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Quantitative Aptitude · Mathematics of Finance

Capital Budgeting: IRR, Payback and Loan Amortization

Updated 1 October 2026 · Fact-checked

Payback period is the time a project takes to recover its initial investment. IRR is the discount rate at which NPV equals zero. EMI is the fixed instalment that repays a loan with interest. To solve: list the cash flows, apply the right formula, and for IRR interpolate between two trial rates.

Understand Capital Budgeting: IRR, Payback and Loan Amortization

Capital budgeting means deciding whether a project is worth the money. You compare what you spend today with the cash it brings back later. Three ideas are tested in this topic: payback, IRR and loan repayment.

The payback period asks one question: how long until I get my money back? You add the yearly cash inflows until they equal the initial investment. It is simple, but it ignores the time value of money and ignores cash flows after the payback point.

The internal rate of return (IRR) is the discount rate at which the present value of inflows equals the initial outlay. At that rate, NPV = 0. If IRR is higher than the required rate of return, the project is acceptable. You usually find IRR by trying two rates and interpolating.

Loan amortization means repaying a loan in equal instalments (EMI). Each instalment has two parts: interest on the outstanding balance and principal repayment. Early instalments are mostly interest. Later ones are mostly principal. An amortization schedule is a table that shows this split period by period.

Key formulas to remember

Payback period (equal annual inflows)
Payback = Initial investment ÷ Annual cash inflow
Use only when every year's inflow is the same.
Payback period (uneven inflows)
Payback = Years fully recovered + (Unrecovered amount ÷ Cash inflow of the next year)
Build a cumulative cash inflow column. This assumes inflows come evenly through the year.
Net present value
NPV = Σ [CFt ÷ (1 + r)^t] − Initial outlay
IRR is the value of r for which this NPV is zero.
IRR by interpolation
IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L)
L is the lower trial rate (positive NPV) and H is the higher trial rate (negative NPV). The answer is an approximation.
EMI
EMI = P × r × (1 + r)^n ÷ [(1 + r)^n − 1]
P is the loan amount. r is the rate per instalment period (monthly rate = annual rate ÷ 12 for monthly EMI). n is the number of instalments.
Loan as present value of EMIs
P = EMI × [1 − (1 + r)^−n] ÷ r
Same relation as the EMI formula, rearranged. Use it to find the loan amount when EMI is given.
Amortization schedule rows
Interest = Opening balance × r; Principal = EMI − Interest; Closing balance = Opening balance − Principal
Closing balance of one period is the opening balance of the next.

How to solve Capital Budgeting: IRR, Payback and Loan Amortization questions

Use this method for any MCQ on payback, IRR or loan repayment.

  1. 1Identify what is asked: payback, IRR, EMI, interest part, principal part or outstanding balance.
  2. 2Write down the initial outlay, the cash flows by year, and the rate and number of periods.
  3. 3Match the rate to the period. For monthly EMI, use monthly rate and number of months. For annual instalments, use the annual rate and years.
  4. 4For payback, build a cumulative inflow column. Find the year where the cumulative total crosses the investment, then add the fraction of that year.
  5. 5For IRR, compute NPV at two trial rates so that one NPV is positive and the other is negative. Then interpolate.
  6. 6For EMI, apply the EMI formula. Calculate (1 + r)^n carefully and keep enough decimals.
  7. 7For schedule questions, compute interest on the opening balance, subtract it from EMI to get principal, and reduce the balance.
  8. 8Check the answer: the balance after the last instalment must be zero, and IRR must lie between your two trial rates.

Quickest way: Option elimination and sanity checks

When to use it: Use under time pressure in the objective paper, especially when options are far apart.

  1. For payback with uneven flows, find the whole year from the cumulative column first. This usually removes two options at once.
  2. For IRR, the answer must lie between the two given trial rates. Eliminate any option outside that range. If NPV at L is much smaller than at H in size, IRR sits closer to L.
  3. For EMI, estimate first. EMI must be more than P ÷ n, because interest is added. Eliminate options below that.
  4. First-period interest is just P × r. If the question asks for first-month interest or principal, skip the full EMI formula when the EMI is given.
  5. Outstanding balance after any instalment must be less than the loan and more than zero. Check that the principal part grows each period.
  6. If a question needs a long power such as (1 + r)^60 and no values are given, skip it and return later. Wrong answers cost 0.25 marks.

Common mistakes in Capital Budgeting: IRR, Payback and Loan Amortization

  • Using the annual rate with the number of months in the EMI formula.

    The question gives the rate per annum but instalments are monthly, and students rush.

    Fix: Convert first: r = annual rate ÷ 12 and n = years × 12. Write r and n before using the formula.

  • Calculating interest on the original loan every period.

    Students confuse amortization with simple interest on the full amount.

    Fix: Interest is always on the opening balance of that period. It falls as the balance falls.

  • Stopping payback at the year when cumulative inflow crosses the investment, without taking the fraction.

    Students think payback must be a whole number of years.

    Fix: Add the unrecovered amount divided by that year's inflow to the completed years.

  • Interpolating IRR with both trial NPVs positive or both negative.

    Students pick trial rates carelessly.

    Fix: Choose rates so one NPV is positive and one is negative. Otherwise you are extrapolating and the answer is unreliable.

  • Subtracting the whole EMI from the balance instead of only the principal part.

    Students forget that EMI includes interest.

    Fix: Closing balance = Opening balance − (EMI − Interest).

  • Treating a higher payback period as better.

    Students mix up the decision rule.

    Fix: A shorter payback is preferred. For IRR, a higher value is better, and it is accepted if it exceeds the required rate.

Worked examples

Example 1

A project needs an investment of ₹5,00,000. Cash inflows are ₹1,50,000, ₹2,00,000, ₹2,00,000 and ₹1,50,000 in years 1 to 4. The payback period is: (A) 2.25 years (B) 2.50 years (C) 2.75 years (D) 3.00 years

Show the solution
  1. Cumulative inflow after year 1 = ₹1,50,000.
  2. Cumulative inflow after year 2 = ₹1,50,000 + ₹2,00,000 = ₹3,50,000.
  3. Cumulative inflow after year 3 = ₹3,50,000 + ₹2,00,000 = ₹5,50,000, which is more than ₹5,00,000. So payback is between 2 and 3 years.
  4. Unrecovered amount after year 2 = ₹5,00,000 − ₹3,50,000 = ₹1,50,000.
  5. Fraction of year 3 needed = ₹1,50,000 ÷ ₹2,00,000 = 0.75.
  6. Payback = 2 + 0.75 = 2.75 years.

Answer: (C) 2.75 years

Example 2

For an investment of ₹1,00,000, the NPV is +₹4,000 at a discount rate of 10% and −₹2,000 at 15%. Using interpolation, the IRR is approximately: (A) 12.00% (B) 12.50% (C) 13.33% (D) 14.00%

Show the solution
  1. L = 10% with NPV = 4,000. H = 15% with NPV = −2,000.
  2. NPV at L − NPV at H = 4,000 − (−2,000) = 6,000.
  3. IRR = 10 + (4,000 ÷ 6,000) × (15 − 10).
  4. IRR = 10 + 0.6667 × 5 = 10 + 3.33 = 13.33%.
  5. Check: the answer lies between 10% and 15%, and is closer to 15% because the positive NPV is larger in size.

Answer: (C) 13.33%

Example 3

A loan of ₹1,00,000 is repaid in two equal annual instalments at 10% p.a., with the first instalment paid at the end of year 1. EMI is ₹57,619 (approximately). The outstanding balance just after the first instalment is: (A) ₹42,381 (B) ₹50,000 (C) ₹52,381 (D) ₹57,619

Show the solution
  1. Check EMI: ₹1,00,000 × 0.10 × (1.1)^2 ÷ [(1.1)^2 − 1] = 12,100 ÷ 0.21 = ₹57,619 (approximately).
  2. Interest for year 1 = ₹1,00,000 × 10% = ₹10,000.
  3. Principal repaid in instalment 1 = ₹57,619 − ₹10,000 = ₹47,619.
  4. Closing balance = ₹1,00,000 − ₹47,619 = ₹52,381.
  5. Check: year 2 interest = ₹5,238 and ₹52,381 + ₹5,238 = ₹57,619, which equals the EMI. So the balance clears fully.

Answer: (C) ₹52,381

Exam tips

  • Read whether the question asks for the EMI, the interest part, the principal part or the closing balance. Many wrong options are values from the same schedule.
  • In IRR questions, check the trial rates given. The answer must lie between them, so use that to cut options before calculating.
  • For uneven cash flows, always write the cumulative column. It takes 20 seconds and avoids guesswork.
  • Check the instalment timing and compounding period (monthly or yearly) in the wording before choosing r and n.
  • Leave long power calculations for the end if no (1 + r)^n value is given. A skipped question costs nothing, a wrong answer costs 0.25 marks.

Practice questions from Mathematics of Finance

Capital Budgeting: IRR, Payback and Loan Amortization: frequently asked questions

How do I calculate EMI in CA Foundation maths?

Use EMI = P × r × (1 + r)^n ÷ [(1 + r)^n − 1]. Here r is the rate per instalment period and n is the number of instalments. For monthly EMI, divide the annual rate by 12 and multiply years by 12.

What is IRR and how is it found in the exam?

IRR is the discount rate at which NPV is zero. In the exam you usually compute NPV at two rates, one positive and one negative, and interpolate. The result is an approximation of the true IRR.

What is the weakness of the payback period method?

It ignores the time value of money and ignores cash flows after the payback point. So it shows how quickly money returns, not how profitable the project is overall.

How does an amortization schedule work?

Each row starts with the opening balance. Interest is opening balance × rate per period. Principal repaid is EMI minus interest, and the closing balance is opening balance minus principal. The balance reaches zero after the last instalment.