3, pp. 425–442.
Stern, N. (2007). Stern Review on the Economics of Climate Change. HM Treasury, United Kingdom.
Swedish Transport Administration (2012). Economic principles and calculation values for the transportation sector: ASEK 5.
Vennemo, H (2011). Systematic uncertainty in practical cost-benefit analyses. Vista Analyse AS. Report 2011/25.
Weitzman, M. L. (1998). Why the Far-Distant Future Should Be Discounted at Its Lowest Possible Rate. Journal of Environmental Economics and Management, November 1998, 36(3), pp. 201-08.
Weitzman, M. L. (2001). Gamma Discounting, American Economic Review (91:1)
Weitzman, M. L. (2012). Rare Disasters, Tail-Hedged Investment, and Risk-Adjusted Discount Rates, NBER Working Paper No. 18496
Footnotes
With diminishing marginal utility, the elasticity of marginal utility of consumption is negative, whilst the Ramsey formula features the numerical value of such elasticity.
This elasticity is also discussed in Chapter 3 on distribution within one and the same generation.
Chapter 3 notes that it is not possible to find a method for estimating the “correct” marginal utility of consumption on the basis of economic theory alone, since neoclassical economic theory uses a utility concept that is not necessarily measurable.
See Box. 8.1 of the NOU 1997: 27 Green Paper for a formal presentation of such a stylised model.
The presentation of the general pricing model and the Capital Asset Pricing Model is, in the main, based on Cochrane (2005).
The optimisation problem may be solved for different sets of concave utility functions and return distributions. If utility is quadratic, no assumptions are required with regard to distribution characteristics because an investor with such risk preferences will under any circumstance be concerned about expectation and variance only. This is the utility function used in the classical derivation of the CAPM. The problem with quadratic utility is that the so-called absolute risk aversion increases with increasing wealth, which is not particularly realistic.
The other assumptions are that investors have constant relative risk aversion and that they live on capital income only. The last one hundred years have seen large growth in per capita consumption and wealth that have not been accompanied by any long-term trend in interest rates and risk premiums. This suggests that the risk aversion of investors is not particularly dependent on wealth. Campbell and Viceira (2003) argue, against this background, that one can and should as a main rule assume that investors have constant relative risk aversion. The introduction of labour income will, according to Campbell and Viceira (2003), under normal circumstances increase the optimal portion invested in equities.
Gollier (2008) shows, with regard to the time structure of the discount rate, that if preferences exhibit constant relative risk aversion, and consumption developments follow a discrete stochastic process in which consumption changes are stochastically independent and identically distributed, the time structure of the interest rate will be flat in the sense that the discount rate will be independent of the time horizon. If, on the other hand, consumption growth is positively correlated over time, consumption uncertainty will, from the perspective of the investment date, increase the further into the future such consumption takes place. This will imply a discount rate that declines over time. If the stochastic process behind consumption developments is subject to so-called mean reversion, thus implying that consumption reverts to a normal level as the result of the business cycle, the reduction in the discount rate over time will be significantly less pronounced.
See footnote 8.
It follows from the Report to the Storting that this concerns return figures computed as geometric means (growth rates).
The arithmetic mean, R_{A}, of the values a_{1},a_{2} and a_{3} er ⅓(a_{1} + a_{2}+a_{3}). The geometric mean, R_{G,}, of the same values is.
If the values a_{1},a_{2} and a_{3} are not identical, the geometric mean will be lower than the arithmetic mean. If successive return figures are statistically independent and identically distributed with a variance equal to σ^{ 2}, we find that R_{G} ≈ R_{A} – 0,5σ^{2}. See for example Johnsen (1996) for a more detailed discussion.
The assumptions include a volatility relating to the real return on equity investments of 15 percent and normally distributed returns (log-normal equity prices). Further assumptions are an average corporation tax rate abroad of 25 percent and a 50-percent equity portion, as well as a 5-percent borrowing interest rate for enterprises.
See for example Dovre Group (2010).
No specific reasons or sources are invoked in this respect in HM Treasury (2003).