Discounting Constant Annual Costs or Benefits

A4.11 Table 3 contains annuity factors, which are convenient for discounting a series of constant annual costs or benefits. For instance, suppose a constant annual payment of £10,000 in real terms has to be paid every year from Year 1 to Year 20. The PV of this cost stream may be calculated by applying the annuity factor for 20 years, which is 14.2339 using a 3.5% discount rate. The PV in this example is £142,339.

A4.12 Annuity Factors may still be used where the constant stream of costs or benefits begins later than Year 1.

Example: What is the PV in the middle of Year 0 of 15 constant real annual costs of £75 starting in the middle of Year 7?

1) First, calculate the PV as if the costs started in Year 1. Using the annuity factor for 15 years and a 3.5% per annum discount rate, the PV is £865 ( = £75 x 11.5281).

2) However, given that the costs start in Year 7, this £865 figure represents the PV at the middle of Year 6.

3) To convert this to the PV at the middle of Year 0, apply the discount factor for Year 6, which is 0.8135.

4) The solution is thus £704 ( = £865 x 0.8135) or ( = £75 x 11.5281 x 0.8135).

TABLE 1: DISCOUNT FACTORS @ 3.5% P.A.

Year	Discount Factor	Year	Discount Factor
0	1.0000	23	0.4533
1	0.9662	24	0.4380
2	0.9335	25	0.4231
3	0.9019	26	0.4088
4	0.8714	27	0.3950
5	0.8420	28	0.3817
6	0.8135	29	0.3687
7	0.7860	30	0.3563
8	0.7594	40	0.2651
9	0.7337	50	0.1973
10	0.7089	60	0.1468
11	0.6849	75	0.0942
12	0.6618	80	0.0833
	0.6394	90	0.0651
14	0.6178	100	0.0508
15	0.5969	125	0.0274
16	0.5767	150	0.0167
17	0.5572	200	0.0062
18	0.5384	250	0.0029
19	0.5202	300	0.0014
20	0.5026	350	0.0009
21	0.4856	400	0.0005
22	0.4692	500	0.0002

TABLE 2: DISCOUNT FACTORS @ 1% TO 10% p.a.

Year	1.0%	2.0%	3.0%	3.5%	4.0%	5.0%	6.0%	7.0%	8.0%	9.0%	10%
0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
1	0.9901	0.9804	0.9709	0.9662	0.9615	0.9524	0.9434	0.9346	0.9259	0.9174	0.9091
2	0.9803	0.9612	0.9426	0.9335	0.9246	0.9070	0.8900	0.8734	0.8573	0.8417	0.8264
3	0.9706	0.9423	0.9151	0.9019	0.8890	0.8638	0.8396	0.8163	0.7938	0.7722	0.7513
4	0.9610	0.9238	0.8885	0.8714	0.8548	0.8227	0.7921	0.7629	0.7350	0.7084	0.6830
5	0.9515	0.9057	0.8626	0.8420	0.8219	0.7835	0.7473	0.7130	0.6806	0.6499	0.6209
6	0.9420	0.8880	0.8375	0.8135	0.7903	0.7462	0.7050	0.6663	0.6302	0.5963	0.5645
7	0.9327	0.8706	0.8131	0.7860	0.7599	0.7107	0.6651	0.6227	0.5835	0.5470	0.5132
8	0.9235	0.8535	0.7894	0.7594	0.7307	0.6768	0.6274	0.5820	0.5403	0.5019	0.4665
9	0.9143	0.8368	0.7664	0.7337	0.7026	0.6446	0.5919	0.5439	0.5002	0.4604	0.4241
10	0.9053	0.8203	0.7441	0.7089	0.6756	0.6139	0.5584	0.5083	0.4632	0.4224	0.3855
11	0.8963	0.8043	0.7224	0.6849	0.6496	0.5847	0.5268	0.4751	0.4289	0.3875	0.3505
12	0.8874	0.7885	0.7014	0.6618	0.6246	0.5568	0.4970	0.4440	0.3971	0.3555	0.3186
13	0.8787	0.7730	0.6810	0.6394	0.6006	0.5303	0.4688	0.4150	0.3677	0.3262	0.2897
14	0.8700	0.7579	0.6611	0.6178	0.5775	0.5051	0.4423	0.3878	0.3405	0.2992	0.2633
15	0.8613	0.7430	0.6419	0.5969	0.5553	0.4810	0.4173	0.3624	0.3152	0.2745	0.2394
16	0.8528	0.7284	0.6232	0.5767	0.5339	0.4581	0.3936	0.3387	0.2919	0.2519	0.2176
17	0.8444	0.7142	0.6050	0.5572	0.5134	0.4363	0.3714	0.3166	0.2703	0.2311	0.1978
18	0.8360	0.7002	0.5874	0.5384	0.4936	0.4155	0.3503	0.2959	0.2502	0.2120	0.1799
19	0.8277	0.6864	0.5703	0.5202	0.4746	0.3957	0.3305	0.2765	0.2317	0.1945	0.1635
20	0.8195	0.6730	0.5537	0.5026	0.4564	0.3769	0.3118	0.2584	0.2145	0.1784	0.1486
21	0.8114	0.6598	0.5375	0.4856	0.4388	0.3589	0.2942	0.2415	0.1987	0.1637	0.1351
22	0.8034	0.6468	0.5219	0.4692	0.4220	0.3418	0.2775	0.2257	0.1839	0.1502	0.1228
23	0.7954	0.6342	0.5067	0.4533	0.4057	0.3256	0.2618	0.2109	0.1703	0.1378	0.1117
24	0.7876	0.6217	0.4919	0.4380	0.3901	0.3101	0.2470	0.1971	0.1577	0.1264	0.1015
25	0.7798	0.6095	0.4776	0.4231	0.3751	0.2953	0.2330	0.1842	0.1460	0.1160	0.0923
26	0.7720	0.5976	0.4637	0.4088	0.3607	0.2812	0.2198	0.1722	0.1352	0.1064	0.0839
27	0.7644	0.5859	0.4502	0.3950	0.3468	0.2678	0.2074	0.1609	0.1252	0.0976	0.0763
28	0.7568	0.5744	0.4371	0.3817	0.3335	0.2551	0.1956	0.1504	0.1159	0.0895	0.0693
29	0.7493	0.5631	0.4243	0.3687	0.3207	0.2429	0.1846	0.1406	0.1073	0.0822	0.0630
30	0.7419	0.5521	0.4120	0.3563	0.3083	0.2314	0.1741	0.1314	0.0994	0.0754	0.0573

TABLE 3: EQUIVALENT ANNUAL COST & ANNUITY FACTORS @ 3.5% p.a.

Years	Equivalent Annual Cost of £1 p.a. ("EAC Factors")		Present Value of £1 p.a. ("Annuity Factors")
1	1.035		0.9662
2	0.5264		1.8998
3	0.3569		2.8019
4	0.2723		3.6735
5	0.2215		4.5158
6	0.1877		5.3296
7	0.1635		6.1162
8	0.1455		6.8762
9	0.1315		7.6107
10	0.1203		8.3206
11	0.1111		9.0066
12	0.1036		9.6696
13	0.0971		10.3013
14	0.0916		10.9296
15	0.0869		11.5281
16	0.0827		12.1067
17	0.0791		12.6659
18	0.0759		13.2064
19	0.0730		13.7289
20	0.0704		14.2339
21	0.0681		14.7221
22	0.0660		15.1940
23	0.0641		15.6502
24		0.0623	16.0913
25		0.0607	16.5176
26		0.0593	16.9298
27		0.0579	17.3283
28		0.0567	17.7136
29		0.0555	18.0861
30		0.0544	18.4462