---
title: "IRR with investment balance diagram"
subject: "Engineering Economics"
chapter: 3
key: "semester_6/engineering_economics:irr-balance-diagram-800"
source: https://www.bctengineer.com/q/semester_6/engineering_economics/irr-balance-diagram-800
site: BCT Engineering
---

# IRR with investment balance diagram

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

- Asked in: 2072 Asoj, 6 marks
- Video: [Find IRR | Engineering economics | Unrecovered investment balance table and graphic form](https://www.youtube.com/watch?v=O5FoZWr4dQU) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/engineering_economics/irr-balance-diagram-800
- Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-3

## Question

Calculate IRR from the following cash flow and draw investment balance diagram.
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cash Flow | -800 | 250 | 300 | 400 | -150 | 600 |

## Short answer

IRR ≈ 20.9% is found by trial and error. The investment balance diagram shows the unrecovered investment remains negative throughout and reaches zero at year 5, confirming the project earns that rate on the outstanding balance.

## Answer

#### Cash flow
```text
 Year :  0     1     2     3     4     5
 CF   : -800  +250  +300  +400  -150  +600
```
There are three sign changes (− + + + − +), so more than one IRR is possible in theory; only one is positive and meaningful here (checked by scanning PW over 0 to 300%).

#### IRR by trial and error
PW(i) = −800 + 250(P/F,i,1) + 300(P/F,i,2) + 400(P/F,i,3) − 150(P/F,i,4) + 600(P/F,i,5)

| i | PW(i) |
|---|---|
| 10% | 245.83 |
| 20% | 16.94 |
| 21% | -1.34 |
| 22% | -18.95 |
| 20.93% | 0.00 |

Interpolation between 20% and 21%: IRR = 20 + 1 × 16.94/(16.94 − (-1.34)) ≈ 20.93%.

**IRR ≈ 20.93%**

#### Investment balance diagram
Unrecovered investment at end of year t: B_t = B_(t−1)(1 + i) + CF_t, with i = 20.93%. The balance must reach exactly zero at year 5.

| Year | B_(t−1)(1+i) | CF | B_t |
|---|---|---|---|
| 1 | -967.4 | +250 | -717.4 |
| 2 | -867.5 | +300 | -567.5 |
| 3 | -686.3 | +400 | -286.3 |
| 4 | -346.2 | -150 | -496.2 |
| 5 | -600.0 | +600 | 0.0 |

```text
 Unrecovered balance (each # = about 40)
 yr0:   -800.0 |-####################
 yr1:   -717.4 |-##################
 yr2:   -567.5 |-##############
 yr3:   -286.3 |-#######
 yr4:   -496.2 |-############
 yr5:     0.0 |0
```
The balance stays negative (investment not yet recovered) until the end of year 5, rising from −800 to −286.3 in year 3, falling back to −496.2 in year 4 because of the −150 outflow, and reaching 0 at year 5. This confirms the IRR of 20.93%: the project recovers the capital and earns exactly 20.93% on the unrecovered balance.

**Answer: IRR ≈ 20.9% (accept if MARR is lower).**

## Check yourself

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

1. What is the IRR found by interpolation?
   A. About 10%
   B. About 22%
   C. About 20.93%
   D. About 25%
   Answer: C. PW is +16.94 at 20% and -1.34 at 21%, giving 20.93%. (Key point: IRR)

2. How is unrecovered balance computed?
   A. B_t = B_(t-1) + CF_t
   B. B_t = B_(t-1)(1 + i) + CF_t at i = IRR
   C. B_t = CF_t / (1+i)
   D. B_t = B_(t-1)(1 - i)
   Answer: B. The balance grows at the IRR and is adjusted by each cash flow. (Key point: Balance recursion)

3. What should the balance be at year 5, and why?
   A. Positive, equal to profit
   B. Equal to -800
   C. Equal to 600
   D. Exactly zero, confirming the IRR
   Answer: D. At the IRR, capital is exactly recovered with return, so B5 = 0; the year-4 outflow of -150 pushes the balance back down. (Key point: Final balance)

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