Time Value of Money (TVM): Compounding, Discounting, Annuities and Present Value Basics

Overview

Time Value of Money (TVM) means a rupee today is worth more than a rupee tomorrow because money can earn interest/returns over time. TVM is the base for almost all finance decisions such as capital budgeting (NPV/IRR), valuation and loan repayments.

This chapter is scoring because formulas are standard—if you show the right working and units (rate + time), you get marks.

Learning objectives

You should be able to:

• Explain why TVM exists.
• Compute FV using compounding and PV using discounting.
• Convert between nominal rate and effective annual rate (EAR).
• Compute PV/FV of annuity (basic).
• Present answers in exam format with bold keywords, formula and a small table.

Why money has time value

Main reasons:

• Earning capacity: money can be invested to earn return.
• Inflation: purchasing power reduces over time.
• Risk/uncertainty: future money is uncertain.
• Preference for liquidity: people prefer money now.

So, we bring future cash flows to present using discounting.

Key terms

• PV (Present Value): value today of a future amount.
• FV (Future Value): value in future of an amount invested today.
• Interest rate (r): return per period (be careful: per year vs per month).
• Number of periods (n): total compounding/discounting periods.
• Discount factor (DF): $DF = \frac{1}{(1+r)^n}$
• Annuity: equal cash flow each period for a fixed number of periods.
• EAR: effective annual rate, actual yearly yield considering compounding frequency.

Future value (FV): compounding basics

Single sum compounding

If PV is invested at rate $r$ for $n$ periods: $FV = PV(1+r)^n$ Meaning: money grows because interest is earned on principal and (in compound interest) on prior interest.

Present value (PV): discounting basics

PV is the reverse of compounding. $PV = \frac{FV}{(1+r)^n} = FV \times DF$ Where:

• $r$ is discount rate per period,
• $n$ is number of periods,
• $DF$ is discount factor.

Interpretation: PV tells “how much you should pay today” to receive a future amount.

Compounding frequency and effective annual rate (EAR)

If a nominal annual rate is compounded $m$ times per year:

• Periodic rate = $\frac{r}{m}$
• Total periods = $n \times m$ (if $n$ years)

Effective annual rate: $EAR = \left(1+\frac{r}{m}\right)^m - 1$ Mini table:

Annuity basics: PV and FV of annuity

Annuity = equal payments each period.

FV of ordinary annuity

$FV_{ann} = A \times \frac{(1+r)^n - 1}{r}$

PV of ordinary annuity

$PV_{ann} = A \times \frac{1 - (1+r)^{-n}}{r}$ Where $A$ = annuity payment per period.

Note: For annuity due (payments at beginning), multiply by $(1+r)$ (basic adjustment).

Perpetuity (very basic)

A perpetuity is an annuity that continues forever. $PV_{perp} = \frac{A}{r}$ (Assuming payments start one period later; keep it basic unless asked.)

Discount factor table method (how to use)

In many exam problems, you use PVIF/PVIFA tables:

• PVIF = $\frac{1}{(1+r)^n}$ for single sum.
• PVIFA = $\frac{1 - (1+r)^{-n}}{r}$ for annuity.

Steps:

1. identify $r$ per period and $n$,
2. read factor from table,
3. multiply by cash flow.

Common exam mistakes (and how to avoid)

• Mismatch of units: annual rate with monthly periods. Fix: convert both to same period.
• Wrong sign: PV vs FV confusion. Fix: write formula first.
• Wrong annuity type: ordinary vs due. Fix: check “end of period” vs “beginning”.
• Using nominal rate as effective: fix by EAR when compounding frequency is given.

Mini worked examples

Example 1: PV of a future amount

Find PV of ₹10,000 received after 3 years at 10% p.a. $PV = \frac{10,000}{(1.10)^3} \approx \frac{10,000}{1.331} \approx 7,513$

Example 2: FV of an annuity (ordinary)

Annual deposit ₹5,000 for 4 years at 8%: $FV = 5,000 \times \frac{(1.08)^4 - 1}{0.08}$ Show the factor and multiply.

Quick recap (1-minute revision)

• TVM: ₹1 today > ₹1 tomorrow due to returns, inflation, risk, liquidity.
• FV: $PV(1+r)^n$; PV: $FV/(1+r)^n$.
• Convert compounding frequency using periodic rate $r/m$; compute EAR if needed.
• Annuity PV/FV use PVIFA/FVIFA formulas; annuity due = ×$(1+r)$.
• Always match the rate period with the number of periods.