Valuation of Bonds, Shares and Businesses

Introduction

Valuation converts expected future economic benefits into a present monetary amount. It underpins investment selection, financing decisions, financial reporting, acquisitions, restructuring and performance assessment. Although valuation models differ across securities, their central principle is consistent: an asset is worth the present value of the cash flows available to its investor, discounted at a rate reflecting timing and risk.

This chapter examines bonds, preference shares, ordinary shares and entire businesses. It integrates discounted cash flow (DCF), dividend discount, free-cash-flow and market-multiple models. Particular attention is given to yield, duration, growth assumptions and sensitivity analysis because small changes in these inputs can materially alter estimated value.

Learning Objectives

After studying this chapter, students should be able to:

• value fixed-rate, zero-coupon and perpetual bonds;
• distinguish coupon rate, current yield and yield to maturity;
• calculate and interpret Macaulay duration and modified duration;
• value preference shares with perpetual or finite dividends;
• apply constant-growth and multistage dividend discount models;
• estimate business value using FCFF and FCFE models;
• reconcile enterprise value with equity value;
• apply and critically evaluate market multiples;
• construct valuation sensitivity and scenario analyses; and
• identify governance, ethical and model-risk issues in valuation.

1. The Economic Foundation of Valuation

The general DCF model is:

where:

• V₀ = value at time 0, in currency units;
• CF_t = expected cash flow at time t;
• r_t = discount rate appropriate to the maturity and risk of that cash flow; and
• n = final forecast period.

If one constant discount rate is appropriate, r_t=r for all periods. This simplification assumes the risk and term structure represented by the rate remain sufficiently stable.

Valuation requires consistency between three elements:

1. Cash-flow claim: Debt cash flows must be discounted at a debt-required return, equity cash flows at an equity-required return, and operating cash flows available to all capital providers at WACC.
2. Currency and inflation: Nominal cash flows require nominal discount rates in the same currency. Real cash flows require real rates.
3. Timing: Annual, semi-annual and monthly cash flows must use compatible periodic discount rates.

Figure: Future cash flows are discounted to their present value using rates consistent with their risk and timing.

Valuation is forward-looking. Historical accounts provide evidence, but value depends on expected future cash flows and risk. Consequently, valuation is an informed estimate rather than an observable fact.

2. Bond Valuation

2.1 Bond cash flows

A conventional fixed-rate bond promises periodic coupon payments and repayment of face value at maturity. For a bond with m coupon payments per year:

where:

• P₀ = bond price;
• F = face or redemption value;
• C=Fc/m = coupon payment per period;
• c = annual coupon rate;
• y = nominal annual yield to maturity compounded m times;
• N=mT = total payment periods; and
• T = years to maturity.

The formula assumes promised cash flows occur on schedule and that one yield discounts all payments. For risky bonds, the yield incorporates benchmark interest rates, credit risk and other premiums.

Worked Example 1: Fixed-rate bond

A five-year bond has a £1,000 face value, a 6% annual coupon and annual payments. Investors require an 8% annual yield.

The annual coupon is C=£1,000×0.06=£60. Therefore:

The bond sells below face value because its 6% coupon rate is lower than the required 8% yield. Subject to credit effects, bond price and market yield move in opposite directions.

2.2 Special bond forms

A zero-coupon bond makes no interim coupon payments:

An irredeemable or perpetual bond paying a constant annual coupon has value:

The perpetuity formula applies only when payments continue indefinitely, the first payment occurs one period later, and y>0.

Floating-rate notes, callable bonds and convertible bonds require additional analysis. A callable bond cannot be valued reliably as a non-callable bond when exercise is probable because the issuer's call option limits price appreciation.

2.3 Coupon rate, current yield and yield to maturity

These measures are not interchangeable:

For Worked Example 1, current yield is:

YTM is 8%, but this is not a guaranteed realised return. The standard interpretation assumes coupons are received as promised and reinvested consistently with the yield calculation.

For bonds traded between coupon dates, the dirty price equals the quoted clean price plus accrued interest. Settlement and accounting analysis must distinguish these price conventions.

2.4 Duration and interest-rate sensitivity

Maturity alone is an incomplete measure of interest-rate exposure because coupon payments return part of the investment before maturity. Macaulay duration is the present-value-weighted average time to receive a bond's cash flows:

Here, D_Mac is measured in years and CF_t is the cash flow in period t. Modified duration converts Macaulay duration into an approximate price sensitivity to annual yield:

For a small parallel yield change ∆y:

The approximation is appropriate for option-free bonds and small yield changes. For the five-year bond, Macaulay duration is approximately 4.44 years and modified duration is approximately 4.11. A 1 percentage-point increase in yield therefore implies an approximate price fall of 4.11%.

The price–yield relationship is curved rather than linear. Convexity improves the approximation for larger yield changes:

Duration is widely used in asset–liability management, immunisation and fixed-income portfolio risk control. It does not, however, fully represent credit-spread changes, embedded options or non-parallel shifts in the yield curve.

3. Preference Share Valuation

Preference shares normally provide a fixed dividend and rank ahead of ordinary shares for dividends and capital distributions. Unlike bond interest, preference dividends are generally not contractual obligations in the same sense, and their omission may not constitute default.

For an irredeemable preference share paying a constant annual dividend D_p:

where k_p is the return required by preference shareholders.

If an £8 annual preference dividend is required to yield 10%:

A redeemable preference share should be valued by discounting each dividend and the redemption amount:

where R_n is the redemption payment. Analysts must also consider cumulative dividend rights, conversion provisions, call terms, tax treatment and liquidity.

4. Ordinary Share Valuation

Ordinary shareholders hold a residual claim. Their returns arise from dividends and changes in share price, but these ultimately depend on the company's capacity to generate distributable cash flows.

4.1 General dividend discount model

The dividend discount model (DDM) values a share as the present value of expected dividends:

where D_t is the expected dividend per share and k_e is the required return on equity.

4.2 Gordon constant-growth model

If dividends are expected to grow indefinitely at a constant rate g:

The model requires k_e>g, the first dividend one period from now, and a sustainable perpetual growth rate. A perpetual growth rate should not normally exceed long-run nominal economic growth in the company's currency indefinitely.

Suppose the current dividend is £1.50, long-run growth is 4% and the required equity return is 10%:

The Gordon model is highly sensitive when k_e-g is small. Growth must also be economically connected to reinvestment. A useful steady-state relationship is:

where b is the earnings retention ratio and ROE is the return on reinvested equity, assuming both remain stable. Retention creates value only when reinvestment earns an adequate return; retaining more earnings does not automatically increase value.

4.3 Multistage dividend growth

Young or restructuring companies rarely enter stable growth immediately. A multistage model forecasts exceptional growth explicitly and then applies a stable terminal model.

Assume a current dividend of £1.00 grows by 15% annually for three years and by 5% perpetually thereafter. The required return is 11%.

The year-4 dividend and terminal value at year 3 are:

Discounting the terminal value and the three dividends gives:

The model is suitable for dividend-paying companies with a defensible payout policy. It is less informative when dividends are absent, discretionary or unrelated to economic capacity. In such cases, free-cash-flow models are often preferable.

4.4 Estimating the required return on equity

The capital asset pricing model (CAPM) is frequently used:

where r_f is the risk-free rate, β_e is equity beta, and E(R_M)-r_f is the expected market risk premium. CAPM estimates are sensitive to the government yield selected, beta period, market proxy and equity-risk-premium method. Country, size or other risk adjustments require careful justification to avoid double counting.

5. Business Valuation Using Free Cash Flow

5.1 Free cash flow to the firm

Free cash flow to the firm (FCFF) is cash generated by operations after tax and necessary reinvestment, but before payments to debt and equity providers:

where EBIT is operating profit before interest and tax, T is the applicable operating tax rate, D\&A is depreciation and amortisation, and ∆NWC is the increase in operating net working capital. Financing cash balances and interest-bearing debt are normally excluded from operating working capital.

FCFF is discounted at the weighted average cost of capital:

where w_e and w_d are market-value capital weights, k_e is the required equity return, and k_d is the pre-tax cost of debt. This conventional formula assumes interest tax shields are usable and debt risk and capital structure are sufficiently stable. Complex or changing financing may require adjusted present value or explicit tax-shield modelling.

Enterprise value is:

Using a perpetual-growth terminal value:

The conditions WACC>g and sustainable stable-state operations are essential.

A simplified enterprise-to-equity bridge is:

Cash is often added as a non-operating asset, while borrowings, unfunded pensions, lease-related claims or minority interests may require adjustments depending on how operating cash flow and enterprise value were defined. Consistency is more important than mechanically applying a standard bridge.

5.2 Free cash flow to equity

FCFE measures cash available directly to ordinary shareholders after debt cash flows:

FCFE is discounted at k_e:

FCFE is useful when leverage can be forecast reliably. FCFF is often more stable when comparing businesses with different financing structures.

6. Applied Case: Orion Components Ltd

This is an illustrative case.

Background and problem

Orion Components Ltd is a mature manufacturer considering a strategic sale. Management forecasts FCFF of £9 million, £10 million and £11 million over the next three years. The base WACC is 9%, perpetual growth is 2.5%, net debt is £30 million and 8 million shares are outstanding. The task is to estimate equity value and assess sensitivity to WACC and growth.

Approach and analysis

The year-4 FCFF is:

The terminal value at the end of year 3 is:

The enterprise value is:

The first two present values are £8.2569 million and £8.4168 million. The year-3 combined cash-flow present value is approximately £142.4382 million. Therefore:

Sensitivity analysis

The table recalculates the complete model for each WACC and perpetual-growth combination. Values are per share.

Outcome

The base estimate is £16.14 per share, but the defensible range changes substantially with modest assumptions. A lower WACC and higher growth rate increase value because they reduce the discounting burden and increase terminal value. The table should not be interpreted as proving that all combinations are equally plausible.

Learning points

• Terminal value can represent a large part of enterprise value.
• WACC and perpetual growth must be assessed jointly.
• Precision in the output does not eliminate uncertainty in the inputs.
• Management forecasts should be tested against capacity, competition and reinvestment needs.
• A valuation conclusion is better expressed as a reasoned range than as an unsupported single figure.

7. Market-Multiple Valuation

Relative valuation applies pricing evidence from comparable businesses or transactions. Common multiples include:

Suppose a company has maintainable EBITDA of £20 million and comparable firms trade at 7 times EV/EBITDA:

If net debt is £35 million and 5 million shares are outstanding:

Comparable companies should resemble the subject in growth, margins, risk, geography, accounting policy and capital intensity. EBITDA is not cash flow because it excludes capital expenditure, working-capital investment and tax. Transaction multiples may include a control premium and deal-specific synergies, so they are not automatically transferable to minority shares.

Multiples are most useful as a triangulation tool alongside intrinsic valuation. If DCF and market multiples disagree, analysts should investigate the reasons rather than average the results mechanically.

8. Sensitivity, Scenarios and Valuation Discipline

8.1 Sensitivity analysis

Sensitivity analysis changes one or two assumptions while holding others constant. Important variables include:

• discount rates;
• revenue growth and margins;
• capital expenditure and working capital;
• explicit forecast length;
• terminal growth; and
• exit multiples.

It reveals model exposure but does not assign probabilities.

8.2 Scenario analysis

Scenario analysis changes internally connected assumptions. A recession scenario may combine lower revenue, reduced margins, higher credit spreads and slower recovery. An upside scenario may combine stronger volume with additional capacity expenditure. Scenarios are more economically coherent than changing isolated inputs, but their probabilities remain judgemental.

8.3 Reverse DCF

A reverse DCF starts with the market price and solves for the growth, margin or return assumptions implied by that price. It helps analysts ask whether market expectations are plausible rather than merely producing another target price.

8.4 Best-practice checks

A defensible valuation should:

1. reconcile forecasts with historical financial statements;
2. separate operating and financing items consistently;
3. use market-value capital weights where observable;
4. link growth to reinvestment and returns on capital;
5. normalise temporary earnings and exceptional items;
6. test terminal assumptions against mature-company economics;
7. reconcile DCF results with trading and transaction evidence; and
8. document sources, judgements and model versions.

9. Limitations, Ethics and Governance

Valuation models can create an illusion of objectivity. Major limitations include forecast error, unstable discount rates, uncertain terminal economics, accounting inconsistency, illiquidity and omitted contingent claims. Models are especially fragile for start-ups, distressed companies, cyclical firms and businesses undergoing structural change.

Ethical problems arise when analysts select assumptions to achieve a desired transaction price, ignore adverse evidence, conceal conflicts of interest or apply unjustified premiums and discounts. Professional integrity requires transparent assumptions, balanced scenarios and clear separation between management representations and independent analyst judgement.

In financial reporting, IFRS 13 defines fair value using a market-participant and exit-price perspective. It prioritises observable inputs through the fair-value hierarchy. A management-specific intrinsic value is therefore not necessarily an IFRS fair value. Purpose and basis of value must be stated before selecting methods and assumptions.

Strong governance commonly includes independent review, spreadsheet controls, source documentation, approval of key assumptions, access restrictions and comparison of prior forecasts with actual outcomes.

10. Technology, AI and Contemporary Developments

Valuation software can integrate market data, automate comparable-company screening and run extensive simulations. Artificial intelligence can assist with extracting lease commitments, segment data and risk disclosures from reports. However, automated outputs may contain classification errors, stale market data or opaque assumptions. Human review remains necessary.

Contemporary valuation challenges include volatile interest rates, inflation uncertainty, intangible-intensive business models, climate-transition risks and changing supply chains. Environmental or technological developments affect value only through their expected effects on cash flow, asset life, investment requirements or risk. Adding an arbitrary “ESG premium” without demonstrating these channels risks double counting.

Chapter Summary

Valuation rests on discounting cash flows at rates consistent with their risk, timing, currency and claim. Bond value equals the present value of coupons and redemption proceeds, while yield and duration describe return and interest-rate exposure. Preference shares may be valued as perpetuities or finite dividend streams.

Ordinary shares can be valued with dividend discount models when payout patterns represent economic capacity. Constant-growth models require sustainable growth below the required return, while multistage models accommodate transitional growth. Business valuation commonly uses FCFF discounted at WACC or FCFE discounted at the cost of equity. Enterprise value must then be reconciled carefully to equity value.

Market multiples provide useful external evidence but depend on genuine comparability. Because discount rates, margins and terminal assumptions materially influence value, sensitivity, scenario and reverse-DCF analyses are essential. Ethical practice requires transparent assumptions, independent judgement, consistent definitions and effective model governance.

Key Terms

Bond: A financial instrument promising specified interest and principal payments.

Clean price: A bond price excluding accrued interest.

Convexity: A measure of curvature in the relationship between bond price and yield.

Current yield: Annual bond coupon divided by current market price.

Dirty price: A bond's clean price plus accrued interest.

Dividend discount model: A model valuing equity as the present value of expected dividends.

Enterprise value: The value of operating assets available to all capital providers.

Equity value: The residual value attributable to ordinary shareholders.

FCFE: Cash flow available to ordinary equity after investment and net debt financing.

FCFF: Operating cash flow available to debt and equity investors after tax and reinvestment.

Gordon growth model: A constant-growth perpetuity model for dividends or free cash flow.

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

Market multiple: A ratio relating market value to an operating or accounting measure.

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

Preference share: A share carrying priority rights, usually including a specified dividend.

Terminal value: The value at a forecast horizon of cash flows occurring thereafter.

WACC: Weighted required return of debt and equity capital providers after relevant tax effects.

Yield to maturity: The discount rate equating a bond's price to its promised cash flows.

Review Questions

Short Answer Questions

1. Why does a bond sell at a discount when its coupon rate is below its required yield?
2. Distinguish current yield from yield to maturity.
3. State the conditions required for the Gordon growth model.
4. Explain the difference between enterprise value and equity value.
5. Why is EBITDA not equivalent to free cash flow?

Long Answer Questions

1. Explain how interest rates, credit risk, maturity and embedded options influence bond valuation.
2. Compare dividend discount, FCFE and FCFF valuation models, including their appropriate applications.
3. Discuss the estimation and limitations of WACC in business valuation.
4. Critically evaluate market multiples as an alternative to intrinsic valuation.
5. Explain the ethical and governance controls required for a professionally defensible valuation.

Analytical / Application Questions

1. A four-year £1,000 bond pays a 5% annual coupon. Calculate its value at a 7% annual required yield and explain whether it trades above or below face value.
2. A company expects next year's dividend to be £2.40. Estimate share value if long-run growth is 3% and the required equity return is 9%. Assess the effect of increasing growth to 4%.
3. Construct a two-variable sensitivity table for a business with forecast FCFF of £6 million next year, using WACC values of 8%, 9% and 10% and perpetual growth rates of 2% and 3%. State the model's assumptions.

Multiple Choice Questions with Answers

Question 1

A five-year £1,000 bond pays a 6% annual coupon. If the annual yield to maturity is 8%, what is its approximate value?

A. £680.58

B. £920.15

C. £1,000.00

D. £1,079.85

Correct Answer: B. £920.15

Question 2

An irredeemable preference share pays an annual dividend of £5. If investors require 8%, what is its value?

A. £40.00

B. £54.00

C. £62.50

D. £67.50

Correct Answer: C. £62.50

Question 3

A share has just paid a dividend of £2.00. If dividends grow perpetually at 3% and the required return is 9%, what is the Gordon-model value?

A. £33.33

B. £34.33

C. £35.33

D. £68.67

Correct Answer: B. £34.33

Question 4

An option-free bond has modified duration of 4.5. For a small 0.5 percentage-point increase in yield, what is the approximate price change?

A. A 0.90% decrease

B. A 2.25% decrease

C. A 4.50% decrease

D. A 9.00% increase

Correct Answer: B. A 2.25% decrease

Question 5

A company reports EBIT of £20m, a 25% tax rate, depreciation of £3m, capital expenditure of £4m and an increase in operating net working capital of £2m. What is FCFF?

A. £10m

B. £11m

C. £12m

D. £16m

Correct Answer: C. £12m

Question 6

A business has enterprise value of £150m, debt of £40m, non-operating cash of £10m and 6m shares. Ignoring other claims, what is the value per share?

A. £16.67

B. £18.33

C. £20.00

D. £25.00

Correct Answer: C. £20.00

Question 7

Which condition is essential when applying a perpetual-growth terminal-value model?

A. Perpetual growth must exceed WACC

B. WACC must exceed perpetual growth

C. Depreciation must equal revenue

D. The firm must have no debt

Correct Answer: B. WACC must exceed perpetual growth

Question 8

In the Orion case, what is the approximate value per share when WACC is 10% and perpetual growth is 2.5%?

A. £12.51

B. £13.46

C. £14.54

D. £16.14

Correct Answer: B. £13.46

Question 9

A company has maintainable EBITDA of £20m and is valued at 7 times EV/EBITDA. If net debt is £35m and 5m shares exist, what is the value per share?

A. £7

B. £20

C. £21

D. £28

Correct Answer: C. £21

Question 10

Which statement about IFRS 13 fair value is correct?

A. It is always identical to management's intrinsic value

B. It excludes all observable market information

C. It uses a market-participant exit-price perspective

D. It requires the Gordon growth model for listed shares

Correct Answer: C. It uses a market-participant exit-price perspective