Chapter six

Economic Criteria for Financial Decisions

Abstract

This chapter provides an explanation of the basic fundamental calculations involved in determining actual cost savings of solar energy alternative systems versus conventional systems. Discrete rate of return factors are tabularized over 30 years for interest rates from 0.5% to 10% for single payment and uniform payment series. Concepts of present and future worth of money are explained with examples. The use of Single Payment Compound Amount Factors, Single Payment Present Worth Factors, and Capital Recovery Factors are discussed with associated examples using discrete rate of return tables. Cash and loan repayment scenarios are addressed. State and Federal tax credits, rebates, and explanations of credits and deductions are provided with associated examples. The relationship between consumer price index and inflation are explained, and the energy inflation rate over a ten-year period is calculated as an example.

Keywords

Consumer price index; CRF (Capital Recovery Factor); Deduction; Discrete rate of return; Equity lines of credit; Equity loans; Future worth; Inflation; IRS; Loan repayment; Payback; Present worth; Rebates; Savings; Single payment; SPCAF (Single Payment Compound Amount Factor); SPPWF (Single Payment Present Worth Factor); Tax credit; Time value of money; Uniform payment series

This Chapter serves as a gateway between the technical and economic guidelines of this book. It connects the basic technical guidelines regarding energy relationships and solar DHW and PV systems presented in Chapters 2 through 5 with the economic guidelines and considerations relative to these alternative energy systems presented in Chapters 7 through 9. The use of Single Payment Compound Amount Factors, Single Payment Present Worth Factors, and Capital Recovery Factors are discussed with associated examples from actual system quotations, including cash and loan repayment scenarios. State and Federal tax credits, rebates, explanations of credits and deductions, equity loans, and the relationships between consumer price index and inflation are all factors that lead to a conceptual understanding of an economic analysis. Understanding the information in this Chapter will provide a supportive foundation in determining the actual payback, break-even costs, and savings associated with the systems discussed in Chapters 7, 8, and 9. Taking all these factors into consideration will determine actual cost savings by using solar DHW and PV systems versus conventional systems.

Before we analyze the investments and resulting payback of using solar as an alternative solution to either heating water or supplementing our electricity demands, let’s discuss a few of the basic financial criteria involved that support making those determinations. This chapter explains the basic fundamental calculations involved so you can determine the actual cost savings of a solar energy alternative system versus a conventional fossil fuel—based energy system. The discrete rate-of-return equations in this chapter may look ominous at first, but they are only multiplication factors. By simply multiplying the appropriate numbers contained in Table 6.1 under the applicable interest rates and periods, we can calculate actual payback times for particular loans as well as determine present and future worth of an initial investment. To accurately assess actual savings and payback period by using solar energy to heat domestic water or to provide electricity, the “time value of money” should be considered. Understanding the present and future worth of money is important to realizing the constantly increasing rates of inflation for the cost of energy. The basic economic discussions in this chapter, therefore, provide a more comprehensive interpretation of independently addressing costs and payback for solar domestic hot water (DHW) and solar photovoltaics (PV).

Table 6.1

Discrete Rate-of-Return Factors

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
0.5% (Interest Rate)
1	1.005	0.9950	1.0050
2	1.010	0.9901	0.5038
3	1.015	0.9851	0.3367
4	1.020	0.9802	0.2531
5	1.025	0.9754	0.2030
6	1.030	0.9705	0.1696
7	1.036	0.9657	0.1457
8	1.041	0.9609	0.1278
9	1.046	0.9561	0.1139
10	1.051	0.9513	0.1028
11	1.056	0.9466	0.0937
12	1.062	0.9419	0.0861
13	1.067	0.9372	0.0796
14	1.072	0.9326	0.0741
15	1.078	0.9279	0.0694
16	1.083	0.9233	0.0652
17	1.088	0.9187	0.0615
18	1.094	0.9141	0.0582
19	1.099	0.9096	0.0553
20	1.105	0.9051	0.0527
21	1.110	0.9006	0.0503
22	1.116	0.8961	0.0481
23	1.122	0.8916	0.0461
24	1.127	0.8872	0.0443
25	1.133	0.8828	0.0427
26	1.138	0.8784	0.0411
27	1.144	0.8740	0.0397
28	1.150	0.8697	0.0384
29	1.156	0.8653	0.0371
30	1.161	0.8610	0.0360
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
0.75% (Interest Rate)
1	1.008	0.9926	1.0075
2	1.015	0.9852	0.5056
3	1.023	0.9778	0.3383
4	1.030	0.9706	0.2547
5	1.038	0.9633	0.2045
6	1.046	0.9562	0.1711
7	1.054	0.9490	0.1472
8	1.062	0.9420	0.1293
9	1.070	0.9350	0.1153
10	1.078	0.9280	0.1042
11	1.086	0.9211	0.0951
12	1.094	0.9142	0.0875
13	1.102	0.9074	0.0810
14	1.110	0.9007	0.0755
15	1.119	0.8940	0.0707
16	1.127	0.8873	0.0666
17	1.135	0.8807	0.0629
18	1.144	0.8742	0.0596
19	1.153	0.8676	0.0567
20	1.161	0.8612	0.0540
21	1.170	0.8548	0.0516
22	1.179	0.8484	0.0495
23	1.188	0.8421	0.0475
24	1.196	0.8358	0.0457
25	1.205	0.8296	0.0440
26	1.214	0.8234	0.0425
27	1.224	0.8173	0.0411
28	1.233	0.8112	0.0397
29	1.242	0.8052	0.0385
30	1.251	0.7992	0.0373
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
1% (Interest Rate)
1	1.010	0.9901	1.0100
2	1.020	0.9803	0.5075
3	1.030	0.9706	0.3400
4	1.041	0.9610	0.2563
5	1.051	0.9515	0.20604
6	1.062	0.9420	0.1725
7	1.072	0.9327	0.1486
8	1.083	0.9235	0.1307
9	1.094	0.9143	0.1167
10	1.105	0.9053	0.10558
11	1.116	0.8963	0.0965
12	1.127	0.8874	0.0888
13	1.138	0.8787	0.0824
14	1.149	0.8700	0.0769
15	1.161	0.8613	0.07212
16	1.173	0.8528	0.0679
17	1.184	0.8444	0.0643
18	1.196	0.8360	0.0610
19	1.208	0.8277	0.0581
20	1.220	0.8195	0.05542
21	1.232	0.8114	0.0530
22	1.245	0.8034	0.0509
23	1.257	0.7954	0.0489
24	1.270	0.7876	0.0471
25	1.282	0.7798	0.04541
26	1.295	0.7720	0.0439
27	1.308	0.7644	0.0424
28	1.321	0.7568	0.0411
29	1.335	0.7493	0.0399
30	1.348	0.7419	0.03875
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
1.25% (Interest Rate)
1	1.013	0.9877	1.0125
2	1.025	0.9755	0.5094
3	1.038	0.9634	0.3417
4	1.051	0.9515	0.2579
5	1.064	0.9398	0.2076
6	1.077	0.9282	0.1740
7	1.091	0.9167	0.1501
8	1.104	0.9054	0.1321
9	1.118	0.8942	0.1182
10	1.132	0.8832	0.1070
11	1.146	0.8723	0.0979
12	1.161	0.8615	0.0903
13	1.175	0.8509	0.0838
14	1.190	0.8404	0.0783
15	1.205	0.8300	0.0735
16	1.220	0.8197	0.0693
17	1.235	0.8096	0.0657
18	1.251	0.7996	0.0624
19	1.266	0.7898	0.0595
20	1.282	0.7800	0.0568
21	1.298	0.7704	0.0544
22	1.314	0.7609	0.0523
23	1.331	0.7515	0.0503
24	1.347	0.7422	0.0485
25	1.364	0.7330	0.0468
26	1.381	0.7240	0.0453
27	1.399	0.7150	0.0439
28	1.416	0.7062	0.0425
29	1.434	0.6975	0.0413
30	1.452	0.6889	0.0402
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
1.5% (Interest Rate)
1	1.015	0.9852	1.0150
2	1.030	0.9707	0.5113
3	1.046	0.9563	0.3434
4	1.061	0.9422	0.2594
5	1.077	0.9283	0.20909
6	1.093	0.9145	0.1755
7	1.110	0.9010	0.1516
8	1.126	0.8877	0.1336
9	1.143	0.8746	0.1196
10	1.161	0.8617	0.10843
11	1.178	0.8489	0.0993
12	1.196	0.8364	0.0917
13	1.214	0.8240	0.0852
14	1.232	0.8118	0.0797
15	1.250	0.7999	0.07494
16	1.269	0.7880	0.0708
17	1.288	0.7764	0.0671
18	1.307	0.7649	0.0638
19	1.327	0.7536	0.0609
20	1.347	0.7425	0.05825
21	1.367	0.7315	0.0559
22	1.388	0.7207	0.0537
23	1.408	0.7100	0.0517
24	1.430	0.6995	0.0499
25	1.451	0.6892	0.04826
26	1.473	0.6790	0.0467
27	1.495	0.6690	0.0453
28	1.517	0.6591	0.0440
29	1.540	0.6494	0.0428
30	1.563	0.6398	0.04164
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
1.75% (Interest Rate)
1	1.018	0.9828	1.0175
2	1.035	0.9659	0.5132
3	1.053	0.9493	0.3451
4	1.072	0.9330	0.2610
5	1.091	0.9169	0.2106
6	1.110	0.9011	0.1770
7	1.129	0.8856	0.1530
8	1.149	0.8704	0.1350
9	1.169	0.8554	0.1211
10	1.189	0.8407	0.1099
11	1.210	0.8263	0.1007
12	1.231	0.8121	0.0931
13	1.253	0.7981	0.0867
14	1.275	0.7844	0.0812
15	1.297	0.7709	0.0764
16	1.320	0.7576	0.0722
17	1.343	0.7446	0.0685
18	1.367	0.7318	0.0652
19	1.390	0.7192	0.0623
20	1.415	0.7068	0.0597
21	1.440	0.6947	0.0573
22	1.465	0.6827	0.0552
23	1.490	0.6710	0.0532
24	1.516	0.6594	0.0514
25	1.543	0.6481	0.0497
26	1.570	0.6369	0.0482
27	1.597	0.6260	0.0468
28	1.625	0.6152	0.0455
29	1.654	0.6046	0.0443
30	1.683	0.5942	0.0431
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
2% (Interest Rate)
1	1.020	0.9804	1.0200
2	1.040	0.9612	0.5151
3	1.061	0.9423	0.3468
4	1.082	0.9238	0.2626
5	1.104	0.9057	0.21216
6	1.126	0.8880	0.1785
7	1.149	0.8706	0.1545
8	1.172	0.8535	0.1365
9	1.195	0.8368	0.1225
10	1.219	0.8203	0.11133
11	1.243	0.8043	0.1022
12	1.268	0.7885	0.0946
13	1.294	0.7730	0.0881
14	1.319	0.7579	0.0826
15	1.346	0.7430	0.07783
16	1.373	0.7284	0.0737
17	1.400	0.7142	0.0700
18	1.428	0.7002	0.0667
19	1.457	0.6864	0.0638
20	1.486	0.6730	0.06116
21	1.516	0.6598	0.0588
22	1.546	0.6468	0.0566
23	1.577	0.6342	0.0547
24	1.608	0.6217	0.0529
25	1.641	0.6095	0.05122
26	1.673	0.5976	0.0497
27	1.707	0.5859	0.0483
28	1.741	0.5744	0.0470
29	1.776	0.5631	0.0458
30	1.811	0.5521	0.04465
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
2.5% (Interest Rate)
1	1.025	0.9756	1.0250
2	1.051	0.9518	0.5188
3	1.077	0.9286	0.3501
4	1.104	0.9060	0.2658
5	1.131	0.8839	0.21525
6	1.160	0.8623	0.1816
7	1.189	0.8413	0.1575
8	1.218	0.8207	0.1395
9	1.249	0.8007	0.1255
10	1.280	0.7812	0.11426
11	1.312	0.7621	0.1051
12	1.345	0.7436	0.0975
13	1.379	0.7254	0.0910
14	1.413	0.7077	0.0855
15	1.448	0.6905	0.08077
16	1.485	0.6736	0.0766
17	1.522	0.6572	0.0729
18	1.560	0.6412	0.0697
19	1.599	0.6255	0.0668
20	1.639	0.6103	0.06415
21	1.680	0.5954	0.0618
22	1.722	0.5809	0.0596
23	1.765	0.5667	0.0577
24	1.809	0.5529	0.0559
25	1.854	0.5394	0.05428
26	1.900	0.5262	0.0528
27	1.948	0.5134	0.0514
28	1.996	0.5009	0.0501
29	2.046	0.4887	0.0489
30	2.098	0.4767	0.04778
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
3% (Interest Rate)
1	1.030	0.9709	1.0300
2	1.061	0.9426	0.5226
3	1.093	0.9151	3535
4	1.126	0.8885	0.2690
5	1.159	0.8626	0.21835
6	1.194	0.8375	0.1846
7	1.230	8131	0.1605
8	1.267	0.7894	0.1425
9	1.305	0.7664	0.1284
10	1.344	0.7441	0.11723
11	1.384	0.7224	0.1081
12	1.426	0.7014	0.1005
13	1.469	0.6810	0.0940
14	1.513	0.6611	0.0885
15	1.558	0.6419	0.08377
16	1.605	0.6232	0.0796
17	1.653	0.6050	0.0760
18	1.702	0.5874	0.0727
19	1.754	0.5703	0.0698
20	1.806	0.5537	0.06722
21	1.860	0.5375	0.0649
22	1.916	0.5219	0.0627
23	1.974	0.5067	0.0608
24	2.033	0.4919	0.0590
25	2.094	0.4776	0.05743
26	2.157	0.4637	0.0559
27	2.221	0.4502	0.0546
28	2.288	0.4371	0.0533
29	2.357	0.4243	0.0521
30	2.427	0.4120	0.05102
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
3.5% (Interest Rate)
1	1.035	0.9662	1.0350
2	1.071	0.9335	0.5264
3	1.109	0.9019	0.3569
4	1.148	0.8714	0.2723
5	1.188	0.8420	0.22148
6	1.229	0.8135	0.1877
7	1.272	0.7860	0.1635
8	1.317	0.7594	0.1455
9	1.363	0.7337	0.1314
10	1.411	0.7089	0.12024
11	1.460	0.6849	0.1111
12	1.511	0.6618	0.1035
13	1.564	0.6394	0.0971
14	1.619	0.6178	0.0916
15	1.675	0.5969	0.08683
16	1.734	0.5767	0.0827
17	1.795	0.5572	0.0790
18	1.857	0.5384	0.0758
19	1.922	0.5202	0.0729
20	1.990	0.5026	0.07036
21	2.059	0.4856	0.0680
22	2.132	0.4692	0.0659
23	2.206	0.4533	0.0640
24	2.283	0.4380	0.0623
25	2.363	0.4231	0.06067
26	2.446	0.4088	0.0592
27	2.532	3950	0.0579
28	2.620	0.3817	0.0566
29	2.712	0.3687	0.0554
30	2.807	0.3563	0.05437
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
4% (Interest Rate)
1	1.040	0.9615	1.0400
2	1.082	0.9246	0.5302
3	1.125	0.8890	0.3603
4	1.170	0.8548	0.2755
5	1.217	0.8219	0.22463
6	1.265	0.7903	0.1908
7	1.316	0.7599	0.1666
8	1.369	0.7307	0.1485
9	1.423	0.7026	0.1345
10	1.480	0.6756	0.12329
11	1.539	0.6496	0.1141
12	1.601	0.6246	0.1066
13	1.665	0.6006	0.1001
14	1.732	0.5775	0.0947
15	1.801	0.5553	0.08994
16	1.873	0.5339	0.0858
17	1.948	0.5134	0.0822
18	2.026	0.4936	0.0790
19	2.107	0.4746	0.0761
20	2.191	0.4564	0.07358
21	2.279	0.4388	0.0713
22	2.370	0.4220	0.0692
23	2.465	0.4057	0.0673
24	2.563	0.3901	0.0656
25	2.666	0.3751	0.06401
26	2.772	0.3607	0.0626
27	2.883	0.3468	0.0612
28	2.999	0.3335	0.0600
29	3.119	0.3207	0.0589
30	3.243	0.3083	0.05783
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
4.5% (Interest Rate)
1	1.045	0.9569	1.0450
2	1.092	0.9157	0.5340
3	1.141	0.8763	0.3638
4	1.193	0.8386	0.2787
5	1.246	0.8025	0.22779
6	1.302	0.7679	0.1939
7	1.361	0.7348	0.1697
8	1.422	0.7032	0.1516
9	1.486	0.6729	0.1376
10	1.553	0.6439	0.12638
11	1.623	0.6162	0.1172
12	1.696	0.5897	0.1097
13	1.772	0.5643	0.1033
14	1.852	0.5400	0.0978
15	1.935	0.5167	0.09311
16	2.022	0.4945	0.0890
17	2.113	0.4732	0.0854
18	2.208	0.4528	0.0822
19	2.308	0.4333	0.0794
20	2.412	0.4146	0.07688
21	2.520	0.3968	0.0746
22	2.634	0.3797	0.0725
23	2.752	0.3634	0.0707
24	2.876	0.3477	0.0690
25	3.005	0.3327	0.06744
26	3.141	0.3184	0.0660
27	3.282	0.3047	0.0647
28	3.430	0.2916	0.0635
29	3.584	0.2790	0.0624
30	3.745	0.2670	0.06139
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
5% (Interest Rate)
1	1.0500	0.95238	1.0500
2	1.1025	0.90703	0.53780
3	1.1576	0.86384	0.36721
4	1.2155	0.82270	0.28201
5	1.2763	0.78353	0.23097
6	1.3401	0.74622	0.19702
7	1.4071	0.71068	0.17282
8	1.4775	0.67684	0.15472
9	1.5513	0.64461	0.14069
10	1.6289	0.61391	0.12950
11	1.7103	0.58468	0.12039
12	1.7959	0.55684	0.11283
13	1.8856	0.53032	0.10646
14	1.9799	0.50507	0.10102
15	2.0789	0.48102	0.09634
16	2.1829	0.45811	0.09227
17	2.2920	0.43630	0.08870
18	2.4066	0.41552	0.08555
19	2.5269	0.39573	0.08275
20	2.6533	0.37689	0.08024
21	2.7860	0.35894	0.07800
22	2.9253	0.34185	0.07597
23	3.0715	0.32557	0.07414
24	3.2251	0.31007	0.07247
25	3.3864	0.29530	0.07095
26	3.5557	0.28124	0.06956
27	3.7335	0.26785	0.06829
28	3.9201	0.25509	0.06712
29	4.1161	0.24295	0.06605
30	4.3219	0.23138	0.06505
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
6% (Interest Rate)
1	1.0600	0.94340	1.0600
2	1.1236	0.89000	0.54544
3	1.1910	0.83962	0.37411
4	1.2625	0.79209	0.28859
5	1.3382	0.74726	0.23740
6	1.4185	0.70496	0.20336
7	1.5036	0.66506	0.17914
8	1.5938	0.62741	0.16104
9	1.6895	0.59190	0.14702
10	1.7908	0.55839	0.13587
11	1.8983	0.52679	0.12679
12	2.0122	0.49697	0.11928
13	2.1329	0.46884	0.11296
14	2.2609	0.44230	0.10758
15	2.3966	0.41727	0.10296
16	2.5404	0.39365	0.09895
17	2.6928	0.37136	0.09545
18	2.8543	0.35034	0.09236
19	3.0256	0.33051	0.08962
20	3.2071	0.31180	0.08719
21	3.3996	0.29416	0.08501
22	3.6035	0.27751	0.08304
23	3.8197	0.26180	0.08128
24	4.0489	0.24698	0.07968
25	4.2919	0.23300	0.07823
26	4.5494	0.21981	0.07690
27	4.8223	0.20737	0.07570
28	5.1117	0.19563	0.07459
29	5.4184	0.18456	0.07358
30	5.7435	0.17411	0.07265
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
7% (Interest Rate)
1	1.0700	0.93458	1.0700
2	1.1449	0.87344	0.55309
3	1.2250	0.81630	0.38105
4	1.3108	0.76290	0.29523
5	1.4026	0.71299	0.24389
6	1.5007	0.66634	0.20980
7	1.6058	0.62275	0.18555
8	1.7182	0.58201	0.16747
9	1.8385	0.54393	0.15349
10	1.9672	0.50835	0.14238
11	2.1049	0.46509	0.13336
12	2.2522	0.44401	0.12590
13	2.4198	0.41496	0.11965
14	2.5785	0.38782	0.11434
15	2.7590	0.36245	0.10979
16	2.9522	0.33873	0.10586
17	3.1588	0.31657	0.10243
18	3.3799	0.29586	0.09941
19	3.6165	0.27651	0.09675
20	3.8697	0.25842	0.09439
21	4.1406	0.24151	0.09229
22	4.4304	0.22571	0.09041
23	4.7405	0.21095	0.08871
24	5.0724	0.19715	0.08719
25	5.4274	0.18425	0.08581
26	5.8074	0.17220	0.08456
27	6.2139	0.16093	0.08343
28	6.6488	0.15040	0.08239
29	7.1143	0.14056	0.08145
30	7.6123	0.13137	0.08059
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
8% (Interest Rate)
1	1.0800	0.92593	1.0800
2	1.1664	0.85734	0.56077
3	1.2597	0.79383	0.38803
4	1.3605	0.73503	0.30192
5	1.4693	0.68058	0.25046
6	1.5869	0.63017	0.21632
7	1.7138	0.58349	0.19207
8	1.8509	0.54027	0.17401
9	1.9990	0.50025	0.16008
10	2.1589	0.46319	0.14903
11	2.3316	0.42888	0.14008
12	2.5182	0.39711	0.13270
13	2.7196	0.36770	0.12652
14	2.9372	0.34046	0.12130
15	3.1722	0.31524	0.11683
16	3.4259	0.29189	0.11298
17	3.7000	0.27027	0.10963
18	3.9960	0.25025	0.10670
19	4.3157	0.23171	0.10413
20	4.6610	0.21455	0.10185
21	5.0338	0.19866	0.09983
22	5.4365	0.18394	0.09803
23	5.8715	0.17032	0.09642
24	6.3412	0.15770	0.09498
25	6.8485	0.14602	0.09368
26	7.3964	0.13520	0.09251
27	7.9881	0.12519	0.09145
28	8.6271	0.11591	0.09049
29	9.3173	0.10733	0.08962
30	10.063	0.09938	0.08883
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
9% (Interest Rate)
1	1.0900	0.91743	1.0900
2	1.1881	0.84168	0.56847
3	1.2950	0.77218	0.39505
4	1.4116	0.70843	0.30867
5	1.5386	0.64993	0.25709
6	1.6771	0.59627	0.22292
7	1.8280	0.54703	0.19869
8	1.9926	0.50187	0.18067
9	2.1719	0.46043	0.16680
10	2.3674	0.42241	0.15582
11	2.5804	0.38753	0.14695
12	2.8127	0.35553	0.13965
13	3.0658	0.32618	0.13357
14	3.3417	0.29925	0.12843
15	3.6425	0.27454	0.12406
16	3.9703	0.25187	0.12030
17	4.3276	0.23107	0.11705
18	4.7171	0.21199	0.11421
19	5.1417	0.19449	0.11173
20	5.6044	0.17843	0.10955
21	6.1088	0.16370	0.10762
22	6.6586	0.15018	0.10590
23	7.2579	0.13778	0.10438
24	7.9111	0.12640	0.10309
25	8.6231	0.11597	0.10181
26	9.3992	0.10639	0.10072
27	10.245	0.09761	0.09974
28	11.167	0.08955	0.09885
29	12.172	0.08216	0.09806
30	13.268	0.07537	0.09734
Table Continued

Years	Single Payment		Uniform Payment Series
(n)	SPCAF	SPPWF	CRF
10% (Interest Rate)
1	1.1000	0.90909	1.1000
2	1.2100	0.82645	0.57619
3	1.3310	0.75131	0.40211
4	1.4641	0.68301	31,547
5	1.6105	0.62092	0.26380
6	1.7716	0.56447	0.22961
7	1.9487	0.51316	0.20541
8	2.1436	0.46651	0.18744
9	2.3579	0.42410	0.17364
10	2.5937	0.38554	0.16275
11	2.8531	0.35049	0.15396
12	3.1384	0.31863	0.14676
13	3.4523	0.28966	0.14078
14	3.7975	0.26333	0.13575
15	4.1772	0.23939	0.13147
16	4.5950	0.21763	0.12782
17	5.0545	0.19784	0.12466
18	5.5599	0.17986	0.12193
19	6.1159	0.16351	0.11955
20	6.7275	0.14864	0.11746
21	7.4003	0.13513	0.11562
22	8.1403	0.12285	0.11401
23	8.9543	0.11168	0.11257
24	9.8497	0.10153	0.11130
25	10.835	0.09230	0.11017
26	11.918	0.08391	0.10916
27	13.110	0.07628	0.10826
28	14.421	0.06934	0.10745
29	15.863	0.06304	0.10673
30	17.449	0.05731	0.10608

Note: SPCAF, Single Payment Compound Amount Factor; SPPWF, Single Payment Present Worth Factor; CRF, Capital Recovery Factor.

6.1. Future Worth of Money

Let’s illustrate this concept with an example. Consider a loan of $8000 (P), a present sum of money, to be paid back with one payment at the end of a 5-year period with an interest rate of 5% a year. The amount actually owed at the end of the first year is the original sum of $8000 plus the $400 interest cost for the use of capital, for a total of $8400. At the end of the second year, the amount owed is $8400 plus the 5% (for the use of capital in the amount of $420 of interest) for a total of $8820. This process of compounding continues as illustrated in Table 6.2 until the end of the 5-year loan period, at which time the original $8000 borrowed actually has cost $10,210.40, representing a combination of the principal and accrued interest.

Table 6.2

Cost of $8000 at 5% Compounded Interest

SPCAF from Table 6.1 (P) × (SPCAF)	Compounded Amount Due	Loan Term in Years
($8000) × (1.0500)	$8400.00	1
($8000) × (1.1025)	$8820.00	2
($8000) × (1.1576)	$9260.80	3
($8000) × (1.2155)	$9724.00	4
($8000) × (1.2763)	$10,210.40	5

The total interest paid of $2210.40 ($10,210.40 − $8000.00) is the rate of return on the money loaned. The lender can say that the “future worth” of the $8000 loaned at 5% over 5 years is $10,210.40. The computation of Table 6.2 can be determined expeditiously using Eqn (6.1).

S=P(1+i)n

(6.1)

Where:

S = a sum of money at a specified future date,

P = a present sum of money,

i = interest rate earned at the end of each period, and

n = the number of interest periods.

The “time value of money” can be displayed graphically as in Figure 6.1. At the end of the first period of time, the time value of P is P + Pi or P(1 + i); at the second interval of time, the time value of P is P(1 + i) + P(1 + i) or P (1 + i)². The sum S at the end of the nth period will result in Eqn (6.1). The factor (1 + i)ⁿ is called the single payment compound amount factor (SPCAF).

FIGURE 6.1 Time scales.

Equation 6.1 can then be rewritten as Eqn (6.2).

S=P(i−nSPCAF)

(6.2)

We now can quickly calculate the compounded amount due over 5 years from Table 6.2 with one computation using the discrete rate-of-return factors from Table 6.1. For example, if $8000 (P) is borrowed at 5% (i), over 5 years (n), the future worth (S), of the initial $8000 can be found using Eqn (6.2) and the SPCAF, which can be obtained from Table 6.1 under the applicable interest rate, as follows:

S=P(i−nSPCAF)

Where:

S = future worth of money,

P = $8000 (the amount of money to be borrowed),

(i − n SPCAF) = (0.5 − 5 SPCAF) = 1.2763, and

S = $8000 (1.2763) = $10,210.40.

6.2. Present Worth of Money

Because of inflation, future money is not as valuable as money at the present (especially if we keep printing paper money) and must be discounted by the factor 1/(1 + i)ⁿ, which is called the single payment present worth factor (SPPWF). Simply stated, the present worth of money is the inverse of Eqn (6.1) and can be written as Eqn (6.3).

P=S(1+i)n

(6.3)

Which can be rewritten as Eqn (6.4):

P=S(i−nSPPWF)

(6.4)

For example, assuming a 5% inflation interest rate, the time value of a future sum of $10,210 occurring 5 years from the initial investment can be found from Eqn (6.4) and the SPPWF from Table 6.1 under the applicable interest rate, as follows:

P=S(i−nSPPWF)

Where:

P = present worth of money,

S = $10,210 (future sum),

(i − n SPPWF) = (0.05 − 5 SPPWF) = 0.78353 (From Table 6.1), and

P = ($10,210) × (0.78353) = $8000.

6.3. The Capital Recovery Factor

It is always convenient to discuss economics in terms of cash. What if a person does not have the cash, however, to purchase a solar DHW or PV system? Is it still cost-effective to finance a solar energy system? To evaluate an actual investment, we need to determine what the total cost of the system will be if the money is borrowed and yearly or monthly payments are made. The future series of end-of-period payments that will just recover a sum “P” over “n” periods with compound interest is illustrated in Figure 6.2. End-of-period payments can be determined using Eqn (6.5).

FIGURE 6.2 Capital recovery scale.

The factor by which a present capital sum “P” is multiplied to find the future repayment series “R” that will exactly recover it with interest is called the capital recovery factor (CRF).

R=P(i−nCRF)

(6.5)

To illustrate the use of this factor, suppose $8000 (P) is borrowed for solar energy system equipment at 5% interest (i) compounded annually for a 5-year (n) term. The series of repayments for each year can be found using Eqn (6.5) and the CRF, which can be obtained from Table 6.1 under the applicable interest rate, as follows:

R=P(i−nCRF).

Where:

R = repayment made at the end of each year,

P = $8000 (present sum),

(i − n CRF) = (0.05 − 5 CRF) = 0.23097, and

R = ($8000) × (0.23097) = $1847.76.

This equates to a repayment of $153.98/month. Table 6.3 illustrates the cost of capital of $8000 at 5% interest with five end-of-year uniform payments for the recovery of capital.

Table 6.3 shows that the money on deposit at the beginning of each period (column 1) earns interest during that period (column 2), and the payment at the end of the period (column 4) repays the interest plus some of the principal (column 6). For example, the unpaid principal at the beginning of year 3 is $5032.09, the interest earned that year at 5% is $251.60, and the payment at the end of that year of $1847.76, consists of $251.60 in interest and $1596 (rounded into whole dollars) in principal.

Table 6.3

Visualizing the Capital Recovery Factor

Year	(1)	(2)	(3) (1) + (2)	(4)	(5) (3) – (4)	(6) (4) – (2)
	Money Owed at Start of Year	Interest Owed at End of Year	Principal and Interest Owed at End of Year	Series of Repayments	Money Owed at End of Year after Repayment	Recovery Capital
1	$8000.00	$400.00	$8400.00	$1847.76	$6552.24	$1448
2	$6552.24	$327.61	$6879.85	$1847.76	$5032.09	$1520
3	$5032.09	$251.60	$5283.69	$1847.76	$3435.93	$1596
4	$3435.93	$171.80	$3607.73	$1847.76	$1759.97	$1676
5	$1759.97	$88.00	$1847.76	$1847.76	$0.00	$1760
					Total	$8000

The actual cost of the solar energy system equipment (in terms of present worth money at 5% interest compounded annually) is the sum of the initial amount of money borrowed plus the present worth of the interest due at the end of each repayment period. In this case, using Eqn (6.4) and the discrete rate-of-return factors from Table 6.1, the present worth of the money invested is shown in Table 6.4.

Table 6.4

Example: Actual Cost of a Solar Energy System − Borrowing $8000 at 5% Interest

(1)	(2)	(3)	(4) (2) × (3)
Number of Years	Interest Owed at End of Year	Single Payment Present Worth Factor (SPPWF)	Actual cost of a solar energy system
1	$400.00	0.95238	$380.95
2	$327.61	0.90703	$297.15
3	$251.60	0.86384	$217.34
4	$171.80	0.82270	$141.34
5	$88.00	0.78353	$68.95
Total present worth cost			$9105.73

Therefore, in a loan-repayment scenario of $8000 at 5% compounded interest with end-of-year repayments, the present worth of money invested for solar energy system equipment would be $9105.73 ($8000 + $1105.73). This is the actual amount of money the solar energy system will cost you considering present worth costs at an interest rate of 5%. A solar energy system should be considered as an investment. As such, if money is borrowed to install such a system, the payments may be worked out with a bank so that repayments are approximately the same as the conventional monthly utility bill for either heating water or supplying electrical energy. Home equity loans or home equity lines of credit are two options to consider if financing a solar energy system.

6.4. Solar Energy Tax Credits

The Energy Tax Act enacted back in November of 1979 originally was passed by Congress as part of the National Energy Act. The objective of that law was to shift from oil and gas supply toward energy conservation and the use of renewable energy sources through taxes and tax credits. The law gave an income tax credit to private residents who adopted solar, wind, or geothermal sources of energy. The credit was equal to 30% of the cost of the equipment up to a maximum of $2000, as well as 20% of costs greater than $2000, up to a maximum of $10,000. Since that time, there have been many changes for allowable alternative tax incentives.

Through 2016, there are federal tax credit incentives in the United States allowing a 30% tax credit with no limit toward a rebate allowance of the cost of either a solar DHW or PV system. In addition, if the tax credit exceeds the amount of federal tax withheld, you could carry that amount over to the following tax year as a continued tax credit. So be sure to check the availability of such tax credits before purchasing a solar energy system to determine your true investment costs. Remember that similar tax credits were introduced back in the early 1980s and then ultimately removed. So it is not unprecedented that such federal incentives could be removed once again. The federal income tax credit can be obtained by filing Internal Revenue Service (IRS) Form 5695 with the yearly personal income tax IRS Form 1040. This credit is based on the cost of equipment and labor of a residential solar DHW or PV system.

Many states have introduced bills that continue to build momentum toward making solar energy more affordable for homeowners and businesses as well as create new jobs in the growing solar energy sector of the economy. Such laws include tax credits for the lease of solar equipment and power purchase agreements, statewide sales tax exemptions, income tax credits, and real property tax abatements for solar installations. Just like federal tax laws, however, such incentives are also subject to change. Instead of attempting to list the many variations in this book, it is recommended you browse the Internet for the most current information under topics, such as “state solar tax rebates” (e.g., www.dsireusa.org). It is important to understand that some state tax rebates may have installation requirements to be eligible for a tax rebate. Solar DHW installations may require the installer or dealer to have a master plumber’s license, a master oil burner technician, or a propane and natural gas technician. PV installations may require the installer or dealer to have a master’s electrician’s license and be certified by the North American Board of Certified Energy Practitioners (NABCEP) or working with someone who is NABCEP certified.

State income taxes also are credited as a percentage return of personal income taxes on a solar investment depending on each state legislation. Savings on income taxes can be realized through the interest paid on borrowed money as a tax deduction from Schedule A of the IRS Form 1040. In some states, increased property taxes may result by adding a solar energy system if the assessed value of the property is increased. In other states, this additional value is exempt from property taxes. The availability of these tax credits and property-assessment exclusions should be investigated to ensure an accurate determination of actual system costs.

There always seems to be a bit of confusion about the difference between tax credits and tax deductions. It is important to recognize and understand this difference. First, a tax credit is not a tax deduction. Deductions are subtracted from income and represent only a percentage of an actual dollar reduction. Credits are subtracted from taxes owed and are a true dollar-for-dollar savings. For example, if you are entitled to a $1000 tax credit, and you owe $100 for income taxes, then you would subtract the total credit from the tax owed, and find you do not owe any taxes that year. In fact, you would carry over an additional $900 as a credit to be applied to the next year’s taxes. Let’s illustrate the difference of tax deductions and credit with another example. Suppose you earn $40,000/year and assume you have a personal earned tax liability of 20% of the income. The total tax liability therefore would be $8000. If you installed solar equipment in the amount of $6000, a 30% tax credit would reduce the $8000 tax liability by $1800. Figures 6.3(a) and 6.3(b) illustrate the tax credit received after completing IRS Form 5695 for a solar energy system cost of $6000. (Note that tax forms may change slightly from year to year.) You then would have a tax liability of $6200 versus $8000. If this $1800 credit was taken incorrectly as a tax deduction, the earned income would be reduced from $40,000 to $38,200, which would result in a tax liability of $7640 versus $6200. As you can see from this simple example, it is important to understand that a solar tax credit results in a tax credit, not a tax deduction, and is worth a lot more in your pocket.

FIGURE 6.3 (a) Example of IRS Form 5695 for tax credit (sheet 1 of 2—tax year 2011) (b) example of IRS Form 5695 for tax credit (sheet 2 of 2—tax year 2011).

6.5. Home Equity Loans and Lines of Credit

Not everyone has the cash available to finance a solar energy system. If you do have the cash available, it makes it that much easier to justify the payback. If you have to borrow the money, then the amount you have to pay in interest will subjugate and diminish the amount saved, but you will need to know to what extent. We will briefly discuss a couple of ways to finance a system, and then in more detail, we will examine the investment savings for both a solar DHW system (Chapter 7) and a PV system (Chapter 8). You should, of course, discuss direct loans with your local banks to determine the best approach. You will find that even if you borrow money to purchase the solar energy system, at today’s low savings account interest rate at 0.5%, you actually will save more money by simply reducing your monthly energy expenditures by cost-effective alternative means.

A home equity line of credit allows you to draw funds from your bank up to a predetermined limit with an option to pay off as much of the line of credit used as you wish. A home equity line of credit normally carries a lower interest rate than a home equity loan, but its rate can fluctuate according to the prime rate. There are also normally no closing costs associated with establishing the line of credit, which is an additional cost factor to consider.

A home equity loan, on the other hand, provides you with a lump sum of money that has a fixed monthly payment over a predetermined period of time. You usually can choose between a variable interest rate of payment or a fixed rate, enabling you to budget a fixed monthly payment. For this type of loan, however, there are also closing costs to consider.

In both cases, the amount borrowed is based on such factors as the value of your home, your income, the remaining balance of your mortgage, and your credit history. It is the interest rates that makes these type of loans appealing because they are almost always lower than conventional bank loans because they are secured against the value of your home. In addition, the interest you pay on either type of loan is often tax deductible if you already meet the filing requirements of Schedule A and itemize deductions with your IRS Form 1040.

6.6. Inflation and Consumer Price Index

Inflation is a rising in the general level of prices of goods and services in an economy, or equivalently of a falling value of money. Inflation lessens the value of savings. As prices rise, the real value of purchasing power deteriorates. Savings accounts, annuities, and other fixed-value assets decline in real value. Inflation is not an easy thing to measure because prices for individual items do not rise evenly or proportionately. Various indexes have been devised to measure different aspects of inflation. The consumer price index (CPI) or the personal consumption expenditures index (PCE) tracks the prices consumers pay for things, and other indexes such as the producer price indexes (PPI) track prices the producers receive for goods and services they provide. The best measure of inflation for a given category depends on the intended use of the data. The CPI for all urban consumers is the most frequently reported statistic in the media and measures inflation as experienced by consumers in their daily living expenses for urban and metropolitan areas in the United States. It is therefore, normally, the better measurement for adjusting payments to consumer’s current purchases and comparing them with those same purchases in an earlier period.

The Bureau of Labor Statistics (BLS) prices everything consumers purchase. Data collectors survey thousands of retail and service establishments to collect price data on thousands of items. All those prices then are categorized and weighted based on the average amount that the consumer spends on those categories. The percentage change from month to month is the rate of inflation expressed as a percentage. Comparing one period’s price statistics against a previous month or previous year provides a crude measure of inflation (if the general level of prices has risen) or deflation (if the general level of prices has fallen). The overall inflation rate represents everything people spend money on, including things such as clothes, medical care, travel, haircuts, and food and energy. The food and energy categories are discarded, however, when calculating what is considered as the “core” inflation rate. Food and energy prices are excluded because they are historically highly volatile. The large changes in food and energy prices can occur because of supply disruptions, such as drought, and Organization of the Petroleum Exporting Countries cutbacks in oil production, respectively. An increase in the price of a single item, such an energy, therefore may cause a price index to rise. For this reason, many measures of core inflation have been developed from the basic price indexes, such as the CPI excluding food and energy and the CPI including only energy as shown in Table 6.5.

Table 6.5

Historical U.S. Inflation Rates from 2002 through 2012

Year	Yearly Average Consumer Price Index (CPI) for All Urban Consumers for All Items Less Food and Energy	Inflation Based on the CPI for All Urban Consumers for All Items Less Food and Energy1	Yearly Average CPI for All Urban Consumers for Energy	Inflation Based on the CPI for All Urban Consumers for Energy2
2002	190.44	2.32%	121.68	−5.77%
2003	193.23	1.46%	136.69	+12.33%
2004	196.64	1.77%	151.46	+10.80%
2005	200.87	2.15%	177.10	+16.93%
2006	205.92	2.51%	196.63	+11.03%
2007	210.73	2.34%	207.77	+5.66%
2008	215.57	2.30%	236.19	+13.68%
2009	219.24	1.70%	193.44	−18.10%
2010	221.34	0.96%	211.78	+9.48%
2011	225.01	1.66%	243.87	+15.16%
2012	229.75	2.11%	246.18	+0.95%

1 Developed from Data Source: FRED, Federal Reserve Economics Data, Federal Reserve Bank of St. Louis; Consumer Price Index for All Urban Consumers: All Items Less Food and Energy (CPILFESL); U.S. Department of Labor; Bureau of Labor Statistics; http://research.stlouisfed.org/fred2/series/CPILFESL; accessed June 10, 2013.

² Developed from Data Source: FRED, Federal Reserve Economics Data, Federal Reserve Bank of St. Louis; Consumer Price Index for All Urban Consumers: Energy (CPIENGSL); U.S. Department of Labor; Bureau of Labor Statistics; http://research.stlouisfed.org/fred2/series/CPIENGSL?rid=10&soid=22; accessed June 10, 2013.

Calculating the inflation rate using the CPI is relatively simple. The BLS surveys thousands of prices all over the country every month to generate a CPI. For instance, the average CPI for energy was 136.69 in 2003 and 151.46 in 2004. To calculate the inflation factor in 2004, subtract the previous year’s CPI (Y1) from the current year’s CPI (Y2) and then divide that number by the previous year’s CPI (Y1). The result is then multiplied by 100 to provide the result as a percentage. So we have the following equation:

Inflationforcurrentyear(2004)=Y2−Y1Y1×(100)

Where:

Y1 (previous year) = 136.69 (year 2003),

Y2 (current year) = 151.46 (year 2004),

Andinflationfor2004=151.46−136.69136.69×(100)=0.1080×(100)=10.8%

As an example of interpreting the effect of energy inflation rates shown in Table 6.5, the cost of energy in terms of 2013 prices is 72% higher than it was 10 years ago. If you add up the individual yearly inflation rates for energy shown in Table 6.5, you will find they equal an overall inflation rate of 72%. Therefore, over a 10-year period, the average energy inflation rate would be 7.2%/year. On the basis of this information and the continued volatility of rising energy prices, we will use a conservative energy inflation rate of 5% in the economic discussions of payback and break-even costs in Chapters 7, 8, and 9.