Question

A new renewable energy generation system has a life-time of 10 years. It is assumed that the salvage value of the system at the end of the 10 years' life-time is negligible. The initial (Present time) investment of the system costs 50,000 HK$. The manufacturer agrees to maintain the system every year over the 10 years' life-time with a uniform annual maintenance fee of 1000 HK$/year. At the end of the 5th year, a key component should be replaced, which will cost 10,000 HK$. Each year, this renewable energy system will generate 5,000 kWh electricity. It is assumed that the electricity price in the first year is 1 HK$/kWh, with an annual escalation rate of 3%. The interest rate is 5%.

Please calculate:

(1) The present value of the total maintenance fee over 10 years' life-time.

(2) The present value of the total saved electricity cost over 10 years' life-time.

(3) The net present value of this system, by considering all the cash flows mentioned in the question statement.

(4) If the replacement cost 10,000 HK$ of the key component at the end of the 5th year is accumulated by depositing a fixed amount of money per year in the first 5 years, how much should be this deposited money per year?

(5) Based on the above questions , please investigate that how the "interest rate" and its variation will affect the results of the above questions, and discuss and comment on their sensitivity. For example, the "interest rate" varies +/- 0~0.03 based on the original 0.05 (i.e. the interest rate varies in the range between 2%~8%).(*two page report is required)

Answer 1

Solution (1)	The present value of the total maintenance fee over 10 years' life-time.
		HK$
	maintenance fee	1000
	PV annuity factory for 10 year	7.7217
	Total present value of tatal maintenance fee	7722
Solution (2)	The present value of the total saved electricity cost over 10 years' life-time.
	= HK$ 45050 (working)

Working
Year	Annual electricity saved	Escalation Rate	Electricity rate	Annual Electricity saved	PV factor @ 5%	Present value of saved electricity cost
0		0	1		1.0000
1	5000	0.03	1.03	5150	0.9524	4905
2	5000	0.03	1.06	5305	0.9070	4811
3	5000	0.03	1.09	5464	0.8638	4720
4	5000	0.03	1.13	5628	0.8227	4630
5	5000	0.03	1.16	5796	0.7835	4542
6	5000	0.03	1.19	5970	0.7462	4455
7	5000	0.03	1.23	6149	0.7107	4370
8	5000	0.03	1.27	6334	0.6768	4287
9	5000	0.03	1.30	6524	0.6446	4205
10	5000	0.03	1.34	6720	0.6139	4125
					Total	45050

Solution (3)	The net present value of the system, by considering all the cash flows mentioned.
		HK$
	Initial investment	50000
	Present value of management fee	7722
	Present vlaue of key component replaced	7835
	(10000*.7835)
	Less: Present value of saved electricity cost	-45050
	Present value of the system (Cost)	20507
Solution (4)	Deposit amount
	where interest rate = 5%
	Annual amount of deposit
	=Replacement amount / Annuity factor of 5 year
	=10000/4.3295
	=2310
Solution (5)	sensititvity analysis - how the "interest rate" and its variation will affect the results?
	Where interest rate decreased by 3% i.e. new interest rate =2%
		HK$
	Initial investment	50000
	Present value of management fee (1000*8.9826)	7722
	Present vlaue of key component replaced	7835
	(10000*.9057)
	Less: Present value of saved electricity cost	-52777
	Present value of the system (Cost)	12780
	Decrease in cost =(20507-12780)*100/20507
	=37.68%

Working -1
Year	Annual electricity saved	Escalation Rate	Electricity rate	Annual Electricity saved	PV factor @ 2%	Present value of saved electricity cost
0		0	1		1.0000
1	5000	0.03	1.03	5150	0.9804	5049
2	5000	0.03	1.06	5305	0.9612	5099
3	5000	0.03	1.09	5464	0.9423	5149
4	5000	0.03	1.13	5628	0.9238	5199
5	5000	0.03	1.16	5796	0.9057	5250
6	5000	0.03	1.19	5970	0.8880	5301
7	5000	0.03	1.23	6149	0.8706	5353
8	5000	0.03	1.27	6334	0.8535	5406
9	5000	0.03	1.30	6524	0.8368	5459
10	5000	0.03	1.34	6720	0.8203	5512
					Total	52777

Where interest rate increase by 3% i.e. new interest rate =8%
	HK$
Initial investment	50000
Present value of management fee (1000*6.7101)	6710.1
Present vlaue of key component replaced	6806
(10000*.6806)
Less: Present value of saved electricity cost	-38883
Present value of the system (Cost)	24633
Increase in cost =(24633-20507)*100/20507
= 20.12%

Working - 2
Year	Annual electricity saved	Escalation Rate	Electricity rate	Annual Electricity saved	PV factor @ 8%	Present value of saved electricity cost
0		0	1		1.0000
1	5000	0.03	1.03	5150	0.9259	4769
2	5000	0.03	1.06	5305	0.8573	4548
3	5000	0.03	1.09	5464	0.7938	4337
4	5000	0.03	1.13	5628	0.7350	4136
5	5000	0.03	1.16	5796	0.6806	3945
6	5000	0.03	1.19	5970	0.6302	3762
7	5000	0.03	1.23	6149	0.5835	3588
8	5000	0.03	1.27	6334	0.5403	3422
9	5000	0.03	1.30	6524	0.5002	3264
10	5000	0.03	1.34	6720	0.4632	3112
					Total	38883

Comment

It can be seen from result due to change in interest rate that if we reduce the interest by 3%, resulted NPV of system is reduced by 37.68% where we increase the interest rate by 3%, the resulted NPV of system by 20.12%. Hence it has convex relationship between value of system and interest rate