← Back to Index

Chapter 2: Time Value of Money

NISM-Series-X-A: Investment Adviser (Level 1) — Module 1

2.1 The Concept of Time Value of Money (TVM)

Money available today is worth more than the same amount in the future, because it has the potential to earn returns. Given a choice between Rs. 100 now and Rs. 100 after a month, every investor prefers it now — because of (a) an instinctive preference for current consumption, and (b) the ability to invest it and earn a return.

Did You Know — Equivalence Example

If Rs. 100 placed in a one-month deposit at 6% p.a. grows to Rs. 100.50, then receiving Rs. 100 now is equivalent to receiving Rs. 100.50 after one month — the investor must be compensated by Rs. 0.50 for waiting.

Similarly, Rs. 1,000 received after 3 years discounted at 5% is worth Rs. 864 today, but at 8% it is worth only Rs. 794 — illustrating that a higher discount rate lowers present value.

Key points

  • Future inflows are discounted by a discount/return/interest rate to find present value.
  • Present inflows are compounded at a compound interest rate to find future value.
  • The further in the future a cash flow occurs, the lower its present value.
  • The higher the discount rate, the lower the present value of future cash flows.

Important parameters in any TVM problem: (a) cash inflows/outflows (single, annuity, even/uneven streams), (b) rate of interest (compounding/discount/reinvestment rate), (c) time period (annual, monthly, quarterly), and (d) frequency of cash flows.

2.2.1 Present Value (PV)

Present value is the amount you would pay today for a cash flow that comes in the future — it brings a future value "down" to today's worth.

PV = FV / (1 + r)^n
For a one-time receipt: PV = C / (1 + r)^n For a regular cash flow (annuity): PV = C * [(1 - (1/(1+r)^n)) / r]

Where FV = Future Value, PV = Present Value, C = regular cash flow, r = rate of return per compounding period, n = number of compounding periods.

Example — Shyam's regular receipts

Shyam will receive Rs. 6,500 a year for 8 years at 7% interest. Summing each year's discounted value:

PV = 6500/1.07¹ + 6500/1.07² + ... + 6500/1.07⁷ = Rs. 38,813.44

Using the regular-receipt formula: PV = 6500 × [(1 − (1/1.07⁷))/0.07] = Rs. 38,813.44 (same answer). In Excel: PV(0.07, 8, -6500).

Example — Single future receipt

A future payment of Rs. 50,000 will be received after 5 years at 6% interest.

PV = 50000 / (1.06)^5 = Rs. 37,362.91

2.2.2 Future Value (FV)

FV = PV * (1 + r)^n

The compounding-period rate must be adjusted for frequency — e.g. 8% p.a. compounded quarterly means r = 8%/4 = 2% per quarter. Greater frequency of compounding means more "interest on interest" and higher returns.

Example — Krishna's Rs. 5 lakh deposit at 8% for 5 years

Scenario 1 (Simple interest, no compounding — interest withdrawn each year):
Interest = Rs. 5,00,000 × 8% × 5 = Rs. 2,00,000

Scenario 2 (Annual compounding, cumulative option):
Maturity value = 500,000 × (1.08)^5 = Rs. 7,34,664
Interest earned = 7,34,664 − 5,00,000 = Rs. 2,34,664

Scenario 3 (Quarterly compounding, cumulative, 20 quarters):
Maturity value = 500,000 × (1 + 8%/4)^20 = 500,000 × (1.02)^20 = Rs. 7,42,974
Interest earned = 7,42,974 − 5,00,000 = Rs. 2,42,974 (highest, due to higher compounding frequency)

Example — Excel FV function

Rs. 5,000 growing at 8% p.a. for 5 years (cash flow at the start of the period, type = 1):
FV(0.08, 5, , -5000, 1) ⇒ Rs. 7,346.64

2.2.3 Rate of Return / CAGR

The Compounded Annual Growth Rate (CAGR) is the underlying compound interest rate that equates the beginning value (PV) of an investment to its ending value (FV) over n years.

PV * (1 + r)^n = FV CAGR = ((End Value / Beginning Value)^(1/n)) - 1

Example — Rs. 100 grows to Rs. 120 in 2 years

120 = 100 × (1+r)^2 ⇒ CAGR = (120/100)^(1/2) − 1 = (1.2)^0.5 − 1 = 1.095 − 1 = 0.095 = 9.5%
In Excel: =RATE(2, , -100, 120) = 0.0954 or 9.54%

Example — Mutual fund: Rs. 10.50 → Rs. 12.25 in 3 years

CAGR = (12.25/10.50)^(1/3) − 1 = 5.27%

Example — Fractional period (450 days)

NAV Rs. 11 redeemed at Rs. 13.50 after 450 days (non-leap year, so n = 450/365 years).
CAGR = (13.50/11)^(365/450) − 1 = 0.1807 = 18.07%

CAGR is the accepted standard measure of investment return, except for periods of less than a year.

2.2.4 Periodic Investments / Pay-outs — PMT (EMI)

PMT(rate, nper, PV) — Excel function for Equated Instalments (e.g. EMI on a loan)

r = periodic interest rate, nper = number of repayment periods, PV = loan amount.

Example — Satish's home loan EMI

Loan = Rs. 30,00,000; rate = 6.5% p.a. (monthly reset, so r = 0.065/12); tenure = 20 years = 240 months.
PMT(0.065/12, 240, -3000000) = Rs. 22,367.19

If the rate falls to 6.25%: PMT(0.0625/12, 240, -3000000) = Rs. 21,927.85 — a fall of about Rs. 439 for a 0.25% rate cut.

2.2.5 Period of the Loan (NPER)

NPER(rate, PMT, PV) — number of periods to repay a loan given a fixed instalment

Example

Loan of Rs. 5,00,000 at 8% p.a. (monthly), EMI = Rs. 12,000.
NPER(0.08/12, -12000, 500000) = 48.97 months — i.e. the loan would be repaid in approximately 49 months. This helps check whether the chosen EMI is matched to an affordable repayment period.

2.2.6 Annuity

An annuity is a sum of money paid at regular intervals (monthly, quarterly, annually) — e.g. a pension. Annuities can be Fixed (fixed returns at regular intervals, e.g. a 5-year FD at 5.5% p.a.) or Floating (returns benchmarked to inflation, an index, or another reference rate as per the agreement).

Ordinary Annuity (payment at the END of each period)

PV = normal Present Value formula, with type = 0 (end of period)

Example 1

Annuity of Rs. 5,000/year at 10% for 4 years:
PV(0.1, 4, -5000), type 0 ⇒ Rs. 15,849.33

Example 2 — using an Annuity Table

Annuity of Rs. 12,000/year for 10 years at 5%:
PV(0.05, 10, -12000) = Rs. 92,660
Annuity-table factor for 10 years @ 5% = 7.7217 ⇒ PV = 12,000 × 7.7217 = Rs. 92,660 (same answer, two methods)

Annuity Due (payment at the START of each period)

PV(rate, nper, -pmt, , type=1) — type entered as 1 instead of 0/blank

Example — same Rs. 5,000/year at 10% for 4 years, paid at the start of each year

PV(0.1, 4, -5000, , 1) ⇒ Rs. 17,434.26

Note this is higher than the ordinary annuity value (Rs. 15,849.33) — receiving money earlier means it can be deployed/invested sooner, so an annuity due is more valuable to a receiver and more costly to a payer.

2.2.7 Perpetuity

A perpetuity is a constant cash flow from an investment that continues forever — e.g. perpetual bonds or pensions payable for a pensioner's lifetime.

PV = C/(1+r)^1 + C/(1+r)^2 + C/(1+r)^3 + ... = C / r

Where PV = Present Value, C = constant cash flow, r = discount rate.

Example — Perpetual bond

A perpetual bond pays Rs. 10,000 interest annually; discount rate = 8%.
PV = C/r = 10,000 / 0.08 = Rs. 1,25,000

Key Takeaways

  • Money has time value — a rupee today is worth more than a rupee tomorrow because it can earn returns; future cash flows are discounted, present amounts are compounded.
  • PV = FV / (1+r)^n and FV = PV(1+r)^n are the two foundational formulas; all others are derived from these.
  • CAGR = (End Value/Beginning Value)^(1/n) − 1 is the standard measure of investment return for periods of one year or more (also works for fractional years).
  • PMT calculates EMI, NPER calculates loan repayment period — both essential for loan planning.
  • Ordinary annuity (end-of-period payments, type=0) is worth less than an annuity due (start-of-period payments, type=1) for the same cash flow, rate and term.
  • Perpetuity PV = C/r — a simplification of an infinite-period annuity present-value series.
  • Always remember to convert annual rates and periods to the relevant compounding frequency (e.g. divide annual rate by 12 and multiply years by 12 for monthly calculations).

Practice Quiz

1. What is the present value of Rs. 50,000 to be received after 5 years, discounted at 6% per annum?

2. An investment of Rs. 100 grows to Rs. 120 in 2 years. What is its CAGR (approximately)?

3. For the same periodic cash flow, rate and number of periods, the present value of an annuity DUE compared to an ordinary annuity will be:

4. A perpetual bond pays Rs. 10,000 as annual interest. If the discount rate is 8%, what is its present value?

5. Krishna invests Rs. 5,00,000 at 8% p.a. for 5 years. Compared to annual compounding, quarterly compounding will result in: