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  1. 083A phone sells for Rs 57,000 cash today, or on a 'no-cost EMI' of Rs 10,000 a month for six months, the first instalment due a month from now. What interest rate is hidden in the EMI?Compounding and time valueHardBank creditCorporate FP&A

    Try it first

    Roughly what annual rate is hidden in the EMI?

    Show the worked solution

    About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.

    Where is the interest, if the EMI says no-cost?

    A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.

    So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.

    The relationship
    57,000=∑k=1610,000(1+r)k=10,000×1−(1+r)−6r  ⇒  r≈1.49%57{,}000 = \sum_{k=1}^{6} \frac{10{,}000}{(1+r)^k} = 10{,}000 \times \frac{1-(1+r)^{-6}}{r} \;\Rightarrow\; r \approx 1.49\%
    rthe monthly interest rate hidden in the instalments
    kthe month each Rs 10,000 instalment is paid
    What it says in wordsThe hidden rate is the one that discounts the six instalments back to the cash price.
    The forgone Rs 3,000 discount is the interest on a no-cost EMICash flows+Rs 57,000: the phone, today-Rs 10,000 at the end of months 1 to 6month 0123456Six EMIs, face value10,00010,00010,00010,00010,00010,000Discounted at 1.49% a month9,8549,7099,5679,4279,2899,153Cash price today57,000Rs 3,000 of interestdiscounted EMIs sum to exactly 57,000Hidden rate: 1.49% a month = 19.4% a year effective (17.8% simple annual)
    Six Rs 10,000 instalments add to Rs 60,000 at face value, but discounted at 1.49% a month they are worth exactly the Rs 57,000 cash price, so the Rs 3,000 discount given up is interest at 19.4% a year effective.

    How do you solve it without a spreadsheet?

    Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.

    A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.

    What changes the answer in real life?

    Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.

    Where candidates lose it

    The quick wrong answer is about 5%: Rs 3,000 on Rs 57,000. Some candidates double it to 10.5% for a year and feel careful. Both ignore that the loan is repaid monthly, so the average balance is only a little over half of Rs 57,000, and the rate on what is actually owed is far higher than the headline.

    The second loss is multiplying the monthly rate by 12 and calling it the annual rate. 17.8% is a nominal rate; money compounding monthly at 1.49% grows 19.4% in a year. Give both and name which is which.

    What the interviewer asks next

    • If the first instalment is paid at purchase, what is the hidden rate? (About 2.1% a month.)
    • The cash discount is only Rs 1,000. What is the hidden rate now, roughly?
    • Who pays the lender when the retailer offers no cash discount at all, and why would the retailer do that?