Q25

2 marksVery Short AnswerSection B

A man takes a personal loan of Rs 2,00,000 at an interest rate of 15% p.a. compounded monthly, to be repaid by equal monthly instalments in 4 years. Calculate the EMI, using reducing balance method. [Given : $(1.0125)^{-48} = 0.55$ ]

Financial Mathematics

EMI by Reducing Balance Method

Official Answer

EMI is calculated using the reducing balance (present value of annuity) formula.

Given data:

Principal, P = Rs 2,00,000
Monthly rate, $i = 15\%/12 = 1.25\% = 0.0125$
Number of instalments, $n = 4 \times 12 = 48$
$(1.0125)^{-48} = 0.55$

Formula (reducing balance):

$\text{EMI} = P \times i / (1 - (1 + i)^{-n})$

Substitution:

$\text{EMI} = (2,00,000 \times 0.0125) / (1 - 0.55)$
$\text{EMI} = 2500 / 0.45$

Final answer:

EMI = Rs 5,555.56 (approximately)

EMIreducing balance methodpresent value of annuitymonthly ratepersonal loancompound interestinstalment0.0125

Marking Scheme

10.5 mark: correct monthly rate $i = 0.0125$ and $n = 48$ .
21 mark: correct use of the reducing-balance EMI formula $\text{EMI} = P \times i / (1 - (1+i)^{-n})$ with proper substitution.
30.5 mark: correct final EMI = Rs 5,555.56 (accept Rs 5,555.55 or Rs 5,556).

Hint

Convert 15% p.a. to a monthly rate (0.0125) and n to 48 months, then use $\text{EMI} = P \times i / (1 - (1+i)^{-n})$ .

Quick Oral Answer

In the reducing balance method interest is charged only on the outstanding principal, so $\text{EMI} = P \times i / (1 - (1+i)^{-n})$ ; here $2500/0.45$ gives an EMI of about Rs 5,555.56.

Analysis & Explanation

This is a direct application of the reducing balance method, which is the realistic way banks compute EMIs on loans.

Concept:

In the reducing balance method, interest each month is charged only on the outstanding (unpaid) principal, not on the original loan. This is captured by the present-value-of-annuity relation $P = \text{EMI} \times (1 - (1 + i)^{-n}) / i$ , which we rearrange to solve for EMI.

Key steps to watch:

The nominal rate 15% p.a. must be converted to a monthly rate by dividing by 12, giving $i = 0.0125$ .
The tenure of 4 years becomes $n = 48$ months, matching the monthly compounding.

Exam trap:

Students often forget to convert the annual rate to monthly, or use $n = 4$ instead of 48. Both give absurd EMIs.

Real-world link:

Every home, car, and personal loan EMI you see on a bank app is generated by exactly this formula. The reducing balance method is why paying an extra instalment early cuts total interest sharply.

Common Mistakes

1Not converting the annual rate 15% to the monthly rate 1.25% (0.0125) before substituting.
2Using $n = 4$ (years) instead of $n = 48$ (months), which does not match monthly compounding.
3Using the flat-rate EMI formula instead of the reducing-balance present-value formula.

Interesting Facts

The reducing-balance EMI formula is mathematically the present value of an annuity, first formalised in actuarial tables in the 17th century for pricing life annuities.

Under RBI norms, retail loans in India (home, auto, personal) must be quoted on a reducing-balance basis, so a 15% reducing rate costs far less than a 15% flat rate for the same tenure.

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Frequently Asked Questions

Why do we use $n = 48$ and not $n = 4$ in this problem?

Because interest is compounded monthly and repayment is monthly, every quantity must be expressed per month. Four years equals 48 months, so $n = 48$ and the monthly rate $i = 15\%/12 = 0.0125$ . Mixing an annual n with a monthly rate is a very common and costly mistake.

How is reducing balance different from the flat rate method?

In the flat rate method interest is charged on the full original principal for the whole tenure, so the EMI is fixed and higher. In the reducing balance method interest is charged only on the outstanding balance, which falls each month, so the effective cost is lower. This question specifically asks for the reducing balance EMI.