---
title: "IRR with varying benefit and O&M; unrecovered balance"
subject: "Engineering Economics"
chapter: 3
key: "semester_6/engineering_economics:irr-500000-unrecovered-balance"
source: https://www.bctengineer.com/q/semester_6/engineering_economics/irr-500000-unrecovered-balance
site: BCT Engineering
---

# IRR with varying benefit and O&M; unrecovered balance

Engineering Economics · Chapter 3: Basic Methodologies of Engineering Economic Analysis

- Asked in: 2065 Chaitra, 10 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/engineering_economics/irr-500000-unrecovered-balance
- Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-3

## Question

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.

## 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.

## Answer

#### 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 |

```text
 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 |

```text
 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.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the net cash flow at year 5?
   A. 1,00,000
   B. 2,20,000
   C. 1,20,000
   D. 1,45,000
   Answer: B. Net annual CF 1,20,000 (145,000 - 25,000) plus salvage 1,00,000 = 2,20,000. (Key point: Net cash flow at year 5)

2. How is the IRR found here?
   A. Divide total inflow by investment
   B. Average the yearly benefits
   C. Use MARR directly
   D. Find i where PW of net cash flows is zero, interpolating between a positive and a negative PW
   Answer: D. IRR is the rate with PW = 0, located by trial rates then interpolation (8% gives +4,191, 10% gives -24,520). (Key point: Trial and error method)

3. What IRR does the answer get?
   A. 12.40%
   B. 6.00%
   C. 10.00%
   D. 8.28%
   Answer: D. Interpolating between 8% and 10% gives about 8.28%. (Key point: IRR result)

4. How is the unrecovered investment balance updated each year?
   A. B_t = CF_t(1+i)
   B. B_t = B_(t-1)(1+i) + CF_t
   C. B_t = B_(t-1)(1-i) - CF_t
   D. B_t = B_(t-1) + CF_t
   Answer: B. Balance grows with interest at IRR, then the year's cash flow is added. (Key point: Unrecovered balance formula)

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