Question

A solar energy system is to deliver 70% of an annual energy requirement of 300 GJ. The solar energy system equipment installed cost is $18,000. The equipment is to be paid for by a 20% down payment and 80% loan at 9%/year over 10 years. The market discount rate is 10%/year. The resale value is expected to be small. This is not an income producing facility. Other costs include: Insurance and O&M (operation and maintenance). These costs collectively are $2550 in the first year, and inflate at 6%/year. The owner’s federal income tax rate is 40% and the installation is in a state where there is no state income tax. Auxiliary energy supplied is 90 GJ/year, its initial cost is $8.80/GJ, and it is expected to rise at 10%/year. Compute the life cycle saving over a 15 year period.

Answer 1

The values provided in question are as follows :

1. Current (purchased) energy cost - $8.80 / GJ ( expected to increase at 10% / year)

2. Solar Energy Equipment

• Cost - $18,000
• Requirement meet - 70%
• Down Payment - 20%
• Loan rate - 9% per annum
• Loan period - 10 years
• Other Costs ( Insurance , O&M ) - $2,550 (Inflation at 6% p.a.)
• Income Tax rate - 40%

Present Value of 15 year costs, when energy is purchased from market :

• First Year Post Tax costs of Energy = GJ required x Cost per GJ x (1-Tax) = 300 GJ x $8.80 x (1-0.4) = $1,584
• PV of First year costs = $1,584/1.1 = $1,440
• Since the inflation rate as well as discount rate are same - 10 %, therefore the present value of the 15 year costs shall be = $1,440 * 15 = $21,600^#

Present Value of 15 year costs, when 70% energy needs are met by Solar System :

• Down Payment = $18,000 x 20% = $3,600 ...(A)

• Present Value of Insurance and O&M Costs  
  • First year post tax cost = $2,550 x (1-Tax) = $2,550 x (1-0.4) = $1,530
  • Present Value of 15 years = $ 16, 305 (Approx)    ...(B)

    Year	FV	PV
    1	1,530	1,390.91
    2	1,622	1,340.33
    3	1,719	1,291.59
    4	1,822	1,244.62
    5	1,932	1,199.37
    6	2,047	1,155.75
    7	2,170	1,113.72
    8	2,301	1,073.23
    9	2,439	1,034.20
    10	2,585	996.59
    11	2,740	960.35
    12	2,904	925.43
    13	3,079	891.78
    14	3,263	859.35
    15	3,459	828.10
    Sum of PVs	16,305.33

• Loan Re-Payment Present Value
  • Loan Amount = $18,000 x 70% = $14,400
  • Annual Payment = Loan Amount / PVAF (9%, 10 years) = $14,400 / 6.4177 = $2,244 (Approx)
  • Present Value of Loan Payments = Annual Installment x PVAF (10%,10 years) = $2,244 x 6.1446 = $13,788 (Approx) ...(C)
  • Loan Repayment Schedule

Year End	Loan Outstanding	Interest	Annual Installment	Principal Repaid
0	14,400	1,296	2244	948
1	13,452	1,211	2244	1,033
2	12,419	1,118	2244	1,126
3	11,292	1,016	2244	1,228
4	10,065	906	2244	1,338
5	8,726	785	2244	1,459
6	7,268	654	2244	1,590
7	5,678	511	2244	1,733
8	3,945	355	2244	1,889
9	2,059*	185	2244	2,059

• Present Value of Auxiliary Energy Costs
  • First year Post Tax Costs = GJ required x rate per GJ * (1-Tax) = 300 GJ x 30% x $8.80 * (1 - 0.4) = $475.2
  • PV of first year costs = $475.2/1.1 = $432
  • PV of 15 years costs = $432 x 15 = $6,480^#   ...(D)

• Present value of Tax Savings due to Depreciation & Interest = $5, 857 ...(E)

• Year	Depreciation	Interest	Total	Tax	PV of Tax
  1	1,200	1,296	2,496	998	908
  2	1,200	1,211	2,411	964	797
  3	1,200	1,118	2,318	927	697
  4	1,200	1,016	2,216	887	606
  5	1,200	906	2,106	842	523
  6	1,200	785	1,985	794	448
  7	1,200	654	1,854	742	381
  8	1,200	511	1,711	684	319
  9	1,200	355	1,555	622	264
  10	1,200	185	1,385	554	214
  11	1,200		1,200	480	168
  12	1,200		1,200	480	153
  13	1,200		1,200	480	139
  14	1,200		1,200	480	126
  15	1,200		1,200	480	115
  Sum	18,000	8,037	26,037	10,415	5,857

Total Present Value of Cost in Solar System Installation option (A+B+C+D-E) = $3,600 + $ 16, 305 + $13,788 + $6,480 - $5, 857 = $34,316

PV of Savings in Purchasing Energy from Market vs generating from Solar System for 15 years = $34,316 - $21,600 = $12,716

Answer = $12,716

Assumptions :

1. Loan is repaid in 10 equal annual installments made at the end of the year
2. Payments for Auxiliary Energy and other costs are made at end of the year

* The values of loan outstanding is adjusted to settle rounding off differences.

# - The figures are not discounted since the growth and dicsount rate are same i.e. 10%, nullifying each other's effect.