Time value of money: NPV, IRR and BCR
Compounding and discounting, present value, and the measures NPV, IRR, benefit-cost ratio and payback, with a worked farm investment appraisal.
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A fruit orchard costs money today and pays back over twenty years. A drip irrigation system, a cold store, a dairy shed or a watershed project all involve spending now for benefits later. A dollar received in five years is worth less than a dollar today, because today's dollar could be invested to earn interest, and because the future is uncertain. To compare costs and benefits that occur at different times, economists discount future values to the present. The resulting measures, net present value, internal rate of return and benefit–cost ratio, are used by farmers, banks and governments to decide which investments are worthwhile.
Project appraisal with discounting is a standard topic in farm management, agricultural finance and project planning exams.
Compounding and discounting
Compounding finds the future value of money invested now at interest rate r:
Discounting reverses this, finding the present value of a future amount:
The factor 1 ÷ (1 + r)n is the discount factor. At 10 per cent, 1,000 dollars received in 5 years has a present value of 1,000 ÷ 1.15 = 1,000 ÷ 1.6105 ≈ 621 dollars.
For a constant amount received every year for n years (an annuity), the present value is the amount multiplied by the annuity factor, the sum of the discount factors: about 3.791 for 5 years at 10 per cent.
The discount rate reflects the opportunity cost of capital, usually the interest rate on loans or the return from the best alternative use of money, sometimes with an allowance for risk. Governments often use a social discount rate set in national guidance.
Undiscounted measures
- Payback period: the number of years until cumulative net benefits repay the initial investment. Simple and useful for judging liquidity and risk, but it ignores benefits after payback and the timing of money.
- Simple rate of return: average annual profit ÷ investment.
Discounted measures
Net present value (NPV)
where Bt and Ct are benefits and costs in year t. Accept the project if NPV > 0: it earns more than the discount rate. NPV gives the absolute gain in present-value terms and is the most reliable criterion for choosing among projects.
Internal rate of return (IRR)
The discount rate at which NPV = 0. It is found by trial and error or by spreadsheet. Accept if IRR exceeds the cost of capital. IRR is easy to communicate as a percentage, but it can mislead when comparing projects of different sizes, and projects whose cash flows change sign more than once can have more than one IRR.
Benefit–cost ratio (BCR)
Accept if BCR > 1. It is widely used in public projects such as irrigation and watershed development. Its value depends on whether operating costs are deducted from benefits or added to costs, so the method should be stated.
All three criteria agree on whether a single project is acceptable: NPV > 0, IRR > r and BCR > 1 occur together.
Appraising a drip irrigation investment
A farmer can install drip irrigation for 10,000 dollars now. It is expected to raise gross income by 5,000 dollars a year for 5 years, with extra operating costs of 2,000 dollars a year, giving net benefits of 3,000 a year. The discount rate is 10 per cent, and the system has no salvage value.
Payback period: 10,000 ÷ 3,000 ≈ 3.3 years.
NPV: present value of net benefits = 3,000 × 3.791 = 11,372; NPV = 11,372 − 10,000 = 1,372 dollars. Positive, so worthwhile.
IRR: the rate at which 3,000 × annuity factor = 10,000, so the annuity factor must be 3.333. At 15 per cent the factor is 3.352; at 16 per cent, 3.274. Interpolating, IRR ≈ 15.2 per cent, above 10 per cent.
BCR: present value of gross benefits = 5,000 × 3.791 = 18,954; present value of costs = 10,000 + 2,000 × 3.791 = 17,582. BCR = 18,954 ÷ 17,582 ≈ 1.08.
Sensitivity: if benefits fall by 10 per cent (to 4,500 a year, net 2,500), NPV = 2,500 × 3.791 − 10,000 = −522: the investment becomes unattractive. A 10 per cent drop in gross income is enough to change the decision, so the farmer should check yield and price assumptions carefully, or seek a subsidy or cheaper system.
Practical points
- Use real values and a real discount rate (net of inflation), or nominal values with a nominal rate; do not mix them.
- Include all costs and benefits, including replacement of components, salvage value at the end, and working capital.
- For public projects, consider social costs and benefits, such as environmental effects and employment, and use shadow prices where market prices are distorted.
- Always run sensitivity analysis: change prices, yields, costs and the discount rate to see how robust the decision is.
Key points
- Money now is worth more than money later; FV = PV × (1 + r)n and PV = FV ÷ (1 + r)n.
- Payback period is simple but ignores timing and later benefits.
- NPV = discounted benefits − discounted costs; accept if positive.
- IRR is the discount rate that makes NPV zero; accept if above the cost of capital.
- BCR = PV of benefits ÷ PV of costs; accept if above 1.
- Sensitivity analysis tests how robust the decision is to errors in assumptions.
What the exams ask
- School-leaving and entrance papers: simple and compound interest, and the idea of present value.
- Degree and research-fellowship entrances: discount and annuity factors, NPV, IRR, BCR and payback calculations, and comparison of criteria.
- Professional and project examinations: appraising farm and public investments, social cost–benefit analysis, sensitivity and risk, and loan appraisal by banks.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Which discount rate should a farmer use?
Usually the interest rate on borrowed money, or the return the farmer could earn on savings or another investment. Riskier projects may warrant a higher rate.
Why does a higher discount rate make long-term projects look worse?
Benefits far in the future are divided by large discount factors, so they count for little. Projects with early costs and late benefits, such as tree crops, are especially sensitive.
Can NPV and IRR disagree?
When comparing mutually exclusive projects of different sizes or timings, they can rank them differently. NPV is generally preferred for choosing between them.
Sources
- Gittinger, J. P. Economic Analysis of Agricultural Projects, 2nd edition. Johns Hopkins University Press for the World Bank, 1982.
- Kay, R. D., Edwards, W. M. and Duffy, P. A. Farm Management, 9th edition. McGraw-Hill, 2020. Chapter on investment analysis.
- Boardman, A. E., Greenberg, D. H., Vining, A. R. and Weimer, D. L. Cost-Benefit Analysis: Concepts and Practice, 5th edition. Cambridge University Press, 2018.
- FAO. Economic and Financial Analysis of Investment Projects. FAO Investment Centre, EASYPol module, 2006.
- Brealey, R. A., Myers, S. C. and Allen, F. Principles of Corporate Finance, 13th edition. McGraw-Hill, 2020.
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