Valuation of Bonds, Equity and Businesses

Introduction

Valuation translates expected future economic benefits into a present value. Although bonds, shares and entire businesses have different contractual and risk characteristics, their valuation rests on a common principle: an asset is worth the present value of the cash flows available to its investor.

Market prices can depart from estimated intrinsic value because investors use different forecasts, discount rates and risk assessments. Consequently, valuation is not merely a mechanical calculation. It requires defensible assumptions about growth, credit risk, capital structure, competitive advantage and terminal performance.

Learning Objectives

After studying this chapter, students should be able to:

1. value coupon bonds and zero-coupon bonds using discounted cash flow;
2. interpret yield to maturity, credit spreads and duration;
3. value equity using dividend and free-cash-flow models;
4. value a business using enterprise discounted cash flow;
5. convert enterprise value into equity value correctly;
6. apply EV/EBITDA and P/E market multiples without confusing their valuation bases;
7. evaluate terminal growth, discount-rate and credit-risk assumptions; and
8. conduct and communicate sensitivity analysis responsibly.

1. The Valuation Principle

For an asset generating cash flows CF_t over n periods, its present value is:

where:

• V₀ = value today, in currency units;
• CF_t = cash flow at the end of period t;
• r_t = appropriate discount rate for maturity t; and
• t = time measured consistently with the discount rate.

If one constant discount rate is appropriate, r_t = r. The discount rate must reflect the timing, currency and risk of the cash flow. Nominal cash flows require nominal discount rates, while real cash flows require real rates. Cash flows in one currency should not be discounted using a rate constructed for another currency without adjustment.

Figure: Cash-flow forecasts and risk assessment jointly determine present value.

2. Bond Valuation

2.1 Bond cash flows

A conventional fixed-rate bond pays contractual coupons and repays principal at maturity. For an annual-coupon bond:

where:

• P₀ = bond price;
• C = annual coupon payment;
• F = face or redemption value;
• y = yield per annual period; and
• n = number of annual periods to maturity.

This formula assumes promised payments occur and that one yield appropriately discounts all payments. A term structure should be used when maturity-specific spot rates differ materially:

Here, s_t is the spot rate for maturity t.

A zero-coupon bond has no intermediate coupons:

2.2 Worked example: annual-coupon bond

A five-year bond has a face value of $1,000, an annual coupon rate of 6% and a yield to maturity of 7%. Thus, C = 0.06 × 1,000 = $60.

The bond sells below face value because its coupon rate is below the required yield. If the coupon rate equalled the yield, the bond would sell at par, assuming redemption at face value.

2.3 Semiannual coupons and yield conventions

For m payments per year, the general price expression is:

where i_p is the effective yield per coupon period.

Dividing an annual quoted yield by two is valid only when the quotation is a nominal annual yield with semiannual compounding. Thus, a nominal annual yield of 8% compounded semiannually implies a six-month rate of 4%.

An effective annual yield must instead be converted:

For an 8% effective annual yield, the six-month rate is (1.08)^1/2 − 1 ≈ 3.923%, not 4%.

2.4 Yield to maturity and realised return

Yield to maturity is the internal rate of return that equates a bond's price with its promised payments. It is not automatically the investor's realised return. Realisation of the quoted yield assumes:

• the issuer makes all promised payments;
• the investor holds the bond to maturity; and
• coupons are reinvested at the yield to maturity.

2.5 Credit risk and credit spreads

Corporate bond yields incorporate a risk-free benchmark and compensation for credit, liquidity and other risks:

This spread is not a pure probability-of-default measure. It can include expected loss, uncertainty, liquidity, tax effects and risk premiums.

Expected credit loss may be expressed conceptually as:

Credit analysis should examine leverage, interest coverage, cash-flow stability, covenant protection, collateral, seniority and recovery prospects. A risky bond should not be valued by discounting promised cash flows at a risk-free rate. Analysts may either discount promised payments at a risky yield or probability-weight expected payments and use a rate consistent with the remaining systematic risk. The two approaches must not be mixed inconsistently.

2.6 Interest-rate sensitivity and duration

Macaulay duration is the present-value-weighted average time to receive a bond's cash flows:

If t counts coupon periods, D_M is measured in coupon periods. It must be divided by the number of coupon periods per year to report duration in years.

Modified duration estimates percentage price sensitivity to a small yield change:

The yield change and modified duration must use the same period convention. Duration is a linear approximation; convexity improves estimates for larger yield changes. Bond prices normally rise when required yields fall and decline when required yields rise.

3. Equity Valuation

Unlike bondholders, shareholders have a residual claim without a contractual maturity date. Equity valuation therefore depends heavily on forecasts of profitability, reinvestment, growth and risk.

3.1 Dividend discount model

The general dividend discount model is:

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

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

This model requires k_e > g. The first dividend, D₁, is next period's dividend—not the dividend just paid. Constant growth must also be economically sustainable.

Example. A company has just paid a dividend of $3.00 per share. Dividends are expected to grow at 4% indefinitely, and the required return is 10%.

The Gordon model is most suitable for mature, dividend-paying companies. A multistage model is preferable when short-term growth differs from sustainable long-run growth.

3.2 Growth and reinvestment

Growth is created by retaining and productively reinvesting funds. Under stable accounting relationships, a useful approximation is:

A higher retention ratio does not necessarily create value. Retained earnings add value only when the return earned on incremental investment justifies the associated cost of capital and risk.

3.3 Free cash flow to equity

Free cash flow to equity (FCFE) is cash available to ordinary shareholders after operations, taxes, reinvestment and net debt financing:

Here, ∆NWC is the increase in operating net working capital. An increase consumes cash. Equity value is:

FCFE is especially useful when dividends do not reflect the company's capacity to distribute cash. However, it can become volatile when borrowing changes sharply.

4. Enterprise Valuation Using Free Cash Flow

4.1 Free cash flow to the firm

Free cash flow to the firm (FCFF) is cash available to all long-term capital providers:

where EBIT is operating profit before interest and taxes and T is the applicable operating tax rate.

Because FCFF is before debt payments, it is discounted at the weighted average cost of capital (WACC):

4.2 Weighted average cost of capital

where:

• E = market value of equity;
• D = market value of interest-bearing debt;
• k_e = cost of equity;
• k_d = pre-tax cost of debt; and
• T = marginal corporate tax rate applicable to interest deductions.

WACC assumes that the forecast cash flows and capital structure are compatible with the chosen weights and risks. Market values, rather than accounting book values, are generally appropriate. If leverage changes materially over time, an adjusted present value or period-specific discount-rate approach may be preferable.

4.3 Terminal value

A continuing-value estimate using perpetual growth is:

with:

The conditions are WACC > g and sustainable long-run operations. The growth rate should normally be consistent with long-term nominal economic growth in the relevant market and currency. It is inappropriate to assume perpetual growth that requires impossible market share, reinvestment or returns.

4.4 From enterprise value to equity value

A simplified bridge is:

Other adjustments may include non-controlling interests, unfunded pension obligations, preferred shares, leases and non-operating assets. Every item should be treated consistently with the cash flows and peer multiples used.

5. Market-Multiple Valuation

Relative valuation estimates value from comparable securities or transactions. A multiple is reliable only when its numerator and denominator represent claims belonging to the same capital providers.

The notation “selected EV/EBITDA ratio” means a chosen enterprise-value-to-EBITDA multiple; it does not mean enterprise value divided by an unspecified “EBITDA multiple.” Similarly, the selected P/E ratio is a price-to-earnings ratio.

Example. If a defensible peer median EV/EBITDA ratio is 8.0× and the subject company's maintainable EBITDA is $45 million:

If the selected P/E ratio is 14.0× and forecast earnings per share are $2.10:

Peer selection should consider industry, geography, growth, margins, leverage, size and accounting policies. Analysts should distinguish historical from forward multiples and remove non-recurring items consistently. Multiples provide a market-based cross-check, but they can reproduce market mispricing.

6. Illustrative Business-Valuation Case

Background

Northstar Payments Ltd is a fictional, established payment-services company. Management is considering a strategic investment, and its board requires an independent valuation. All figures are in millions of dollars except per-share values.

Problem

The analyst must estimate enterprise and equity value while identifying the effect of long-run growth and WACC assumptions.

Forecasts and approach

Forecast FCFF is:

The base assumptions are:

• WACC: 9%;
• perpetual growth after Year 5: 3%;
• debt: $110 million;
• excess cash: $20 million;
• non-controlling interest: $5 million; and
• ordinary shares: 10 million.

Base-case analysis

The present value of the explicit FCFF is:

Terminal value at the end of Year 5 is:

Its present value is:

Therefore:

The equity bridge gives:

Verified sensitivity analysis

The following table recalculates enterprise value for each WACC–growth combination while holding the five explicit FCFF forecasts unchanged.

For example, at 8% WACC and 2% growth:

At 9% WACC and 4% growth:

These checks show that enterprise value increases with growth and decreases with WACC. The relationship is nonlinear because the denominator WACC − g becomes small as growth approaches WACC.

Market cross-check and outcome

If comparable firms trade at a selected EV/EBITDA ratio of 8.0× and Northstar's maintainable EBITDA is $45 million, the market-based enterprise value is $360 million. This is moderately below the base DCF estimate of $374.3 million.

The board should not simply average the estimates. It should investigate whether the difference reflects optimistic cash-flow forecasts, superior operating quality, peer-selection differences or temporary market conditions. A defensible conclusion might present a range around the base estimate rather than a false point estimate.

Learning points

1. Terminal value can represent a large proportion of enterprise value.
2. Growth and WACC must be economically consistent.
3. Enterprise value requires a separate bridge to shareholder value.
4. Market multiples are useful cross-checks, not substitutes for analysis.
5. Sensitivity tables must be reproducible from stated forecasts.

7. Assumption Risk, Governance and Professional Practice

7.1 Common valuation errors

Frequent errors include:

• discounting FCFF at the cost of equity rather than WACC;
• discounting FCFE at WACC;
• using the current dividend rather than next year's dividend in the Gordon model;
• assuming g ≥ WACC in a perpetual-growth model;
• mixing nominal cash flows with real discount rates;
• using book-value capital weights without justification;
• combining an enterprise-value multiple with an equity-level denominator;
• ignoring debt-like obligations or non-controlling interests; and
• reporting sensitivity results that do not reproduce from the model.

7.2 Ethical and governance considerations

Valuations influence lending, mergers, insurance investments, financial reporting and shareholder transactions. Analysts may face pressure to select assumptions that support a desired outcome. Professional conduct requires:

• independence and disclosure of conflicts;
• clear identification of data sources;
• consistent treatment of favourable and unfavourable evidence;
• documentation of model changes;
• scenario and sensitivity analysis; and
• communication of uncertainty rather than spurious precision.

Regulated financial institutions must also observe applicable capital, suitability, fair-value, model-risk and disclosure requirements. The precise rules depend on jurisdiction and purpose.

7.3 Technology and artificial intelligence

Modern valuation systems can obtain market data, automate comparable-company screening and generate scenarios rapidly. AI may assist with document extraction and anomaly detection. However, automated tools can propagate stale data, inappropriate peers, hidden assumptions and fabricated outputs. Human review remains essential for accounting adjustments, business-model interpretation and verification of formulas.

Best practice includes version control, independent model review, input-validation rules, audit trails and reconciliation to source financial statements.

8. Contemporary Valuation Challenges

Interest-rate volatility directly affects bond yields, WACC and valuation multiples. Inflation complicates the separation of nominal growth from real operating improvement. Intangible assets, platform economics and subscription models can make historical earnings less representative of future cash generation. Climate transition, cyber risk and regulation may alter capital expenditure, insurance costs, growth and terminal assumptions.

These factors should be incorporated through cash flows, discount rates or scenarios without double-counting the same risk. For example, reducing expected cash flows for a specific regulatory cost and adding an excessive discount-rate premium for the same cost may understate value twice.

Chapter Summary

Valuation is based on discounting expected cash flows at rates consistent with their risk, timing, currency and claimholder. Bond value reflects coupons, principal, market yields, credit risk and interest-rate sensitivity. Semiannual yields must be interpreted according to their quotation convention, and Macaulay duration must be reported in appropriate time units.

Equity can be valued through dividends or FCFE, while an operating business is commonly valued through FCFF discounted at WACC. Terminal value requires sustainable growth below the discount rate. Enterprise value must be reconciled to equity value by adjusting for debt, cash and other claims. Market multiples provide useful external evidence when comparable companies and numerator–denominator relationships are selected correctly. Because results are highly assumption-sensitive, professional valuation requires transparent forecasts, reproducible calculations, governance and ethical independence.

Key Terms

Bond: A debt security promising specified payments to investors.

Coupon: Periodic interest payment made by a bond issuer.

Credit spread: Yield difference over a comparable benchmark, reflecting credit and other risks.

Dividend discount model: Equity model based on the present value of expected dividends.

Duration: Measure of a bond's cash-flow timing and interest-rate sensitivity.

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

Equity value: Residual value attributable to ordinary shareholders.

FCFE: Cash flow available to equity investors after reinvestment and net borrowing.

FCFF: Cash flow available to debt and equity capital providers.

Gordon growth model: Constant-growth perpetuity model for dividends or cash flows.

Intrinsic value: Value estimated from expected economic benefits and required returns.

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

Perpetual growth rate: Sustainable constant growth assumed beyond the explicit forecast period.

Present value: Current equivalent of a future cash flow discounted for time and risk.

Spot rate: Discount rate applicable to a single cash flow at a specified maturity.

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

WACC: Weighted average required return of debt and equity capital providers.

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

Review Questions

Short Answer Questions

1. Why does a bond sell below face value when its coupon rate is below its required yield?
2. Distinguish enterprise value from equity value.
3. State the condition required by the Gordon growth model.
4. Why is a credit spread not a pure measure of expected default loss?
5. When is dividing an annual yield by two appropriate for semiannual bond valuation?

Long Answer Questions

1. Explain the assumptions and limitations of yield to maturity as an investment-return measure.
2. Compare the dividend discount, FCFE and FCFF valuation approaches.
3. Explain how WACC should be estimated and when a constant WACC may be inappropriate.
4. Critically evaluate the use of EV/EBITDA and P/E multiples in equity research.
5. Discuss the ethical, governance and model-risk issues arising in professional valuation.

Analytical / Application Questions

1. Value a four-year, $1,000 bond paying a 5% annual coupon when the required annual yield is 6.5%. Explain whether it trades above or below par.
2. A company expects FCFF of $12 million next year, growing at 3% indefinitely. If WACC is 9%, estimate enterprise value and evaluate the sensitivity to a decline in WACC to 8%.
3. Revalue Northstar Payments using a 10% WACC and 3% perpetual growth, then calculate equity value per share using the stated balance-sheet adjustments.

Multiple Choice Questions with Answers

Question 1

A five-year annual-coupon bond has a face value of $1,000, a 6% coupon rate and a 7% yield. Its approximate price is:

A. $959

B. $1,000

C. $1,041

D. $1,300

Correct Answer: A. $959

Question 2

An 8% effective annual yield corresponds to which effective six-month yield?

A. 3.000%

B. 3.923%

C. 4.000%

D. 8.000%

Correct Answer: B. 3.923%

Question 3

When Macaulay duration is calculated with time measured in semiannual coupon periods, the initial result is measured in:

A. Calendar years

B. Percentage points

C. Semiannual periods

D. Currency units

Correct Answer: C. Semiannual periods

Question 4

A company has just paid a dividend of $3.00. If dividends grow at 4% perpetually and the required equity return is 10%, the Gordon model value is:

A. $30.00

B. $50.00

C. $52.00

D. $78.00

Correct Answer: C. $52.00

Question 5

Which discount rate is normally used for FCFF when capital structure is stable?

A. Pre-tax cost of debt

B. Cost of equity

C. Risk-free rate

D. WACC

Correct Answer: D. WACC

Question 6

Which condition is necessary for a perpetual-growth terminal-value formula?

A. Growth must exceed WACC

B. WACC must exceed growth

C. Growth must equal WACC

D. Debt must equal equity

Correct Answer: B. WACC must exceed growth

Question 7

A selected EV/EBITDA ratio is 8.0× and maintainable EBITDA is $45 million. The implied enterprise value is:

A. $5.625 million

B. $53 million

C. $360 million

D. $450 million

Correct Answer: C. $360 million

Question 8

An enterprise has EV of $374 million, debt of $110 million, excess cash of $20 million and non-controlling interest of $5 million. Its equity value is:

A. $239 million

B. $279 million

C. $309 million

D. $509 million

Correct Answer: B. $279 million

Question 9

Holding forecast cash flows and perpetual growth constant, an increase in WACC normally causes enterprise value to:

A. Increase

B. Remain unchanged

C. Decrease

D. Equal book value

Correct Answer: C. Decrease

Question 10

Which practice is correct in relative valuation?

A. Apply an EV/EBITDA ratio to earnings per share

B. Apply a P/E ratio to total debt

C. Apply an EV/EBITDA ratio to maintainable EBITDA

D. Apply a P/E ratio to FCFF

Correct Answer: C. Apply an EV/EBITDA ratio to maintainable EBITDA