- 2070 Bhadra · 6+5+5 marks
U/B diagram, BCR with PW, discounted payback
Short answer
Unrecovered balance becomes positive in year 4. Conventional B/C = 200,000/(100,000+75,816−6,209)=1.18 (acceptable). Discounted payback = 3.51 years > 3 years, so project fails payback test.
Initial Investment = Rs. 100,000; Salvage Value = 0; Annual O&M Cost = Rs. 20,000; Useful Life = 5 years; Annual Benefit = 60,000 at the end of first year, thereafter decreases by 4,000 each year for the remaining years. a) Draw U/B diagram. b) Evaluate conventional BCR using PW formulation. Take salvage value = 10,000. c) Evaluate Discounted Payback Period. Take standard (cut off) Payback Period = 3 years.
The MARR is not given; assumption: MARR = 10% per year.
Net cash flow
| Year | Benefit | O&M | Net CF |
|---|---|---|---|
| 0 | −1,00,000 | ||
| 1 | 60,000 | 20,000 | 40,000 |
| 2 | 56,000 | 20,000 | 36,000 |
| 3 | 52,000 | 20,000 | 32,000 |
| 4 | 48,000 | 20,000 | 28,000 |
| 5 | 44,000 | 20,000 | 24,000 |
a) Unrecovered balance (U/B) diagram (salvage = 0, i = 10%)
B_t = B_(t−1)(1.10) + net CF_t
| Year | B_(t−1) × 1.10 | Net CF | B_t |
|---|---|---|---|
| 1 | -110,000 | 40,000 | -70,000 |
| 2 | -77,000 | 36,000 | -41,000 |
| 3 | -45,100 | 32,000 | -13,100 |
| 4 | -14,410 | 28,000 | 13,590 |
| 5 | 14,949 | 24,000 | 38,949 |
Unrecovered balance (each # = about 4,000)
yr0: -100,000 |-#########################
yr1: -70,000 |-##################
yr2: -41,000 |-##########
yr3: -13,100 |-###
yr4: 13,590 |+###
yr5: 38,949 |+##########
The balance is negative until between years 3 and 4, where it crosses zero (about 3.5 years), and ends at +38,949 at year 5, which is the future worth of the project's surplus.
b) Conventional BCR using PW (S = 10,000)
- PW(B) = 60,000 (P/F,10%,1) + 56,000 (P/F,10%,2) + 52,000 (P/F,10%,3) + 48,000 (P/F,10%,4) + 44,000 (P/F,10%,5) = Rs 200,000 (check: 60,000 (P/A,10%,5) − 4,000 (P/G,10%,5) = 227,447 − 27,447)
- PW(O&M) = 20,000 × (P/A, 10%, 5) = 20,000 × 3.7908 = Rs 75,816
- PW(S) = 10,000 × (P/F, 10%, 5) = 10,000 × 0.6209 = Rs 6,209
B/C = PW(B) / [I + PW(O&M) − PW(S)] = 200,000 / (1,00,000 + 75,816 − 6,209) = 200,000 / 169,607 = 1.179
Since B/C > 1, the project is acceptable.
c) Discounted payback period (cut-off = 3 years, S = 0)
| Yr | Benefit | Net CF | (P/F,10%,t) | PW of net CF | Cumulative PW − 1,00,000 |
|---|---|---|---|---|---|
| 1 | 60,000 | 40,000 | 0.9091 | 36,364 | -63,636 |
| 2 | 56,000 | 36,000 | 0.8264 | 29,752 | -33,884 |
| 3 | 52,000 | 32,000 | 0.7513 | 24,042 | -9,842 |
| 4 | 48,000 | 28,000 | 0.6830 | 19,124 | 9,282 |
| 5 | 44,000 | 24,000 | 0.6209 | 14,902 | 24,184 |
After 3 years, the unrecovered amount is 9,842. Year 4 recovers PW = 19,124.
θ_d = 3 + 9,842 / 19,124 = 3.51 years
Answer: B/C ≈ 1.18 (> 1, accept); discounted payback = 3.51 years > 3 years, so the project fails the discounted payback test.
✓ Answer checked by Claude Opus 5.5 (high effort)
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