Appendix 2 - Discounted present value 1/

1/Adapted from - Guidelines: land evaluation for rainfed agriculture. Soils Bulletin 52. FAO, Rome, 1984.

In projects which require land improvements, it is necessary to incur capital expenditure in the first year or early years in return for benefits, in the form of increased production and profits, that will be received in future years. In irrigation schemes and many other agricultural projects, initial capital expenditure leads up to a steady state of increased production after a number of years. Cash flow discounting is a way of setting initial capital expenditure against future benefits or, more generally, of balancing costs incurred and benefits received at different periods in the future.

Money invested in the present earns interest, and acquires a higher value in future years. If the interest rate is 10%, $100 invested this year becomes $110 in one year's time, $121 in two years, or in the general case, 100 x (1 + r)ⁿ in n year's time, where r is the interest rate expressed as a fraction, i.e. 10% as 0.1. Thus the money value of expenditure incurred now increases in the future because the capital spent on a land improvement could alternatively have been placed in some interest-earning investment.

It would be possible to compare expenditure and benefits at different periods by adding compound interest and bringing all the values to some common date in the future. However, because the decision to invest is made now, it is better to carry out the process in reverse and bring all costs and benefits to their equivalents at the present time, called their present value. Discounting can be regarded as the reverse of addition of interest. Taking a discount rate r of 0.1 (10%), expenditure or cost of $100 in one year's time has a present value of 100/(1 + 0.1) = $90.9. The present value of $100 spent or received two years hence is 100/(1 + 0.1)² = $82.6; or another way of looking at this is to say that a foreseen expenditure of $100 in two years' time could be met by setting aside $82.6 now in an investment earning 10% compound interest. The discounting procedure is exactly the same whether dealing with a cost or a benefit. In the general case, a cost incurred or benefit received of $p in n year's time has a present value of:

The value 1/(1 + r)ⁿ is called the discount factor, used to multiply any actual cost or benefit to give its present value (Table B.1).

After an initial period, maintenance costs and benefits often even out to a steady amount each year. A short cut to the calculations is possible using tables of cumulative discount factors. For example, at a discount rate of 10%, $100 received in years 1 to 5 inclusive has a present value of 90.9 + 82.6 + 75.1 + 68.3 + 62.1 = $379. The cumulative discount factor is thus 3.79. To calculate the present value of a cost or benefit in years 5 to 20 inclusive, take the multiplier for 20 years and subtract that for 5 years (Table B.2).

The procedures are the same whether a commercial rate of interest (and thus discounting), currently of the order of 15% in many countries, is assumed or whether the calculation is done in terms of an assumed lower 'social' rate of interest.

The factors in Table B.2, Calculation of the Present Value of a Future Constant Annual Cost or Benefit in Years 1 to n Inclusive can also be adapted to the purpose of amortizing (spreading) an investment. The uniform periodic payment required is calculated by dividing the sum to be amortized by the factor appropriate to the number of years and the interest rate.

The uniform annual payment required to amortize $1 000 over 20 years at 10% interest is obtained by dividing 1 000 by the factor 8.51. The periodic payment is $117.51.

DISCOUNT FACTORS

Table B.1 CALCULATION OF THE PRESENT VALUE OF A FUTURE COST OR BENEFIT IN YEAR n

Year	1%	3%	5%	6%	8%	10%	12%	15%	20%
1	.990	.971	.952	.943	.926	.909	.893	.870	.833
2	.980	.943	.907	.890	.857	.826	.797	.756	.694
3	.971	.915	.864	.840	.794	.751	.712	.658	.579
4	.916	.888	.823	.763	.735	.683	.636	.572	.482
5	.951	.863	.784	.747	.681	.621	.567	.497	.402
6	.942	.837	.746	.705	.630	.564	.507	.432	.335
7	.933	.813	.711	.665	.583	.513	.452	.376	.279
8	.923	.789	.677	.627	.540	.467	.404	.327	.233
9	.914	.766	.645	.592	.500	.424	.361	.284	.194
10	.905	.744	.614	.558	.463	.386	.322	.247	.162
11	.896	.722	.585	.527	.429	.350	.287	.215	.135
12	.887	.701	.557	.497	.397	.319	.257	.187	.112
13	.879	.681	.530	.469	.368	.290	.229	.163	.093
14	.870	.661	.505	.442	.340	.263	.205	.141	.078
15	.861	.642	.481	.417	.315	.239	.183	.123	.065
20	.820	.554	.377	.312	,215	.149	.104	.061	.026
30	.742	.412	.231	.174	.099	.057	.033	.015	.004
40	.672	.307	.142	.097	.046	.022	.011	.004	.001
50	.608	.228	.087	.054	.021	.009	.003	.001	.000

Table B.2 CALCULATION OF THE PRESENT VALUE OF A FUTURE CONSTANT ANNUAL COST OR BENEFIT IN YEARS 1 TO n INCLUSIVE

Year	1%	3%	5%	6%	8%	10%	12%	15%	20%
1	0.99	0.97	0.95	0.94	0.93	0.91	0.89	0.87	0.83
2	1.97	1.91	1.86	1.83	1.78	1.74	1.69	1.63	1.53
3	2.94	2.83	2.72	2.62	2.58	2.49	2.40	2.28	2.11
4	3.90	3.72	3.54	3.46	3.31	3.17	3.04	2.85	2.59
5	4.85	4.58	4.33	4.21	3.99	3.79	3.61	3.35	2.99
6	5.80	5.42	5.08	4.92	4.62	4.36	4.11	3.78	3.33
7	6.73	6.23	5.79	5.58	5.21	4.87	4.56	4.16	3.60
8	7.65	7.02	6.46	6.20	5.75	5.33	4.97	4.49	3.84
9	8.57	7.79	7.11	6.80	6.25	5.76	5.33	4.77	4.03
10	9.47	8.53	7.72	7.36	6.71	6.14	5.65	5.02	4.19
12	11.26	9.95	8.86	8.38	7.54	6.81	6.19	5.42	4.44
15	13.87	11.94	10.38	9.71	8.56	7.61	6.81	5.85	4.68
20	18.05	14.88	12.46	11.47	9.82	8.51	7.47	6.26	4.87
30	25.81	19.60	15.37	13.76	11.26	9.43	8.06	6.57	4.98
40	32.84	23.12	17.16	15.05	11.92	9.78	8.24	6.64	5.00
50	39.20	25.73	18.26	15.76	12.23	9.91	8.30	6.66	5.00