Why time has a value

Would you rather have ₹100 today or ₹100 a year from now? Today, obviously. You could invest the ₹100 right now and have more than ₹100 by next year. That intuition — that money today is worth more than the same amount in the future — is the time value of money (TVM). It underpins everything in finance: loan EMIs, bond pricing, capital budgeting, retirement planning, even insurance products.

Future Value of a single amount

If you invest ₹1,000 today at 10% per year, in five years you’ll have ₹1,000 × (1.10)^5 = ₹1,611. The formula:

• PV — present value (today’s amount)
• r — rate per period
• n — number of periods

The exponent does the heavy lifting. ₹1,000 at 10% for 20 years = ₹6,727. At 30 years = ₹17,449. Compounding gets dramatic with time.

Present Value of a single amount

The reverse question: how much do I need today to have a target amount in the future? Discount the future amount back at the same rate:

Need ₹10,00,000 for a child’s higher education in 12 years? At 10% return, you need ₹10,00,000 ÷ (1.10)^12 = ₹3,18,631 invested today. The longer you wait, the more you need to save; the higher the return, the less.

Ordinary annuity — recurring payments

An annuity is a series of equal payments at regular intervals. SIP investments, EMI loans, rent payments — all annuities. Two variants:

• Ordinary annuity — payments at the END of each period. EMIs typically work this way.
• Annuity due — payments at the BEGINNING of each period. Rent typically works this way.

The formulas:

Three TVM scenarios you’ll actually use

Scenario 1 — Retirement corpus. Investing ₹10,000/month for 30 years at 12% annual (1% per month).

FV = 10,000 × [((1.01)^360 − 1) ÷ 0.01] = ₹3.5 crore. The same ₹36 lakh of contributions grows to ₹3.5 crore through compounding.

Scenario 2 — Loan EMI. ₹50 lakh home loan, 8.5% per annum, 20 years (240 months at 0.708% per month).

EMI = 50,00,000 ÷ [(1 − (1.00708)^-240) ÷ 0.00708] = ₹43,391. Total paid over 20 years ≈ ₹1.04 crore — more than double the principal.

Scenario 3 — Bond pricing. Bond pays ₹50 every six months for 10 years (20 coupons) and ₹1,000 at maturity. Market yield 12% per year (6% per period).

Price = PV of coupons + PV of face value = 50 × [PVA factor at 6%, 20 periods] + 1,000 × [PV factor at 6%, 20 periods] = 50 × 11.470 + 1,000 × 0.312 = ₹885.50. Bond trades at a discount because coupon (10%) is below market (12%).

Three rules of thumb

• Rule of 72. Money doubles in roughly 72 ÷ r years. At 12%, doubles every 6 years. At 8%, every 9 years.
• Higher rate = bigger gap between PV and FV. 5% over 10 years multiplies by 1.63. 15% over 10 years multiplies by 4.05.
• Frequency matters. Monthly compounding beats annual compounding at the same nominal rate. Always check whether quoted rates are annual, semi-annual, monthly, or daily compounded.

Bonus: perpetuity

An annuity that lasts forever. Used to value preferred shares (fixed dividend forever), terminal value in DCF, certain real-estate cash flows.

₹100 forever, discounted at 10%, is worth ₹1,000 today. Notice how compact the formula is — that’s because n → infinity collapses the geometric series.

Lesson recap

• Money today is worth more than money tomorrow because of opportunity to earn return.
• FV = PV × (1+r)^n compounds forward; PV = FV ÷ (1+r)^n discounts back.
• Annuities are streams of equal payments; ordinary = end of period, due = start.
• Rule of 72 estimates doubling time; perpetuity uses PMT ÷ r.
• Always confirm the compounding frequency before applying any formula.