Time value of money

Commerce (General / Accounting and Finance / Banking and Insurance) (B.Com) · Finance and banking · free study notes for exams, placements and interviews

In short

Time value of money (TVM) is the principle that a rupee today is worth more than a rupee later, because today's rupee can be invested to earn a return, and because inflation and risk erode future receipts. Every cash flow is therefore moved to a common date — compounded forward to a future value or discounted back to a present value — before amounts of different dates are compared or added. TVM is the engine under almost all of finance: NPV and IRR in capital budgeting, bond and share valuation, EMI and loan amortisation, SIP and retirement planning, lease-versus-buy decisions and insurance pricing. It is a guaranteed scoring chapter in B.Com Financial Management papers, and placement tests at banks, NBFCs, Big-4 and analytics firms test it through EMI, NPV, CAGR and effective-rate sums.

Explained

Why money has a time value

Investment opportunity (the core reason): cash in hand can be deployed at a positive return, so receiving it later means a lost return. This holds even with zero inflation and zero risk.
Inflation: purchasing power falls over time; ₹100 buys less a year later. India's CPI inflation target under the RBI's flexible inflation-targeting framework is 4% with a ±2% band (reviewed periodically, most recently extended in 2026 — check the current notification).
Risk and uncertainty: a promised future receipt may not arrive; the premium for bearing that risk raises the discount rate.
Preference for present consumption: people are impatient; they must be paid to wait.
Consequence: cash flows at different dates are not comparable until adjusted. "₹1 lakh today versus ₹1.2 lakh in 3 years" cannot be answered without a rate.

Compounding and discounting

Compounding moves money forward: interest earns interest. Discounting moves money backward; the two are mathematical inverses (PV = FV / (1+i)ⁿ).
Simple interest pays interest only on principal: SI = P × r × n. Used for short-term bills, some overdraft calculations, savings-bank interest on daily balances (paid out, not compounded within the quarter).
Compound interest pays interest on principal plus accumulated interest — the basis of all TVM work.
Frequency of compounding: annual, semi-annual, quarterly, monthly, daily, continuous. For the same nominal rate, more frequent compounding gives a higher effective return. Indian bank term deposits usually compound quarterly; Indian G-secs pay semi-annual coupons (many corporate bonds pay annually); loan EMIs compound monthly.
Nominal (stated/annual percentage) rate vs effective annual rate (EAR): the EAR is what you actually earn/pay over a year. Banks must disclose the annualised yield on deposits and the annual percentage rate (APR) on retail loans under RBI fair-practice and Key Fact Statement (KFS) rules — the KFS, mandatory for retail and MSME loans from October 2024, must show the all-inclusive APR and the amortisation schedule.
Discount factor (DF) = 1/(1+i)ⁿ; present value tables (PVF, PVAF, FVF, FVAF) are supplied in most university exams — read the right table and the right row/column.
Real vs nominal rate (Fisher relation): (1 + nominal) = (1 + real)(1 + inflation). Discount nominal (money) cash flows at the nominal rate and real cash flows at the real rate — never mix.

Annuities, annuity due and perpetuity

Annuity: a series of equal cash flows at equal intervals. Ordinary (deferred) annuity — flows at the end of each period (EMIs on most loans, bond coupons). Annuity due — flows at the beginning of each period (rent paid in advance, most SIP instalments, lease rentals payable in advance, LIC premiums).
Conversion rule: any annuity-due value = corresponding ordinary-annuity value × (1 + i). One extra period of compounding/one less of discounting.
Perpetuity: an annuity that never ends; PV = A / i. Used for irredeemable preference shares, perpetual (AT-1) bonds, no-growth equity, and terminal value in valuation.
Growing annuity / growing perpetuity: cash flow grows at g per period. Growing perpetuity PV = A₁/(i − g), valid only if i > g — the Gordon growth model's backbone.
Deferred annuity: flows start after m periods — value it as an annuity, then discount that lump sum back m periods (or use PVAF(n+m) − PVAF(m)).
Uneven cash flows: no shortcut — discount each flow separately and add (the basis of NPV).

Loan amortisation, EMI and sinking funds

EMI is the annuity payment that exactly repays principal plus interest over the tenure, derived by setting PV of EMIs = loan amount.
Amortisation schedule: each EMI = interest on the opening balance + principal repayment. Early EMIs are mostly interest; later EMIs mostly principal. This is why prepaying early saves far more interest than prepaying late, and why a 20-year home loan's total interest can approach or exceed the principal.
Floating-rate loans in India: retail floating loans are benchmarked to an external benchmark (usually the RBI repo rate) under the EBLR regime since October 2019, or to the bank's MCLR for older/other loans. When the benchmark moves, banks reset either the EMI or the tenure — RBI's 2023 framework requires borrowers be given the option and be told the impact.
Sinking fund factor: the periodic deposit needed to accumulate a target sum (debenture redemption, asset replacement). A = FV × i/[(1+i)ⁿ − 1].
Capital recovery factor: the periodic amount a present sum can support (annual equivalent cost, loan EMI, pension drawdown). A = PV × i/[1 − (1+i)⁻ⁿ].
Prepayment / moratorium: during a moratorium, interest usually continues to accrue and is capitalised, so the outstanding grows — a standard interview trap.

Applying TVM: NPV, IRR and yields

NPV = Σ CFt/(1+k)^t − initial outlay; accept if NPV > 0. The discount rate k is the opportunity cost of capital.
IRR is the rate at which NPV = 0 — found by trial and error with interpolation in exams, by XIRR/IRR in Excel in practice.
Doubling shortcuts: Rule of 72 — years to double ≈ 72/r (r in %); Rule of 69 (69/r + 0.35) is closer for continuous compounding.
CAGR = (End/Begin)^1/n − 1: the single compound rate that links two values — the standard growth metric in Indian annual reports and equity research.
Where accounting standards force TVM: Ind AS 109 (amortised cost and effective interest rate on financial assets/liabilities), Ind AS 116 (lease liability = PV of lease payments), Ind AS 19 (present value of defined-benefit/gratuity obligations), Ind AS 37 (discounting long-term provisions), Ind AS 36 (value in use = PV of future cash flows). AS 19 (finance leases), AS 15 (employee benefits) and AS 28 (impairment) play similar roles for companies still on Indian GAAP.
Taxation note: TVM comparisons should use post-tax cash flows and post-tax rates; interest income is taxable at slab rates and TDS applies on bank interest above the prescribed threshold (thresholds change in Finance Acts — verify the current limit).

Limitations and practical cautions

The answer is only as good as the discount rate — a small change in k swings long-dated NPVs sharply.
Assumes cash flows and their timing are known; real projects are uncertain, so sensitivity, scenario and probability-adjusted analysis are layered on top.
Mid-period flows: textbook formulas assume year-end flows; practice uses mid-year convention or monthly discounting.
Reinvestment assumption: IRR implicitly assumes intermediate flows are reinvested at the IRR itself — MIRR corrects this.
TVM ignores non-financial factors (strategic fit, statutory compliance, social value) that still decide investments.

Key points

Time value of money — a rupee today is worth more than a rupee tomorrow, because of return, inflation and risk.
Compounding — moving a cash flow forward: FV = PV(1+i)ⁿ; interest earns interest.
Discounting — moving a cash flow backward: PV = FV/(1+i)ⁿ; the inverse of compounding.
Discount factor — 1/(1+i)ⁿ; always less than 1 for positive i, and falls as n or i rises.
Nominal vs effective rate — EAR = (1 + r/m)^m − 1; quarterly compounding of 12% gives 12.55%, monthly 12.68%.
Ordinary annuity — equal flows at period ends; most loan EMIs and bond coupons.
Annuity due — equal flows at period starts; multiply the ordinary-annuity answer by (1+i).
Perpetuity — PV = A/i; irredeemable preference shares, perpetual bonds, terminal value.
Growing perpetuity — PV = A₁/(i − g), only when i > g; basis of the Gordon dividend model.
Sinking fund factor — i/[(1+i)ⁿ − 1]; deposit needed per period to reach a target sum.
Capital recovery factor — i/[1 − (1+i)⁻ⁿ]; the EMI per rupee of loan, reciprocal of PVAF.
Amortisation — EMI split into interest on opening balance plus principal; interest share falls over time.
Rule of 72 — money doubles in about 72/r years; at 12%, about 6 years.
CAGR — (End/Begin)^1/n − 1; the compound rate linking two dated values.
Fisher relation — (1+nominal) = (1+real)(1+inflation); never discount real flows at a nominal rate.
NPV rule — accept if PV of inflows exceeds outflow at the opportunity cost of capital.
Ind AS link — Ind AS 116 lease liabilities, Ind AS 109 effective interest rate and Ind AS 19 gratuity obligations are all PV computations.
Deferred annuity — value as an annuity at its start date, then discount that value back to today.

Formulas and examples

FV of a single sum: FV = PV × (1 + i)ⁿ — PV = present amount (₹), i = interest rate per period (decimal), n = number of periods
PV of a single sum: PV = FV / (1 + i)ⁿ = FV × (1 + i)⁻ⁿ
Non-annual compounding: FV = PV × (1 + r/m)^m×n — r = nominal annual rate, m = compoundings per year, n = years
Effective annual rate: EAR = (1 + r/m)^m − 1; continuous: EAR = e^r − 1, FV = PV × e^r×n
FV of ordinary annuity: FVA = A × [(1 + i)ⁿ − 1] / i — A = equal cash flow per period (₹)
PV of ordinary annuity: PVA = A × [1 − (1 + i)⁻ⁿ] / i
Annuity due: FV(due) = FVA × (1 + i); PV(due) = PVA × (1 + i)
Perpetuity: PV = A / i; Growing perpetuity: PV = A₁ / (i − g), i > g
Sinking fund deposit: A = FV × i / [(1 + i)ⁿ − 1]
Capital recovery / EMI: EMI = P × i × (1 + i)ⁿ / [(1 + i)ⁿ − 1] — i = monthly rate = annual rate/12, n = months
Outstanding balance after k EMIs: B(k) = EMI × [1 − (1 + i)^−(n−k)] / i
NPV: NPV = Σ (CFt / (1 + k)^t) − C₀; IRR: the k that makes NPV = 0
CAGR: CAGR = (FV / PV)^1/n − 1; Rule of 72: doubling years ≈ 72 / r (r in % per year)
Fisher: (1 + nominal) = (1 + real) × (1 + inflation)

Example 1 — Compounding, effective rate and Rule of 72 ₹2,00,000 is deposited for 3 years at 10% nominal, compounded quarterly.

i = 10%/4 = 0.025 per quarter; n = 12 quarters.
(1.025)¹² = 1.344889 → FV = 2,00,000 × 1.344889 = ₹2,68,977.80.
EAR = (1.025)⁴ − 1 = 1.103813 − 1 = 10.38% (against 10% annual compounding, which would give only ₹2,66,200).
Doubling at 10.38% effective: ln2/ln(1.1038) ≈ 7.0 years; Rule of 72 estimate = 72/10.38 ≈ 6.9 years — close enough for mental maths.

Example 2 — SIP: annuity due future value (and why "due" matters) ₹5,000 invested at the beginning of every month for 10 years, expected return 12% p.a. compounded monthly.

i = 0.01 per month, n = 120. (1.01)¹²⁰ = 3.300387.
Ordinary annuity FV = 5,000 × (3.300387 − 1)/0.01 = 5,000 × 230.0387 = ₹11,50,193.
Annuity due FV = 11,50,193 × 1.01 = ₹11,61,695 — about ₹11,500 more just from paying at month-start.
Total invested = ₹6,00,000, so the gain is ₹5,61,695 — the compounding effect, not a guaranteed return.

Example 3 — Home loan EMI, total interest and amortisation Loan ₹40,00,000 at 9% p.a. for 20 years (monthly rests).

i = 0.09/12 = 0.0075; n = 240. (1.0075)²⁴⁰ = 6.009152.
EMI = 40,00,000 × 0.0075 × 6.009152 / (6.009152 − 1) = 1,80,274.56 / 5.009152 = ₹35,989.04.
Total paid = 35,989.04 × 240 = ₹86,37,370 → total interest ₹46,37,370, more than the principal borrowed.
First month: interest = 40,00,000 × 0.0075 = ₹30,000; principal repaid = 35,989.04 − 30,000 = ₹5,989.04 only.
Balance after 10 years (120 EMIs): (1.0075)¹²⁰ = 2.451357 → B = 35,989.04 × [1 − 1/2.451357]/0.0075 = 35,989.04 × 78.9417 = ₹28,41,034 — only about 29% of principal repaid in half the tenure.

EMI no.	Opening balance (₹)	Interest @0.75% (₹)	Principal (₹)	Closing balance (₹)
1	40,00,000	30,000	5,989	39,94,011
2	39,94,011	29,955	6,034	39,87,977
3	39,87,977	29,910	6,079	39,81,898
120	28,55,606	21,417	14,572	28,41,034
240	35,721	268	35,721	0

Example 4 — PV of uneven flows, NPV and IRR A machine costs ₹10,00,000 and yields ₹3,00,000, ₹4,00,000, ₹4,00,000 and ₹3,00,000 at the end of years 1-4. Cost of capital 12%.

Year	Cash flow (₹)	DF @12%	PV (₹)	DF @18%	PV (₹)
1	3,00,000	0.8929	2,67,870	0.8475	2,54,250
2	4,00,000	0.7972	3,18,880	0.7182	2,87,280
3	4,00,000	0.7118	2,84,720	0.6086	2,43,440
4	3,00,000	0.6355	1,90,650	0.5158	1,54,740
		Total PV	10,62,120	Total PV	9,39,710

NPV @12% = 10,62,120 − 10,00,000 = +₹62,120 → accept.
NPV @18% = 9,39,710 − 10,00,000 = −₹60,290.
IRR by interpolation = 12 + 6 × 62,120/(62,120 + 60,290) = 12 + 6 × 0.5075 = ≈15.0% > 12%, so accept. (Interpolation is approximate because NPV is non-linear in k.)

Example 5 — Deferred annuity, perpetuity and real-rate check A retirement plan pays ₹2,00,000 a year for 15 years, the first payment 6 years from today. Discount rate 8%.

PVAF(8%, 15) = [1 − 1.08⁻¹⁵]/0.08; 1.08¹⁵ = 3.172169 → [1 − 0.315242]/0.08 = 8.5595.
Value at year 5 (one period before the first payment) = 2,00,000 × 8.5595 = ₹17,11,900.
Discount 5 years: 1.08⁵ = 1.469328 → PV = 17,11,900/1.469328 = ₹11,65,090.
If instead the ₹2,00,000 were perpetual from year 6, value at year 5 = 2,00,000/0.08 = ₹25,00,000 → PV = 25,00,000/1.469328 = ₹17,01,458.
With 5% inflation, the real rate = 1.08/1.05 − 1 = 2.857%: the same nominal ₹2,00,000 stream in today's purchasing power is far smaller, which is why pension planning uses inflation-indexed (growing) annuities.

Factor	Formula	Use
FVF(i,n)	(1+i)ⁿ	Grow one rupee
PVF(i,n)	(1+i)⁻ⁿ	Discount one rupee
FVAF(i,n)	[(1+i)ⁿ − 1]/i	Grow ₹1 per period
PVAF(i,n)	[1 − (1+i)⁻ⁿ]/i	Discount ₹1 per period
Sinking fund	i/[(1+i)ⁿ − 1]	Deposit to reach ₹1
Capital recovery	i/[1 − (1+i)⁻ⁿ]	EMI per ₹1 of loan

Where it is used

Retail banking and NBFCs: EMI quotation, Key Fact Statement APR disclosure, amortisation schedules, prepayment and foreclosure calculations, deposit maturity values (quarterly compounding) and loan-restructuring NPV tests — roles like credit officer, loan processing, product pricing.
Corporate finance and FP&A: capital budgeting (NPV, IRR, payback, discounted payback), lease-versus-buy, equipment replacement using equivalent annual cost — at Tata, L&T, Reliance, Maruti and most manufacturing treasuries.
Investment banking, equity research and valuation: DCF models, terminal value as a growing perpetuity, bond pricing and yield-to-maturity, dividend discount models — analyst roles at Kotak, ICICI Securities, Axis Capital, Big-4 valuation teams.
Insurance and actuarial work: premium and surrender-value calculation, annuity/pension product pricing (LIC Jeevan Akshay-type immediate annuities), reserving — all built on PV of expected payouts.
Personal finance and wealth management: SIP and goal planning, EPF/PPF/NPS accumulation, Sukanya Samriddhi and Senior Citizens' Savings Scheme maturity values, retirement corpus and withdrawal (SWP) planning; CAGR and XIRR for portfolio returns.
Accounting and audit: Ind AS 116 lease liabilities, Ind AS 109 effective interest rate and amortised cost, Ind AS 19 gratuity valuations, Ind AS 36 impairment testing — core work in statutory audit and financial reporting.

Interview questions

Q: Why is a rupee today worth more than a rupee a year later? A: Because today's rupee can be invested to earn a return, inflation reduces the purchasing power of the future rupee, and the future receipt carries risk of non-payment.
Q: What is the difference between an ordinary annuity and an annuity due? A: Ordinary annuity cash flows occur at the end of each period, annuity due at the beginning; the annuity-due value equals the ordinary-annuity value multiplied by (1 + i).
Q: A bank quotes 8% compounded quarterly and another 8.2% compounded annually — which is better for a depositor? A: 8% quarterly gives EAR = (1.02)⁴ − 1 = 8.24%, marginally better than 8.2%, so compare on effective rates, never nominal.
Q: Why is the interest component of an EMI highest in the first month? A: Interest is charged on the outstanding principal, which is largest at the start; as principal is repaid the interest shrinks and the principal component of the fixed EMI grows.
Q: If a borrower gets a 3-month moratorium, does it come free? A: No — interest normally accrues and is added to the outstanding, so either the EMI or the tenure increases and total interest rises.
Q: What discount rate should be used for a project? A: The opportunity cost of capital for cash flows of that risk — typically the WACC for a project of average firm risk, adjusted upward for riskier projects.
Q: How do you handle inflation in a TVM calculation? A: Be consistent — discount nominal cash flows at the nominal rate, or inflation-adjusted (real) cash flows at the real rate given by (1+nominal)/(1+inflation) − 1.
Q: Why must i be greater than g in a growing perpetuity? A: If g ≥ i the series does not converge and the formula gives a negative or infinite value — growth above the discount rate cannot be sustained forever.
Q: What is the practical difference between CAGR and IRR? A: CAGR uses only the start and end values of a single investment, while IRR accounts for every intermediate cash flow and its timing; for irregular dates XIRR is used.
Q: Prepaying ₹5 lakh in year 2 or year 15 of a 20-year home loan — which saves more interest? A: Year 2, because the prepaid principal would otherwise have attracted interest for 18 more years; late prepayment avoids far less interest.
Q: Where does TVM appear in financial statements? A: In lease liabilities (Ind AS 116), amortised cost and effective interest rate (Ind AS 109), gratuity and pension obligations (Ind AS 19), long-term provisions (Ind AS 37) and impairment value in use (Ind AS 36).
Q: Two projects have the same NPV but one recovers cash earlier — which do you prefer? A: The earlier-recovery project, as it has lower liquidity and reinvestment risk; this is why discounted payback and project duration are reported alongside NPV.

Common mistakes

Using the annual rate with monthly periods (or vice versa) in EMI and SIP sums — always convert rate and n to the same period.
Treating an annuity due as an ordinary annuity, losing the (1 + i) adjustment — common in SIP, rent and lease questions.
Discounting a deferred annuity by n periods instead of to the period just before the first cash flow, then discounting that value back.
Confusing simple and compound interest, or comparing a quarterly-compounded deposit with an annually-compounded one on nominal rates.
Mixing real cash flows with a nominal discount rate, which double-counts or ignores inflation.
Reading the wrong table — PVF instead of PVAF (or using PVAF for a single lump sum) — and rounding four-decimal factors too early, which distorts large-rupee answers.
Forgetting that outflows are negative in NPV, or including the initial outlay again inside the discounted stream.