Avoiding simplistic assumptions in discounting cash flows for private decisions

School of Agricultural and Resource Economics, University of Western Australia

There is a great depth of knowledge of discounting and present value methods in the finance literature. The literature covers a diversity of complexities and difficulties that may arise in an investment analysis, including issues of tax, risk, capital gains, liquidity, and consistency. On the other hand, many applied economists are not familiar with this literature. Those who undertake economics degrees and learn about discounting and present value models in the context of public benefit:cost analysis (BCA) may not be exposed to these issues in private investment analysis. This knowledge gap is potentially serious if economists are employed in positions that involve giving advice on the desirability or otherwise of private investments. Some of the issues addressed in the finance literature also have relevance to analyses of public investments.

The aim of this chapter is to bring to the attention of applied economists a number of issues that are well known in finance circles. The issues presented here are not difficult to represent in numerical investment analyses, but they do require adjustments to the standard approaches and/or formulae of present value models, as included, for example, in microcomputer spreadsheet software. Specific methods to achieve this are presented.

A single chapter cannot cover the full range of non-standard issues that arise in private investment analysis. Robison and Barry’s 1996 text on the subject is 661 pages. Rather, a selection of issues will be addressed, and the reader encouraged to adopt attitudes that (a) the simple standard approach to present value calculation is not necessarily sufficient, and (b) the alternatives are not necessarily difficult. An application of discounting using non-standard assumptions is presented to demonstrate the potential impact on results.

Numerous texts include the standard formula for net present value (NPV)

where t is the time period, N is the total number of time periods, B is a benefit, C is a cost and r is the discount rate. The internal rate of return (IRR) is the discount rate at which NPV equals zero. Notwithstanding a number of problems in interpreting the IRR (e.g. Gould, 1972), the IRR of the existing planned investment strategy conveniently provides the discount rate for calculating NPVs of alternative or new investment options (Robison and Barry, 1996).

The assumptions that are inherently and unavoidably common to all present value calculations are few:

• Benefits and costs can be identified, predicted and quantified in financial terms.
• The appropriate discount rate for each period can be identified.

In practice, a number of additional specific assumptions are commonly made in applications of the discounting technique, including the following:

• The discount rate is the actual or implied rate of interest on a financial instrument, commonly a bank account.
• The discount rate is constant over time.
• Tax is not relevant.
• Risk is not relevant, or is included in the discount rate.
• Inflation rates on prices of inputs and outputs are identical and constant.
• Productivity growth over time is zero.

Apart from the obvious defense of simplicity, some of these assumptions might be justified by a lack of information to support any alternative (e.g. constancy of discount rate and inflation rates). On the other hand, if the assumptions are violated, then application of the simple discounting model will lead to the risk of sub-optimal investment decisions being made. To choose a single example from my own area of specialisation, in formulating long term management advice to farmers in Australia the following conditions would apply:

• Tax directly affects the appropriate discounting procedure because, for example, tax in Australia is paid on the nominal rate of interest earned, not the real rate.
• Historically, inflation rates for prices of farm inputs have been higher than for sale prices of outputs (e.g. Wonder and Fisher, 1990). It might reasonably be expected that this trend of declining real output prices will continue.
• Agriculture has well documented trends of rising yields and productivity over time (e.g. Alston and Pardey, 1996). Indeed, differences in productivity growth can be distinguished for different types of farm enterprises (e.g. crops versus livestock).

Although it is not difficult to include complexities like these in investment analyses, it is rarely done in practice. In the next section, methods by which one may do so are presented. Following this, a numerical example is presented to illustrate the consequences for management decisions and financial outcomes of using simple or more realistic assumptions.

There are numerous taxation systems around the world, and even within a country the taxation arrangements for individual firms are diverse. Here I will only illustrate a single case, aiming to illustrate that taxation can affect the appropriate procedure for discounting and the results achieved, and that including taxation in the calculations need not be difficult. Robison and Barry (1996) cover a wider range of taxation-related issues in present value analysis.

In most industrialised countries, tax on income and business profits is payable on the nominal level of interest earned, with no allowance for inflation. Given this arrangement, if one wishes to allow for taxation in an investment analysis, it is most practical to represent the cash flow in nominal terms. This departs from the common practice in benefit:cost analysis of representing all flows of benefits and costs in real terms. Conducting the analysis in nominal terms has the additional advantage of being easy to explain to non-economists. Explaining that maximising the NPV is equivalent to maximising the net final value (NFV) (analogous to maximising the final balance of a bank account) makes the concepts easy to understand and helps to make clear the rationale for the whole approach.

The usual procedure is to apply a discount factor to net benefits in each year, and sum the present values of those net benefits (as in equation 1). In principle, one may proceed similarly in this tax example, with the discount factor appropriately adjusted for tax. However, the correct calculation of tax payable depends on interest earned, which depends on the cumulative cash balance from the investment, and so the analyst will, as part of the calculation, need to represent a full cash balance (in nominal terms) for the investment over its full life. Having done this, the analyst may take a short cut to the calculation of net present value by taking the final cash balance (i.e. the NFV) and converting it to the NPV.

where T is the marginal tax rate (assumed constant for simplicity) and r_n is the nominal discount rate.

In summary, then, one may proceed as follows:

1. Represent a full cash balance (in nominal terms) for the investment over its full life
2. Calculate nominal interest earned (or paid) each year based on the cash balance
3. Calculate tax payable on net income, including interest
4. Calculate the final cash balance for the investment (the NFV).
5. Calculate the NPV using equation (2).

The inclusion of taxation payments also affects the calculation of an annuity (an equivalent constant annual amount) for the stream of net benefits. Let the function A(r, N, PV) be the standard function for calculating the annuity (as provided as standard in, for example, microcomputer spreadsheet programs). Then A_T, the nominal annuity in the case of taxation being payable on nominal interest, is calculated by equation (3).

For those used to working in real terms when applying discounting, the reintroduction of nominal benefits and costs into the process brings with it the need for care in converting between real and nominal discount rates. Note that the nominal discount rate r_n is calculated not as the real discount rate (r) plus the inflation rate (i) but rather as follows.

so

If tax is deducted on nominal interest, then the conversion from real to nominal discount rates is more complicated.

This conversion is necessary to meet the logical requirement that the NPV should be the same whether calculations are done in nominal or real terms.

The other conditions mentioned above that depart from common simple application of discounted cash flow analysis are trends in productivity and prices. These require no special adjustment to discounting formulae, but simply adjustments to the streams of benefits and costs. An example is provided in the next section.

Weed management by farmers has economic consequences that can persist over several years, so that the selection of a long-term weed management strategy requires a discounted cash-flow analysis. The long-term nature of weed management is exemplified in cases where there is a threat of the weeds developing herbicide resistance (Pannell and Zilberman 2001). A decision support system for farmers was developed to allow economic comparison of long-term weed management systems (Pannell et al. 2004). The decision support system incorporates a discounted cash flow analysis and this was used to illustrate the issues presented above.

In this case study, three weed management systems are compared. System 1 is based on continuous wheat cropping, system 2 on alternations between growing pasture for livestock feed one year and wheat cropping the next, and system 3 is again continuous wheat cropping but with a radical and costly weed control treatment ("green manuring") implemented in the first year. Each system has a range of additional weed control options included, selected from the options described by Pannell et al. (2004).

Table 1 shows the net profits per unit area (hectare) per year under the three systems. These results are presented in real terms and make the usual simplistic assumptions described earlier (no taxation, no productivity growth, constant real prices). Under system 1, profits increase over time, under system 2, profits decrease over time, and under system 3, profits are markedly negative in year 1, maximal in year 2 and then relatively stable around a moderately high level. Because of these differences in profit trends over time, the introduction of more realistic assumptions to the investment analysis will differently affect their relative performances.

Table 1. Annual profit per unit area (A$ ha^-1) for the three farming systems pre-tax, assuming no productivity growth and constant real prices.

Table 2 shows the results of investment analysis of these profit series. To calculate the results in nominal terms, the data in Table 1 were inflated at a constant annual rate of three per cent. Discount rates used for 5% real or 8.15% nominal (based on equation 4b).

Table 2. Investment analysis results for usual simple assumptions (A$ ha^-1)

For the assumptions used, system 2 has the highest NPV and annuity. Systems 1 and 3 perform similarly. The NPV is, of course, the same whether calculated in real or nominal terms, but the annuity is higher in nominal terms, since the nominal value of a constant real annuity rises over time. Loan repayments are often specified as annuities in nominal terms, so the less common nominal approach to calculating annuities in investment analysis is probably more readily understandable by non-specialist decision makers.

Table 3 shows the results of introducing more realistic assumptions to the analysis. The annuities are all calculated on a nominal basis. The first three sets of results are for changes to individual assumption, and the fourth combines the other three.

Table 3. Investment analysis results for alternative assumptions about the profit series (A$ ha^-1).

It is notable that the rankings of the systems vary markedly between the analyses conducted under different assumptions. Both Systems 2 and 3 vary between top ranking and bottom ranking depending on the assumptions. System 1 was lowest ranked in Table 2, but is middle ranked under most of the assumptions in Table 3. Under what, for this issue, is the most realistic analysis (with tax, increasing crop yields and decreasing real output prices), the relative performances of the three systems are substantially different from the results in Table 2.

The results of the case study illustrate an example where the simplistic assumptions commonly used in applied investment analysis would produce misleading results. The highest-ranked alternative in the simple analysis is revealed to be the lowest ranked alternative under a more realistic analysis. Of course no generalisable conclusion about the importance of the analytical approach can be drawn from this single case study, other than that it may be very important in particular cases. The analyst and the decision maker would not necessarily be aware that the results of a simple analysis were misleading, unless the results were compared to an analysis conducted under more realistic assumptions. Given the ease of including the sorts of realistic complexities discussed here, it seems important that they should be included in analyses where they are judged to be relevant.

Taxation is not considered in the calculation of NPV in social benefit:cost analysis, because tax payments are merely transfers of money, not losses to society as a whole. Therefore, the above discussion of adjustments for taxation is of limited relevance in social benefit:cost analysis. However the point about changing real prices and changing productivity can be highly relevant. Information on historical price and productivity trends, if available for relevant industries or sectors, can be very useful in contributing to the framing of assumptions for social BCAs.

The message of this chapter is short and simple. Investment and benefit:cost analysts should critically evaluate their assumptions, and should not shy from deviating from the most simple, convenient and usual assumptions where appropriate. In particular, trends in relative prices and industry productivity are often present over the long term and should be recognised in applied analyses of both public and private investments. This should be considered "best practice" and hopefully can become "common practice" amongst economists and financial analysts.

Alston, J.M. and Pardey, P.G. (1996). Making Science Pay: The Economics of Agricultural R&D Policy, AEI Press, Washington D.C.

Gould, J.R. (1972). On investment criteria for mutually exclusive projects, Economica 39(153): 70-77.

Pannell, D.J., Stewart, V., Bennett, A., Monjardino, M., Schmidt, C. and Powles, S.B. (2004). RIM: A Bioeconomic Model for Integrated Weed Management of Lolium rigidum in Western Australia. Agricultural Systems (in press).

Pannell, D.J. and Zilberman, D. (2001). Economic and sociological factors affecting growers’ decision making on herbicide resistance. In: D.L. Shaner and S.B. Powles (eds.) Herbicide Resistance and World Grains, CRC Press, Boca Raton, pp. 251-277.

Robison, L.J. and Barry, P.J. (1996). Present Value Models and Investment Analysis, The Academic Page, Northport Alabama.

Schall, L.D. and Haley, C.W. (1983). Introduction to Financial Management, 3^rd edition, McGraw-Hill, New York.

Wonder, B. and Fisher, B. (1990). Agriculture in the economy, In: D.B. Williams (ed.) Agriculture in the Australian Economy, 3^rd edition, Sydney University Press, Sydney.

Citation: Pannell, D.J. (2004). Avoiding simplistic assumptions in discounting cash flows for private decisions, In: D. Pannell and S. Schilizzi (eds.), Discounting and Discount Rates in Theory and Practice, Edward Elgar, (forthcoming).  http://www.general.uwa.edu.au/u/dpannell/dp0405.htm

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Last revised: June 16, 2013.