CS Professional · Banking and Insurance - Laws and Practice

Calculation of Interest and Annuities: formula sheet

Full chapter guide

Key formulas

Simple interest: SI = P × r × t; r is the annual rate as a decimal (10% = 0.10); t is in years. For months, use t = months ÷ 12; for days, days ÷ 365.
Amount under simple interest: A = P + SI = P(1 + r t); Amount is the principal plus interest.
Amount under compound interest: A = P(1 + r/m)^(m t); m is the number of compounding periods per year. For annual compounding, m = 1 and A = P(1 + r)^t.
Compound interest: CI = A − P; Find the amount first, then subtract the principal.
Effective rate of interest: Effective rate = (1 + r/m)^m − 1; r is the nominal annual rate as a decimal. Multiply by 100 for a percentage. For annual compounding, effective equals nominal.
Rate and number of periods per compounding: Periodic rate = r ÷ m; number of periods = m × t; Use these two values in A = P(1 + i)^n, where i is the periodic rate and n is the number of periods.
Compound interest versus simple interest for two years: CI − SI = P × (r)² (annual compounding, 2 years); Works only for exactly two years with annual compounding. The difference is the interest on the first year's interest.
Simple interest: I = P × R × T ÷ 100 (T in years); Use T = days ÷ 365 when the period is in days.
Product (balance × days): Product = Balance × Number of days the balance remained unchanged; Count the day of deposit but not the day of withdrawal, or follow the convention stated in the question.
Interest by product method: Interest = Σ Products × R ÷ (100 × 365); Use 366 for a leap year if the question says so. Rate is annual.
Compound interest (FD with periodic rest): A = P × (1 + r ÷ n)^(n×t), where r is annual rate in decimals and n is compounding periods per year; Quarterly rest means n = 4 and rate per quarter = annual rate ÷ 4.
Recurring deposit maturity (quarterly compounding): M = Σ of each instalment × (1 + i)^(remaining quarters), with i = annual rate ÷ 4; a simple-interest RD uses I = P × n(n+1) ÷ 24 × R ÷ 100; Here P is the monthly instalment, n the number of months, R the annual rate in %. The simple version assumes instalments at the start of each month.
Future value of a single sum (annual compounding): FV = P × (1 + r)^n; P = amount today, r = rate per year as a decimal, n = number of years.
Present value of a single sum: PV = FV ÷ (1 + r)^n; Same r and n as the compounding formula, used in reverse.
Discount factor: DF = 1 ÷ (1 + r)^n; PV = FV × DF. Questions often give DF values in a table.
Compounding more than once a year: FV = P × (1 + r ÷ m)^(m × n); m = compounding periods per year. r is the nominal annual rate.
Effective annual rate: EAR = (1 + r ÷ m)^m − 1; Use it to compare rates with different compounding frequencies.
Present value of several future sums: PV = Σ [CFt ÷ (1 + r)^t]; Discount each cash flow separately, then add.
Future value of ordinary annuity: FV = P × [(1 + i)ⁿ − 1] ÷ i; P = payment per period, i = rate per period, n = number of periods. Payments at end of each period.
Present value of ordinary annuity: PV = P × [1 − (1 + i)⁻ⁿ] ÷ i; Gives today's value of n end-of-period payments.
Annuity due (future value): FV(due) = P × [(1 + i)ⁿ − 1] ÷ i × (1 + i); Ordinary annuity value multiplied by (1 + i).
Annuity due (present value): PV(due) = P × [1 − (1 + i)⁻ⁿ] ÷ i × (1 + i); Ordinary annuity present value multiplied by (1 + i).
Present value of perpetuity: PV = P ÷ i; Applies when payments are made at the end of each period forever. For a perpetuity due, add the first payment: PV = P + P ÷ i.
Sinking fund deposit: P = F × i ÷ [(1 + i)ⁿ − 1]; F = target sum needed at the end of n periods; deposits made at the end of each period.
Rate and period matching: i = annual rate ÷ m; n = years × m; m = number of payments per year.
Monthly rate: r = annual rate ÷ 12 ÷ 100; Convert the annual rate to a monthly decimal. 12% p.a. gives r = 0.01. n is the number of months, so 3 years gives n = 36.
EMI (reducing balance): EMI = P × r × (1+r)^n ÷ [(1+r)^n − 1]; P is the loan amount. This is the annuity formula rearranged to find the payment.
Loan as present value of EMIs: P = EMI × [1 − (1+r)^(−n)] ÷ r; Use it to find the loan amount from a given EMI, or to check an answer.
Interest for a month: Interest = opening balance × r; Always charged on the balance at the start of the month.
Principal and closing balance: Principal repaid = EMI − interest; closing balance = opening balance − principal repaid; The closing balance of one month is the opening balance of the next.
Total interest: Total interest = (EMI × n) − P; Holds when all n EMIs are paid as scheduled. The last EMI may differ slightly because of rounding.
Flat rate EMI: EMI = [P + (P × annual rate × years)] ÷ (years × 12); Interest is on the original principal for the whole term. No reduction for repayments.

Quick revision

Simple interest = P × r × t, with r as the annual rate in decimal and t in years.
Compound amount = P(1 + r/m)^(mt), where m is the number of compounding periods a year.
Effective annual rate = (1 + r/m)^m − 1, and it is higher than the nominal rate when m > 1.
Future value of a single sum = PV × (1 + i)^n; present value = FV ÷ (1 + i)^n.
Always use the rate per period and the number of periods in the same unit.
Ordinary annuity: payments at the end of each period; annuity due: at the start.
Future value of an ordinary annuity = A × [(1 + i)^n − 1] ÷ i.
Present value of an ordinary annuity = A × [1 − (1 + i)^−n] ÷ i.
An annuity due is worth (1 + i) times the matching ordinary annuity.
EMI = P × i × (1 + i)^n ÷ [(1 + i)^n − 1], with i as the monthly rate and n as the number of months.
In each EMI, interest = opening balance × monthly rate; principal = EMI − interest.
Show formula, substitution and result in rupees for every sum.

Common mistakes

Using the annual rate and years directly when interest is compounded quarterly or monthly. Fix: Always compute i = r ÷ m and n = m × t first. For 12% quarterly for 2 years, i = 3% and n = 8.
Giving the amount when the question asks for compound interest. Fix: Underline what is asked. Write CI = A − P as the final line.
Applying the rate to the balance without multiplying by days Fix: Always convert to a per-day basis. Interest = product × rate ÷ (100 × 365).
Days in the table do not add up to the period Fix: Add the days column and compare it with the total days in the period before computing.
Dividing when you should multiply, or the reverse. Fix: Future value is always larger than today's sum, present value always smaller (for a positive rate). Check this before you write the answer.
Using the annual rate with half-yearly or quarterly compounding. Fix: Divide the rate by m and multiply the years by m before using the formula.
Using the annual rate with monthly payments Fix: Divide the annual rate by the number of payments per year and multiply the years by the same number.
Treating an annuity due as an ordinary annuity Fix: Underline the payment timing. If payments are at the start, multiply the ordinary annuity result by (1 + i).
Using the annual rate with the number of months in the formula. Fix: Convert both to months first: r = annual rate ÷ 12 ÷ 100 and n = years × 12.
Charging interest on the original loan every month in a reducing-balance schedule. Fix: Interest each month = opening balance × r. Opening balance falls every month by the principal repaid.

Exam tips

Write the given data (P, r, m, t) in a small list before calculating. Examiners award marks for correct identification of values even if arithmetic slips.
Show the formula, the substitution and the result in separate lines. Descriptive papers reward visible method.
When asked to compare two products, compute effective rates and end with a one-line conclusion for the depositor or borrower.
Be ready for short theory questions: define nominal and effective rate, and state why compounding frequency changes the result.
State assumptions, such as a 365-day year for day counts, when the question does not specify.
Draw the balance, days and product table every time. Marks are often given for the method and the table.
State your assumptions (365 days, simple interest on products, day of withdrawal excluded) in one line if the question is silent.
Check that days add up to the period before you multiply anything.