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Worked Examples · Example 15

Q.Mr X took a loan of ₹2,000 for 6 months. Lender deducts ₹200 as interest while lending. Find the effective rate of interest charged by lender.

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The lender hands over only ₹1,800 (after deducting ₹200 up front) but is repaid ₹2,000 after 6 months, so the half-yearly rate is i=2001800=19i=\dfrac{200}{1800}=\dfrac19. Compounding this over the two half-years of a year gives the effective annual rate reff=(1+i)2−1=(1+19)2−1≈23.45%r_{\text{eff}}=(1+i)^2-1=\left(1+\dfrac19\right)^2-1\approx 23.45\%.

This is a discounted (front-end interest) loan: the ₹200 interest is taken out before the money is handed over, so the borrower actually receives ₹1,800 yet must repay the full ₹2,000.

  1. Interest rate for the 6-month period. The ₹200 is interest on the ₹1,800 actually in the borrower's hands for 6 months:

i=2001800=19≈0.1111.i=\frac{200}{1800}=\frac19\approx 0.1111.

  1. Convert to the effective annual rate. An effective rate is the actual rate compounded once a year. Since each half-year multiplies the money by (1+i)(1+i), a full year (two half-years) multiplies it by (1+i)2(1+i)^2, so

reff=(1+i)2−1=(1+19)2−1=10081−1=0.23456.r_{\text{eff}}=(1+i)^2-1=\left(1+\frac19\right)^2-1=\frac{100}{81}-1=0.23456.

  1. Express as a percentage. reff≈23.45%r_{\text{eff}}\approx 23.45\% per annum.
✓Final answer

The effective rate of interest charged by the lender is reff=(1+19)2−1≈23.45%r_{\text{eff}}=\left(1+\dfrac19\right)^2-1\approx \mathbf{23.45\%} per annum.

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