Time Value of Money: formula sheet

Key formulas

Future value of a single sum (compounding)

FV = PV × (1 + r)^n

r is the interest rate per period and n is the number of periods. It moves money forward in time.

Present value of a single sum (discounting)

PV = FV ÷ (1 + r)^n

It moves a future amount back to today. r is called the discount rate here.

Link between compounding and discounting

PV = FV × [1 ÷ (1 + r)^n]

The bracket is the present value factor. It is the reverse of the compounding factor (1 + r)^n.

Nominal rate, real rate and inflation (approximate)

Nominal rate ≈ Real rate + Expected inflation + Risk premium

Use this only to explain the components of an interest rate in theory answers. It is an approximation, not an exact relationship.

Simple interest

SI = P × r × n

r is the annual rate as a decimal (10% = 0.10) and n is time in years. Interest is the same every year.

Amount under simple interest

A = P + SI = P(1 + r × n)

Growth is a straight line over time.

Amount under compound interest (yearly)

A = P(1 + r)^n

Compound interest = A − P.

Amount with m compounding periods a year

A = P(1 + r/m)^(m × n)

m = 2 for half-yearly, 4 for quarterly, 12 for monthly. Use the periodic rate r/m and the number of periods m × n.

Effective rate of interest

EAR = (1 + r/m)^m − 1

r is the nominal annual rate as a decimal. If m = 1, the effective rate equals the nominal rate.

Compound interest

CI = P[(1 + r/m)^(m × n) − 1]

Gives interest directly without finding the amount first.

Present sum from amount

P = A ÷ (1 + r/m)^(m × n)

Used when the final amount is known and the starting sum is needed.

Future value of a single sum

FV = P × (1 + i)^n = P × FVIF(i, n)

P is the amount today, i is the rate per period, n is the number of periods.

Future value of an ordinary annuity

FVA = A × [((1 + i)^n − 1) ÷ i] = A × FVIFA(i, n)

A is the equal payment at the end of each period. Use when payments fall at period end.

Future value of an annuity due

FVA(due) = A × [((1 + i)^n − 1) ÷ i] × (1 + i)

Each payment is at the start of the period, so the ordinary annuity value is multiplied by (1 + i).

Compounding more than once a year

FV = P × (1 + r ÷ m)^(m × t)

r is the annual rate, m is compounding periods per year, t is years. Use i = r ÷ m and n = m × t.

Effective annual rate

EAR = (1 + r ÷ m)^m − 1

Use it to compare nominal rates with different compounding frequencies.

Present value of a single sum

PV = FV ÷ (1 + r)^n = FV × PVIF(r, n)

PVIF(r, n) = 1 ÷ (1 + r)^n. r is the rate per period and n is the number of periods.

Present value of an ordinary annuity

PV = A × [1 − (1 + r)^−n] ÷ r = A × PVIFA(r, n)

A is the equal payment at the end of each period.

PVIFA from PVIF

PVIFA(r, n) = Σ PVIF(r, t) for t = 1 to n

Adding the PVIF values of years 1 to n gives the PVIFA. Useful when only one table is given.

Present value of an annuity due

PV = A × PVIFA(r, n) × (1 + r)

Equivalent method: A + A × PVIFA(r, n − 1). Payments are at the start of each period.

Present value of uneven cash flows

PV = Σ CFt × PVIF(r, t)

Discount each year's cash flow with its own factor, then add.

Rate per period when interest is compounded more than once a year

r = annual rate ÷ m and n = years × m

m is the number of compounding periods in a year.

Present value of a perpetuity

PV = C ÷ r

C is the fixed cash flow per period, received at the end of each period starting one period from now. r is the discount rate per period.

Present value of a growing perpetuity

PV = C1 ÷ (r − g)

C1 is the cash flow at the end of the first period. Valid only when r > g and growth is constant forever.

Next cash flow from current cash flow

C1 = C0 × (1 + g)

Use this when the question gives the cash flow just paid (C0), not the next one.

Perpetuity due

PV = C + C ÷ r

Use when the first payment is made today. It is the first payment plus an ordinary perpetuity.

Implied rate or cash flow

r = C ÷ PV ; C = PV × r

Rearrange the formula when the price is known and the rate or cash flow is asked.

Future value annuity factor

FVIFA = [(1 + r)ⁿ − 1] ÷ r

r is the rate per period and n the number of periods. Payments are at the end of each period (ordinary annuity).

Sinking fund deposit

A = FV × r ÷ [(1 + r)ⁿ − 1] = FV ÷ FVIFA

The term r ÷ [(1 + r)ⁿ − 1] is the sinking fund factor.

Present value annuity factor

PVIFA = [1 − (1 + r)⁻ⁿ] ÷ r

Use it to value a series of equal end-of-period payments.

Loan instalment (EMI)

A = P ÷ PVIFA = P × r ÷ [1 − (1 + r)⁻ⁿ]

The term r ÷ [1 − (1 + r)⁻ⁿ] is the capital recovery factor.

Amortisation schedule rows

Interest = Opening balance × r; Principal = Instalment − Interest; Closing balance = Opening balance − Principal

The closing balance of one period is the opening balance of the next.

Relation of the two factors

Capital recovery factor = Sinking fund factor + r

Useful as a cross-check when both tables are given.

Exact doubling period

n = ln 2 ÷ ln(1 + r)

r is the annual rate as a decimal, with annual compounding. Use when a calculator or log values are given.

Rule of 72

Doubling period (years) ≈ 72 ÷ rate (% p.a.)

Best for annual compounding at about 6% to 10%. Rate is entered as a whole number, such as 8, not 0.08.

Rule of 69 (continuous compounding)

Doubling period (years) ≈ 69.3 ÷ rate (% p.a.)

Exact for continuous compounding, because ln 2 ≈ 0.693.

Rule of 69 adjusted for annual compounding

Doubling period (years) ≈ 69.3 ÷ rate (% p.a.) + 0.35

Gives a closer estimate for annual compounding than the plain 69.3 version.

Rate needed to double

Rate (%) ≈ 72 ÷ number of years

Reverse use of the Rule of 72.

Doubling under simple interest

Doubling period = 100 ÷ rate (% p.a.)

Interest is only on principal, so total interest equals principal after this time.

Tripling period (rule of thumb)

Tripling period ≈ 110 ÷ rate (% p.a.)

Use only if the question asks for it or you are told this rule.

Quick revision

• Compound amount: A = P × (1 + r)^n, where r is the rate per period and n is the number of periods.
• Simple interest: SI = P × r × n; it is not earned on earlier interest.
• Present value of a single sum: PV = FV ÷ (1 + r)^n.
• Future value of an ordinary annuity: A × [(1 + r)^n − 1] ÷ r, with payments at the end of each period.
• Present value of an ordinary annuity: A × [1 − (1 + r)^−n] ÷ r.
• An annuity due has payments at the start of each period; its value is the ordinary annuity value × (1 + r).
• Perpetuity: PV = A ÷ r; growing perpetuity: PV = A1 ÷ (r − g), valid only when r > g.
• Sinking fund payment = target amount ÷ FV annuity factor; capital recovery payment = loan ÷ PV annuity factor.
• When interest is compounded m times a year, use rate ÷ m and years × m.
• Rule of 72: doubling years ≈ 72 ÷ rate in per cent; Rule of 69 is a closer approximation for continuous or frequent compounding.
• Always match the rate period to the cash flow period.

Common mistakes

• Writing that money is worth more today only because of inflation. Fix: Give all reasons. Even with zero inflation, money today can earn a return and future cash carries risk.
• Confusing compounding with discounting. Fix: Ask where the money is going. Forward in time means multiply. Back to today means divide.
• Using the annual rate for half-yearly or quarterly compounding. Fix: Always divide the annual rate by m and multiply the years by m before using the formula.
• Giving the amount when the question asks for compound interest. Fix: Re-read the last line of the question. Subtract P to get interest.
• Using the ordinary annuity formula for an annuity due. Fix: Underline the timing of payments. If it is the start of the period, multiply the ordinary annuity result by (1 + i).
• Using the annual rate with half-yearly or quarterly compounding without adjusting. Fix: Divide the rate by the number of compounding periods per year and multiply n by the same number.
• Using PVIFA for flows that are not equal. Fix: Check that every amount is identical. If even one year differs, use PVIF for each year, or split the flow into an annuity plus a separate amount.
• Treating an annuity due as an ordinary annuity. Fix: Underline the timing. For payments at the start, multiply the ordinary annuity PV by (1 + r).
• Using C0 instead of C1 in the growing perpetuity formula Fix: Read the wording. If the cash flow has already been paid, multiply by (1 + g) to get C1 before dividing by (r − g).
• Applying the growing perpetuity formula when g is equal to or more than r Fix: Always compare r and g first. The formula needs r > g. Otherwise state that the formula is not applicable.

Exam tips

• Theory questions often ask for reasons or importance. Use bullet points with a one-line explanation each.
• Always add a small rupee example. It shows understanding and earns marks.
• Mention the interest rate as the link between present and future values, and name compounding and discounting.
• In numerical parts, show the formula, the values substituted and the final rupee answer with units.
• Link the concept to financial management uses such as capital budgeting and valuation in the closing line.
• Write the formula, then the substituted values, then the answer. Marks are given for method even if arithmetic slips.
• Always state the periodic rate and the number of periods in a separate line. This shows the examiner you handled compounding correctly.
• If the question gives no tables, keep numbers simple and compute step by step. Do not leave powers unsolved.