Equivalent Annual Cost and Annuity

EXERCISE 01

Machine A costs 450, lasts three years, and costs 200 per year to maintain. Machine B costs 600, lasts two years, and costs 150 per year. The discount rate is 10%; both can be replaced repeatedly.

A.Calculate equivalent annual costs and choose a machine.

A.

EAC=\frac{PV(\mathrm{costs})}{[1-(1+r)^{-n}]/r}

EAC_A=450\frac{0.10}{1-1.10^{-3}}+200

EAC_B=600\frac{0.10}{1-1.10^{-2}}+150

EAC(A) = 380.95; EAC(B) = 495.71. Choose A.

Equivalent Annual Cost and Annuity

EXERCISE 02

Choose between two ovens that provide the same production capacity. Replacement on equivalent terms is possible when each oven wears out. Use a 12% opportunity cost of capital.

Oven	Initial cost ($)	Annual cost ($)	Life
A	34,000	6,000	4 years
B	24,000	8,000	3 years

A.Calculate each equivalent annual cost.
B.Choose an oven and explain the comparison.

A.

Oven A

a_4=\frac{1-1.12^{-4}}{0.12}

PV_A=34{,}000+6{,}000a_4

EAC_A=34{,}000/a_4+6{,}000\approx17{,}193.97

Oven B

a_3=\frac{1-1.12^{-3}}{0.12}

PV_B=24{,}000+8{,}000a_3

EAC_B=24{,}000/a_3+8{,}000\approx17{,}992.38

EAC=\frac{PV(\mathrm{costs})}{[1-(1+r)^{-n}]/r}

Oven	PV of costs ($)	EAC ($/year)
A	52,224.10	17,193.97
B	43,214.65	17,992.38

B.

Choose A: its equivalent annual cost is lower.

Comparing the present values alone would favour the shorter-lived machine unfairly. EAC makes the service periods comparable under the replacement assumption.

Equivalent Annual Cost and Annuity

EXERCISE 03

A.When is equivalent annual value or cost useful instead of directly comparing project NPVs?

A.

EAA=NPV\frac{r}{1-(1+r)^{-n}}

It is useful for mutually exclusive alternatives with unequal lives when they can be repeated on equivalent terms and provide a comparable service. Convert each NPV into an equivalent annual amount using its own life and a consistent opportunity cost. For a one-off project, automatic repetition is not justified; compare relevant cash flows over the actual decision horizon.

Equivalent Annual Cost and Annuity

EXERCISE 04

Initial equipment investment is 48,000.00. The project lasts 5 years. The tax rate is 40.000%, required return 8.360% and terminal sale proceeds 6,500.00. Initial working capital is 6,000.00. Marketing costs are 5,000, 4,000, 3,000, 2,000 and 1,000 and are included in cash costs. A one-off operating-margin loss of 500 occurs at time 0. Assuming an immediately usable 40% tax deduction, its after-tax cash cost is 300. NWC funds 15% of next-year revenue from time 0.

Year	1	2	3	4	5
Revenue	40,000.00	44,000.00	48,400.00	53,240.00	58,564.00
Cash operating costs	29,000.00	30,400.00	32,040.00	33,944.00	36,138.40
Depreciation	8,000.00	8,000.00	8,000.00	8,000.00	8,000.00
Additional capital expenditure	0.00	0.00	0.00	0.00	0.00
Closing NWC	6,600.00	7,260.00	7,986.00	8,784.60	0.00

A.Build the incremental cash-flow schedule and calculate NPV.
B.Compare this repeatable project with a seven-year frame project having NPV 6,000.

A.

FCF_t=(R_t-C_t-Dep_t)(1-T)+Dep_t-\Delta NWC_t-Capex_t

S_{net}=S-T(S-BV)

Terminal book value = 8,000.00; after-tax disposal proceeds = 7,100.00.

FCF_0=-48{,}000-300-6{,}000=-54{,}300

OCF_1=(40{,}000-29{,}000-8{,}000)(1-0.4)+8{,}000=9{,}800

\Delta NWC_1=6{,}600-6{,}000=600

FCF_1=9{,}800-(600)-0=9{,}200

BV_N=48{,}000+0-40{,}000=8{,}000

S_{net}=6{,}500-0.4(6{,}500-8{,}000)=7{,}100

Year	1	2	3	4	5
EBIT	3,000.00	5,600.00	8,360.00	11,296.00	14,425.60
Operating tax	1,200.00	2,240.00	3,344.00	4,518.40	5,770.24
Operating cash flow	9,800.00	11,360.00	13,016.00	14,777.60	16,655.36
Change in NWC	600.00	660.00	726.00	798.60	-8,784.60
Discount factor	0.922850	0.851652	0.785947	0.725311	0.669353

Year	0	1	2	3	4	5
Free cash flow	-54,300.00	9,200.00	10,700.00	12,290.00	13,979.00	32,539.96
Discounted cash flow	-54,300.00	8,490.22	9,112.67	9,659.28	10,139.12	21,780.71

PV_t=\frac{FCF_t}{(1+0.0836)^t}

NPV=FCF_0+\sum_{t=1}^{5}PV_t\approx -54{,}300+59{,}182=4{,}882

NPV = 4,882.00. Accept.

B.

EAA=NPV\frac{r}{1-(1+r)^{-n}}

EAA_{gears}=4{,}882.000503\frac{0.0836}{1-1.0836^{-5}}

EAA_{frames}=6{,}000\frac{0.0836}{1-1.0836^{-7}}

Gears EAA = 1,234.35; frames EAA = 1,166.66.

Choose the larger EAA because the projects are mutually exclusive, repeatable, and have the same risk.

Equivalent Annual Cost and Annuity

EXERCISE 05

E-Power Alfa is evaluating a 5-year project. Equipment costs 2,500.00, annual straight-line depreciation is 500.00, and expected terminal sale proceeds are 400.00. The corporate tax rate is 30.00%. Operating working capital equals 5.00% of sales, is invested at each year-end and fully recovered in the last year. Tax losses can offset other taxable profits immediately. Exclude the 50 already spent on design. Debt is 30% of financing at 5%, equity costs 10%, giving WACC = 0.30 × 5% × 0.70 + 0.70 × 10% = 8.05%. Cannibalisation starts at 80 and grows 10% each year. Alternative repeatable project Beta has cash flows −2,200, 910, 1,010 and 1,040 at years 0–3.

Year	1	2	3	4	5
Revenue	1,000.00	1,200.00	1,440.00	1,728.00	2,073.60
Cash operating costs	400.00	480.00	576.00	691.20	829.44
Additional incremental costs	80.00	88.00	96.80	106.48	117.13

A.Calculate the discount rate and evaluate the project.
B.Calculate Alfa’s IRR and PI, then select between the repeatable alternatives.

A.

The appropriate discount rate is 8.05%. Exclude the 50 already spent on design. Debt is 30% of financing at 5%, equity costs 10%, giving WACC = 0.30 × 5% × 0.70 + 0.70 × 10% = 8.05%. Cannibalisation starts at 80 and grows 10% each year. Alternative repeatable project Beta has cash flows −2,200, 910, 1,010 and 1,040 at years 0–3.

FCF_t=EBIT_t(1-T_c)+Dep_t-\Delta NWC_t-CapEx_t

Year	1	2	3	4	5
EBIT	20.00	132.00	267.20	430.32	627.03
Tax	-6.00	-39.60	-80.16	-129.10	-188.11
Operating cash flow	514.00	592.40	687.04	801.22	938.92
Change in NWC	50.00	10.00	12.00	14.40	-86.40

CF_{disposal}=S-T_c(S-BV)

CF_{disposal}=400-0.3(400-(2500-500\times 5))=280.00

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3	4	5
Free cash flow	-2,500.00	464.00	582.40	675.04	786.82	1,305.32
Discount factor	1.0000	0.9255	0.8565	0.7927	0.7337	0.6790
Present value	-2,500.00	429.43	498.85	535.12	577.27	886.33

NPV=(-2{,}500)+\frac{464}{1.0805}+\frac{582.40}{1.0805^{2}}+\frac{675.04}{1.0805^{3}}+\frac{786.82}{1.0805^{4}}+\frac{1{,}305.32}{1.0805^{5}}

NPV = 427.00. Accept.

B.

0=\sum_{t=0}^5\frac{FCF_{A,t}}{(1+IRR)^t}

Alfa IRR = 13.3557%; PI = 1.1708.

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3
Free cash flow	-2,200.00	910.00	1,010.00	1,040.00
Discount factor	1.0000	0.9255	0.8565	0.7927
Present value	-2,200.00	842.20	865.11	824.44

NPV=(-2{,}200)+\frac{910}{1.0805}+\frac{1{,}010}{1.0805^{2}}+\frac{1{,}040}{1.0805^{3}}

NPV = 331.75. Accept.

EAA=\frac{NPV}{a_{n|r}}

EAA_A=\frac{427.004162}{a_{5|0.0805}}=107.09

EAA_B=\frac{331.753623}{a_{3|0.0805}}=128.85

Choose Beta for indefinite like-for-like repetition: its equivalent annual value is higher.