Valuation of Bonds, Shares and Businesses

Introduction

Valuation converts expected future economic benefits into a present value. It underpins investment selection, financing decisions, financial reporting, mergers and acquisitions, and corporate strategy. Although bonds, shares and businesses have different contractual and economic characteristics, their values follow the same principle: expected future cash flows must be discounted at a rate reflecting timing and risk.

Market price and intrinsic value are not necessarily equal. Market price is the amount currently quoted or negotiated; intrinsic value is an estimate produced from cash-flow, growth and risk assumptions. A difference between them may indicate mispricing, but it may also reflect model error, omitted information, liquidity constraints or different risk assessments.

Learning Objectives

After studying this chapter, students should be able to:

• value coupon bonds, zero-coupon bonds and perpetual bonds;
• distinguish coupon rate, current yield and yield to maturity;
• calculate Macaulay duration and modified duration;
• explain the inverse, convex relationship between bond prices and yields;
• value redeemable and irredeemable preference shares;
• apply dividend-discount models to ordinary shares;
• use price and enterprise-value multiples appropriately;
• estimate enterprise and equity value using free cash flow;
• evaluate growth, discount-rate and terminal-value assumptions; and
• identify professional, governance and ethical issues in valuation.

1. The Valuation Principle

If an asset generates expected cash flows CF_t at the end of periods t = 1,2,…,n, its present value is:

where:

• V₀ = value at time 0, in currency units;
• CF_t = expected cash flow in period t;
• r_t = risk-adjusted discount rate applicable to maturity t;
• n = number of forecast periods.

When one constant discount rate is appropriate, r_t = r. This simplification assumes a flat relevant yield curve and no material differences in risk among cash flows. In practice, spot rates may be preferable for contractual cash flows of different maturities.

Three inputs dominate most valuations:

1. Cash flows: their amount, timing and probability.
2. Growth: the expected development of cash flows over time.
3. Required return: compensation for the time value of money and risk.

A valuation is therefore conditional, not an unquestionable fact. Analysts should disclose assumptions and examine sensitivity to changes in them.

2. Bond Valuation

A bond normally promises periodic coupons and repayment of face value at maturity. The issuer is the borrower, while the investor is the lender.

2.1 Basic terminology

If a bond has face value F, annual coupon C, maturity n years and required annual yield y, its price immediately after a coupon payment is:

This formula assumes annual end-of-year payments and that the promised cash flows are discounted at one yield. For a default-risky bond, expected cash flows or a suitably risk-adjusted discount rate should be used rather than treating promised payments as certain.

For m coupon payments per year:

Here C_a is the annual coupon amount and y^(m) is the nominal annual yield compounded m times per year. Payment frequency and yield convention must be consistent.

Worked Example 1: Coupon-bond price

A five-year bond has a face value of $1,000, an 8% annual coupon and a required yield of 10%. The annual coupon is C = 0.08 × 1,000 = $80.

The bond sells below par because its 8% coupon rate is lower than the market-required yield of 10%.

2.2 Zero-coupon and perpetual bonds

A zero-coupon bond pays only face value at maturity:

A perpetual bond paying a constant annual coupon forever has value:

The perpetuity formula requires a constant payment, a positive discount rate and no maturity date.

2.3 Bond yields

The coupon rate is fixed at issue, but market yields and prices change.

The current yield measures annual coupon income relative to current price:

It ignores capital gains or losses and the time value of money. Yield to maturity, or YTM, is the internal rate of return on promised cash flows if the bond is held to maturity and coupons are reinvested at that yield.

Except in simple cases, YTM is found numerically. It is a promised yield, not necessarily the investor's realized return, because default, sale before maturity and reinvestment rates can differ from assumptions.

Worked Example 2: Solving YTM from market price

A three-year bond has face value $100, an annual coupon of $8 and a market price of $95.03 immediately after a coupon payment. Its YTM solves:

Testing y = 10% gives:

Therefore, the YTM is approximately 10.00%. A financial calculator or spreadsheet can solve the equation more precisely. The result is sensible because the bond trades below par and its YTM exceeds its 8% coupon rate.

2.4 Price–yield relationship

Bond price and required yield move in opposite directions. The relationship is also convex: for equal-sized yield changes, the price increase caused by a yield decrease is generally greater than the price decrease caused by an equal yield increase.

Figure: A bond's price declines at a decreasing rate as its yield rises, producing economic convexity.

Longer-maturity bonds, lower-coupon bonds and bonds with lower initial yields generally display greater price sensitivity to yield changes.

2.5 Duration

Macaulay duration is the present-value-weighted average time until a bond's cash flows are received:

where D_M is measured in years when t is measured in years.

Modified duration estimates the percentage price response to a small yield change:

For annual compounding:

Here ∆y is expressed as a decimal; for example, 50 basis points equals 0.005. The approximation is most reliable for small, parallel yield changes.

Worked Example 3: Macaulay and modified duration

Use the three-year bond from Worked Example 2 at a 10% YTM.

If the yield rises by 0.50 percentage points:

The estimated price decline is 1.262%, or approximately $95.03 × 0.01262 = $1.20. Duration is a first-order approximation; convexity explains why the exact price change will differ slightly.

2.6 Clean and dirty prices

Between coupon dates, a quoted clean price excludes accrued interest. The dirty, or invoice, price includes it:

Under a simple proportional convention:

Actual calculations follow the bond's contractual day-count convention, such as actual/actual or 30/360.

3. Preference-Share Valuation

Preference shares commonly pay a fixed dividend and rank ahead of ordinary shares for dividends and liquidation proceeds, but behind debt. Dividends are normally distributions rather than legally enforceable interest obligations.

An irredeemable preference share paying constant annual dividend D_p has value:

where k_p is the required return on preference equity. This perpetuity model assumes the dividend remains constant indefinitely.

Worked Example 4: Irredeemable preference share

A preference share pays an annual dividend of $6 and investors require 9%:

A redeemable preference share is valued like a bond, using expected dividends and redemption proceeds:

where R is the redemption amount. Analysts must consider whether dividends are cumulative, callable or convertible.

4. Ordinary-Share Valuation

Ordinary shareholders receive residual cash flows after contractual claims. Their returns arise from dividends and changes in share price.

4.1 Dividend-discount model

The general dividend-discount model is:

where E(D_t) is the expected dividend in period t and k_e is the required return on ordinary equity.

For constant perpetual dividends:

For dividends growing perpetually at constant rate g, the Gordon growth model is:

The model requires k_e > g, a stable payout policy and a sustainable long-run growth rate. The numerator is next period's dividend, not the dividend just paid.

Worked Example 5: Gordon growth valuation

A company has just paid a dividend of $2.00 per share. Dividends are expected to grow at 4% indefinitely, and shareholders require 10%.

If the market price is $40, the difference does not automatically prove overvaluation. The market may expect higher growth, lower risk or different future distributions.

4.2 Multistage dividend valuation

Companies rarely enter stable growth immediately. A two-stage model values separately the high-growth period and the stable-growth terminal value:

Here g_s is the perpetual stable-growth rate after year N. It should generally not exceed expected long-run nominal economic growth when cash flows and discount rates are nominal.

4.3 Earnings, retention and growth

Under restrictive stable-state assumptions, sustainable dividend growth may be approximated by:

where b is the earnings retention ratio and ROE is return on equity earned on retained funds. The relationship is not automatic: growth creates value only when reinvested funds earn an adequate return.

5. Relative Valuation

Relative valuation estimates value by comparing the subject asset with similar listed companies or transactions.

A central consistency rule is that equity-value numerators must be matched with equity-level denominators, while enterprise-value numerators must be matched with pre-financing operating measures.

If comparable companies trade at a median P/E of 12 and the subject company's sustainable EPS is $2.50, the indicated share value is:

Relative valuation is simple but not assumption-free. The selected peers, accounting policies, growth prospects, business risk, leverage and one-off items all affect comparability. A high multiple may reflect superior growth rather than overvaluation.

6. Business Valuation Using Free Cash Flow

6.1 Enterprise and equity value

Enterprise value, or EV, represents the value of operations available to all capital providers. A simplified bridge is:

In practice, analysts may also adjust for leases, pension deficits, minority interests, non-operating investments and contingent liabilities.

Free cash flow to the firm, or FCFF, is commonly estimated as:

where:

• EBIT = earnings before interest and tax;
• T = applicable operating tax rate;
• ∆NWC = increase in operating net working capital.

FCFF is discounted at the weighted average cost of capital, or WACC, provided operating risk and capital structure assumptions are consistent.

For perpetual growth after year N:

The conditions WACC > g and sustainable long-run operating assumptions are essential.

Worked Example 6: Enterprise valuation and growth scenarios

An illustrative company is expected to generate FCFF of $10 million at each year-end for five years. Thereafter FCFF grows by 2% perpetually. WACC is 9%. The company has debt of $50 million, no excess cash and 20 million shares.

The present value of the explicit cash flows is:

Year-six FCFF is $10 × 1.02 = $10.20 million, so terminal value at year 5 is:

Its present value is:

Therefore:

Consider a distinct alternative scenario in which FCFF is $10 million in year 1 and grows at 3% every year from then onward, including during the explicit period. Because the same 3% perpetual growth pattern begins immediately, the value can be checked directly as a growing perpetuity:

The equivalent five-year DCF gives approximately $41.09 million for explicit cash flows and $125.58 million for terminal value. Thus:

This alternative changes the entire cash-flow path, not merely the terminal-growth input. If only terminal growth changed from 2% to 3% while years 1–5 remained at $10 million, EV would instead be approximately $150.50 million. Scenario labels must therefore state exactly which assumptions change.

6.2 Terminal-value risk

Terminal value often represents a large proportion of total enterprise value. Small changes in WACC or perpetual growth can materially affect the result. Analysts should:

• use a mature-state growth rate;
• ensure capital expenditure supports assumed growth;
• normalize margins and working capital;
• test multiple WACC and growth combinations;
• compare implied terminal multiples with market evidence; and
• avoid presenting one point estimate as certain.

7. Illustrative Case: Valuing Meridian Components Ltd.

Background

Meridian Components Ltd. is a fictional manufacturer considering an initial public offering. Management has requested both DCF and relative valuations. Forecast FCFF is $12 million next year, expected to grow by 5% annually through year 5. From year 6 onward, growth is expected to stabilize at 2.5%. WACC is 9%. Debt is $60 million, excess cash is $10 million, and 25 million shares will be outstanding.

Comparable listed manufacturers trade at a median EV/EBITDA of 7.0. Meridian's maintainable next-year EBITDA is $24 million.

Problem

The board must estimate a reasonable equity-value range while avoiding overreliance on optimistic management forecasts.

DCF analysis

Forecast FCFF is:

Year-six FCFF is approximately 14.586 × 1.025 = $14.951 million.

Relative valuation

Approach and outcome

The DCF indicates approximately $6.03 per share, while the comparable-company method indicates $4.72. The difference may reflect Meridian's assumed growth, the selected peer multiple or differing business risks. The board should not average the figures mechanically. It should investigate whether forecast margins, reinvestment and growth are supportable and whether peers genuinely match Meridian's scale and risk.

A defensible valuation range could use the relative estimate as a market cross-check and the DCF as an assumption-driven central case, supplemented by downside scenarios.

Learning points

• DCF and relative valuation answer related but different questions.
• The enterprise-to-equity bridge must include debt and excess cash consistently.
• Terminal value can dominate a DCF estimate.
• Differences between methods should be explained rather than concealed.

8. Limitations and Professional Practice

Valuation models are sensitive to forecasts that cannot be observed directly. Key limitations include uncertain cash flows, changing interest rates, model risk, accounting differences, illiquidity and behavioural biases.

Good professional practice includes:

1. reconciling forecasts with historical performance and industry economics;
2. matching nominal cash flows with nominal discount rates and real cash flows with real rates;
3. matching currency-specific cash flows with discount rates for that currency;
4. avoiding double counting of risk in both cash flows and discount rates;
5. separating operating and non-operating assets;
6. using sensitivity, scenario and reverse-valuation analysis; and
7. documenting data sources, model changes and independent review.

Spreadsheets and valuation software improve speed but can propagate errors. Formula auditing, version control, automated data validation and independent replication are important controls. AI can assist with peer screening, document extraction and scenario generation, but outputs require human verification, especially where data definitions differ.

9. Ethical, Governance and Regulatory Considerations

Valuations may be influenced by incentives. Managers seeking financing may favour optimistic growth; acquiring firms may justify excessive bid prices; and advisers may face conflicts when fees depend on transaction completion.

Ethical valuation requires objectivity, professional competence, transparency and confidentiality. Analysts should disclose conflicts, distinguish management assumptions from independent assumptions, and avoid selectively choosing comparables. Material uncertainty should be communicated through ranges and scenarios.

Financial-reporting valuations may also be governed by accounting standards concerning fair value, impairment, business combinations and financial instruments. Regulatory definitions of fair value may differ from investment value or strategic value. Analysts must identify the applicable purpose and valuation basis before selecting a method.

Chapter Summary

Valuation is based on discounting expected cash flows at a rate consistent with their timing and risk. Bond values equal the present value of coupons and principal. Bond prices move inversely and convexly with yields; duration provides a first-order measure of interest-rate sensitivity. Preference shares may be valued as perpetuities or finite dividend streams.

Ordinary shares can be valued using dividend-discount models, including constant-growth and multistage approaches. Relative valuation applies market multiples but requires comparable firms and consistent numerators and denominators. Business DCF valuation discounts FCFF at WACC to estimate enterprise value, which is then reconciled to equity value. Growth, terminal value and discount-rate assumptions must be sustainable, transparent and tested. Ethical and well-governed valuation practice emphasizes evidence, consistency, sensitivity analysis and independent review.

Key Terms

Accrued interest: Coupon interest earned since the previous payment date.

Clean price: Bond price excluding accrued interest.

Convexity: Curvature in the bond price–yield relationship.

Coupon rate: Annual contractual coupon divided by face value.

Dirty price: Bond price including accrued interest.

Dividend-discount model: Share valuation based on the present value of expected dividends.

Enterprise value: Value of operating assets attributable to all capital providers.

Equity value: Residual value attributable to ordinary shareholders.

Face value: Principal amount repayable on a bond.

Free cash flow to the firm: Cash flow available to debt and equity providers after operating investment.

Gordon growth model: Perpetual dividend or cash-flow model with constant growth.

Macaulay duration: Present-value-weighted average time to receive a bond's cash flows.

Modified duration: Approximate percentage price sensitivity to a change in yield.

Relative valuation: Valuation using market multiples from comparable assets or firms.

Terminal value: Value of cash flows occurring after the explicit forecast period.

Weighted average cost of capital: Required return used to discount operating cash flows to all capital providers.

Yield to maturity: Discount rate equating a bond's market price with promised cash flows.

Zero-coupon bond: Bond that pays no coupon before maturity.

Review Questions

Short Answer Questions

1. Distinguish intrinsic value from market price.
2. Why does a bond sell below par when its coupon rate is below its required yield?
3. What is the difference between current yield and YTM?
4. State the consistency rule for enterprise-value and equity-value multiples.
5. Why must perpetual growth be lower than the discount rate?

Long Answer Questions

1. Explain the valuation principle and discuss how cash-flow, growth and discount-rate assumptions interact.
2. Compare Macaulay duration, modified duration and convexity as measures of bond interest-rate risk.
3. Evaluate the strengths and limitations of the Gordon growth model.
4. Compare DCF valuation with relative valuation and explain when each is most useful.
5. Discuss ethical, governance and model-control issues that arise in professional valuation.

Analytical / Application Questions

1. A four-year bond has face value $1,000, a 6% annual coupon and a required yield of 8%. Calculate its price and explain whether it trades at a premium or discount.
2. A company has just paid a dividend of $1.50. Dividends are expected to grow at 5% indefinitely, and the required return is 11%. Estimate intrinsic value and assess the effect of reducing growth to 4%.
3. Forecast FCFF for years 1–3 is $8 million, $9 million and $10 million. From year 4 it grows at 2.5% perpetually. With WACC of 9%, debt of $30 million, cash of $5 million and 10 million shares, estimate enterprise value, equity value and value per share.

Multiple Choice Questions with Answers

Question 1

A bond's coupon rate is 6%, while its required yield is 8%. Assuming normal positive cash flows and no special options, the bond should trade:

A. Above face value

B. At face value

C. Below face value

D. At zero value

Correct Answer: C. Below face value

Question 2

A perpetual preference share pays an annual dividend of $5 and has a required return of 10%. Its value is:

A. $25

B. $50

C. $55

D. $500

Correct Answer: B. $50

Question 3

A company has just paid a dividend of $3. Dividends will grow at 4% indefinitely and the required return is 10%. Using the Gordon model, the share value is:

A. $31.20

B. $50.00

C. $52.00

D. $75.00

Correct Answer: C. $52.00

Question 4

Which measure estimates a bond's percentage price change for a small change in yield?

A. Current yield

B. Modified duration

C. Coupon rate

D. Face value

Correct Answer: B. Modified duration

Question 5

A bond has modified duration of 4.0. If its yield rises by 0.25 percentage points, the duration estimate of the price change is:

A. A 0.10% increase

B. A 1.00% increase

C. A 0.10% decrease

D. A 1.00% decrease

Correct Answer: D. A 1.00% decrease

Question 6

Which valuation multiple is internally consistent?

A. Enterprise value divided by EBITDA

B. Enterprise value divided by earnings per share

C. Share price divided by total company revenue

D. Equity value divided by EBIT before interest

Correct Answer: A. Enterprise value divided by EBITDA

Question 7

FCFF is $12 million next year and will grow at 3% perpetually. If WACC is 9%, enterprise value under a growing perpetuity is:

A. $100 million

B. $133.33 million

C. $200 million

D. $400 million

Correct Answer: C. $200 million

Question 8

An enterprise is valued at $180 million and has debt of $50 million and excess cash of $10 million. Ignoring other adjustments, equity value is:

A. $120 million

B. $140 million

C. $220 million

D. $240 million

Correct Answer: B. $140 million

Question 9

Which statement about bond convexity is correct?

A. Bond price increases linearly as yield increases

B. Equal yield increases and decreases always produce equal price changes

C. Bond price normally declines at a decreasing rate as yield rises

D. Convexity means bond price is independent of yield

Correct Answer: C. Bond price normally declines at a decreasing rate as yield rises

Question 10

A quoted clean bond price is $980 and accrued interest is $15. The dirty price is:

A. $965

B. $980

C. $995

D. $1,015

Correct Answer: C. $995