NISM Certifications · NISM-Series-X-A: Investment Adviser (Level 1)

Time Value of Money: formula sheet

Full chapter guide

Key formulas

Future value (single sum): FV = PV × (1 + r)^n; r is the rate per period and n is the number of periods. Rate and period must match.
Present value (single sum): PV = FV ÷ (1 + r)^n; Discounting is the reverse of compounding. A higher r or longer n gives a lower PV.
Approximate real rate: Real rate ≈ nominal rate − inflation rate; A quick approximation. The exact form is (1 + nominal) ÷ (1 + inflation) − 1.
Core rule: Compare cash flows only at the same point in time; Never add or compare rupees from different dates directly.
Simple interest: SI = P × r × t; r is the annual rate as a decimal and t is in years. Convert months to years (9 months = 0.75 year).
Amount under simple interest: A = P × (1 + r × t); A is principal plus interest.
Compound amount (annual compounding): A = P × (1 + r)^t; Compound interest = A − P.
Compound amount (m times a year): A = P × (1 + r ÷ m)^(m × t); m = 2 for half-yearly, 4 for quarterly, 12 for monthly. Periodic rate is r ÷ m and number of periods is m × t.
Effective annual rate: EAR = (1 + r ÷ m)^m − 1; r is the nominal annual rate as a decimal. EAR is greater than r when m > 1 and r > 0.
Continuous compounding: A = P × e^(r × t); Limit as m becomes very large. e ≈ 2.71828. EAR = e^r − 1.
Future value of a lump sum: FV = PV × (1 + r)ⁿ; r is the rate per period as a decimal. n is the number of periods.
Present value of a lump sum: PV = FV ÷ (1 + r)ⁿ; Also written PV = FV × (1 + r)⁻ⁿ. The rate r is the discount rate per period.
Non-annual compounding: FV = PV × (1 + r ÷ m)^(m × t); r is the annual rate, m is compounding periods per year, t is years.
Solving for rate: r = (FV ÷ PV)^(1 ÷ n) − 1; Gives the rate per period. Use it for implied return questions.
Solving for number of periods: n = ln(FV ÷ PV) ÷ ln(1 + r); Rarely needed in full. Usually you test the options in the formula.
Future value of ordinary annuity: FV = PMT × [((1 + r)ⁿ − 1) ÷ r]; PMT is the payment per period, r the rate per period, n the number of periods. Payments at end of each period.
Present value of ordinary annuity: PV = PMT × [1 − (1 + r)⁻ⁿ] ÷ r; Gives value today of n equal end-of-period payments.
Annuity due conversion: FV (due) = FV (ordinary) × (1 + r); PV (due) = PV (ordinary) × (1 + r); Use the same PMT, r and n. Due value is always higher for r > 0.
Present value of perpetuity: PV = PMT ÷ r; First payment is one period from now. r must be the rate per payment period.
Growing perpetuity: PV = PMT₁ ÷ (r − g); PMT₁ is the first payment, one period away. Valid only if r > g.
Periodic rate: r = annual rate ÷ number of periods per year; For monthly payments, divide by 12 and use n in months.
Solving for the rate: r = (FV ÷ PV)^(1/n) − 1; For a single lump sum with annual compounding. This is also the CAGR formula.
Solving for the number of periods: n = ln(FV ÷ PV) ÷ ln(1 + r); Exam questions usually give options, so test them instead of using logs.
CAGR: CAGR = (Ending value ÷ Beginning value)^(1/years) − 1; n is the number of years between the two values, not the number of data points.
Net present value: NPV = Σ [CFt ÷ (1 + r)^t] − Initial outlay; Use the same r for all flows. Accept if NPV > 0.
IRR condition: NPV at IRR = 0; Accept if IRR is greater than the required rate of return.
Rule of 72: Years to double ≈ 72 ÷ rate (in %); Approximate. Rearranged, rate ≈ 72 ÷ years.
Real rate of return (Fisher): (1 + real) = (1 + nominal) ÷ (1 + inflation); The simple approximation is real ≈ nominal − inflation. Use the exact form when asked.

Quick revision

Money today is worth more than the same amount later because it can earn a return.
Simple interest: SI = P × r × t. Interest is only on the principal.
Compound interest: interest is also earned on earlier interest.
FV = PV × (1 + r)ⁿ; PV = FV ÷ (1 + r)ⁿ.
Higher rate or longer time means higher FV and lower PV.
For non-annual compounding, divide the rate by periods per year and multiply the years by periods per year.
Ordinary annuity: payments at the end of each period. Annuity due: at the start, so its value is higher.
Perpetuity PV = payment ÷ discount rate.
NPV = sum of discounted cash flows minus initial investment. Positive NPV means accept.
IRR is the discount rate at which NPV equals zero.
Rule of 72: years to double ≈ 72 ÷ annual rate in %. It is only an approximation.
X-A negative marking is 25% of the marks of the question, so avoid blind guesses on 2-mark items.

Common mistakes

Thinking TVM is only about inflation. Fix: Remember there are three drivers: inflation, opportunity cost and risk. Even with zero inflation, money today is worth more because it can earn a return.
Adding rupees received in different years. Fix: Bring each amount to the same date first, then add or compare.
Using the nominal rate directly when compounding is half-yearly, quarterly or monthly. Fix: Always divide the rate by m and multiply the years by m. Monthly at 12% means 1% for each of 12 periods per year.
Treating compound interest as the amount, or the amount as the interest. Fix: Read the last line of the question. If it asks for interest, subtract P from A.
Using the annual rate and number of years when compounding is half-yearly or quarterly. Fix: Divide the rate by m and multiply the years by m before using the formula.
Multiplying PV by (1 + r) × n instead of raising to power n. Fix: Compound means repeated multiplication. Use (1 + r)ⁿ. Simple interest uses 1 + r × n.
Using the annual rate with monthly payments. Fix: Divide the annual rate by 12 and use n as the number of months.
Treating an annuity due as ordinary. Fix: Underline the timing words. For due, multiply the ordinary result by (1 + r).
Using n as the number of data points instead of the number of years in CAGR Fix: Count the gaps. Values at the end of years 0 to 5 mean n = 5.
Forgetting to subtract 1 after raising to the power 1/n Fix: Always subtract 1 and then multiply by 100 to get the percentage.

Exam tips

Expect direct questions asking for the reasons behind TVM. Learn the three: inflation, opportunity cost and risk.
Watch for options with words like always or guaranteed. TVM depends on the rate, so these are often traps.
In numerical questions, check that the rate and the number of periods use the same time unit.
On a 2-mark caselet question a wrong answer costs more under negative marking, so eliminate by direction (forward or back) before guessing.
Look for the words "compounded" and the frequency in the question. A missed frequency is the most common trap.
Expect EAR comparison questions. The option with more frequent compounding at the same nominal rate has the higher EAR.
Treat the rate as annual unless stated otherwise, and check whether time is in months or years.
Wrong answers cost marks on X-A, and 2-mark caselet questions cost twice as much as 1-mark ones. Use the floor check to avoid guessing on calculation questions.