flowchart LR T[Total Risk<br/>Variance / Std Dev] --> S[Systematic Risk<br/>Market-related<br/>Captured by β] T --> U[Unsystematic Risk<br/>Firm-specific<br/>Diversifiable] S -. priced .-> CAPM[CAPM] U -. not priced .-> CAPM style T fill:#FCE4EC,stroke:#AD1457 style S fill:#FFEBEE,stroke:#C62828 style U fill:#E8F5E9,stroke:#2E7D32
45 Value and Returns
45.1 What is Value?
In finance, the value of an asset is the present value of the cash flows it is expected to generate, discounted at a rate that reflects their risk. The classical statement comes from John Burr Williams’s Theory of Investment Value (1938): “a stock is worth the present value of all the dividends ever to be paid upon it” (williams1938?). Brealey-Myers-Allen put it crisply: “the value of any financial asset is the present value of the cash flows the asset is expected to generate” (brealeymyers2020?).
Three categories of “value” recur in NTA stems:
TipCommon Concepts of Value
| Concept | What it captures |
|---|---|
| Book value | Accounting carrying value — historical cost less accumulated depreciation |
| Market value | Price at which the asset trades in an active market |
| Intrinsic / Fair value | The PV of expected future cash flows at the appropriate discount rate |
| Liquidation value | What the asset would fetch in a forced sale |
| Replacement value | What it would cost to replace the asset today |
The financial-management view emphasises intrinsic value — what the asset is worth given its cash flows — as opposed to its price, which can deviate.
45.2 Bond Valuation
A bond is a debt instrument that promises a stream of fixed coupon payments and the repayment of face value at maturity. The value of a coupon bond:
\[P_0 = \sum_{t=1}^{n} \frac{C}{(1+k_d)^t} + \frac{F}{(1+k_d)^n}\]
where \(C\) = annual coupon, \(F\) = face value, \(n\) = years to maturity, \(k_d\) = required yield (discount rate).
TipBond Pricing — Three Cases
| Coupon vs Yield | Price vs Face | Bond is said to trade at |
|---|---|---|
| Coupon rate > Yield to maturity | Price > Face value | Premium |
| Coupon rate = Yield to maturity | Price = Face value | Par |
| Coupon rate < Yield to maturity | Price < Face value | Discount |
45.2.1 Yield to Maturity (YTM)
The YTM is the internal rate of return on a bond held to maturity — the discount rate that equates the current market price to the present value of remaining cash flows.
45.2.2 Bond yields and interest-rate risk
Bond prices and interest rates move inversely. The sensitivity is captured by duration (Macaulay; Modified) and convexity — the textbook interest-rate risk metrics. Modified duration approximates the % change in price for a 1% change in yield.
45.3 Share Valuation — Equity
The fundamental relationship: price = present value of expected dividends.
45.3.1 Constant-growth (Gordon) model
If dividends grow at a constant rate \(g\) forever and \(g < k_e\):
\[P_0 = \frac{D_1}{k_e - g}\]
where \(D_1\) is next year’s dividend, \(k_e\) is the cost of equity, and \(g\) is the constant growth rate. Rearranging gives the cost of equity used in topic 42.
45.3.2 Walter’s model
J.E. Walter’s classic model (1956) holds that dividend policy affects share price when the firm’s return on retention \(r\) differs from the cost of equity \(k_e\) (walter1956?):
\[P_0 = \frac{D + \frac{r}{k_e}(E - D)}{k_e}\]
where \(D\) = dividend per share, \(E\) = EPS. Three implications:
TipWalter’s Three Cases
| Firm type | Condition | Optimal pay-out |
|---|---|---|
| Growth firm | r > k_e | 0% pay-out — retain everything |
| Normal firm | r = k_e | Indifferent |
| Declining firm | r < k_e | 100% pay-out — distribute everything |
45.3.3 Gordon’s model (with retention)
Myron Gordon’s model (1959) makes the same point through a bird-in-hand lens — investors prefer current dividends to uncertain future capital gains (gordon1959?):
\[P_0 = \frac{E(1-b)}{k_e - br}\]
where \(b\) = retention ratio, \(r\) = return on retention. The model has the same prescription as Walter’s for growth and declining firms.
45.3.4 Multi-stage models
Real-world firms rarely have constant growth forever. The two-stage and H-model dividend-discount models combine an initial high-growth phase with a stable long-run phase. Damodaran’s textbook is the standard reference for these extensions (damodaran2012?).
45.4 Returns
The return on an investment is the income plus capital appreciation, expressed as a percentage of the amount invested.
TipThree Return Concepts
| Concept | What it captures |
|---|---|
| Holding-Period Return (HPR) | Income + capital gain over the holding period |
| Realised return | Actually earned in the past |
| Expected return | Anticipated for the future |
45.4.1 Holding-period return
For a single period:
\[R = \frac{D + (P_1 - P_0)}{P_0}\]
where \(D\) is dividend received during the period.
45.4.2 Expected return
Expected return on a risky investment is the probability-weighted average of possible returns:
\[E(R) = \sum_i p_i \cdot R_i\]
45.4.3 Annualised returns
Common methods:
TipCommon Annualised Return Measures
| Measure | What it is | Best for |
|---|---|---|
| Arithmetic mean | Simple average of periodic returns | Forecasting one-period return |
| Geometric mean (CAGR) | Compound growth rate | Multi-period performance |
| Money-weighted return (IRR) | Reflects timing of cash flows | Investor-specific performance |
| Time-weighted return | Strips out timing of cash flows | Comparing fund managers |
45.5 Risk
Risk is the variability of return around its expected value. Risk and return are inseparable in finance.
45.5.1 Measures of risk
TipCommon Measures of Risk
| Measure | What it captures |
|---|---|
| Variance / Standard deviation (σ) | Total dispersion of returns |
| Beta (β) | Systematic (market-related) risk |
| Coefficient of variation (σ / E(R)) | Risk per unit of return |
| Value at Risk (VaR) | Maximum loss expected at a confidence level |
| Downside risk / semi-deviation | Variability below the target |
45.5.2 Systematic vs unsystematic risk
TipTwo Components of Total Risk
| Type | Source | Diversifiable? |
|---|---|---|
| Systematic / Market risk | Macroeconomic, regulatory, interest-rate, political | No |
| Unsystematic / Specific risk | Firm-specific — strike, fire, management change | Yes — through diversification |
The CAPM says only systematic risk is priced — investors are not rewarded for risks they could have diversified away.
45.6 The Risk-Return Trade-off
The Capital Market Line (CML) plots expected return against total risk for efficient portfolios:
\[E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m} \cdot \sigma_p\]
The Security Market Line (SML), derived from CAPM, plots expected return against systematic risk (beta):
\[E(R_i) = R_f + \beta_i \cdot (E(R_m) - R_f)\]
The SML extends to all securities and portfolios — efficient or not.
TipCML vs SML
| Feature | CML | SML |
|---|---|---|
| Risk on x-axis | Total risk (σ) | Systematic risk (β) |
| Plots | Efficient portfolios only | All securities and portfolios |
| Implication | Total risk is priced for efficient portfolios | Only systematic risk is priced for individual securities |
45.7 Practice Questions
Q 01
Bond Pricing
Medium
A bond's coupon rate is 12% and its yield to maturity is 10%. The bond is most likely trading at:
View solution
Correct Option: C
Coupon > YTM → bond trades at a premium (price > face value). Coupon < YTM → discount; coupon = YTM → par.
Q 02
Gordon
Medium
Under the Gordon constant-growth model, a share is valued as:
View solution
Correct Option: A
P₀ = D₁ ÷ (k_e − g). Valid when g < k_e.
Q 03
Walter
Medium
In Walter's model, a "growth firm" — where r > k_e — should ideally have:
View solution
Correct Option: B
Walter: when r > k_e, retain and re-invest — 0 per cent pay-out. When r < k_e, pay it all out. When r = k_e, indifferent.
Q 04
Beta
Medium
Beta (β) measures:
View solution
Correct Option: C
β = sensitivity to market moves = systematic risk. Total risk is variance / standard deviation.
Q 05
HPR
Medium
A share bought at ₹100 paid a dividend of ₹4 and was sold at ₹110 after one year. The holding-period return is:
View solution
Correct Option: B
HPR = (4 + 110 − 100) ÷ 100 = 14 ÷ 100 = 14%.
Q 06
Diversifiable
Medium
Which of the following risks cannot be diversified away?
View solution
Correct Option: C
Macroeconomic / regulatory risks are systematic — they affect all firms and cannot be diversified away. The other three are firm-specific.
Q 07
CML vs SML
Medium
The Security Market Line (SML) plots expected return against:
View solution
Correct Option: B
SML: x-axis is β. CML: x-axis is total risk σ — but plots only efficient portfolios.
Q 08
Williams
Medium
"A stock is worth the present value of all the dividends ever to be paid upon it" — the foundational dividend-discount idea — is most associated with:
View solution
Correct Option: B
John Burr Williams's Theory of Investment Value (1938) is the foundational text. Gordon and Shapiro later formalised the constant-growth case.
ImportantQuick recall
- Value = PV of expected cash flows discounted at appropriate rate (Williams, 1938).
- Concepts of value: book · market · intrinsic · liquidation · replacement. FM emphasises intrinsic value.
- Bond price: discounted PV of coupons + face value. Coupon > YTM → premium; coupon = YTM → par; coupon < YTM → discount.
- Gordon’s constant growth: P₀ = D₁ / (k_e − g), valid when g < k_e.
- Walter’s model: dividend matters when r ≠ k_e. Growth firm (r > k_e) → 0% pay-out; declining (r < k_e) → 100%.
- Returns: HPR, expected, realised. Annualised: arithmetic, geometric (CAGR), money-weighted (IRR), time-weighted.
- Risk = variability. Total = systematic (β, market) + unsystematic (firm-specific, diversifiable). CAPM prices only systematic risk.
- CML plots return vs total risk (σ) for efficient portfolios; SML plots return vs β for all securities.