- 2065 Chaitra · 10 marks
IRR with varying benefit and O&M; unrecovered balance
Short answer
IRR ≈ 8.28% found by trial-and-error interpolation between 8% and 10% PW values. Unrecovered investment balance decreases annually to zero at year 5, confirming IRR. Net cash flows: Rs 1.00, 1.05, 1.10, 1.15, 2.20 lakh with Rs 5 lakh initial investment.
Find IRR of the following project with initial investment of Rs 5,00,000 and Salvage value of Rs 1,00,000 at the end of 5 year. The annual benefit and operation and maintenance cost are as following.
| End of Year | Benefit | Operation and Maintenance |
|---|---|---|
| 1 | 105000 | 5000 |
| 2 | 115000 | 10000 |
| 3 | 125000 | 15000 |
| 4 | 135000 | 20000 |
| 5 | 145000 | 25000 |
| Draw unrecovered investment balance diagram also. |
Net cash flow
Net annual CF = benefit − O&M; salvage Rs 1,00,000 is added at year 5.
| Year | Benefit | O&M | Net | + Salvage | Net CF |
|---|---|---|---|---|---|
| 0 | −5,00,000 | ||||
| 1 | 1,05,000 | 5,000 | 1,00,000 | 1,00,000 | |
| 2 | 1,15,000 | 10,000 | 1,05,000 | 1,05,000 | |
| 3 | 1,25,000 | 15,000 | 1,10,000 | 1,10,000 | |
| 4 | 1,35,000 | 20,000 | 1,15,000 | 1,15,000 | |
| 5 | 1,45,000 | 25,000 | 1,20,000 | 1,00,000 | 2,20,000 |
Net: -5.00L +1.00L +1.05L +1.10L +1.15L +2.20L
yr : 0 1 2 3 4 5
(Equivalently the gradient is 5,000 per year on a base of 1,00,000, plus salvage.)
IRR by trial and error
PW(i) = −5,00,000 + 1,00,000(P/F,i,1) + 1,05,000(P/F,i,2) + 1,10,000(P/F,i,3) + 1,15,000(P/F,i,4) + 2,20,000(P/F,i,5)
| i | PW(i) |
|---|---|
| 6% | +35,635 |
| 8% | +4,191 |
| 10% | -24,520 |
Interpolating between 8% and 10%: IRR = 8 + 2 × 4191 / (4191 − (-24520)) = 8.29%; the exact root is 8.28%.
Unrecovered investment balance (i = 8.28%)
B_t = B_(t−1)(1 + i) + CF_t
| Year | B_(t−1)(1+i) | CF | B_t |
|---|---|---|---|
| 1 | -541,405 | 100,000 | -441,405 |
| 2 | -477,958 | 105,000 | -372,958 |
| 3 | -403,842 | 110,000 | -293,842 |
| 4 | -318,175 | 115,000 | -203,175 |
| 5 | -220,000 | 220,000 | 0 |
Unrecovered balance (each # = about 20,000)
yr0: -500,000 |-#########################
yr1: -441,405 |-######################
yr2: -372,958 |-###################
yr3: -293,842 |-###############
yr4: -203,175 |-##########
yr5: 0 |0
The unrecovered amount is reduced year by year and becomes exactly zero at year 5. This confirms that IRR = 8.28%: each year's net inflow first pays the interest (8.28% of the unrecovered balance) and the rest reduces the capital.
Answer: IRR ≈ 8.28%. (No MARR is given, so accept only if MARR is below this rate.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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