---
title: "Monthly deposits, quarterly compounding, balance at year 8"
subject: "Engineering Economics"
chapter: 2
key: "semester_6/engineering_economics:equal-monthly-deposits-balance-8-years"
source: https://www.bctengineer.com/q/semester_6/engineering_economics/equal-monthly-deposits-balance-8-years
site: BCT Engineering
---

# Monthly deposits, quarterly compounding, balance at year 8

Engineering Economics · Chapter 2: Interest and Time Value of Money

- Asked in: 2076 Shrawan, 3 marks
- Video: [Compounding More Frequent than Payments - Engineering Economics Lightboard](https://www.youtube.com/watch?v=kEcGEXlK5oY) — Engineering Economics Guy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/engineering_economics/equal-monthly-deposits-balance-8-years
- Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-2

## Question

If you make equal monthly deposits of Rs 2000 into the bank for five years, saving accounts that pays interest rate of 12% compounded quarterly. Find the balance at the end of 8 year?

## Short answer

Monthly rate from 12% compounded quarterly: (1.03)^(1/3)-1 = 0.99016%. FV after 5 years of Rs 2,000/month: Rs 1,62,824. Compound this for 3 years at 3% quarterly: F8 = 1,62,824 × (1.03)^12 = Rs 2,32,148.

## Answer

#### Approach
Deposits are monthly but interest is compounded quarterly (12% nominal). Convert to an equivalent monthly rate, find the amount after 5 years, then compound it for 3 more years.

- Quarterly rate = 12%/4 = 3%
- Monthly rate: i_m = (1.03)^(1/3) − 1 = **0.99016% per month**

```text
 2000 each month (60 deposits)
 |------------------|---------------|
 0                  5 yr            8 yr
                    F5 = ?          F8 = ?
```

#### Step 1: Amount at end of year 5
(F/A, 0.99016%, 60) = [(1.03)²⁰ − 1] / 0.0099016 = (1.80611 − 1)/0.0099016 = 81.41
F5 = 2,000 × 81.41 = Rs 162,824

#### Step 2: Compound for 3 years (12 quarters)
F8 = F5 (F/P, 3%, 12) = 162,824 × 1.42576 = **Rs 232,148**

**Answer: Balance at the end of year 8 = about Rs 232,150.**

## Check yourself

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

1. What monthly rate is used for 12% compounded quarterly?
   A. 1%
   B. 3%
   C. (1.03)^(1/3) - 1 = 0.99016%
   D. 12/12 = 1%
   Answer: C. Equivalent monthly rate from the quarterly 3%. (Key point: Monthly rate)

2. What is computed first?
   A. 2,000 x 96 months
   B. Amount at year 8 directly with the annuity factor
   C. Present worth at year 0
   D. Amount at year 5: 2,000 (F/A, 0.99016%, 60) = Rs 162,824
   Answer: D. Deposits run for 5 years only. (Key point: Step 1)

3. How is the balance carried from year 5 to 8?
   A. F8 = F5 (F/P, 3%, 12)
   B. F5 (F/P, 3%, 3)
   C. F5 (F/A, 3%, 12)
   D. F5 + 2,000 x 36
   Answer: A. 3 years = 12 quarters at 3%. (Key point: Step 2)

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