29 Cost of capital and time value of money

29.1 Two Foundational Ideas

This topic combines two of the most-tested ideas in business finance: the time value of money — the foundation of every valuation in finance — and the cost of capital — the discount rate at which those valuations are made. A rupee tomorrow is worth less than a rupee today because (a) it could have been invested to earn a return, (b) inflation erodes its purchasing power, and (c) uncertainty about whether it will materialise. The discount rate that reflects these considerations is, for the firm, its cost of capital.

29.2 Time Value of Money — Compounding and Discounting

29.2.1 Future Value (Compounding)

If ₹P is invested at an annual rate $r$ for $n$ years: \[FV = P (1 + r)^n\]

With m compoundings per year: \[FV = P \left(1 + \frac{r}{m}\right)^{mn}\]

Under continuous compounding: $FV = P e^{rn}$.

29.2.2 Present Value (Discounting)

Reverse the process: \[PV = \frac{FV}{(1 + r)^n}\]

The fraction $\frac{1}{(1+r)^n}$ is the present value factor (PVF) at rate $r$ for $n$ periods.

29.2.3 Annuity

An annuity is a series of equal cash flows for a fixed period.

Annuity Formulae

Type	Formula	Notes
PV of ordinary annuity	$PV = A \times \frac{1 - (1+r)^{-n}}{r}$	Cash at end of period
FV of ordinary annuity	$FV = A \times \frac{(1+r)^n - 1}{r}$	Cash at end of period
PV of annuity due	$PV_{\text{due}} = PV_{\text{ord}} \times (1 + r)$	Cash at start of period
PV of perpetuity	$PV = \frac{A}{r}$	Cash flow forever
PV of growing perpetuity	$PV = \frac{A}{r - g}$, with $r > g$	Gordon growth model

29.2.4 Effective vs Nominal Rate

If nominal rate is $r$ with $m$ compoundings per year, the effective annual rate (EAR) is:

\[\text{EAR} = \left(1 + \frac{r}{m}\right)^m - 1\]

29.3 Cost of Capital — Concept

The cost of capital is “the minimum rate of return the firm must earn on its investments to maintain the market value of equity” (Solomon Ezra). Two equivalent definitions:

Definitions

From investor side — required rate of return given the risk class.
From firm side — opportunity cost of using capital.
Operationally — the discount rate used in NPV calculations.

29.4 Cost of Specific Sources

29.4.1 Cost of Debt

Cost of Debt (K_d) Formulae

Before tax: $K_d = I / P$ where I = interest, P = market price (or issue price).
After tax: $K_d (\text{post-tax}) = K_d (1 - T)$ where T = tax rate.
Issued at premium/discount: $K_d = \frac{I + (F - P)/n}{(F + P)/2}$ (yield-to-maturity approximation).

The post-tax cost is what matters because interest is tax-deductible — debt creates a tax shield.

29.4.2 Cost of Preference Shares

\[K_p = \frac{D_p + (F - P)/n}{(F + P)/2}\]

where D_p = preference dividend; F = face value; P = net proceeds; n = redemption period.

Note: preference dividend is not tax-deductible; some jurisdictions levy Dividend Distribution Tax — to be added.

29.4.3 Cost of Equity

Three Methods for Cost of Equity

Method	Formula	Notes
Dividend Yield (no growth)	$K_e = D / P_0$	Suitable when dividends constant
Gordon Growth Model	$K_e = (D_1 / P_0) + g$	Constant growth at rate g
CAPM	$K_e = R_f + \beta (R_m - R_f)$	Sharpe (1964); risk-adjusted
Bond yield + Risk premium	$K_e = K_d + $ Risk premium	Rough approximation
Earnings Price (E/P)	$K_e = E_1 / P_0$	When no growth and full payout

29.4.4 Cost of Retained Earnings

$K_r$ = $K_e (1 - t_p)(1 - b)$ where $t_p$ = personal tax rate of shareholder; $b$ = brokerage. In practice $K_r ≈ K_e$ — opportunity cost of foregone external alternative.

29.5 Weighted Average Cost of Capital (WACC)

The firm’s overall cost of capital weights each source by its market-value proportion:

\[\text{WACC} = w_e K_e + w_p K_p + w_d K_d (1 - T) + w_r K_r\]

with $\sum w_i = 1$.

WACC — Practical Notes

Use market values, not book values, for weights.
Use post-tax K_d.
WACC is the appropriate discount rate for projects of the same risk as the firm; not for projects of different risk classes.
A target capital structure may be used in place of current weights.

Distractor warning

PYQs often ask: which sources of capital have a tax shield? Only debt interest has a tax shield. Preference dividends and equity dividends do not.

29.6 Marginal Cost of Capital

The marginal cost of capital (MCC) is the cost of raising one more rupee of capital. As the firm crosses certain “break points” (e.g., exhausts cheap retained earnings, has to issue new equity with flotation cost, or moves to higher-coupon debt), MCC rises in steps. The graph relating MCC to cumulative capital raised is the MCC schedule.

29.7 Time Value Worked Numerical Examples

Three Quick Computations

Problem	Working	Answer
₹10,000 invested at 8 % for 3 years	10,000 × (1.08)^3	₹12,597
PV of ₹50,000 in 5 years at 10 %	50,000 / (1.10)^5	₹31,046
PV of perpetual cash flow of ₹6,000 at 12 %	6,000 / 0.12	₹50,000

flowchart LR
  TVM[Time Value of Money] --> FV[Future Value<br/>Compounding]
  TVM --> PV[Present Value<br/>Discounting]
  TVM --> AN[Annuities &<br/>Perpetuities]
  TVM --> EAR[Effective Annual Rate]
  COC[Cost of Capital] --> KD[K_d post-tax]
  COC --> KP[K_p]
  COC --> KE[K_e — DG, CAPM]
  COC --> WACC[WACC<br/>market-value weights]
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

29.8 Importance of Cost of Capital

Why Cost of Capital Matters

Investment decisions — accept projects where IRR > WACC; reject otherwise.
Capital-structure decisions — choose mix that minimises WACC.
Dividend decisions — retain or pay out depending on alternative-use return vs K_e.
Performance evaluation — Economic Value Added (EVA) = NOPAT − (WACC × Capital Employed).
Project appraisal — discount rate in NPV.

29.9 Practice Questions

Q 01 FV Easy

₹1,000 invested at 10 % per annum compounded annually for 2 years grows to:

A₹1,100
B₹1,200
C₹1,210
D₹1,260

View solution

Correct Option: C

1,000 × (1.10)^2 = 1,000 × 1.21 = **₹1,210**.

Q 02 PV Easy

PV of ₹11,000 receivable in 1 year at 10 % is:

A₹10,000
B₹9,000
C₹11,000
D₹12,100

View solution

Correct Option: A

PV = 11,000 / 1.10 = **₹10,000**.

Q 03 Perpetuity Medium

PV of a perpetuity of ₹500 at 10 % discount rate is:

A₹500
B₹5,000
C₹50,000
D∞

View solution

Correct Option: B

PV = A/r = 500/0.10 = **₹5,000**.

Q 04 EAR Medium

A nominal rate of 10 % compounded *quarterly* gives an effective annual rate of approximately:

A10.00 %
B10.25 %
C10.38 %
D11.00 %

View solution

Correct Option: C

EAR = (1 + 0.10/4)^4 − 1 = 1.0254^4 − 1 ≈ **10.38 %**.

Q 05 Kd Medium

A debenture pays 10 % interest; corporate tax rate is 30 %. Post-tax cost of debt is:

A10 %
B7 %
C13 %
D3 %

View solution

Correct Option: B

K_d × (1 − T) = 10 % × (1 − 0.30) = **7 %**.

Q 06 Gordon Medium

A share with expected dividend ₹4, market price ₹50, expected growth 6 %. Cost of equity (Gordon):

A8 %
B10 %
C12 %
D14 %

View solution

Correct Option: D

K_e = D_1 / P_0 + g = 4/50 + 0.06 = 8 % + 6 % = **14 %**.

Q 07 CAPM Medium

Under CAPM, if R_f = 6 %, R_m = 14 %, β = 1.5, cost of equity is:

A12 %
B15 %
C18 %
D21 %

View solution

Correct Option: C

K_e = 6 + 1.5 × (14 − 6) = 6 + 12 = **18 %**.

Q 08 CAPM Author Medium

The Capital Asset Pricing Model (CAPM) was developed by:

AMarkowitz
BWilliam Sharpe (1964)
CModigliani-Miller
DFama

View solution

Correct Option: B

**William Sharpe (1964)** — CAPM; also Lintner, Mossin independently. Nobel 1990 (Sharpe + Markowitz + Miller).

Q 09 WACC Hard

Equity 60 %, K_e = 15 %; Debt 40 %, K_d post-tax = 7 %. WACC:

A9 %
B10.6 %
C11.8 %
D13 %

View solution

Correct Option: C

WACC = 0.6 × 15 + 0.4 × 7 = 9 + 2.8 = **11.8 %**.

Q 10 Weights Medium

Weights in WACC should ideally be based on:

ABook values
BMarket values
CCost values
DLiquidation values

View solution

Correct Option: B

**Market values** — reflect investors' current required returns.

Q 11 Tax shield Medium

Tax shield benefit applies to:

AEquity dividend
BPreference dividend
CInterest on debt
DAll of these

View solution

Correct Option: C

Only **interest** is tax-deductible; dividends are not.

Q 12 Author Medium

"Cost of capital is the minimum rate of return that a firm must earn to maintain the market value of its equity." This definition is by:

ASolomon Ezra
BModigliani-Miller
CWilliam Sharpe
DFisher

View solution

Correct Option: A

**Solomon Ezra**'s classic definition.

Q 13 EVA Medium

Economic Value Added (EVA) is computed as:

ANOPAT − (WACC × Capital Employed)
BPAT − Dividends
CRevenue − Variable cost
DEPS × P/E

View solution

Correct Option: A

EVA = NOPAT − (WACC × Capital Employed). Stern Stewart trademark.

Q 14 Annuity Hard

An *annuity due* differs from an *ordinary annuity* in that:

ACash flow at end of period
BCash flow at beginning of period
CMid-period cash flow
DNo cash flow

View solution

Correct Option: B

**Annuity due** — cash flow at *start* of period; ordinary — at *end*.

Q 15 Risk Medium

In CAPM, β measures:

ATotal risk
BSystematic (market) risk
CUnsystematic (firm-specific) risk
DInflation risk

View solution

Correct Option: B

**β** = systematic / non-diversifiable risk relative to the market.

Q 16 Marginal Medium

Marginal cost of capital is:

ACost of total capital
BCost of raising the next/additional rupee of capital
CCost of equity only
DCost of historical capital

View solution

Correct Option: B

**MCC** = cost of *additional* capital — rises in steps as the firm exhausts cheaper sources.

Q 17 Continuous Hard

Under continuous compounding, the future value of ₹P at rate r for n years is:

AP (1 + r)^n
BP × e^(rn)
CP × (1 − r)^n
DP × log(rn)

View solution

Correct Option: B

Continuous compounding: **FV = P × e^(rn)**.

Q 18 Growing Hard

PV of a growing perpetuity, with first cash flow A, growth g, discount rate r (r > g):

AA / r
BA / (r − g)
CA / (r + g)
DA × g / r

View solution

Correct Option: B

**Gordon growth formula** — PV = A / (r − g).

Q 19 Use of WACC Medium

In project appraisal, WACC is used as:

AA measure of profit
BThe discount rate / hurdle rate for projects of the same risk class
CThe interest rate on cash
DA measure of dividend payout

View solution

Correct Option: B

WACC = the discount rate for NPV; the *hurdle* an IRR must exceed.

Q 20 Doubling Hard

By the *Rule of 72*, money invested at 9 % p.a. doubles approximately in:

A5 years
B6 years
C8 years
D9 years

View solution

Correct Option: C

72 / 9 = **8 years** — Rule of 72 approximation.

29.10 Quick Recall

Quick recall

Time Value of Money — FV = P(1+r)^n; PV = FV/(1+r)^n; continuous FV = Pe^(rn).
Annuity: PV = A × [1 − (1+r)^−n]/r; FV = A × [(1+r)^n − 1]/r.
Perpetuity PV = A/r; growing perpetuity PV = A/(r − g) (r > g — Gordon).
EAR = (1 + r/m)^m − 1.
Cost of Debt post-tax = K_d (1 − T) — interest is tax-shield.
Cost of Equity: Dividend yield (D/P), Gordon (D_1/P_0 + g), CAPM K_e = R_f + β(R_m − R_f) — Sharpe 1964; Bond yield + risk premium.
WACC = w_e K_e + w_p K_p + w_d K_d(1−T); use market-value weights; for projects of same risk class.
MCC rises in steps as cheaper sources exhausted.
Definitional: Solomon Ezra — cost of capital as minimum rate to maintain market value of equity.
EVA = NOPAT − (WACC × Capital Employed).
Rule of 72: doubling time ≈ 72 / interest rate %.