46  Value and Returns

46.1 The Valuation Principle

The valuation principle of corporate finance: the value of any asset = the present value of expected future cash flows discounted at the appropriate risk-adjusted rate. This single idea underlies bond valuation, equity valuation, derivative pricing, capital budgeting, M&A valuation, and even human-capital valuation.

46.2 Time Value of Money — Recap

TipTime value of money — core formulas
Concept Formula
Future Value (FV) FV = PV × (1 + r)ⁿ
Present Value (PV) PV = FV / (1 + r)ⁿ
FV of Annuity FVA = A × [((1+r)ⁿ − 1) / r]
PV of Annuity PVA = A × [(1 − (1+r)⁻ⁿ) / r]
PV of Annuity Due PVA × (1 + r)
Perpetuity P = A / r
Growing Perpetuity P = A / (r − g), r > g
Effective Annual Rate (EAR) (1 + r/m)^m − 1
Continuous Compounding FV = PV × e^(rt)
TipRules of thumb
  • Rule of 72 — Doubling period ≈ 72 / r (%); approx. for r between 6-10 %.
  • Rule of 69 — Doubling period = 0.35 + 69 / r (%); more accurate.
  • Rule of 70 — half-life decay; approximation for r small.

46.3 Concepts of Value

TipTypes of value
Type Meaning
Face / Par Value Stated on the instrument (e.g., ₹10 share, ₹1000 bond)
Book Value Accounting value; (Assets − Liabilities) ÷ Shares for equity
Market Value Current trading price in the market
Intrinsic / Fundamental Value PV of future expected cash flows; what the asset should be worth
Liquidation Value Value if assets are sold and liabilities settled
Replacement Value Cost to replace the asset today
Going-Concern Value Value as an operating business
Fair Value (Ind AS 113) Price in an orderly arm’s-length transaction

46.4 Bond Valuation

46.4.1 Bond Pricing — The Core Formula

\[P_0 = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^n}\]

where C = annual coupon, FV = face value, r = required yield, n = years to maturity. The price is the PV of all coupons (an annuity) plus the PV of the face value (a single sum).

46.4.2 Yield Measures

TipFive bond yield measures
Measure Formula Notes
Nominal / Coupon Yield Coupon / Face Value The stated rate
Current Yield Coupon / Market Price Ignores capital gain
Yield to Maturity (YTM) IRR of bond’s cash flows Most-used yield measure
Yield to Call (YTC) IRR to first call date For callable bonds
Realised Yield Actual return earned Ex-post
NoteBond Pricing Theorems — Malkiel (1962)

Burton Malkiel (1962) identified the famous bond-price theorems:

  1. Bond prices move inversely to interest rates.
  2. Longer maturity → larger price change for a given rate change.
  3. Price sensitivity diminishes at a decreasing rate with maturity.
  4. Lower coupons → greater interest-rate sensitivity.
  5. Yield decrease moves price more than equal yield increase (convexity).

46.4.3 Duration and Convexity

TipDuration and convexity
  • Macaulay Duration — weighted average time to receive cash flows; measures interest-rate sensitivity.
  • Modified Duration = Macaulay Duration / (1 + YTM); % price change for 1 % yield change.
  • Convexity — curvature of price-yield relationship; corrects for duration’s linear approximation.
  • PVBP / DV01 — price change for a 1-basis-point yield change.
  • Duration is lower for higher coupons and higher yields.
  • A zero-coupon bond’s duration = its maturity.

46.4.4 Bond Price-Yield Relationships

TipThree relationships
If Then
Coupon rate = YTM Bond at par (face value)
Coupon rate > YTM Bond at premium
Coupon rate < YTM Bond at discount

46.5 Equity Valuation

46.5.1 Dividend Discount Models (DDM)

John Burr Williams (1938) — The Theory of Investment Value — established the dividend discount approach. Myron J. Gordon (1959, 1962) gave the popular constant-growth form.

TipDDM variants
Model Formula
Single-Period DDM P₀ = (D₁ + P₁) / (1 + Ke)
Multi-Period DDM P₀ = Σ Dₜ / (1+Ke)^t
No-Growth (Zero Growth) P₀ = D / Ke
Gordon (Constant Growth) P₀ = D₁ / (Ke − g)
Two-Stage High-growth phase + stable phase
Three-Stage / H-Model Growth declines linearly to stable rate

46.5.2 Gordon Growth Model Assumptions

TipGordon assumptions
  • Constant growth in dividends forever.
  • Ke > g (else value is infinite).
  • Stable retention and ROE.
  • Constant cost of equity.
  • Dividend policy is relevant to value.
Noteg — Sustainable Growth Rate

g = b × ROE where b = retention ratio, ROE = return on equity. Growth without external financing comes from retained earnings × the rate at which retained earnings earn.

46.5.3 Other Equity-Valuation Approaches

TipOther equity-valuation methods
Method Idea
Earnings Capitalisation P = EPS / Ke (no growth)
PE Ratio Method P = EPS × P/E (relative multiple)
Book Value Method P = Book Value per Share × P/B
Free Cash Flow to Equity (FCFE) PV of FCFE discounted at Ke
Free Cash Flow to Firm (FCFF) PV of FCFF discounted at WACC → less debt
Residual Income PV of (NI − Ke × Equity)
Asset-Based Net asset value
EVA-based NOPAT − (WACC × Capital Employed)
Tobin’s Q Market value / Replacement cost
Relative Valuation Multiples — P/E, EV/EBITDA, EV/Sales, P/B

46.5.4 FCFF and FCFE

TipFCFF vs FCFE
Measure Formula Discount rate
FCFF EBIT(1−t) + Depreciation − CapEx − Δ WC WACC
FCFE NI + Depreciation − CapEx − Δ WC + Net Borrowings Cost of Equity

46.6 Returns — Concept and Types

TipComponents of return
  • Income return — dividends / coupons / rent.
  • Capital appreciation — change in price.
  • Total return = Income + Capital appreciation.

46.6.1 Holding-Period Return (HPR)

TipHPR formula

\[\text{HPR} = \frac{D + (P_1 - P_0)}{P_0}\]

Where D = dividend (or coupon), P₀ = beginning price, P₁ = ending price.

46.6.2 Multi-Period Return Measures

TipReturn measures
Measure Formula
Arithmetic Mean Return Σ Rₜ / n
Geometric Mean Return (CAGR) [(1+R₁)(1+R₂)…(1+Rₙ)]^(1/n) − 1
IRR / Money-Weighted Return Discount rate that equates inflows to outflows
Time-Weighted Return Geometric average across sub-periods
Real Return (1 + Nominal) / (1 + Inflation) − 1 ≈ Nominal − Inflation
Effective Annual Rate (EAR) (1 + r/m)^m − 1
Continuously Compounded ln(1 + R)
NoteArithmetic vs Geometric — Why it matters

Geometric mean ≤ Arithmetic mean always (equal only when all returns are identical). Geometric is the correct measure of realised growth in wealth; arithmetic overstates due to compounding asymmetry. The gap widens with volatility (variance drag).

46.6.3 Real vs Nominal Returns

Fisher Effect (Irving Fisher 1930) — Nominal interest rate ≈ Real rate + Expected inflation.

\[(1 + r_{\text{nominal}}) = (1 + r_{\text{real}}) \times (1 + i)\]

46.7 Risk-Adjusted Return Measures

TipRisk-adjusted return measures
Measure Formula Risk metric
Sharpe Ratio (Rp − Rf) / σp Total risk (SD)
Treynor Ratio (Rp − Rf) / βp Systematic risk (β)
Jensen’s Alpha Rp − [Rf + βp(Rm − Rf)] Excess over CAPM
Information Ratio Active Return / Tracking Error vs benchmark
Sortino Ratio (Rp − Rf) / Downside Deviation Downside risk only
M² (Modigliani²) Risk-adjusted to market σ Total risk
NoteSharpe vs Treynor

Sharpe (William Sharpe 1966) uses total risk — useful for undiversified portfolios. Treynor (Jack Treynor 1965) uses systematic risk — useful when the portfolio is part of a diversified holding.

46.8 Risk and Return — Investment Triangle

flowchart TB
  R[Return]
  RI[Risk]
  L[Liquidity]
  R --- T[Investment<br/>Trade-Off]
  RI --- T
  L --- T
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

The investor’s trilemma: an investment can offer at most two of high return, low risk and high liquidity.

46.9 Market Efficiency — EMH

Eugene Fama (1970) — Efficient Market Hypothesis (EMH). Prices reflect available information in three forms:

TipThree forms of EMH
Form Information reflected Implication
Weak Past prices and volumes Technical analysis useless
Semi-Strong All public information Fundamental analysis on public info useless
Strong All info — public + private Even insider trading wouldn’t help

Fama received the Nobel Prize 2013 along with Shiller and Hansen.

46.10 Behavioural Finance Critique

Daniel Kahneman, Amos Tversky, Richard Thaler challenged EMH with biases:

TipCommon behavioural biases
  • Loss aversion — losses hurt more than equal gains.
  • Overconfidence — overestimating one’s prediction.
  • Anchoring — over-reliance on first piece of information.
  • Herding — following the crowd.
  • Representativeness — judging by stereotypes.
  • Availability bias — over-weighting recent or memorable events.
  • Mental accounting — treating money differently by source.
  • Disposition effect — selling winners too early, holding losers too long.
  • Confirmation bias.
  • Sunk cost fallacy.

Kahneman won Nobel 2002; Thaler won Nobel 2017.

46.12 Practice Questions

Q 01 Valuation principle Easy

The value of any asset equals:

  • AIts book value
  • BPV of expected future cash flows
  • CHistorical cost
  • DLiquidation value
View solution
Correct Option: B
Present value of expected future cash flows discounted at the appropriate risk-adjusted rate.
Q 02 Rule of 72 Medium

At 8 % p.a., approximately how long does it take money to double?

  • A5 years
  • B7 years
  • C9 years
  • D12 years
View solution
Correct Option: C
Rule of 72 → 72 / 8 = 9 years.
Q 03 Gordon Medium

The Gordon constant-growth equity valuation formula is:

  • AP = D / Ke
  • BP = D₁ / (Ke − g)
  • CP = EPS × P/E
  • DP = NI / Ke
View solution
Correct Option: B
Gordon Growth Model: P = D₁ / (Ke − g), Ke > g.
Q 04 YTM Medium

If coupon rate equals YTM, the bond trades at:

  • APremium
  • BDiscount
  • CPar
  • DAbove par by tax shield
View solution
Correct Option: C
When coupon = YTM, bond trades at par (face value).
Q 05 Inverse Easy

Bond prices and interest rates have:

  • ADirect positive relationship
  • BInverse relationship
  • CNo relationship
  • DQuadratic relationship
View solution
Correct Option: B
First Malkiel theorem — bond prices move inversely to yields.
Q 06 Duration Hard

The duration of a zero-coupon bond equals:

  • AHalf its maturity
  • BTwice its maturity
  • CIts maturity
  • DZero
View solution
Correct Option: C
Single cash flow at maturity → Macaulay Duration = maturity.
Q 07 Williams Hard

The Dividend Discount approach was founded in 1938 by:

  • AJohn Burr Williams
  • BMyron Gordon
  • CWilliam Sharpe
  • DEugene Fama
View solution
Correct Option: A
John Burr Williams, *The Theory of Investment Value* (1938).
Q 08 Sustainable g Medium

Sustainable growth rate equals:

  • Ab × ROE
  • Bb × Ke
  • CPayout × ROA
  • DKe − Rf
View solution
Correct Option: A
g = b × ROE, where b = retention ratio.
Q 09 HPR Medium

An investor buys a share at ₹100, receives ₹5 dividend, and sells at ₹110. HPR is:

  • A5 %
  • B10 %
  • C15 %
  • D20 %
View solution
Correct Option: C
(5 + 110 − 100) / 100 = 15 %.
Q 10 Geo vs Arith Medium

For a series of varying returns, the relationship between arithmetic and geometric means is:

  • AArithmetic > Geometric
  • BArithmetic < Geometric
  • CAlways equal
  • DCannot say
View solution
Correct Option: A
Arithmetic ≥ Geometric always; gap widens with volatility (variance drag).
Q 11 Sharpe Medium

Sharpe Ratio uses which measure of risk?

  • ABeta
  • BStandard deviation
  • CTracking error
  • DDownside deviation
View solution
Correct Option: B
Sharpe uses total risk (SD); Treynor uses β; Sortino uses downside.
Q 12 Fisher Medium

The Fisher Effect relates:

  • ANominal rate, real rate, inflation
  • BBeta and return
  • CCost of equity and Ke
  • DSharpe and Treynor
View solution
Correct Option: A
(1 + nominal) = (1 + real) × (1 + inflation). Irving Fisher (1930).
Q 13 EMH Medium

Under the semi-strong form of EMH:

  • AOnly past prices are reflected
  • BAll public information is reflected
  • CAll information including private is reflected
  • DNo information is reflected
View solution
Correct Option: B
Semi-strong → all public info; fundamental analysis on public data does not yield abnormal returns.
Q 14 Fama Hard

Efficient Market Hypothesis is associated with:

  • AEugene Fama
  • BRobert Shiller
  • CBurton Malkiel
  • DDaniel Kahneman
View solution
Correct Option: A
Eugene Fama (1970); Nobel 2013.
Q 15 Behavioural Medium

"Loss aversion" — the idea that losses hurt about twice as much as equal gains — is from:

  • AKahneman & Tversky
  • BMarkowitz
  • CSharpe
  • DModigliani
View solution
Correct Option: A
Kahneman and Tversky — Prospect Theory (1979); Kahneman Nobel 2002.
Q 16 FCFF Hard

FCFF should be discounted at:

  • ACost of equity
  • BWACC
  • CRisk-free rate
  • DCost of debt
View solution
Correct Option: B
FCFF is unlevered → discount at WACC; FCFE at Ke.
Q 17 Convexity Hard

Bond convexity:

  • ALinearises duration
  • BCorrects the linear approximation of duration
  • CMeasures coupon yield
  • DAlways negative
View solution
Correct Option: B
Convexity corrects duration's linear approximation; price-yield curve is convex.
Q 18 Perpetuity Easy

A perpetual annuity paying ₹500 forever at 10 % discount rate is worth:

  • A₹5,000
  • B₹500
  • C₹50,000
  • DInfinite
View solution
Correct Option: A
P = A/r = 500/0.10 = ₹5,000.
Q 19 Treynor Hard

Treynor ratio is preferable to Sharpe when:

  • APortfolio is stand-alone
  • BPortfolio is part of a diversified holding
  • CReturns are negatively skewed
  • DPortfolio has zero risk
View solution
Correct Option: B
Treynor uses β (systematic risk) → suited when unsystematic risk is already diversified away.
Q 20 Match value Hard

Match the concept with its description:

(i) Intrinsic Value (a) Trading price
(ii) Book Value (b) PV of future cash flows
(iii) Market Value (c) Stated on instrument
(iv) Face Value (d) Net assets / shares
  • A(i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
  • B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
  • C(i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
  • D(i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
View solution
Correct Option: A
Intrinsic — PV; Book — net assets/shares; Market — trading; Face — stated.

46.12.1 Advanced Format Questions

AR 1Assertion-ReasonHard

A: Bond price falls when interest rates rise.
R: Bond price is the PV of future cash flows discounted at YTM.

  • ABoth true; R explains A
  • BBoth true; R does not explain A
  • CA true, R false
  • DA false, R true
View solution
Correct Option: A
S 1Statement-basedMedium

Bond valuation inputs: (i) Coupon. (ii) Face value. (iii) YTM. (iv) Maturity.

  • AAll four
  • B(i) and (ii) only
  • C(iii) and (iv) only
  • D(i), (ii), (iii) only
View solution
Correct Option: A
N 1NumericalMedium

Dividend ₹2; Growth 5%; Required return 12%. Gordon model price:

  • A₹30 (approx)
  • B₹20
  • C₹40
  • D₹50
View solution
Correct Option: A
P = D₁/(r−g) = 2 × 1.05 / (0.12 − 0.05) = 2.10/0.07 = ₹30.
N 2NumericalHard

Bought share at ₹100; sold at ₹120; dividend ₹5. Holding-period return:

  • A25 %
  • B20 %
  • C5 %
  • D30 %
View solution
Correct Option: A
(20 + 5)/100 = 25%.

46.13 Quick Recall

ImportantQuick recall
  • Valuation Principle: Value = PV of expected future cash flows at risk-adjusted rate.
  • TVM: FV/PV/Annuity/Perpetuity/Growing Perpetuity/EAR/Continuous compounding.
  • Rules of thumb: 72/r (doubling), 69/r (more accurate).
  • Types of value: Face · Book · Market · Intrinsic (PV) · Liquidation · Replacement · Going-Concern · Fair Value (Ind AS 113).
  • Bond pricing: P = Σ C/(1+r)ᵗ + FV/(1+r)ⁿ.
  • Yield measures: Nominal · Current · YTM (IRR) · YTC · Realised.
  • Malkiel bond theorems (1962): inverse · longer maturity = larger change · diminishing rate · lower coupons = greater sensitivity · convexity.
  • Duration: Macaulay = weighted time; Modified = Macaulay/(1+YTM); Zero-coupon duration = maturity; PVBP/DV01.
  • Bond at: par (coupon = YTM); premium (coupon > YTM); discount (coupon < YTM).
  • Equity DDM: John Burr Williams (1938); Gordon (1959, 1962): P = D₁ / (Ke − g).
  • Sustainable growth: g = b × ROE.
  • Other equity methods: Earnings cap · PE · Book Value · FCFE (at Ke) · FCFF (at WACC) · Residual Income · EVA · Tobin’s Q · Relative multiples.
  • HPR = (D + P₁ − P₀) / P₀.
  • Returns: Arithmetic mean ≥ Geometric mean (CAGR); IRR; Time-weighted; Real (Fisher); EAR; Continuously compounded.
  • Fisher Effect: (1 + nominal) = (1 + real)(1 + inflation).
  • Risk-adjusted measures: Sharpe (SD) · Treynor (β) · Jensen Alpha · Information ratio · Sortino · M².
  • Investment trilemma: at most 2 of high return + low risk + high liquidity.
  • EMH — Fama (1970, Nobel 2013): Weak (past) · Semi-Strong (public) · Strong (private).
  • Behavioural finance — Kahneman (Nobel 2002), Thaler (Nobel 2017): loss aversion · overconfidence · anchoring · herding · representativeness · availability · mental accounting · disposition effect · confirmation · sunk cost.
  • Modern trends: ESG valuation · real options · Monte Carlo · ML · crypto valuation · platform-firm valuation · sustainability premium · DLOM/DLOC · TSR · climate scenarios.