Chapter 5: Security Valuation
Exhaustive Master Notes covering Equity Valuation (Dividend Discount Models - Gordon, Walter, Multi-Stage & H-Model; Free Cash Flow Models - FCFF & FCFE; Market Multiples & Rights Issues), Preference Shares, Fixed Income Bond Valuation (The Five Bond Theorems, YTM, Macaulay Duration, Modified Duration, Convexity Adjustment, and Immunization), and ICAI Valuation Standards.
1. Overview of Valuation & Value Concepts
Security Valuation is the quantitative determination of the theoretical fair market worth (intrinsic value) of an asset or liability. In CA Final AFM, valuation is the cornerstone of corporate finance, mergers & acquisitions, fairness opinions, and strategic portfolio allocations.
Intrinsic Value
The present value of all expected future cash flows that an asset will generate over its life, discounted at a rate commensurate with its underlying operational and financial risk.
Fair Value (Ind AS 113)
The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date.
Book Value
The net balance sheet worth of equity (Assets minus Liabilities) based strictly on historical accounting records and depreciation conventions.
Liquidation Value
The net cash realizable if the company terminates operations immediately and sells off all physical assets in piecemeal distress sales.
2. Return Concepts & Equity Risk Premium (ERP)
2.1 Holding Period Return (HPR)
• \(P_0\) = Beginning acquisition price
• \(P_1\) = Ending realization or market price
• \(D_1\) = Cash dividend received during the holding period
2.2 Equity Risk Premium (ERP)
The Equity Risk Premium is the incremental return demanded by rational investors over and above the risk-free rate of return (\(R_f\)) to compensate for the higher uncertainty and volatility of investing in equity equities:
$$\text{Equity Risk Premium (ERP)} = E(R_m) - R_f$$Techniques for estimating ERP include:
- Historical Premium Approach: Computes the historical geometric or arithmetic mean difference between equity index returns and sovereign bond yields over 50–100 years.
- Forward-Looking / Survey Approach: Surveys institutional fund managers, CFOs, and equity research economists.
- Country Risk Premium (CRP): In emerging markets like India, foreign investors add an additional sovereign default spread and relative equity market volatility multiplier: $$\text{CRP} = \text{Sovereign Default Spread} \times \left( \frac{\sigma_{\text{Equity}}}{\sigma_{\text{Govt Bond}}} \right)$$
3. Discount Rate Selection & Cash Flow Matching
The fundamental rule of financial valuation is the Rule of Internal Consistency:
| Cash Flow Type | Appropriate Discount Rate | Resulting Value Generated |
|---|---|---|
| Free Cash Flow to Firm (FCFF) | WACC (Weighted Average Cost of Capital) | Enterprise / Firm Value (\(V_0\)) |
| Free Cash Flow to Equity (FCFE) | Cost of Equity (\(K_e\)) | Equity Value (\(E_0\)) |
| Dividends (DDM) | Cost of Equity (\(K_e\)) | Equity Value Per Share (\(P_0\)) |
| Nominal Cash Flows | Nominal Discount Rate (\(r_n\)) | PV adjusted for inflation via nominal rate |
| Real Cash Flows | Real Discount Rate (\(r_r\)) | PV adjusted via real rate: \((1 + r_n) = (1 + r_r)(1 + i)\) |
Estimating Cost of Equity (\(K_e\))
ICAI specifies three standard methodologies for determining \(K_e\):
- Capital Asset Pricing Model (CAPM): $$K_e = R_f + \beta (R_m - R_f)$$
- Build-Up Method: $$K_e = R_f + \text{Equity Risk Premium} + \text{Size Premium} + \text{Company Specific Risk Premium}$$
- Bond Yield Plus Risk Premium: $$K_e = Y_d + \text{Equity Spread (typically 3% to 4%)}$$
4. Dividend Discount Models (DDM)
The Dividend Discount Model views the value of an equity share as the present value of all expected future cash dividends in perpetuity.
4.1 Zero-Growth Dividend Model (Perpetuity)
4.2 Constant Growth Model (Gordon Growth Model)
• \(P_0\) = Current intrinsic price per share
• \(D_0\) = Dividend just paid (last historical dividend)
• \(D_1 = D_0(1 + g)\) = Dividend expected at the end of Year 1
• \(K_e\) = Cost of equity / required rate of return (\(K_e > g\) must hold)
• \(g\) = Constant growth rate in perpetuity
• \(b\) = Retention ratio \(= 1 - \frac{\text{DPS}}{\text{EPS}} = 1 - \text{Payout Ratio}\)
• \(r\) = Return on Equity (ROE) earned on reinvested profits
1. The required rate of return must strictly exceed the growth rate (\(K_e > g\)), otherwise the denominator becomes zero or negative, rendering the model meaningless.
2. The retention ratio \(b\) and rate of return on investment \(r\) remain constant indefinitely.
3. The firm relies entirely on internal equity financing (retained earnings); no external equity is issued.
5. Walter Model & Optimum Dividend Policies
Formulated by Prof. James E. Walter, this model demonstrates that dividend policy directly impacts share price by evaluating the relationship between internal rate of return (\(r\)) and the market cost of capital (\(K_e\)).
• \(P\) = Market price per share
• \(D\) = Dividend per share
• \(E\) = Earnings per share (EPS)
• \((E - D)\) = Retained earnings per share
• \(r\) = Internal rate of return on retained earnings (ROE)
• \(K_e\) = Cost of equity / opportunity cost of capital
5.1 Walter's Decision Rules on Dividend Policy
| Firm Classification | Condition | Economic Rationale | Optimum Payout Ratio | Effect on Share Price \(P\) |
|---|---|---|---|---|
| Growth Firm | \(r > K_e\) | Firm can earn higher returns than shareholders can earn in the open market. Reinvestment maximizes wealth. | 0% (100% Retention) | Price \(P\) is maximized when payout is 0%. Any dividend payout reduces share price. |
| Declining Firm | \(r < K_e\) | Shareholders can earn higher returns elsewhere than the firm's internal projects. Retention destroys wealth. | 100% (0% Retention) | Price \(P\) is maximized when 100% of earnings are distributed as dividends. |
| Normal Firm | \(r = K_e\) | Firm generates exactly what the capital market demands. Reinvestment and distribution are equivalent. | Indifferent (0% to 100%) | Dividend policy has zero impact on share price \(P\); price remains constant across all payout ratios. |
6. Multi-Stage Dividend Models & The H-Model
6.1 Two-Stage Dividend Growth Model
Appropriate for rapidly expanding firms experiencing a temporary phase of supernormal growth (\(g_s\)) for \(n\) years, followed by stable long-term growth (\(g_n\)) in perpetuity:
6.2 The H-Model (Linear Transition Model)
In reality, high growth does not crash abruptly into stable growth overnight; it declines gradually. Developed by Fuller and Hsia, the H-Model assumes growth begins at a high rate (\(g_S\)) and declines linearly over a transition period of \(2H\) years to a stable long-term rate (\(g_L\)):
• \(D_0\) = Current dividend just paid
• \(g_S\) = Initial short-term supernormal growth rate
• \(g_L\) = Long-term sustainable normal growth rate
• \(H = \frac{t}{2}\) = Half-life of the transition period (e.g. if transition takes 10 years, \(H = 5\))
• \(K_e\) = Required rate of return on equity
• Component 1: Represents the base stable value of the firm growing at \(g_L\).
• Component 2: Represents the extra value contributed by the supernormal growth windfall during the transition.
7. Earnings-Based & Market Multiple Valuation
7.1 Earnings Capitalization Model
When a company does not reinvest or earns exactly its cost of capital (\(r = K_e\)), intrinsic price equals normalized earnings capitalized at the cost of equity:
$$P_0 = \frac{E_1}{K_e}$$7.2 Price-to-Earnings (P/E) Multiple
The most widely deployed market multiple in equity research:
$$\text{Target Share Price} = \text{Estimated Forward EPS } (E_1) \times \text{Peer Group / Benchmark P/E Multiple}$$7.3 Price-to-Book Value (P/BV) Ratio
• If \(\text{ROE} > K_e\), the firm creates economic value and trades at a substantial premium to book value (\(P_0 / \text{BV}_0 > 1.0\)).
• If \(\text{ROE} < K_e\), the firm destroys economic value and trades at a discount to book value.
7.4 Price-to-Sales (P/S) Ratio
Calculated as \(\frac{\text{Market Price per Share}}{\text{Sales per Share}}\) or \(\frac{\text{Market Capitalization}}{\text{Total Revenue}}\). Highly reliable for cyclical firms facing temporary accounting losses, turnaround candidates, or early-stage tech firms with zero or negative net profits, because sales revenue is immune to accounting depreciation and inventory adjustments.
8. Cash Flow-Based Valuation: FCFF & FCFE
Modern institutional corporate finance prefers Discounted Cash Flow (DCF) models based on Free Cash Flow rather than accounting earnings or dividends, because earnings are distorted by accrual policies and dividends are subject to managerial discretion.
8.1 Free Cash Flow to Firm (FCFF)
FCFF represents the cash generated by operations that is freely available to all capital providers (both equity shareholders and debt holders) after meeting operating expenses, corporate taxes, working capital investments, and capital expenditures.
8.2 Free Cash Flow to Equity (FCFE)
FCFE represents the residual cash flow freely available exclusively to equity shareholders after meeting capital expenditures, working capital additions, and all debt service obligations (interest and principal repayments).
9. Enterprise Value (EV) & Multiples
Enterprise Value (EV) represents the total economic theoretical acquisition price of a company, viewing the firm as an entire operating business regardless of how it is financed.
Popular Enterprise Multiples
- EV / EBITDA Multiple: The gold standard in leveraged buyouts and cross-border M&A. Because EBITDA is pre-interest, pre-tax, and pre-depreciation, EV/EBITDA is immune to differences in capital structure (debt vs equity) and differing depreciation accounting methods across countries.
- EV / Sales Multiple: Indispensable for cyclical companies or fast-growing startups operating at an EBITDA loss.
10. Valuation of Rights Issues
When an existing company raises fresh equity capital by offering existing shareholders the right to subscribe to new shares in proportion to their current holdings at a concession price, it constitutes a Rights Issue.
• \(P_0\) = Cum-rights market price per share
• \(S\) = Subscription / offer price for the right share (always \(S < P_0\))
• \(N\) = Number of existing shares required to entitle the holder to purchase 1 right share (e.g. in a 1-for-4 rights issue, \(N = 4\))
• \(P_{ex}\) = Market price per share immediately after the stock goes ex-rights
• \(V_R\) = Value of right attaching to 1 existing share: \(\frac{V_R \text{ per right share}}{N}\)
Under frictionless market assumptions, a rights issue does not change a shareholder's total wealth, whether they choose to exercise their rights (wealth preserved via ownership of additional shares at concession) or renounce and sell their rights in the market (wealth preserved via cash proceeds received from sale of rights offsetting the drop from cum-rights to ex-rights price).
12. Bond Valuation & Pricing Mechanics
A bond represents a contractual debt agreement where the issuer promises periodic fixed coupon interest payments and principal redemption at maturity.
• \(P_0\) = Intrinsic market value / clean price of the bond today
• \(I\) = Periodic annual coupon interest payment \(= \text{Face Value} \times \text{Coupon Rate}\)
• \(k_d\) = Required yield / market interest rate on equivalent risk bonds (YTM)
• \(RV\) = Redemption value / maturity value (par or premium)
• \(n\) = Number of years to maturity
Semi-Annual Coupon Bonds Adjustment
Most institutional bonds pay interest semi-annually. In such cases:
$$\text{Periodic Coupon } = \frac{I}{2}, \quad \text{Periodic Yield } = \frac{k_d}{2}, \quad \text{Total Periods } = 2n$$ $$P_0 = \left( \frac{I}{2} \right) \times \text{PVAF}_{\frac{k_d}{2}, 2n} + RV \times \text{PVF}_{\frac{k_d}{2}, 2n}$$Clean Price
The quoted market price of the bond excluding any accrued interest accumulated since the last coupon date.
Dirty Price (Invoice Price)
The actual cash settlement price paid by the buyer: \(\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}\).
13. The Five Bond Value Theorems
Formulated by Burton Malkiel, these five theorems govern the mathematical relationship between bond prices, yields, maturities, and coupon rates:
| Theorem No. | Principle | Mathematical & Economic Significance |
|---|---|---|
| Theorem 1 | Inverse Price-Yield Relationship | Bond price moves inversely to interest rates (yields). When yield rises, price falls; when yield falls, price rises. • If \(\text{Coupon} = \text{Yield} \implies P_0 = \text{Par Value}\). • If \(\text{Coupon} < \text{Yield} \implies P_0 < \text{Par Value}\) (Discount Bond). • If \(\text{Coupon} > \text{Yield} \implies P_0 > \text{Par Value}\) (Premium Bond). |
| Theorem 2 | Maturity Effect | For a given change in yield, a longer-term bond experiences a larger price percentage change than a shorter-term bond. Long maturity amplifies price volatility. |
| Theorem 3 | Diminishing Elasticity | Although price sensitivity increases with maturity, it does so at a diminishing rate. The percentage price change from year 20 to 30 is far smaller than from year 1 to 10. |
| Theorem 4 | Coupon Effect | For a given change in yield, lower-coupon bonds are more price-sensitive than higher-coupon bonds. A zero-coupon bond possesses the maximum price sensitivity for a given maturity. |
| Theorem 5 | Convexity Asymmetry | For an equal magnitude shift in yield (+Δy vs -Δy), the capital gain resulting from a yield decrease is strictly greater than the capital loss resulting from a yield increase. Bond price-yield curves are convex to the origin! |
14. Yield to Maturity (YTM)
Yield to Maturity (YTM) is the internal rate of return (IRR) earned by an investor who buys the bond at its current market price, holds it until maturity, and reinvests all coupon cash flows at the YTM rate.
• \(I\) = Annual coupon interest payment
• \(RV\) = Redemption value at maturity
• \(P_0\) = Current market purchase price of the bond
• \(n\) = Number of years to maturity
• Note: In ICAI exams, use this approximation to find trial discount rates, and then apply linear interpolation between two trial rates for exact precision: $$\text{Exact YTM} = L + \frac{\text{PV at } L - P_0}{\text{PV at } L - \text{PV at } H} \times (H - L)$$
15. Macaulay Duration, Modified Duration & Convexity
15.1 Macaulay Duration (\(D\))
Developed by Frederick Macaulay in 1938, Macaulay Duration measures the weighted average maturity of a bond's cash flows, where the weights are the present values of each cash flow divided by the total bond price.
• \(t\) = Period in which the cash flow is received
• \(C_t\) = Cash flow at period \(t\) (coupon interest or maturity redemption)
• \(Y\) = Yield to Maturity (YTM) per period
• \(P_0\) = Current market price of the bond (\(P_0 = \sum \text{PV}(C_t)\))
• Zero-Coupon Bond: Duration equals its full time to maturity (\(D = n\)) because there are no intermediate coupon payments.
• Coupon-Carrying Bond: Duration is strictly less than its maturity (\(D < n\)) because intermediate coupons return cash early.
• Inverse Relations: Duration decreases as coupon rate increases; Duration decreases as YTM increases.
15.2 Modified Duration (\(MD\))
Modified Duration directly quantifies the percentage price volatility of a bond for a 100 basis point (1%) shift in yield:
• \(m\) = Compounding frequency per year (\(m = 1\) for annual, \(m = 2\) for semi-annual)
• \(\Delta Y\) = Change in market yield in decimal form (e.g. +50 bps = \(+0.0050\))
• Negative sign confirms inverse relationship between price and yield.
15.3 Bond Convexity Adjustment
Modified duration assumes a straight tangent line for price changes. But since the true price-yield curve is convex, duration alone underpredicts price increases when yields fall and overpredicts price drops when yields rise. Adding Convexity restores perfect accuracy:
16. Bond Immunization Strategy
Immunization is an active portfolio management technique designed to insulate a bond portfolio from interest rate risk so that its value at a targeted future horizon date matches or exceeds a promised liability, regardless of what happens to interest rates in the interim.
When interest rates shift, two opposing forces interact:
1. Price Risk: Rising interest rates cause bond market prices to drop (hurting returns).
2. Reinvestment Risk: Rising interest rates allow coupon cash flows to be reinvested at higher yields (helping returns).
The Immunization Condition: By matching the Macaulay Duration of the bond portfolio to the target investment horizon (\(D_A = H\)), price risk exactly offsets reinvestment risk!
17. Term Structure of Interest Rates & Yield Curves
The Yield Curve plots bond yields against differing maturities for debt securities of identical credit quality (typically risk-free government G-Secs).
Normal (Upward Sloping)
Long-term yields exceed short-term yields, reflecting healthy economic growth expectations and liquidity premiums.
Inverted (Downward Sloping)
Short-term yields exceed long-term yields. Historically the most reliable precursor of impending economic recession.
Flat Yield Curve
Short, medium, and long-term maturities offer identical yields; represents economic transition phases.
Humped Yield Curve
Intermediate maturities offer higher yields than both ultra-short and ultra-long maturities.
Three Theories of Term Structure
- Pure Expectations Theory: Forward rates are unbiased predictors of expected future short-term spot rates: $$(1 + r_{0, 2})^2 = (1 + r_{0, 1})(1 + f_{1, 2})$$
- Liquidity Preference Theory: Investors prefer liquid short-term debt and demand an extra liquidity premium to hold volatile long-term bonds. Hence, yield curves naturally slope upward.
- Market Segmentation Theory: Institutional investors (banks prefer short-term; pension funds prefer long-term) are confined to specific maturity segments, with yields set strictly by supply and demand in each independent segment.
18. Convertible Debentures, Warrants & Bond Refunding
18.1 Convertible Debentures
18.2 Zero Coupon Bonds (Deep Discount Bonds)
Bonds carrying zero coupon payments sold at a deep discount to face value and redeemed at par:
$$P_0 = \frac{RV}{(1 + k_d)^n}$$18.3 Bond Refunding Decision
A capital budgeting appraisal where an issuer evaluates whether to exercise the call option on existing high-coupon bonds and reissue lower-coupon debt in a declining interest rate environment:
$$\text{NPV of Refunding} = \text{PV of Post-Tax Interest Savings} + \text{PV of Tax Shield on Call Premium \& Unamortized Costs} - \text{Net Initial Cash Outlay}$$19. Money Market Instruments & Yield Formulas
Money market instruments are short-term, highly liquid debt instruments with maturities under 1 year:
| Instrument | Issuer | Maturity & Characteristics | Yield Formula |
|---|---|---|---|
| Treasury Bills (T-Bills) | Government of India (RBI) | Issued at discount to face value in tenors of 91, 182, and 364 days. Zero default risk. | $$\text{Yield} = \frac{F - P}{P} \times \frac{365}{M} \times 100$$ |
| Commercial Paper (CP) | Highly rated corporate borrowers | Unsecured short-term promissory notes issued at discount. Tenor: 7 days to 1 year. | $$\text{Yield} = \frac{F - P}{P} \times \frac{365}{\text{Days}} \times 100$$ |
| Certificate of Deposit (CD) | Commercial Banks / All-India Financial Institutions | Negotiable term deposits issued to bulk corporate depositors at market-determined rates. | $$\text{Yield} = \frac{\text{Maturity Value} - P}{P} \times \frac{365}{\text{Days}} \times 100$$ |
20. ICAI Valuation Standards (IVS) & Registered Valuers
Under Section 247 of the Companies Act, 2013, valuations required under company law, insolvency (IBC 2016), and corporate restructuring must be conducted by Registered Valuers registered with the Insolvency and Bankruptcy Board of India (IBBI).
| Standard Code | Standard Title | Core Scope & Mandate |
|---|---|---|
| IVS 101 | Definitions | Defines foundational concepts: Base of Value, Premise of Value, Valuation Date, Valuation Approach. |
| IVS 102 | Valuation Bases | Establishes valuation bases: Fair Value, Participant Specific Value, and Liquidation Value. |
| IVS 103 | Valuation Approaches & Methods | Codifies the three recognized approaches: Market Approach, Income Approach, and Cost Approach. |
| IVS 201 | Scope of Work, Analyses and Evaluation | Mandates terms of engagement, scope of investigations, data validation, and documentation. |
| IVS 202 | Valuation Reporting | Mandates minimum contents of a formal Valuation Report (opinion, methodology, assumptions, disclaimers). |
| IVS 301 | Business Valuation | Applies to the valuation of businesses, business ownership interests, and corporate equity securities. |
21. Comprehensive Solved ICAI Practical Problems
Solved Problem 1: Book Value, ROE, Retention Ratio: Gordon vs Walter Model (ICAI TYK 1)
Problem Statement: A company has a book value per share of ₹137.80. Its return on equity (ROE) is 15% and it follows a policy of retaining 60% of its earnings. If the Opportunity Cost of Capital is 18%, compute the price of the share today using both the Dividend Growth Model and Walter's Model.
Step 1: Compute Basic Parameters:
• \(\text{Book Value per Share} = ₹137.80\)
• \(\text{Earnings Per Share (EPS)} = \text{Book Value} \times \text{ROE} = 137.80 \times 15\% = ₹20.67\)
• \(\text{Retention Ratio } b = 60\% = 0.60\)
• \(\text{Dividend Payout Ratio } (1 - b) = 40\% = 0.40\)
• \(\text{Current Dividend } D_0 = \text{EPS} \times 40\% = 20.67 \times 0.40 = ₹8.268\)
• \(\text{Sustainable Growth Rate } g = b \times r = 0.60 \times 15\% = 9\% = 0.09\)
• \(\text{Cost of Capital } K_e = 18\% = 0.18\)
Step 2: Valuation under Dividend Growth Model (Gordon Model):
Expected Dividend \(D_1 = D_0 (1 + g) = 8.268 \times (1 + 0.09) = 8.268 \times 1.09 = ₹9.012\)
Step 3: Valuation under Walter's Model:
Under Walter's Model, the formula utilizes current dividend \(D = ₹8.268\), \(E = ₹20.67\), \(r = 0.15\), \(K_e = 0.18\):
Summary: Share price under Gordon Model is ₹100.13 and under Walter's Model is ₹103.35.
Solved Problem 2: Rights Issue Ex-Rights Price, Value of Right & Wealth Invariance (ICAI TYK 2)
Problem Statement: ABC Limited's shares are currently selling at ₹13 per share. There are 10,00,000 shares outstanding. The firm is planning to raise ₹20 lakhs to finance a new project.
Required: What are the ex-rights price and value of a right if:
(i) The firm offers one right share for every two shares held (1-for-2)?
(ii) The firm offers one right share for every four shares held (1-for-4)?
(iii) How does the wealth of a shareholder holding 100 shares change from (i) to (ii)? Does a rights issue increase shareholders' wealth?
Case (i): Rights offered on 1-for-2 basis:
• Number of right shares issued \(= \frac{10,00,000}{2} = 5,00,000 \text{ shares}\).
• Funds to be raised \(= ₹20,00,000\).
• Subscription Price per right share \(S = \frac{20,00,000}{5,00,000} = ₹4.00\).
• Ex-rights price \(P_{ex}\):
$$P_{ex} = \frac{N \times P_0 + S}{N + 1} = \frac{2 \times 13 + 4}{2 + 1} = \frac{26 + 4}{3} = \frac{30}{3} = ₹10.00$$
• Value of Right \(V_R = P_{ex} - S = 10.00 - 4.00 = ₹6.00\) (or \(\frac{P_0 - S}{N + 1} = \frac{13 - 4}{3} = ₹3.00\) per existing share).
Case (ii): Rights offered on 1-for-4 basis:
• Number of right shares issued \(= \frac{10,00,000}{4} = 2,50,000 \text{ shares}\).
• Subscription Price per right share \(S = \frac{20,00,000}{2,50,000} = ₹8.00\).
• Ex-rights price \(P_{ex}\):
$$P_{ex} = \frac{N \times P_0 + S}{N + 1} = \frac{4 \times 13 + 8}{4 + 1} = \frac{52 + 8}{5} = \frac{60}{5} = ₹12.00$$
• Value of Right \(V_R = P_{ex} - S = 12.00 - 8.00 = ₹4.00\) (or \(\frac{P_0 - S}{N + 1} = \frac{13 - 8}{5} = ₹1.00\) per existing share).
Case (iii): Shareholders' Wealth Analysis (Holder of 100 shares):
• Initial Wealth: \(100 \text{ shares} \times ₹13 = ₹1,300\).
• Under Option (i) (1-for-2):
- Entitled to 50 right shares at ₹4 = Cash outflow ₹200.
- Value of 150 shares after issue at \(P_{ex} = ₹10\) \(= 150 \times 10 = ₹1,500\).
- Net Wealth \(= \text{Total Value} - \text{Cash Outflow} = 1,500 - 200 = ₹1,300\).
• Under Option (ii) (1-for-4):
- Entitled to 25 right shares at ₹8 = Cash outflow ₹200.
- Value of 125 shares after issue at \(P_{ex} = ₹12\) \(= 125 \times 12 = ₹1,500\).
- Net Wealth \(= \text{Total Value} - \text{Cash Outflow} = 1,500 - 200 = ₹1,300\).
• Conclusion: Shareholders' wealth remains strictly unchanged at ₹1,300 in both cases. A rights issue per se does NOT increase shareholders' wealth; it merely reallocates equity ownership unless the newly financed project generates positive Net Present Value (NPV).
Solved Problem 3: Two-Stage Dividend Valuation (ICAI TYK 3)
Problem Statement: MNP Ltd. has declared and paid an annual dividend of ₹4 per share (\(D_0\)). It is expected to grow @ 20% for the next two years and 10% thereafter indefinitely. The required rate of return of equity investors is 15%. Compute the current intrinsic price at which equity shares should sell.
Given PVIF @ 15%: Year 1 = 0.8696; Year 2 = 0.7561.
Step 1: Compute Expected Dividends for Supernormal Period:
• Year 1 Dividend \(D_1 = D_0 \times (1 + 0.20) = 4.00 \times 1.20 = ₹4.80\).
• Year 2 Dividend \(D_2 = D_1 \times (1 + 0.20) = 4.80 \times 1.20 = ₹5.76\).
Step 2: Compute PV of Supernormal Dividends:
• \(\text{PV of } D_1 = 4.80 \times 0.8696 = ₹4.174\)
• \(\text{PV of } D_2 = 5.76 \times 0.7561 = ₹4.355\)
• \(\text{Total PV of Dividends (Years 1–2)} = 4.174 + 4.355 = ₹8.529\)
Step 3: Compute Terminal Value at End of Year 2 (\(P_2\)):
Stable growth begins in Year 3 with \(g_n = 10\%\):
• Year 3 Dividend \(D_3 = D_2 \times (1 + 0.10) = 5.76 \times 1.10 = ₹6.336\).
$$P_2 = \frac{D_3}{K_e - g_n} = \frac{6.336}{0.15 - 0.10} = \frac{6.336}{0.05} = ₹126.72$$
Step 4: Discount Terminal Value to Present Value:
$$\text{PV of } P_2 = 126.72 \times \text{PVIF}_{15\%, 2} = 126.72 \times 0.7561 = ₹95.813$$
Step 5: Compute Current Intrinsic Price (\(P_0\)):
$$P_0 = \text{PV of } D_1 + \text{PV of } D_2 + \text{PV of } P_2 = 4.174 + 4.355 + 95.813 = ₹104.34$$
The stock should currently sell at ₹104.34.
Solved Problem 4: Macaulay Duration, Modified Duration & Price Volatility Table
Problem Statement: A ₹1,000 par value bond carries an 8% annual coupon and has 3 years to maturity. The current required market yield (YTM) is 10%.
Required:
(i) Calculate the current market price of the bond.
(ii) Calculate the Macaulay Duration of the bond.
(iii) Calculate the Modified Duration and estimate the percentage price change if yield increases by 75 basis points (+0.75%).
Step 1: Cash Flow & Duration Computation Table:
| Year (\(t\)) | Cash Flow (\(C_t\)) | PV Factor @ 10% | PV of Cash Flow (\(\text{PV}_t\)) | Weight (\(w_t = \text{PV}_t / P_0\)) | Weighted Time (\(t \times w_t\)) |
|---|---|---|---|---|---|
| 1 | ₹80 | 0.9091 | ₹72.73 | 0.0765 | 0.0765 |
| 2 | ₹80 | 0.8264 | ₹66.11 | 0.0696 | 0.1392 |
| 3 | ₹1,080 | 0.7513 | ₹811.40 | 0.8539 | 2.5617 |
| Total Market Price (\(P_0\)) | ₹950.24 | 1.0000 | \(D = 2.7774 \text{ years}\) | ||
Step 2: Macaulay Duration:
$$\text{Macaulay Duration } (D) = \mathbf{2.78 \text{ years}}$$
(Notice that \(D = 2.78 \text{ years} < 3 \text{ years}\), confirming Theorem 4!)
Step 3: Modified Duration & Price Sensitivity:
$$MD = \frac{D}{1 + Y} = \frac{2.7774}{1 + 0.10} = \frac{2.7774}{1.10} = \mathbf{2.5249 \text{ years}}$$
If yield increases by \(\Delta Y = +0.75\% = +0.0075\):
$$\frac{\Delta P}{P_0} \approx -MD \times \Delta Y = -2.5249 \times (+0.0075) = -0.01894 = \mathbf{-1.89\%}$$
$$\text{Estimated Price Drop} = -1.89\% \times 950.24 = \mathbf{-₹17.96}$$
$$\text{Estimated New Bond Price} = 950.24 - 17.96 = \mathbf{₹932.28}$$
22. Self-Test Practice Questions (ICAI TYK)
In a coupon-carrying bond, the investor receives cash inflows before the final maturity date in the form of intermediate annual or semi-annual coupon payments. Because positive weights are assigned to periods prior to maturity (\(t = 1, 2, \dots, n-1\)), the weighted average maturity must mathematically be strictly less than the final redemption year (\(n\)).
Only in the case of a Zero-Coupon Bond, where 100% of the cash flow is received strictly at maturity with zero intermediate cash flows, does duration equal maturity (\(D = n\)).
• Valuation Formula: \(P_0 = \frac{RV}{(1 + k_d)^n}\).
• Zero Reinvestment Risk: Because there are no intermediate coupon flows to reinvest, an investor holding a ZCB to maturity faces zero reinvestment rate risk, locking in the promised yield unconditionally.
• Maximum Duration & Price Volatility: For a given maturity, ZCBs exhibit the maximum Macaulay duration (\(D = n\)) and the highest price sensitivity to interest rate fluctuations among all debt instruments.
• DDM: Measures cash flows actually paid out to shareholders in dividends. It reflects managerial dividend discretion, conservative payout policies, and signaling motives.
• FCFE: Measures cash flows available to be paid out to equity holders after funding capital expenditures, debt repayments, and working capital needs.
• Identical Valuation Condition: DDM and FCFE yield identical equity valuations when either:
(1) Dividends paid equal 100% of FCFE in every period; or
(2) Any excess cash retained (\(\text{FCFE} - \text{Dividends}\)) is invested in liquid marketable securities that earn a return exactly equal to the cost of equity (\(K_e\)).
Economic Rationale: The true mathematical price-yield function of a standard option-free bond is strictly convex to the origin. The slope of the curve flattens as yields rise (limiting capital losses) and steepens as yields fall (accelerating capital gains). In financial terms, positive convexity is a built-in protective buffer for bondholders, amplifying upside during bull rallies while softening downside during selloffs.
23. Master Quick Revision Summary
| Model / Valuation Metric | Core Formula | Crucial Exam Takeaway |
|---|---|---|
| Gordon Constant Growth | $$P_0 = \frac{D_1}{K_e - g}$$ | Growth rate \(g = b \times r\). Must satisfy \(K_e > g\). |
| Walter Model | $$P = \frac{D + \frac{r}{K_e}(E - D)}{K_e}$$ | If \(r > K_e \implies\) Payout 0%; If \(r < K_e \implies\) Payout 100%; If \(r = K_e \implies\) Indifferent. |
| H-Model | $$P_0 = \frac{D_0(1 + g_L) + D_0 \cdot H \cdot (g_S - g_L)}{K_e - g_L}$$ | \(H = t / 2\) (half-life of linear transition). |
| FCFF | $$\text{EBIT}(1 - t) + \text{Dep} - \text{Capex} - \Delta \text{NWC}$$ | Discounted at WACC to obtain Enterprise Value. |
| FCFE | $$\text{NI} + \text{Dep} - \text{Capex} - \Delta \text{NWC} + \text{Net Borrowing}$$ | Discounted at \(K_e\) to obtain Equity Value directly. |
| Ex-Rights Price | $$P_{ex} = \frac{N \times P_0 + S}{N + 1}$$ | Value of Right \(V_R = P_{ex} - S\). Shareholder wealth invariant. |
| YTM Approximation | $$\text{YTM} \approx \frac{I + \frac{RV - P_0}{n}}{\frac{RV + P_0}{2}}$$ | Interpolate between two discount rates for exact YTM. |
| Macaulay Duration | $$D = \frac{\sum t \cdot \text{PV}(C_t)}{P_0}$$ | Duration of ZCB \(= n\); Duration of coupon bond \(< n\). |
| Modified Duration | $$MD = \frac{D}{1 + Y/m}, \quad \frac{\Delta P}{P} \approx -MD \cdot \Delta Y$$ | Quantifies percentage price change per 1% change in yield. |
| Immunization Condition | $$\text{Portfolio Duration } D_A = \text{Target Horizon } H$$ | Price risk perfectly cancels reinvestment risk. |