Time Value of Money: formula sheet

Key formulas

Future value (compounding, single sum)

FV = PV × (1 + r)^n

r is the rate per period and n is the number of periods. Match the rate to the period, for example half-yearly rate for half-yearly periods.

Present value (discounting, single sum)

PV = FV ÷ (1 + r)^n

The discount factor is 1 ÷ (1 + r)^n. It is below 1 for any positive r and n.

Approximate required rate of return

Nominal rate ≈ Real rate + Expected inflation + Risk premium

A quick way to show how the reasons for TVM build up the rate. The exact link between real and nominal rates is (1 + nominal) = (1 + real) × (1 + inflation).

Real rate of return

Real rate = [(1 + nominal rate) ÷ (1 + inflation rate)] − 1

Use this when a question asks for the return after allowing for inflation.

Simple interest

SI = P × r × n

r is the annual rate as a decimal and n is in years. Convert months to years (9 months = 0.75).

Amount under simple interest

A = P × (1 + r × n)

Amount is principal plus interest.

Compound amount (annual)

A = P × (1 + r)^n

Compound interest = A − P.

Compound amount (m times a year)

A = P × (1 + r/m)^(m × n)

Rate per period is r/m and number of periods is m × n.

Effective annual rate

EAR = (1 + r/m)^m − 1

r is the nominal annual rate. For annual compounding EAR equals r.

Nominal rate from EAR

r = m × [(1 + EAR)^(1/m) − 1]

Use when the effective rate is given and you need the periodic or nominal rate.

Continuous compounding

A = P × e^(r × n); EAR = e^r − 1

e ≈ 2.71828. Use only when the question says continuous.

Future value of a single sum

FV = PV × (1 + r)^n

r is the rate per period and n is the number of periods. (1 + r)^n is the FV factor.

Present value of a single sum

PV = FV ÷ (1 + r)^n = FV × PV factor

PV factor = 1 ÷ (1 + r)^n. Always smaller than 1 for r > 0.

Relation between the factors

PV factor = 1 ÷ FV factor

Use it to get one factor when only the other is given in the table.

More than one compounding a year

FV = PV × (1 + r ÷ m)^(m × n)

r is the annual nominal rate, m is the number of compounding periods per year, n is years.

Effective annual rate

EAR = (1 + r ÷ m)^m − 1

Use it to compare rates compounded at different frequencies.

Future value of ordinary annuity

FV = A × [(1 + i)ⁿ − 1] ÷ i

A = payment per period, i = rate per period, n = number of periods. The bracket is the FV annuity factor.

Present value of ordinary annuity

PV = A × [1 − (1 + i)⁻ⁿ] ÷ i

The bracket is the PV annuity factor. Payments are at the end of each period.

Annuity due conversion

FV (due) = FV (ordinary) × (1 + i); PV (due) = PV (ordinary) × (1 + i)

Use it only when payments are at the start of each period.

Sinking fund payment

A = FV × i ÷ [(1 + i)ⁿ − 1]

i ÷ [(1 + i)ⁿ − 1] is the sinking fund factor, the reciprocal of the FV annuity factor.

Loan instalment (EMI)

A = P × i ÷ [1 − (1 + i)⁻ⁿ]

P = loan amount. The instalment is P divided by the PV annuity factor. This is the capital recovery factor method.

Rate and period matching

i = annual rate ÷ periods per year; n = years × periods per year

Example: 12% p.a. paid monthly gives i = 1% and n = 12 × years.

Present value of a perpetuity

PV = C ÷ r

C = fixed cash flow per period, r = discount rate per period. First cash flow comes at the end of period 1.

Present value of a growing perpetuity

PV = C1 ÷ (r − g)

C1 = cash flow at end of year 1, g = constant growth rate. Valid only when r > g.

Cash flow in year n of a growing perpetuity

Cn = C1 × (1 + g)^(n − 1)

Use it to find a later cash flow, or to get C1 from a given C0 using C1 = C0 × (1 + g).

Perpetuity due

PV = C + C ÷ r

Used when the first cash flow occurs immediately (at time 0).

Perpetuity starting later

PV today = [C ÷ r] ÷ (1 + r)^k

If the first flow comes at the end of year k + 1, the formula gives value at end of year k. Discount that value back k years.

Rule of 72

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

Use for annual compounding, best for rates of about 6% to 10%.

Rule of 69 (69.3)

Doubling period (years) ≈ 69.3 ÷ interest rate (% p.a.) ; or 0.693 ÷ r

Accurate for continuous compounding. For annual compounding it is a good approximation only at low rates, and it underestimates the period as the rate rises.

Exact doubling period

n = ln 2 ÷ ln(1 + r) = log 2 ÷ log(1 + r)

Comes from (1 + r)ⁿ = 2. Use when the question wants an exact figure.

Rate needed to double

Rate (%) ≈ 72 ÷ years

Reverse use of the Rule of 72.

Tripling period (extension)

n = ln 3 ÷ ln(1 + r)

Same logic with 3 in place of 2. A common rule is about 114 ÷ rate.

Present value factor

PVF = 1 ÷ (1 + r)^n

r is the rate per period and n is the number of periods. Use the table value if the question gives one.

PV of uneven cash flows

PV = C1 ÷ (1 + r)^1 + C2 ÷ (1 + r)^2 + ... + Cn ÷ (1 + r)^n

Each flow gets its own factor. Add the discounted values.

Net present value

NPV = PV of cash inflows − initial outlay

For outflows later in the project, subtract their present values too.

Interpolation for rate

r = r1 + [NPV1 ÷ (NPV1 − NPV2)] × (r2 − r1)

r1 gives positive NPV1 and r2 gives negative NPV2. Use the NPV values with their signs.

Interpolation using PV and outlay

r = r1 + [(PV1 − Outlay) ÷ (PV1 − PV2)] × (r2 − r1)

PV1 is at the lower rate r1 and PV2 is at the higher rate r2. Outlay lies between PV1 and PV2.

Interpolation for number of periods

n = n1 + [(PV1 − Target) ÷ (PV1 − PV2)] × (n2 − n1)

Use two whole-number periods whose PVs sit on either side of the target value.

Quick revision

• Time value of money: a rupee today is worth more than the same rupee later because it can earn a return.
• Simple interest = P × r × n; interest is only on the original principal.
• Compound interest: FV = PV × (1 + r)^n; interest is earned on interest too.
• With m compounding periods a year, use rate r ÷ m and periods n × m.
• Present value: PV = FV ÷ (1 + r)^n.
• Ordinary annuity: cash flows at period end; annuity due: cash flows at period start.
• FV of ordinary annuity = A × [((1 + r)^n − 1) ÷ r].
• PV of ordinary annuity = A × [1 − (1 + r)^−n] ÷ r.
• Perpetuity PV = A ÷ r; growing perpetuity PV = A1 ÷ (r − g), valid only when r > g, where A1 is the next cash flow.
• Rule of 72: doubling time ≈ 72 ÷ rate in percent; Rule of 69 (more precise for continuous compounding) uses 69 ÷ rate in percent.
• For uneven cash flows, discount each one separately and add the present values.
• Always check: rate and period must be in the same unit.

Common mistakes

• Listing only inflation as the reason for time value of money. Fix: Money would have time value even with zero inflation because it can earn interest. Always give all four reasons: interest, inflation, risk, liquidity.
• Adding cash flows from different years directly. Fix: Convert every amount to one date first. Only then add or compare.
• Using the annual rate with the number of periods in quarters, or the reverse. Fix: Always write r/m and m × n first. For 12% for 2 years compounded quarterly, use 3% and 8 periods.
• Giving the amount when compound interest is asked. Fix: Subtract the principal at the end. Compound interest = A − P.
• Using the annual rate and years when interest is compounded half-yearly or quarterly. Fix: Divide the rate by the number of compounding periods per year and multiply the years by the same number before looking up the factor.
• Multiplying by the PV factor when FV is asked, or the reverse. Fix: Decide the direction first. FV needs a factor above 1, and PV needs a factor below 1. Do the sense check at the end.
• Treating an annuity due as an ordinary annuity Fix: Circle the timing words first. For due, multiply the ordinary annuity answer by (1 + i).
• Using the annual rate with monthly payments Fix: Divide the rate by the number of periods per year and multiply the years by the same number before using any factor.
• Using the wrong cash flow in the growing perpetuity formula (using C0 instead of C1). Fix: Whenever the question says 'just paid' or 'current', multiply by (1 + g) first to get C1.
• Using the formula when growth rate is equal to or higher than the discount rate. Fix: Always check r > g before applying PV = C1 ÷ (r − g). Otherwise the formula does not give a valid value.

Exam tips

• For a 'why does money have time value' question, write four headed points with a one-line explanation each. This earns step marks.
• Add a small rupee example in theory answers. It shows application and makes the answer stand out.
• In MCQs, watch for options that say the time value exists only because of inflation. That is incomplete.
• In numericals, write the formula, the substitution and the final value with the unit. Do not skip the formula line.
• Link the concept to later chapters in one sentence if space allows: NPV, IRR and valuation all depend on discounting.
• Write the periodic rate and number of periods on separate lines before any calculation. Examiners give step marks for the setup even if arithmetic slips.
• In MCQs, check the frequency word (half-yearly, quarterly, monthly) first. Wrong options are usually built from the annual-compounding answer.
• When a question gives compounding frequency for one option and not for another, compare them on EAR and show both EARs.