Time Value of Money and Annuity - Simple and Compound Interest: formula sheet

Key formulas

Simple interest

SI = P × R × T ÷ 100

P is principal in ₹, R is rate per annum in %, T is time in years.

Amount

A = P + SI = P × (1 + R × T ÷ 100)

Amount is the total repaid at the end of the period.

Principal

P = SI × 100 ÷ (R × T)

Use when interest, rate and time are known. If amount is given, first find SI = A − P or use P = A ÷ (1 + RT ÷ 100).

Rate

R = SI × 100 ÷ (P × T)

Answer is in % per annum when T is in years.

Time

T = SI × 100 ÷ (P × R)

Answer is in years. Multiply by 12 for months.

Time conversion

Months ÷ 12 = years; Days ÷ 365 = years

Use the conversion the question implies. Most exam problems use clean values like 6 months = 0.5 year.

Compound amount (yearly compounding)

A = P(1 + r/100)^n

P is the principal, r is the annual rate in % and n is the number of years.

Compound amount (m times a year)

A = P(1 + r/(100 × m))^(m × n)

m = 2 for half-yearly, 4 for quarterly, 12 for monthly. Rate per period = r ÷ m. Periods = m × n.

Compound interest

CI = A − P

Always subtract the principal if the question asks for interest.

Varying rates

A = P(1 + r1/100)(1 + r2/100)(1 + r3/100)

Use one factor per year with that year's rate. Do not add the rates.

CI minus SI for 2 years (yearly compounding)

CI − SI = P(r/100)²

Valid only for 2 years with yearly compounding. It is a quick way to find the principal when the difference is given.

Effective annual rate

E = (1 + i ÷ m)^m − 1

i = nominal annual rate as a decimal (12% = 0.12). m = compounding periods per year. Multiply E by 100 for a percentage.

Rate per compounding period

Periodic rate = i ÷ m

Half-yearly m = 2, quarterly m = 4, monthly m = 12. Example: 12% quarterly gives 3% per quarter.

Half-yearly shortcut

E = i + (i ÷ 2)²

Exact for m = 2. Example: i = 0.10 gives 0.10 + 0.0025 = 0.1025, that is 10.25%.

Nominal rate from effective rate

i = m × [(1 + E)^(1 ÷ m) − 1]

Use when the effective rate is given and you must find the nominal rate for a stated compounding frequency.

Amount using effective rate

A = P × (1 + E)^n

n = number of years. This gives the same amount as using (1 + i ÷ m)^(m × n).

Future value (compound interest)

A = P × (1 + r)^n

P is the present sum, r is the rate per period as a decimal, n is the number of periods.

Present value

P = A ÷ (1 + r)^n

Discounting. The present value is always less than the future sum when r > 0.

Compounding more than once a year

A = P × (1 + r/m)^(m × n)

r is the annual rate, m is the compounding periods per year (half-yearly 2, quarterly 4). Use rate r/m and m × n periods.

Depreciation (reducing balance)

V = C × (1 − d)^n

C is the cost, d is the yearly rate as a decimal, V is the value after n years. Total depreciation = C − V.

Growth at a fixed rate

Pₙ = P₀ × (1 + g)^n

Used for population, sales, or output growing at g per period.

Straight-line depreciation (for contrast)

Yearly depreciation = (Cost − Scrap value) ÷ n

This is a fixed amount each year. It is not the compound formula.

Future value of annuity immediate

A = P × [(1 + i)ⁿ − 1] ÷ i

P is the payment per period, i the rate per period (as a decimal), n the number of periods. Payments at the end of each period.

Present value of annuity immediate

PV = P × [1 − (1 + i)⁻ⁿ] ÷ i

Gives today's value of n end-of-period payments.

Annuity due (future or present value)

Due value = Immediate value × (1 + i)

Applies to both future and present value, because each payment is one period earlier.

Present value of perpetuity (immediate)

PV = P ÷ i

First payment is one period from now.

Present value of perpetuity due

PV = P + P ÷ i = P × (1 + i) ÷ i

First payment is made today, so add one payment to the immediate value.

Sinking fund instalment

P = F × i ÷ [(1 + i)ⁿ − 1]

F is the target amount. Uses the future value formula solved for P.

EMI (loan instalment)

EMI = Loan × i ÷ [1 − (1 + i)⁻ⁿ]

Uses the present value formula solved for P. Loan is the present value.

Quick revision

• Simple interest: SI = P × r × t ÷ 100; amount = P + SI.
• Compound amount: A = P(1 + r/100)^n; CI = A − P.
• For compounding m times a year, use rate ÷ m per period and n × m periods.
• At the same rate, CI is greater than SI for periods longer than one compounding period. For one compounding period (e.g. one year with yearly compounding), CI = SI.
• Effective rate = (1 + i/m)^m − 1, where i is the nominal annual rate as a decimal.
• More frequent compounding gives a higher effective rate than the nominal rate.
• Present value = Future value ÷ (1 + r)^n.
• Reducing balance depreciation: value = cost × (1 − d)^n.
• Immediate annuity payments are made at the end of each period; annuity due at the start.
• Annuity due value = immediate annuity value × (1 + r).
• Perpetuity present value = payment ÷ rate, with the rate as a decimal.
• Always check whether the question asks for amount, interest, or rate.

Common mistakes

• Using months directly as T Fix: Divide by 12 first. 9 months = 0.75 year.
• Giving SI when the question asks for the amount Fix: Underline the word 'amount' or 'total repaid'. Add the principal to the SI.
• Using the annual rate and the number of years directly when interest is compounded half-yearly or quarterly. Fix: Always write m first. Divide the rate by m and multiply the years by m before putting numbers into the formula.
• Reporting the amount as the compound interest. Fix: Underline whether the question asks for amount or interest. If it asks for interest, subtract P.
• Using the nominal rate directly in each period, for example 12% per quarter. Fix: Always compute i ÷ m first. Quarterly 12% means 3% per quarter.
• Forgetting to subtract 1 and giving 1.1255 as the answer. Fix: Effective rate = growth factor − 1. Then convert to percent.
• Putting r = 10 instead of 0.10 in the formula. Fix: Always write 10% as 0.10 before bringing it into (1 + r).
• Multiplying instead of dividing when finding present value. Fix: Ask which way the money moves. A future sum brought to today is divided by (1 + r)^n.
• Using the immediate formula when the payments are made at the start of each period. Fix: Underline the timing phrase first. If payments are in advance, multiply the immediate result by (1 + i).
• Not adjusting the rate and number of periods for half-yearly or monthly payments. Fix: Always make i and n match the payment gap: i = annual rate ÷ periods per year, n = years × periods per year.

Exam tips

• Convert time and rate to years and per annum before anything else. Most wrong answers come from units.
• For questions with two amounts at two times, subtract them to get the yearly interest, then work back to the principal.
• Cancel zeros before multiplying. With round numbers, the answer should come in under 30 seconds.
• Check whether the option list contains the interest as a trap next to the amount.
• There is no negative marking, so if you are short on time, eliminate options that are clearly too small or too large and still mark an answer.
• Write the rate per period and the number of periods on your rough sheet before any multiplication. Most wrong options are built from errors in these two numbers.
• Read the last line of the question. It may ask for the amount, the interest, or the difference between CI and SI.
• Learn the powers of 1.1, 1.05 and 1.02 for 2 and 3 periods. They save time because the numbers in this paper are kept clean.