Time Value of Money

1Why money has time value

Money today is worth more than the same amount later because:

• people instinctively prefer consumption now; and
• money today can be invested to earn a return.

Example ₹100 in a one-month deposit at 6% a year becomes ₹100.50. Receiving ₹100 now and ₹100.50 in a month are equivalent; ₹100 in a month is worth less.

Key points:

• Future cash flows are discounted at a discount rate to find present value; present amounts are compounded to find future value.
• The later a cash flow, the lower its present value.
• The higher the discount rate, the lower the present value: ₹1,000 in 3 years is worth about ₹864 today at 5%, but only about ₹794 at 8%.

The parameters in any TVM problem: cash flows (single, annuity, even or uneven), the rate (compounding, discount or reinvestment rate), the time period and the frequency of cash flows. Problems usually solve for one of these. (Interest on loans of seeds and animals goes back to around 5000 BC.)

Remember

• Later cash flow or higher discount rate → lower present value.

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2Present value

Present value (PV) is what a future cash flow is worth today.

Formula PV = FV ÷ (1 + r)ⁿ, where r = rate per compounding period and n = number of periods

Formula PV of a regular cash flow C = C × [1 − 1 ÷ (1 + r)ⁿ] ÷ r

Example ₹50,000 in 5 years at 6%: PV = 50,000 ÷ 1.06⁵ ≈ ₹37,363.

Example ₹6,500 a year for 8 years at 7%: PV = 6,500 × [1 − 1 ÷ 1.07⁸] ÷ 0.07 ≈ ₹38,813. In Excel: PV(rate, nper, pmt, [fv], [type]), i.e. PV(0.07, 8, −6500), with type 0 for end-of-period payments.

Example A client needs ₹5,00,000 in 5 years and can earn 10%: invest 5,00,000 ÷ 1.1⁵ ≈ ₹3,10,461 today.

Remember

• PV discounts; FV compounds.

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3Future value and compounding frequency

Formula FV = PV × (1 + r)ⁿ

Adjust the rate and periods for the compounding frequency: 8% a year compounded quarterly means 2% per quarter and 4 periods a year. (The Post Office Monthly Income Scheme pays monthly; the Senior Citizens' Savings Scheme quarterly.) More frequent compounding earns more interest on interest.

Example ₹5 lakh for 5 years at 8%.

• Interest paid out and spent (simple interest): 5,00,000 × 8% × 5 = ₹2,00,000.
• Compounded annually (cumulative): 5,00,000 × 1.08⁵ = ₹7,34,664 → interest ₹2,34,664.
• Compounded quarterly: 5,00,000 × 1.02²⁰ = ₹7,42,974 → interest ₹2,42,974.

In Excel: FV(rate, nper, [pmt], [pv], [type]): FV(0.08, 5, , −5000) = ₹7,346.64.

Example ₹1,00,000 at 8% for 3 years: 1,00,000 × 1.08³ ≈ ₹1,25,971.

The effective annual rate captures compounding: EAR = (1 + r ÷ m)^m − 1, where m = compounding periods per year. 12% compounded quarterly gives (1.03)⁴ − 1 ≈ 12.55%.

Remember

• More frequent compounding → higher maturity value.
• EAR = (1 + r/m)^m − 1.

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4Rate of return (CAGR)

The compound annual growth rate (CAGR) is the rate that grows the beginning value to the end value, assuming interim returns are reinvested.

Formula CAGR = (End value ÷ Beginning value)^(1/n) − 1

Example ₹100 becomes ₹120 in 2 years: (1.2)^(1/2) − 1 ≈ 9.5%. In Excel: RATE(2, , −100, 120) = 9.54%.

Example NAV ₹10.50 to ₹12.25 in 3 years: (12.25 ÷ 10.5)^(1/3) − 1 ≈ 5.27%.

Example NAV ₹11 to ₹13.50 in 450 days: (13.5 ÷ 11)^(365/450) − 1 ≈ 18.07% (fractional years work too).

CAGR is the standard measure of returns, except for periods under a year.

Goals must allow for inflation: a cost today grows at the inflation rate, FV = cost × (1 + inflation)ⁿ. A ₹10 lakh course rising 8% a year costs about 10 × 1.08¹⁰ ≈ ₹21.6 lakh in 10 years. The precise real rate of return is (1 + nominal) ÷ (1 + inflation) − 1: 10% nominal with 6% inflation gives 1.10 ÷ 1.06 − 1 ≈ 3.77% (not simply 4%).

Remember

• CAGR = (End ÷ Start)^(1/n) − 1.
• Real return = (1 + nominal) ÷ (1 + inflation) − 1.

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5Periodic payments (EMI)

The EMI on a loan is found with Excel's PMT(rate, nper, pv), using the monthly rate and number of months.

Example ₹30 lakh for 20 years at 6.5%: PMT(0.065/12, 240, −30,00,000) ≈ ₹22,367 a month. If the rate falls to 6.25%, the EMI falls to about ₹21,928.

This shows clients whether a loan is affordable and how rate changes affect payments.

Remember

• Divide the annual rate by 12 and multiply years by 12 for monthly loans.

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6Loan period (NPER)

NPER(rate, pmt, pv) gives the number of payments needed to repay a loan.

Example ₹5 lakh at 8% with an EMI of ₹12,000: NPER(0.08/12, −12000, 500000) ≈ 49 months.

The period is then matched with an affordable EMI.

Remember

• NPER solves for time; PMT solves for the instalment.

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7Annuities

An annuity is a series of regular payments (monthly, quarterly, yearly), such as a pension.

• Fixed annuity: fixed payments (a deposit paying 5.5% for 5 years).
• Floating annuity: payments linked to inflation, an index or another benchmark.

Annuities are widely used to provide retirement income.

Ordinary annuity: payments at the end of each period (most annuities, such as annual bond interest).

Example ₹5,000 a year for 4 years at 10%: PV(0.1, 4, −5000) ≈ ₹15,849. ₹12,000 a year for 10 years at 5%: annuity factor 7.7217 → 12,000 × 7.7217 ≈ ₹92,660.

Annuity due: payments at the start of each period (type = 1 in Excel), so each payment is received a period earlier and is worth more.

Example ₹5,000 a year for 4 years at 10%, paid at the start: PV(0.1, 4, −5000, , 1) ≈ ₹17,434, compared with ₹15,849 for the ordinary annuity.

An annuity due is better for the receiver (money comes earlier) and costlier for the payer. PV of an annuity due = PV of the ordinary annuity × (1 + r).

Remember

• Ordinary annuity: end of period. Annuity due: start of period, higher value.

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8Perpetuity

A perpetuity pays a constant cash flow forever, as with perpetual bonds (and, loosely, lifelong pensions).

Formula PV of a perpetuity = C ÷ r

Example A perpetual bond paying ₹10,000 a year at an 8% discount rate is worth 10,000 ÷ 0.08 = ₹1,25,000.

Remember

• Perpetuity value = cash flow ÷ discount rate.

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