Every bank product, from a savings account in Multan to a car loan in Karachi to a bill discounted in London, rests on a few formulas about how money grows and shrinks with time. Once you understand simple interest, compounding, effective rates and discounting, you can compare offers, check a bank's figures and solve almost any banking question in a BBA, MBA, ACCA or IBP exam. This article builds those tools step by step.
Simple Interest
With simple interest, profit is earned only on the original principal \( P \):
$$ I = P r t, \qquad A = P(1 + rt) $$
where \( r \) is the annual rate (as a decimal) and \( t \) is time in years. PKR 500,000 at 12% simple interest for 2 years earns \( 500{,}000 \times 0.12 \times 2 = 120{,}000 \), giving PKR 620,000. Simple interest is used for short-term instruments, many savings certificates that pay profit periodically, and day-count calculations on deposits.
Compound Interest
With compounding, interest is added to the balance and itself earns interest. If the annual nominal rate \( r \) is compounded \( m \) times a year for \( t \) years,
$$ A = P\left(1 + \frac{r}{m}\right)^{mt}. $$
Worked Example 1: A Monthly-Compounded Deposit
Ayesha deposits PKR 500,000 at 12% per annum, compounded monthly, for 2 years. Find the maturity value and the effective annual rate.
Step 1: Rate per period and number of periods.
$$ \frac{r}{m} = \frac{0.12}{12} = 0.01, \qquad mt = 12 \times 2 = 24 $$
Step 2: Maturity value.
$$ A = 500{,}000 \times (1.01)^{24} = 500{,}000 \times 1.269735 \approx \text{PKR } 634{,}867.32 $$
Step 3: Interest earned. \( 634{,}867.32 - 500{,}000 = \text{PKR } 134{,}867.32 \), compared with PKR 120,000 under simple interest and PKR 127,200 under annual compounding (\( 500{,}000 \times 1.12^2 = 627{,}200 \)).
Step 4: Effective annual rate (EAR). The EAR is the annually compounded rate that gives the same growth:
$$ \text{EAR} = \left(1 + \frac{r}{m}\right)^m - 1 = (1.01)^{12} - 1 \approx 0.126825 = 12.68\% $$
Sanity check: \( 500{,}000 \times 1.126825^2 \approx 634{,}867 \), matching Step 2. The EAR is the fair way to compare offers: a 12% monthly-compounded deposit beats a 12.5% annually-compounded one.
The Rule of 72
A quick mental shortcut: money doubles in roughly \( 72 / (\text{rate in \%}) \) years. At 12%, doubling takes about 6 years; the exact figure is 6.12 years, since \( 1.12^6 \approx 1.974 \).
Present Value and Discounting
Discounting reverses compounding. The present value of an amount \( F \) due in \( t \) years at annual rate \( r \) is
$$ PV = \frac{F}{(1 + r)^t}. $$
A payment of PKR 1,000,000 due in 3 years, discounted at 12%, is worth \( 1{,}000{,}000 / 1.12^3 \approx \text{PKR } 711{,}780.25 \) today. Present value is how banks price bonds, how businesses evaluate projects and how courts value future compensation.
Bank Discount (Bill Discounting)
When a bank discounts a trade bill, it deducts interest in advance on the face value. This is different from ordinary interest, which is charged on the amount actually lent. For a bill with face value \( F \), discount rate \( d \) and \( n \) days to maturity (on a 365-day basis, as is common for rupee instruments):
$$ D = F \times d \times \frac{n}{365}, \qquad \text{Proceeds} = F - D $$
Worked Example 2: Discounting an Export Bill
A Sialkot sports-goods exporter holds a bill of PKR 1,000,000 due in 90 days. The bank discounts it at 15% per annum. Find the proceeds and the true interest rate the exporter pays.
Step 1: Discount.
$$ D = 1{,}000{,}000 \times 0.15 \times \frac{90}{365} \approx \text{PKR } 36{,}986.30 $$
Step 2: Proceeds.
$$ 1{,}000{,}000 - 36{,}986.30 = \text{PKR } 963{,}013.70 $$
Step 3: Equivalent simple interest rate. The exporter effectively borrows PKR 963,013.70 and repays PKR 1,000,000 after 90 days:
$$ r = \frac{36{,}986.30}{963{,}013.70} \times \frac{365}{90} \approx 0.1558 = 15.58\% $$
Interpretation: a 15% discount rate is equivalent to about 15.58% interest, because the charge is calculated on the larger face value. Exam questions often test this distinction.
Loan Instalments (Equated Monthly Instalments)
Most consumer and business loans are repaid in equal monthly instalments. For principal \( P \), monthly rate \( i \) and \( n \) months,
$$ \text{EMI} = \frac{P\, i}{1 - (1 + i)^{-n}}. $$
Worked Example 3: A KIBOR-Linked Business Loan
A shop owner borrows PKR 2,000,000 for 5 years. The rate is 14% per annum (for illustration, KIBOR of 11% plus a 3% spread), with monthly instalments.
Step 1: \( i = 0.14/12 \approx 0.011667 \), \( n = 60 \).
Step 2:
$$ \text{EMI} = \frac{2{,}000{,}000 \times 0.011667}{1 - (1.011667)^{-60}} \approx \text{PKR } 46{,}536.50 $$
Step 3: Total cost. Total repaid is \( 46{,}536.50 \times 60 \approx 2{,}792{,}190 \), so total interest is about PKR 792,190.
Step 4: First instalment split. Interest in month 1 is \( 2{,}000{,}000 \times 0.011667 \approx 23{,}333.33 \), so only PKR 23,203.17 reduces the principal. Early instalments are interest-heavy; later ones mostly repay principal.
A useful check on any loan schedule is that the outstanding balance after the final instalment is zero (allowing for rounding). If a bank statement shows a different total cost, ask whether processing fees, insurance or a different compounding basis has been included.
Because KIBOR-linked loans are usually repriced periodically, the EMI is recalculated on the outstanding balance whenever the benchmark changes.
Profit Rates in Islamic Banking
Islamic banks in Pakistan quote expected profit rates rather than interest rates. On a Mudarabah savings account, the actual profit depends on the bank's pool earnings and the agreed profit-sharing ratio, so the declared rate can change month to month. The arithmetic of converting monthly profit to an annual rate, or of comparing payout frequencies, uses exactly the same compounding and EAR formulas shown above. For example, a declared rate of 10% per annum on PKR 1,000,000, paid monthly, yields about PKR 8,333 per month before any tax deduction.
Did You Know?
- Fibonacci's Liber Abaci (1202) included some of the earliest European worked problems on interest and partnership profits.
- The constant \( e \approx 2.718 \) was discovered by Jacob Bernoulli in 1683 while studying what happens when interest is compounded more and more often.
- KIBOR, the Karachi Interbank Offered Rate, is the main benchmark for conventional rupee lending in Pakistan.
Key Takeaways
- Simple interest grows linearly; compound interest grows exponentially.
- Compare offers using the effective annual rate, not the nominal rate.
- Present value discounts future cash to today using \( F/(1+r)^t \).
- Bank discount is charged on face value, so the true interest rate is higher than the quoted discount rate.
- EMIs follow the annuity formula, and early payments are mostly interest.
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