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Corp 3 – Cost of Capital_ Advanced Topics

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

The weighted average cost of capital is the return an issuer must earn on its assets to satisfy every provider of capital at once. Written across the three capital types a listed company typically uses, it is a weighted blend of the after-tax cost of debt, the cost of preferred equity and the cost of common equity.

The three weights are all non-negative and sum to 1.0.

The formula is trivial. Populating it is not. Three features of the task make a WACC estimate a judgement rather than a measurement.

Each component cost can be estimated by several different methods, and no single method is the right one. Two competent analysts applying different methods to the same company will disagree.
The weights are supposed to reflect a long-term target capital structure, which may or may not resemble the capital structure the company happens to run today.
The marginal tax rate has to be estimated, and it may differ from the company average or effective tax rate reported in the financial statements.

Because every one of those choices feeds through to a discount rate, and every discount rate feeds through to a valuation, the assumptions behind a WACC estimate matter as much as the arithmetic. This reading works through each component in turn and, at every stage, names the choice the analyst is making.

The two component costs, in outline

Debt sits ahead of equity in the capital structure and therefore carries lower risk and a lower cost. Before the tax deductibility of interest is considered, the cost of debt can be decomposed into a benchmark risk-free rate plus a credit spread that pays investors for the risk in that particular security.

The credit spread is where company-specific information enters. It reflects the riskiness of the business model, expected profitability and growth, the applicable tax rates, the protective covenants written into the debt, the leverage policy the company follows and any signalled change to it, the maturity and callability of the issue, and the nature and liquidity of the assets and operations standing behind it.

The cost of equity is harder, because it is not observable. A recent bond issue gives a usable read on the pre-tax cost of debt; nothing equivalent exists for equity, so the required return of equity investors must be modelled. The single-factor capital asset pricing model remains the most common approach in practice. It says that expected return is the risk-free rate plus the equity risk premium scaled by the asset sensitivity to the market factor.

The ERP is the extra return equity investors demand over the risk-free rate; beta scales it to the individual security.

Preferred equity sits between the two. It normally carries a stated dividend rate and a claim on assets ranking ahead of common equity, so the risk premium investors demand on a preferred issue is smaller than the premium on the same company common equity. Preferred equity is therefore cheaper than common equity and dearer than debt. That ordering is a useful sanity check, and Section 10 shows a case where a published model estimate violates it.

Figure 1: Where a cost of capital estimate comes from

Every arrow is a main channel, which is why all four are drawn the same way: each driver set feeds both component costs. Top-down conditions set the risk-free benchmark that both estimates start from, the general level of credit spreads and the ERP; bottom-up characteristics set the issuer credit spread on the debt side and the beta on the equity side. The two component costs are then blended at target weights to give the WACC.

Financial theory says companies should seek the mix of debt and equity that minimises WACC and therefore maximises shareholder wealth. Because business risk and tolerance for financial risk differ across companies, the resulting capital structures, debt costs and equity costs differ too. Some of that variation is not about the company at all: it is about where the company is domiciled and what the market is doing. Those are the top-down factors, and they show up in the risk-free rate, in aggregate credit spreads and in the market-wide equity risk premium.

Exhibit 1 summary: the two families of cost of capital factors

External, set by the environment	Internal, set by the issuer
How much capital the market is supplying	Swings in revenue, earnings and cash flow
Prevailing rates, inflation and spread levels	What the assets are, and how saleable
Legal tradition, regulation and country risk	Profitability, coverage and gearing
The tax regime the issuer files in	Options written into the securities

Capital availability

The simplest determinant is how much capital is on offer in the market, region or country where the issuer is raising it. Where capital is plentiful, issuers get better terms and a lower cost of capital.

Developed economies generally supply that abundance. Their capital markets are more established and more liquid, their currencies more stable, property rights better protected and the rule of law stronger than in developing economies. Investors perceive less risk in a company operating inside such a system, and lower perceived risk translates directly into narrower credit spreads, a lower ERP and a lower cost of capital. Where corporate debt markets are thin, companies fall back on bank loans or on the shadow banking system, meaning lending by financial institutions that are not regulated as banks.

Market conditions

Interest rates, inflation and the general macroeconomic setting move the cost of capital continuously. Credit spreads and equity risk premiums embed both issuer-specific risk and the prevailing appetite of debt and equity investors. When spreads and premiums widen, capital providers are signalling that they see more risk and want to be paid more for supplying funds.

Inflation enters through the benchmark rate. A higher relative rate of inflation lifts the risk-free rate and therefore raises the cost of capital for every issuer in that economy. Spreads follow the cycle, widening in recessions and tightening in expansions, and so does the ERP, which investors push up in downturns and allow to fall in expansions. In developed economies, monetary policy that is more predictable and more transparent reduces uncertainty about rates and inflation, which lowers the cost of capital across the board.

The cycle also changes issuance behaviour. As spreads tighten in an expansion, debt gets cheaper and companies borrow more, funding growth or refinancing older and dearer debt. When spreads widen in a recession, borrowing slows. One further channel is the currency: where exchange rates are volatile and currency risk is correspondingly high, companies face a higher cost of capital.

Legal and regulatory considerations, and country risk

Empirical work links the state of a capital market to the legal tradition of the country it sits in. Countries whose legal systems are based on common law tend to have more mature markets and stronger enforcement of investor rights than countries with civil law systems. Stronger investor protection supports deeper capital markets, because investors feel more secure about what they own. Where those protections exist, investors demand smaller credit spreads and a smaller ERP, and issuers enjoy a lower cost of capital.

Direct regulation matters as well. Government and related bodies set policies that drive capital structure, payout policy and even pricing decisions. Financial institutions and utilities are the standard examples of heavily regulated issuers whose financing choices are constrained from outside.

Tax jurisdiction

The final external factor is the marginal income tax rate the company faces. In many jurisdictions interest expense is tax deductible, which lowers the after-tax cost of debt by the value of the tax saving. The relationship runs in an order that catches candidates out: the higher the marginal tax rate, the larger the tax benefit of carrying debt, and therefore the more attractive debt becomes in the capital structure. A tax increase, considered in isolation and holding everything else constant, lowers the after-tax cost of debt.

Example 1 · Worked

GW, a junior analyst, is researching two companies in the same industry that are headquartered in, and raising capital in, different countries. GW collects the following description of each capital market.

Capital market features by country

Market characteristic	Country A	Country B
Spread levels	Wide	Narrow
Interest rate volatility	High	Low
Inflation	High	Low
Supply of capital	Low	High
Marginal corporate tax rate	15%	25%

1. Which country is more likely to offer corporate issuers a lower cost of capital, and why?

Solution. Country B, on every one of the five features.

Start with the tax rate, because the sign is the one most often reversed. Country B levies 25% against 15% in Country A. Provided interest is deductible and the company has taxable income to shelter, the higher rate produces a larger tax shield and therefore a lower after-tax cost of debt, which makes debt more attractive in the capital structure.

The other four features point the same way. Narrow credit spreads, low interest rate volatility and a low inflation rate all reduce the cost of both debt and equity relative to a market with high rates, high volatility and wide spreads. And a larger supply of available capital produces more favourable terms for issuers, which again lowers the cost of capital.

2. Country A is considering legislation that would raise its corporate income tax rate from 15% to 25%. What effect would that have on the cost of debt for issuers in Country A?

Solution. It would lower the after-tax cost of debt. Raising the rate from 15% to 25% increases the value of each currency unit of deductible interest. So long as the company has enough taxable income before interest to absorb the deduction, the benefit of deductibility is larger and the after-tax cost of debt falls.

3. What assumption underlies that conclusion?

Solution. That interest is in fact deductible for tax purposes. If interest is not deductible, the tax change has no effect on WACC at all. The conclusion holds only where interest is deductible and the company has taxable earnings against which to deduct it.

Holding the external environment fixed, two companies in the same country will still face different costs of capital. The WACC should ultimately reflect the riskiness of the expected cash flow stream, and four families of company characteristic drive that: the volatility of revenue, earnings and cash flow; the nature and liquidity of the assets; financial strength, profitability and leverage; and the features embedded in the securities themselves.

Revenue, earnings and cash flow volatility

Subscription-based businesses, telecommunications and media streaming among them, generate recurring revenue that produces stable and predictable earnings and cash flows. Investors treat a high proportion of recurring revenue as a positive, because the revenue line is less exposed to the swings of the macroeconomy. Cyclical businesses sit at the other end. Industrial equipment manufacturers and pay-per-use models produce volatile revenues, earnings and cash flows that track the cycle closely.

Three specific mechanisms link business and financial risk to that volatility:

Companies with greater sales risk, meaning uncertainty about both the price and the number of units sold, face greater potential revenue volatility.
Companies that draw most of their revenue from a handful of customers carry customer concentration risk, which is itself a form of sales risk.
Companies with higher operating and financial leverage, that is a higher proportion of fixed costs and a heavier debt burden, convert any given revenue swing into a larger earnings swing.

For a given level of debt, the company with more predictable revenues, earnings and cash flows has a lower probability of default, a narrower credit spread and therefore a lower cost of both debt and equity.

Environmental, social and governance risk belongs in the same category. A company in a carbon-intensive industry that is visibly not acting to mitigate its environmental impact invites investors to demand a higher return, because the externality carries financial consequences: mitigation costs, boycotts or shifting consumer preference that reduce sales, and litigation. Known employee safety problems raise the risk of lawsuits and reputational damage in the same way. Weak governance carries its own cost: anti-takeover provisions may deter a bid, but they also entrench management. Analysts have a choice here. Rather than raising the discount rate for ESG risk, they can instead adjust the forecast cash flows in the valuation model.

Exhibit 2 summary: volatility characteristics and the cost of capital

Revenue, earnings, cash flow volatility	Effect on cost of capital
Higher stability of revenues, earnings, and cash flows	Lower
Higher revenue concentration	Higher
Higher earnings predictability	Lower
Higher operating leverage	Higher
Higher financial leverage	Higher
Higher ESG risks	Higher

Asset nature and liquidity

Tangible assets are physical: property, plant and equipment, and inventory. Intangible assets, such as goodwill, patents, intellectual property rights and a skilled and stable workforce, have no physical form. Companies whose asset base is mostly tangible can usually access debt capital more cheaply, because those assets can be pledged as collateral.

Fungibility and liquidity work the same way. Assets that are interchangeable with other units of the same kind and easily sold, such as cash and marketable securities, support cheaper capital than specialised, illiquid plant. There is one subtlety worth holding on to: pledging tangible assets as collateral lowers the cost of debt but can raise the cost of equity, because creditors then hold a prior claim on those assets in a liquidation and the residual claim gets riskier.

Exhibit 3 summary: asset type and the cost of capital

Asset type	Effect on cost of capital
Higher proportion of fungible, tangible assets	Lower
Higher proportion of liquid assets	Lower

Financial strength, profitability and leverage

Projected financial strength is the next determinant. Weakening profitability, poor cash generation, a low interest coverage ratio or tight liquidity all raise the cost of capital, because they signal deterioration. When a company comes to raise new capital, the price it pays depends heavily on the debt it already carries. Holding business risk constant, a company with a higher total debt to EBITDA ratio, a higher debt to equity ratio or a lower interest coverage ratio faces wider credit spreads and a higher probability of default, because its capacity to service additional debt is already reduced.

Exhibit 4 summary: financial strength and the cost of capital

Financial strength measure	Effect on cost of capital
Higher profitability	Lower
Higher cash flow generation	Lower
Higher interest coverage, liquidity	Lower
Higher leverage ratios	Higher

Security features

Embedded options change the cost of capital at issuance, and the rule is always the same: a feature that benefits the investor lowers the issuance yield, and a feature that benefits the issuer raises it. What complicates the picture is that the initial effect can reverse later.

Callability. A call feature benefits the issuer. If rates fall, the issuer refinances at the lower prevailing rate and calls the expensive debt away from investors. Investors, disadvantaged, demand a higher yield at issuance than on an otherwise similar option-free bond. So callable debt costs more initially, although the cost may fall later if rates do drop and the call is exercised.
Putability. A put feature benefits the investor, who can sell the bond back to the issuer before maturity. That is valuable when rates rise, because the investor recovers the principal and reinvests at the higher yield. It also allows investors to escape company events, a leveraged buyout or an acquisition for instance, that would raise the risk of the bond and depress its price. In exchange, investors accept a lower yield at issuance. The cost can rise later if rates rise and the issuer has to refinance at a higher rate to fund the bonds put back to it.
Convertibility. The conversion option lets the investor exchange the bond for common stock at a stated ratio, so investors accept a lower return than on an option-free bond. The later cost is not a higher interest rate but equity dilution if the bonds are converted.
Cumulative versus non-cumulative preferred. Cumulative preferred stock obliges the company to pay any missed dividends in full before paying common shareholders. Non-cumulative preferred carries no such obligation; the only constraint is that common dividends cannot be reinstated unless preferred dividends are currently being paid. In a liquidation, preferred shareholders may have a claim for unpaid dividends ahead of any distribution to common holders. Investors accept a lower return on cumulative preferred than on otherwise similar non-cumulative preferred.
Share class. Where a company issues multiple classes of common stock, one class typically receives superior voting rights, superior cash flow rights, or both. The cost of common equity is higher for the shares that carry the inferior rights.

Exhibit 5 summary: security features and the cost of capital

Security	Feature	Effect on cost of capital
Debt	Callability	Higher
Debt	Putability	Lower
Debt	Convertibility	Lower
Preferred equity	Cumulative dividends	Lower
Common equity	Inferior cash flow or voting rights	Higher

Example 2 · Worked

GW now gathers a common size balance sheet and selected operating data for the two companies.

Common size balance sheet and selected information

Item	Company 1	Company 2
Cash and near-cash items	5%	10%
Securities readily marketable	15%	7%
Trade receivables	12%	19%
Inventories	3%	2%
Remaining current assets	4%	4%
Net property, plant and equipment	46%	29%
Goodwill and other intangibles	10%	24%
All other assets	5%	5%
Net debt to EBITDA	2.1	2.5
Interest coverage ratio	12.6	7.9
Operating leverage level	Low	High
Share of sales from the largest five customers	15%	27%
Embedded feature in the existing debt	Put	Call

Which company is more likely to have a lower cost of capital? Justify the answer against each factor family.

Solution. Company 1, and every factor family points the same way.

Asset liquidity. Company 1 holds 20% of assets in cash, equivalents and marketable securities (5% + 15%) against 17% for Company 2 (10% + 7%).

Asset tangibility. Company 1 carries 10% in intangibles and goodwill against 24% for Company 2, and 46% in net property, plant and equipment against 29%. Tangible and liquid asset bases support cheaper debt and equity, and the larger property, plant and equipment balance can be pledged as collateral.

Leverage. Company 1 has a lower net debt to EBITDA ratio, 2.1 against 2.5, and a higher interest coverage ratio, 12.6 against 7.9. Both indicate more capacity to service additional debt.

Earnings stability. Company 1 operates with lower operating leverage, so a smaller proportion of fixed costs, and a more diversified customer base, with the top five customers representing 15% of sales against 27% for Company 2. Less operating leverage and less customer concentration mean steadier earnings and cash flows.

Security features. Company 1 debt carries embedded put options, which benefit investors and therefore lower the yield the company pays. Company 2 debt carries embedded call options, which benefit the issuer and therefore raise the yield investors demand. Putable bonds are cheaper at issuance than otherwise comparable callable bonds.

An analyst checklist

Whether the approach to WACC is built from the top down, from the bottom up, or from a combination of the two, the same list of assumptions has to be settled. It is worth holding the whole list in view before starting.

Exhibit 6 summary: checklist for a WACC determination

Top-down, external	Bottom-up, company specific
Is debt and equity capital on offer at all	Sales risk
State of the debt market, spreads above all	Leverage, both operating and financial
State of the equity market, the ERP above all	Debt terms: interest type, collateral, embedded options
Position in the cycle, expansion against recession	Equity terms: ranking, voting rights
Legal and regulatory maturity, country risk, common law against civil law	Exposure to ESG risk
Where the issuer is taxed	How tangible and how liquid the assets are
	Whether interest is deductible

There is no single method for the cost of debt, and the right one depends on what kind of debt the company actually has. Four questions settle the choice:

Type of debt. Is it publicly traded, non-traded or private, bank debt, or a lease?
Debt liquidity. How liquid and marketable is the issue?
Credit rating. Does the debt carry a rating?
Debt currency. In what currency is the debt denominated?

Figure 2: Choosing a method for the cost of debt

The left-hand outcomes are the ones reached by answering Yes; each No pushes the analyst one step further down into estimation. Bank debt and leases sit outside the bond route entirely and are priced from the terms of the borrowing itself.

Traded debt

Where a company has publicly traded debt with no embedded options, known as straight debt, the yield to maturity on the longest dated issue is a reasonable estimate of the cost of issuing new straight debt. There is an exception worth remembering: if a shorter-dated bond is more liquid and trades far more frequently than the longest dated one, the shorter bond yield may be the more reliable estimate, because it reflects genuine current pricing rather than a stale quote. Either way, the yield to maturity is the current market interest rate on that debt, which is what issuing similar new debt would cost today.

Non-traded debt

Most private companies, and some public ones, have debt that is illiquid or does not trade at all. A quoted yield to maturity then either does not exist or is unreliable, because it contains a large liquidity premium.

The first fallback is to check whether the debt is rated. If it is, the analyst can take the yields to maturity on bonds of other companies with similar maturities and the same rating and apply matrix pricing to infer a yield for the subject company. Section 10 works this through in full.

If no rating exists, the analyst infers one. A synthetic credit rating is deduced from the fundamentals of the borrower itself, typically interest coverage and leverage ratios, using a model that maps ratio ranges into rating classes. Statistical models built on proprietary data covering bond ratings, features and rating classes can perform that classification. Once a rating is inferred, the analyst either takes the yield to maturity on bonds with a similar maturity and that rating, or takes the current credit spread for that rating and maturity and adds it to the benchmark risk-free rate.

One caution applies throughout. The credit rating of the issuer may differ from the ratings on its individual securities, and a single issuer may have several issues rated differently because of their features. A company can carry both AA rated and A rated debt at once, the AA issue offering more protection through collateral, seniority, convertibility or some other feature. The task is to estimate a cost of debt that best represents the risk profile of the company.

Example 3 · Worked

Analysts at the Brunswix Firm examined a large number of rated manufacturers and derived the likely range of ratios for each credit rating class.

Exhibit 7: Rating classes and leverage ratios

Rating class	Interest coverage	D/E
AAA	IC > 10 times	D/E < 35%
AA	8 < IC < 10	35% < D/E < 40%
A	5 < IC < 8	40% < D/E < 42%
BBB	3 < IC < 5	42% < D/E < 44%
BB	2 < IC < 3	44% < D/E < 50%
B	1.4 < IC < 2.0	50% < D/E < 60%
CCC	1.0 < IC < 1.4	60% < D/E < 70%
CC	0.6 < IC < 1.0	70% < D/E < 80%
C	0.3 < IC < 0.6	80% < D/E < 100%
D	IC < 0.3	D/E > 100%

A Lee wants to use the model to predict a rating for Gamma Company, a manufacturer with non-traded debt. Gamma has an interest coverage ratio of 1.5 and a D/E ratio of 43%.

1. What rating class should Lee assign, and why?

Solution. No single class can be assigned, because the two ratios conflict. An interest coverage ratio of 1.5 falls inside the B band (1.4 < IC < 2.0). A D/E ratio of 43% falls inside the BBB band (42% < D/E < 44%). The two indicators are three notches apart, so the model alone does not settle the rating.

2. What else should Lee do to reach a synthetic rating?

Solution. Look at the same ratios over time for Gamma and examine whether they are trending. A trend indicates the direction the ratios are likely to take next, which lets judgement break the tie. If the interest coverage ratio has been improving, for instance, Lee might use judgement to assign a BB rating rather than the B rating implied by the current coverage level.

Bank debt

In many countries, banks supply most corporate borrowing and nearly all small business funding. Bank facilities carry fixed or floating rates, and they may amortise in full, amortise in part, or not amortise at all. Amortising loans generally carry a lower cost of debt, because repaying principal over the life of the loan reduces default risk. Non-amortising loans, which repay the whole principal at maturity in the manner of a bullet bond, carry more default risk and a higher cost.

To estimate the cost of bank debt, the analyst should try to establish the interest rate the company pays on new bank borrowing. A recently arranged loan is a good estimate provided that the rate reflects current market conditions and that the risk profile of the company has not materially changed since the loan was arranged. Any doubt on either point makes the estimate unreliable.

Leases

Companies lease property, aircraft and other large capital assets. A finance lease, also called a capital lease, is an amortised loan in substance: the lessee uses the asset, makes payments, and either owns the asset at the end of the term or has an option to buy it. Operating leases are different in treatment, being expensed with no capitalisation of the property on the lessee statements. Because a lease is a form of secured borrowing, it usually costs a company less than unsecured borrowing in the capital markets to buy the same asset outright.

Under IFRS 16 and ASC 842, the rate implicit in the lease is the discount rate at which the present value of the lease payments plus the present value of the residual value equals the fair value of the leased asset plus the direct initial costs of the lessor.

The discount rate that solves this equality is the rate implicit in the lease.

In practice the lessee often does not know the residual value or the lessor direct costs. Where that is so, the incremental borrowing rate can be used instead: the rate the company would pay to borrow on a collateralised basis over the same term. If even that is unavailable, the analyst falls back on the non-traded debt methods above. Most public company filings do disclose the interest rates on lease liabilities. Finally, some tax jurisdictions treat a finance lease as a purchase by the lessee and a sale by the lessor, in which case the interest component is deductible and the cost of the lease should be put on an after-tax basis.

Example 4 · Worked

G&S Airlines is deciding whether to borrow, to use cash on hand, or to lease a new aircraft. The unsecured incremental borrowing rate is 6% and the cost of equity is 11%. The lease as negotiated runs for 15 years, paying EUR 9.0 million annually in arrears. Fair value of the asset under lease is EUR 100 million. The lessor would incur EUR 5 million of direct costs at the time of the agreement. The residual value of the asset after 15 years is EUR 10 million.

Net cash flows on the lease (EUR millions)

Item	Year 0	Years 1 to 14	Year 15
Lease payment		9.0	9.0
Residual value			10.0
Fair value of leased asset	−100.0
Lessor direct costs	−5.0
Net cash flow	−105.0	9.0	19.0

1. What is the implied interest cost of the lease?

Solution. Solve for the discount rate that sets the present value of the inflows equal to the initial net outflow of EUR 105.0 million. The inputs are N = 15, PV = −105.0, PMT = 9.0, FV = 10.0. The solution is 4.08%.

Read the cash flow line carefully. The initial outflow is 105.0, not 100.0, because the lessor direct costs of 5.0 form part of what the lease has to recover. The final year inflow is 19.0, being the 9.0 payment plus the 10.0 residual value.

2. What should G&S weigh in deciding between buying outright and leasing?

Solution. At 4.08%, leasing is both cheaper and lower risk than the unsecured incremental borrowing rate of 6%, which is unsurprising given that the lease is effectively secured on the aircraft. It is also cheaper than funding the purchase with equity at 11%, which would additionally be dilutive. Leasing also avoids the risks that come with owning the asset. Against that, the lease increases the leverage of G&S, since the finance lease appears as a liability.

International considerations

The cost of debt should reflect the currency in which the cash flows of the company actually arise. For an entity in a less mature foreign market, one approach is to add a country risk premium to the yield, using a country risk rating. A country risk rating scores a country on economic conditions, political risk, exchange rate risk, and the development and regulation of its securities markets.

Two terms need separating. Sovereign risk is the likelihood that the country defaults on its own debt obligations. Country risk is broader and includes what sits beyond sovereign risk: political stability, economic competitiveness and human development. Ratings may follow a letter scale like credit ratings, or a numeric range such as 0 to 10 or 0 to 100 measured against a benchmark country. For each rating class or score, a median interest rate is calculated, and comparing that median with the benchmark country rate produces the country risk premium.

Exhibit 8: Country risk premiums against a benchmark

Country	Rating (1 = least risk, 10 = most risk)	Median interest rate	Country risk premium
A	1	4.0%	0.0%
B	5	7.0%	3.0%
C	2	4.5%	0.5%
D	8	15.5%	11.5%
E	7	9.5%	5.5%
F	6	7.5%	3.5%

Country A is the benchmark, so its premium is zero by construction. Every other premium is the median rate less the 4.0% benchmark rate. Country C, rated 2 with a median rate of 4.5%, carries a premium of 0.5%. Note that the premiums do not rise smoothly with the rating: Country F, rated 6, carries 3.5% while Country E, rated 7, carries 5.5% and Country D, rated 8, jumps to 11.5%.

A limit on the interest deduction

One practical point closes the section. If a jurisdiction caps the monetary amount of interest that can be deducted, and the company has already reached that cap, then the cost of debt is not adjusted for tax at all. The cost of debt in a WACC is the cost of raising the next unit of debt, and no further tax benefit is available on it.

The equity risk premium is the extra return investors demand for holding risky equity securities rather than a risk-free asset. Formally, it is what equities are expected to return less the return on a benchmark riskless asset. Even in long-established developed markets, estimating it is difficult and subject to estimation error, which is why two analysts using the same market data can reach materially different investment conclusions.

Two broad approaches exist. The historical approach, also called the ex post approach, uses backward-looking data. The forward-looking approach, also called the ex ante approach, uses expectational data. Both are used in practice, and both carry limitations that show up in the resulting estimate.

What the historical approach assumes

A historical ERP estimate is normally the mean of the difference between the return on a broad equity index and the return on government debt, used as a proxy for the risk-free asset, over some sample period. The approach is appropriate where reliable long-term return data exist.

Using a historical average as a forward estimate embeds two assumptions: that returns are stationary, meaning that the parameters generating returns are the same in the future as in the past, and that markets are relatively efficient, so that realised average returns over the long run are an unbiased estimate of what investors expected to earn. Before applying the method, the analyst has to ask whether history in this particular market actually carries information about the future.

The four decisions

Building a historical ERP requires four choices, and the estimate is only as defensible as the four together.

Which equity index best represents equity market returns?
What time period should the estimate cover?
Which measure of mean return should be used?
Which proxy for the risk-free rate is best?

Index selection. The index should represent the returns a typical equity investor in that market earns. Broad-based, market-value-weighted indexes are the usual choice: the S&P 500 Index, the Russell 3000 Index, the MSCI EAFE Index, the Australia All Ordinaries and the Shanghai Composite Index are all examples.

Time period. This is a trade-off with no clean answer. Using the longest reliable series available maximises the data, but the distant past may not represent the current market environment, and research shows substantial evidence that underlying return volatility is not constant across time in many equity markets. Fluctuating volatility matters less in a long series, but only if the ERP itself has not shifted permanently. A shorter period keeps the estimate closer to today, at the cost of more noise. A period covering only part of a business cycle, or dominated by a disruption such as the global financial crisis or the COVID-19 pandemic, may not be robust enough to forecast anything; in that case a period excluding the disruption is needed. The mirror-image problem is a run of strong returns, which inflates the historical mean and makes the estimate likely to overstate the forward-looking ERP. On balance analysts tend to favour a longer period, because the standard error of the estimate falls as the estimation window lengthens.

Mean type. The choice is between an arithmetic and a geometric mean of the difference between the index return and the benchmark rate.

Exhibit 9: Arithmetic against geometric mean return

Mean type	Advantages	Disadvantages
Arithmetic mean	Simple to compute, and every observation in the series enters it	Extreme observations move it heavily, and it overstates the terminal wealth an investor should expect
Geometric mean	Every observation enters it, outliers carry less weight, and it estimates the terminal wealth an investor should expect

The arithmetic mean is the average single-period return and therefore best represents the mean return in a single period. Since the popular required-return models, the capital asset pricing model and multifactor models, are single-period models, the arithmetic mean is the model-consistent choice. The geometric mean is the compound growth rate that carries the beginning value of one unit of money invested through to its ending value. It is generally preferred, because it is less sensitive to outliers and is consistent with expected terminal wealth. Both are used in practice, and the choice moves the answer: an arithmetic mean is almost always the higher of the two.

Risk-free proxy. The choice lies between a short-term government debt rate, such as a US dollar or euro Treasury bill rate, and the yield to maturity on a long-term government bond. Government issues are preferred over even the highest rated corporate bonds because their default risk is near zero.

Exhibit 10: Short-term against long-term risk-free proxy

Risk-free proxy	Advantage	Disadvantage
Short-term government bill rate	Measures a riskless return precisely, provided there is no default	Its duration is nothing like that of a perpetual equity claim
Long-term government bond YTM	Duration sits far closer to that of a perpetual equity claim	Not a wholly riskless return when bought, since the rates at which coupons will be reinvested are unknown

Some analysts prefer a very short-term government rate, for example a three-month benchmark rate, on the grounds that such an instrument is typically a zero-coupon bond whose return is known at purchase and does not depend on reinvesting coupons. The stated yield is genuinely the return the investor receives, assuming no default. The weakness is duration: three months is nothing like the duration of an infinite-life equity security or of most investment horizons.

Industry practice has tended the other way, toward a long-term government bond yield. The return actually earned on a long bond is not known at purchase, because it depends in part on the rates at which coupons are reinvested over the life of the bond. That is a genuine disadvantage. Analysts nonetheless use the current yield to maturity as an approximation for the expected return on the bond.

Limitations

Two limitations of the historical approach need stating plainly.

ERPs vary over time. If the premium has shifted permanently to a different level in recent years, an estimate built from a long historical series does not represent the future ERP.
Survivorship bias inflates the estimate. Equity index data suffer from the removal of poorly performing or defunct companies from index membership, so index performance reflects the relative winners only. The measured historical premium is therefore biased upward.

Example 5 · Worked

D Smith and J Müller are equity analysts at Odyssey Investments. They estimate the ERP for the same market using different assumptions.

Historical ERP assumptions and data

Assumption	Smith	Müller
Index used	Russell 3000	S&P 500
Length of the sample	35 years	65 years
Proxy for the riskless rate	30-year Treasury bond	3-month Treasury bill
Averaging method	Arithmetic	Geometric
Mean index return across the sample	12.96%	11.23%
Mean riskless rate across the sample	6.25%	3.11%

1. Calculate the ERP under each set of assumptions.

Solution. The historical ERP is the mean index return less the mean risk-free rate.
Smith: 12.96% − 6.25% = 6.71%.
Müller: 11.23% − 3.11% = 8.12%.

2. Explain why both estimates could be valid.

Solution. The two analysts made different choices on the four key decisions, and none of those choices is wrong. Smith used a broader index over a shorter window with a long-bond proxy and an arithmetic mean; Müller used a narrower index over a much longer window with a bill-rate proxy and a geometric mean. The gap of 1.41 percentage points is entirely a product of assumptions. That is the lesson: differences in underlying assumptions produce different ERP estimates and, through them, different valuations.

3. Give one justification for using a very short-term government bond rate in a historical ERP, and the disadvantage.

Solution. The justification is that a short-term government bond is typically a zero-coupon instrument with a return known up front and independent of coupon reinvestment, so the stated yield is genuinely the investor return, assuming no default. That is not true of the yield to maturity on a long-term government bond. The disadvantage is that the short-term rate does not match the duration of an infinite-life equity security.

4. What key assumption must an analyst make to justify using a historical ERP inside an asset pricing model?

Solution. That returns are stationary, meaning the parameters describing the return-generating process are the same going forward as they were in the sample period.

5. Why might the geometric mean be preferred over the arithmetic mean here?

Solution. Estimates built on geometric means are less sensitive to outliers. In addition, compounding wealth forward with a geometric mean estimates the expected terminal value of wealth, which the arithmetic mean overstates.

A forward-looking approach follows from a simple observation: investor returns depend only on the expected future cash flows of the investment, so the ERP should depend only on expectations for the economic and financial variables that drive those cash flows. Ex ante estimates use current information and current expectations. Three methods are in common use: survey-based estimates, dividend discount models, and macroeconomic modelling.

Survey-based estimates

The most direct method is to ask. Surveys sample a group of people, frequently experts, about their expectations for the ERP itself or for capital market expectations from which an ERP can be inferred. Such surveys consistently show a much higher ERP in developing markets than in developed markets. The main weakness is that survey estimates are sensitive to recent market returns, so the answers move with the last few quarters of performance rather than with anything fundamental.

Dividend discount model estimates

The second method starts from the value of a stock as the present value of expected future dividends. Under a constant expected earnings growth rate this reduces to the Gordon growth model.

Rearranging for the required return on equity gives the expected dividend yield plus the expected earnings growth rate.

Applied to a broad index rather than a single stock, both inputs are obtainable. Indexes publish a dividend yield, and the year-ahead dividend for an index is reasonably predictable. The expected earnings growth rate can be taken from consensus analyst expectations for the index, built either top down or bottom up. Subtracting the current risk-free rate from the expected market return gives a forward-looking ERP.

One assumption is buried in this and it matters. The constant growth model implies that earnings, dividends and prices all grow at the same rate, which means a constant price to earnings ratio. If the analyst does not believe that, an adjustment for expected P/E expansion or contraction is required. The logic is mechanical: from a given market level associated with a given level of earnings and a given P/E, the return from capital appreciation cannot exceed the earnings growth rate unless the P/E rises, and cannot fall short of it unless the P/E falls. A P/E change itself comes from a change in the expected earnings growth rate or from a change in risk. Some analysts also add the aggregate spend on buybacks by index constituents to the dividend yield term, to capture total payout. Anyone doing that should also consider how buybacks alter growth in earnings and dividends, since the two adjustments are not independent.

For a rapidly growing economy a single growth rate is implausible, and the analyst may model several stages: a fast growth stage for the aggregate of index companies, a transition stage in which growth declines, and a mature stage growing at a moderate sustainable rate. The required return is then the internal rate of return that equates the sum of the present values of the three stages to the index price.

Solve for the internal rate of return, then subtract the risk-free proxy to obtain the ERP.

Example 6 · Worked

A forward-looking ERP for the UK market is being built from the FTSE 100 Index.

FTSE 100 inputs and analyst forecast ranges

Input	FTSE 100 Index	Analyst forecast range
Dividend yield, one year forward	1.94%	1.5% to 3.5%
Earnings growth expected over the long run	5.0%	4.0% to 6.0%
Yield on the long Gilt	1.63%	1.5% to 2.5%

1. Estimate the ERP using the constant growth DDM.

Solution. ERP = 1.94% + 5.0% − 1.63% = 5.31%.

2. What happens to the estimate at the extremes of the forecast ranges?

Solution. Take the least favourable combination and then the most favourable.
Low: 1.5% + 4.0% − 2.5% = 3%.
High: 3.5% + 6.0% − 1.5% = 8%.
The ERP ranges from 3% to 8%, a spread of five percentage points around a point estimate of 5.31%. That is the sensitivity the analyst is carrying. A sensible next step is to simulate the dividend yield and the long bond yield across this range, using information about the distribution of forecasts among the analysts as part of the simulation.

Macroeconomic modelling

The third method builds the ERP from forecast economic variables such as inflation and expected growth in real earnings per share, using the relationships between macroeconomic and financial variables that appear in equity valuation models. These models are more reliable where public equities are a large share of the economy, which is generally true of developed markets and generally not true of developing ones.

The standard example is the Grinold–Kroner (2001) decomposition of the return on equity. In outline, the ERP is the dividend yield plus the expected capital gain, less the expected risk-free rate.

The dividend yield is the expected income component. The repricing term captures expected changes in the P/E ratio of the market being evaluated, which is exactly the adjustment the constant growth DDM leaves out. Earnings growth per share is itself decomposed into expected inflation plus real economic growth less the percentage change in shares outstanding.

Empirical work finds that the dilution effect, the change in shares outstanding, varies significantly across countries for a variety of reasons. Setting it aside, the full model is a five-factor build of the expected market return.

Grinold–Kroner factors and their common proxies

Factor	Symbol	Common proxy
Income the market is expected to pay out	DY	Dividend yield on a broad market index
Drift expected in the market P/E	ΔP/E	An analyst overlay for mispricing of the market, usually set at zero
Inflation expected over the horizon	i	Nominal minus real yield on matched-maturity paper
Real growth expected in earnings per share	g	Growth in real GDP
Net change expected in the share count	ΔS	Varies by market and by era

Expected inflation can be read out of the government bond market directly, by comparing nominal and real yields on similar maturity benchmark bonds. Comparing a US Treasury bond with a Treasury Inflation-Protected Security of the same maturity gives the implied rate.

Figure 3: Building an ERP from the Grinold–Kroner factors

Scenario 1 of Example 7 drawn to scale. The repricing term is zero here, so it has no width. Both bars end at the same expected return of 7.5%, and the ERP is simply what is left after the 2.5% risk-free rate is removed.

Example 7 · Worked

1. A 10-year Treasury bond yields 2.3%. An inflation-protected Treasury of matching maturity yields 0.66%. What inflation rate does that imply?

Solution. Use the ratio form rather than the subtraction shortcut:
i = (1.023 ÷ 1.0066) − 1 = 0.016, or 1.6%.
The subtraction approximation would give 2.3% − 0.66% = 1.64%, close enough for most purposes but not identical.

An analyst is now estimating a forward-looking ERP for a market under two scenarios.

Grinold–Kroner inputs by scenario

Input	Scenario 1	Scenario 2
r_f	2.5%	3%
i	1.6%	3%
g	3%	2%
Δ(P/E)	0	1%
DY	2.2%	2%
ΔS	−0.7%	0

2. Calculate the ERP under each scenario using the Grinold–Kroner model.

Solution. Substitute into ERP = [DY + Δ(P/E) + i + g − ΔS] − E(r_f). Note the double negative in Scenario 1: shares outstanding are shrinking, so subtracting a negative ΔS adds to earnings growth per share.

Scenario 1: {2.2 + 0 + [1.6 + 3.0 − (−0.7)]} − 2.5 = 2.2 + 5.3 − 2.5 = 5.0%.
Scenario 2: {2.0 + 1.0 + [3.0 + 2.0 − 0.0]} − 3.0 = 3.0 + 5.0 − 3.0 = 5.0%.

The two scenarios reach the same premium of 5.0% from quite different inputs, which compensates investors for average market risk given the expectations for inflation, real earnings growth, P/E growth and income.

A note on the source: the CFA Institute text introduces the Scenario 1 answer with the figure 3.6% and then sets out the equation that produces 5.0%. The equation and its inputs are reproduced here exactly, and they give 5.0%.

3. How does the ERP change when expected inflation rises? When expected income falls?

Solution. A rise in expected inflation generally leaves the ERP unchanged, because i is added and r_f is subtracted, and both rise by the same amount. A fall in expected income reduces the ERP directly, by reducing DY.

4. When is the Grinold–Kroner model not appropriate?

Solution. Where listed equity forms too small a slice of a developing economy for the aggregates to connect to it. The model ties equity returns to economy-wide variables, and that tie is weak when quoted companies account for only a fraction of activity.

Why an inflation shock passes straight through the ERP

The Grinold–Kroner model is a nominal expected return with a nominal risk-free rate subtracted from it. Expected inflation enters the expected return through earnings growth per share, and it enters the risk-free rate through the Fisher relationship. If both move together, the premium between them is untouched. This is a useful discipline: the ERP is a relative quantity, so a variable that shifts both the expected equity return and the risk-free rate by the same amount cannot move it. Only a variable that hits one side alone, such as the dividend yield or the repricing term, changes the answer.

Limitations of the forward-looking approach

Ex ante estimates are less exposed to non-stationarity and data biases than historical estimates, but each method has its own weakness.

Limitations by forward-looking method

Approach	Limitation
Surveys	Subject to sampling and response biases, and to behavioural biases such as recency bias, placing more weight on recent events, and confirmation bias, attending to information that supports an existing opinion and ignoring the rest
DDM	Assumes a constant P/E. Where earnings, dividends and prices grow at different rates, an adjustment for P/E expansion or contraction is needed
Macroeconomic models	Financial and economic models can carry modelling errors or behavioural biases in the underlying forecasts

Example 8 · Worked

Estimate the ERP for a market using both approaches from the following information.

Market inputs

Input	Value
Inflation expected	1.9%
P/E drift expected	−1.2%
Income component expected	1.8%
Real earnings per share growth expected	2.7%
Share count change expected	0.0%
Three-month government yield today	0.96%
Geometric mean return on the equity index, long run	9.96%
Geometric mean return on short government paper, long run	3.15%

Calculate the ERP under the historical approach and under the forward-looking approach.

Solution.
Historical. Take the mean difference between the equity index return and the government debt return: 9.96% − 3.15% = 6.81%.

Forward-looking. Apply Grinold–Kroner, remembering that the repricing term is negative here:
ERP = {1.8 − 1.2 + (1.9 + 2.7 − 0.0)} − 0.96 = 5.20 − 0.96 = 4.24%.

The two answers differ by 2.57 percentage points. The forward-looking figure is lower mainly because the expected P/E contraction of 1.2% subtracts from the expected return, and because the current three-month yield of 0.96% is far below the 3.15% long-run average government return used in the historical calculation.

With an ERP in hand, the analyst can move to the required return on equity for a specific company. Three families of method are available: the dividend discount model, the bond yield plus risk premium build-up, and risk-based factor models. Private and international companies add further complications, treated in the two sections that follow.

The dividend discount model

The constant growth DDM used to build a market ERP can be pointed at a single company instead. Given a forecast next dividend, an expected perpetual growth rate in dividends and the current share price, the required return follows directly.

For Company X, with a current share price of EUR 40, an expected future dividend of EUR 1.04 and an expected perpetual dividend growth rate of 4%, the cost of equity is:

r_e = (1.04 ÷ 40.00) + 0.04 = 0.026 + 0.04 = 0.066, or 6.6%.

The method is straightforward and rests on defensible logic: the share price is the present value of future dividends, and the dividend is the relevant cash flow to an equity holder. It also carries two hard requirements. The shares must be publicly traded, and the company must pay dividends that are stable and predictable. Neither holds for a great many companies.

Equity analysts more often build a multiyear forecast that ends with a forecast share price. The required return is then the rate that discounts the forecast dividends and the terminal share price back to the current price.

Suppose dividends of USD 1.00, USD 1.25, USD 1.35 and USD 1.50 are forecast for years 1 to 4, the current share price is USD 40.00 and the forecast share price at the end of year 4 is USD 45.00. Solving for the internal rate of return on the cash flow series −40.00, 1.00, 1.25, 1.35 and 46.50 gives 6.015%. Note that the year 4 flow is 46.50, the final dividend of 1.50 plus the terminal price of 45.00. This estimate reflects the near-term dividend forecast and the forecast share price, which is why it differs from a single-stage answer.

The bond yield plus risk premium approach

The bond yield plus risk premium approach, BYPRP for short, estimates the required return on equity by starting from the cost of debt of the same company and adding a premium for the additional risk equity investors bear relative to the debt investors of that company.

r_d is normally proxied by the YTM on the long-term debt of the company.

The difficulty is entirely in the risk premium. One common route takes the average historical gap between the returns on an equity index and those on a corporate bond index, which measures how much more equity holders have earned than the bondholders of comparable issuers. That is the same logic as a historical ERP, applied one rung further up the capital structure.

Exhibit 11: The BYPRP approach, advantages and disadvantages

Advantages	Disadvantages
Estimating the cost of debt of the company gives a starting point grounded in the return its own debt investors demand	Determination of the risk premium is relatively arbitrary
	The approach requires the company to have traded debt
	Where several traded issues exist with different features, there is no rule for which yield to select; common practice is the long-term bond YTM

Example 9 · Worked

The yield on the bonds of a company is put at 4.3%. History suggests equity holders have earned 6.1% above the yields on long-dated corporate bonds.

1. Estimate the required return on equity using the BYPRP approach.

Solution. r_e = 0.043 + 0.061 = 0.1040, or 10.40%.

2. What considerations attach to this method?

Solution. Two. Using historical data is inappropriate if risk premiums are not stationary, and the sample period is doing all the work in that case. And the company may have no traded debt at all, or several traded issues carrying different yields, in which case the starting point is itself a choice.

Risk-based models

Risk-based models express the required return as compensation for the time value of money plus compensation for bearing risk. What separates one model from another is how the second term is built. Factor models are the main class, and the capital asset pricing model and the Fama–French models are the ones covered here. Other factor models include theoretically derived models, statistical factor models, fundamental factor models and macroeconomic factor models.

CAPM

The single-factor CAPM needs a beta estimate, a risk-free rate and an ERP.

Beta is normally estimated with the market model, which replaces the expected returns of the company and the market with their actual historical returns and regresses the excess returns of the stock on the excess returns of an equity market index.

The estimated slope coefficient is used as the proxy for beta. A variation runs the regression on raw returns without subtracting the risk-free rate from either side.

Three questions attach to any beta produced this way. Which equity market index is most appropriate? What period was used, balancing the need for enough data against the risk that data from too far back no longer represents the company? And which risk-free proxy was used? That last question is not cosmetic. With a normal upward-sloping yield curve, using the short-term benchmark bill rate produces a meaningfully lower cost of equity estimate than using the long-term government bond yield, and the steeper the curve, the larger the gap.

A company does not have to be listed for CAPM to be usable. The beta of a comparable listed company with similar business risk can be unlevered to give the beta of a company with no debt, then re-levered at the leverage of the subject company. Section 11 carries out that operation in full.

Fama–French models

The Fama–French models extend the CAPM by adding factors. In the three-factor model, returns are explained by the market factor plus a size factor, measured by market capitalisation, and a value factor, measured by the relationship between the book value and the equity value of the company.

SMB is the size premium, the average return difference between small and large capitalisation companies. HML is the value premium, the average return difference between companies with high and low book-to-market ratios.

The five-factor model adds a profitability factor and an investment factor.

RMW is the profitability premium, the average return difference between robust and weak profitability. CMA is the investment premium, the average return difference between conservative and aggressive investment policies.

Estimation follows the CAPM pattern: regress the excess equity returns of the company on the factors to obtain factor betas, then combine those betas with estimates of the factor risk premiums and the risk-free rate.

Three cautions apply to all of these models. Different risk factor models often yield different results for the same company. The beta on the market factor normally differs between the single-factor CAPM and a multifactor model, because the additional factors absorb part of what the market factor was carrying alone. And the excess returns used to fit the factor betas are frequently computed against a short-dated riskless rate, which understates that rate whenever the curve slopes upward. That understatement can be remedied by using a different risk-free rate series, properly adjusted for periodicity, in the regression.

Example 10 · Worked

An analyst estimates the required return on equity using the Fama–French five-factor model. The risk-free rate is 3.82%.

Estimated betas and factor risk premiums

Factor	Estimated beta	Risk premium
Market	1.2	6.5%
Size (SMB)	0.10	1.8%
Value (HML)	−0.20	4.0%
Profitability (RMW)	0.5	2.0%
Style (CMA)	0.2	1.0%

Calculate the required return on equity.

Solution. Multiply each beta by its premium and add to the risk-free rate. The value beta is negative, so that term subtracts.

r_e = 0.0382 + (1.2 × 0.065) + (0.10 × 0.018) + (−0.2 × 0.04) + (0.5 × 0.02) + (0.2 × 0.01)
r_e = 0.0382 + 0.078 + 0.0018 − 0.008 + 0.01 + 0.002 = 0.1220, or 12.2%.

Example 11 · Worked

An analyst has gathered the following for a single company. The Fama–French three-factor coefficients were estimated using the same risk-free rate used in the CAPM.

CAPM and Fama–French three-factor inputs

Input	Value
Riskless rate, 10-year government bond	6%
Market return expected	10%
Beta on the ERP	0.8
SMB premium assumed	5%
HML premium assumed	2%
Regression intercept	0.01
Market factor coefficient	0.75
SMB factor coefficient	0.15
HML factor coefficient	0.05

1. What is the required return under the CAPM?

Solution. r_e = 0.06 + 0.8(0.10 − 0.06) = 0.092, or 9.2%.

2. What is the required return under the Fama–French three-factor model?

Solution. Work with excess return, then add the risk-free rate back.
r_e − 0.06 = 0.01 + [0.75(0.10 − 0.06)] + [0.15(0.05)] + [0.05(0.02)] = 0.0485
r_e = 0.0485 + 0.06 = 0.1085, or 10.85%.

3. Why do the two answers differ?

Solution. The Fama–French model admits more drivers of return, whereas the CAPM confines them to the single market factor. Here the SMB and HML terms add [0.15(0.05)] + [0.05(0.02)] = 0.0085, or 0.85 percentage points. The remainder of the 1.65 percentage point gap comes from the different market beta, 0.75 against 0.8, and from the regression intercept of 0.01.

4. A separate three-factor regression on another stock gives r_i − r_f = 0.003 + 1.2 F1 − 0.4 F2 + 0.2 F3. Assume a 2% expected riskless rate, with factor premiums F1 = 0.05, F2 = 0.01 and F3 = 0.04. Find the cost of equity.

Solution. Apply the coefficients to the premiums, keeping the sign on the second factor.
r_e − r_f = 0.003 + (1.2 × 0.05) − (0.4 × 0.01) + (0.2 × 0.04) = 0.003 + 0.06 − 0.004 + 0.008 = 0.067
r_e = 0.067 + 0.02 = 0.087, or 8.7%.

5. A company currently pays a dividend of USD 2.50, its share price is USD 50 and dividends are expected to grow at 5% per year in perpetuity. Using the DDM, what is the required return on equity, and what happens to it if the growth rate is revised upward or the share price falls?

Solution. The dividend given is the current one, so grow it one period first.
r_e = [2.50(1.05) ÷ 50] + 0.05 = (2.625 ÷ 50) + 0.05 = 0.0525 + 0.05 = 0.1025, or 10.25%.

An upward revision to g raises the required return twice over, because it lifts both the dividend yield term D₁/P₀ and the growth term itself. A fall in the share price raises the dividend yield term and therefore also raises the required return.

Everything so far has assumed observable prices. Private companies break that assumption, and three features make them harder to price than listed peers.

Security prices and returns are not available, so the CAPM and Fama–French models cannot be applied directly. They can, however, be adapted and applied indirectly.
Private companies tend to be smaller, earlier in the corporate life cycle, managed by their owners, and controlled by a concentrated ownership structure.
They are less liquid and typically disclose less information relevant to an investor.

The required return for a private company therefore commonly includes three additional premiums: a size premium, an industry risk premium and a specific-company risk premium.

Smaller size is associated with greater risk, arising from more difficulty raising capital, more uncertain growth prospects and riskier operations. An industry premium can be added where the company operates in a relatively risky industry. The specific-company premium is a catch-all for factors that are not easy to diversify away, such as geographic concentration risk or key-person risk.

Illiquidity is the notable exception to this pattern. It is a genuine risk of private ownership, but it is conventionally not handled as an addition to the required return. Instead it appears as a reduction in the estimated value of the equity interest, called a discount for lack of marketability. Adding an illiquidity premium to the discount rate as well would double-count it.

The expanded CAPM

The expanded CAPM adapts the standard model by adding a premium for small size and for other company-specific risks, using a beta drawn from listed peers.

Four steps produce the estimate:

Estimate an industry beta from a peer group of publicly traded companies in the same industry as the subject private company.
Given a risk-free rate and an ERP, compute a CAPM estimate of the required return.
Decide whether additional premiums for size and for other company-specific factors are warranted.
If they are, add them to reach the final estimate.

The size premium is usually assumed to be inversely related to company size, so smaller companies attract a larger premium. Where the estimate is properly based on the lowest market capitalisation decile of public companies, which is often appropriate because many private businesses are small relative to listed ones, the result corresponds to the return on a micro-cap public equity issue of average systematic risk.

Caution is needed with a historical size premium. The population of small capitalisation companies includes formerly larger companies that have fallen into financial distress. Where that is so, a historical premium may need adjusting downward before it is applied to a small but financially healthy private company, because part of the measured premium is compensation for distress rather than for size.

Estimating the specific-company premium is the least standardised part of the exercise and draws on both qualitative and quantitative evidence.

Exhibit 12: Company-specific premium, qualitative against quantitative factors

Qualitative factors	Quantitative factors
The industry in which the business operates	Financial and operational leverage
Competitive position within the industry	Volatility in cash flows and earnings
Experience and expertise of management	Earnings predictability
Customer and supplier concentration	Pricing power
Geographic concentration of the business
Governance model of the company
Asset nature and type, tangible against intangible

These factors are assessed relative to a peer group of publicly traded or other private companies in the same industry. The larger the company-specific risks identified, the larger the premium.

The build-up approach

The second method dispenses with beta altogether. The build-up approach starts at the risk-free rate and adds premiums for each risk consideration in turn.

The ERP here is estimated from equity indexes of publicly traded companies and is not beta adjusted. The largest capitalisation companies dominate the value of such indexes, and a beta of one is implicitly being applied to the premium, so adding the riskless rate to the ERP yields, in effect, what a large-capitalisation listed issue of average systematic risk must return. Everything added after that is an adjustment away from that starting point.

As with the expanded CAPM, a size premium is normally added for the smaller scale of most private companies, and the premium is typically beta adjusted, meaning adjusted for the difference in betas between small-cap and large-cap stocks so as to isolate the size effect from the market-sensitivity effect. A specific-company premium may then be added for risks incremental to those already captured. The build-up approach is the right tool when a set of comparable public companies is either unavailable or of questionable comparability.

Figure 4: The build-up ladder

Each tread is a benchmark in its own right. Stopping after the second tread gives the return on an average-risk large capitalisation issue; stopping after the third gives a micro-capitalisation return. The last two treads are the part that is specific to the subject company, and they are the part carrying the most judgement.

Exchange rates, inflation, data availability and model reliability all complicate a cost of equity estimate for a company operating across borders. A locally focused CAPM, for instance, may simply not work in an emerging market where the local index is narrow and the return history short. Risk premium estimation for emerging markets is the hardest case, and two supplementary approaches are covered here: the country spread model and the country risk rating model.

The country spread model

Under the country spread model, investors require an additional premium, the country risk premium or country spread premium, for the extra risk of investing in another country, often referred to as the local country. That extra risk may come from economic conditions, the risk of expropriation, political risk or other sources. For an emerging equity market, the ERP becomes:

The lambda term is the level of exposure of the company to the local country.

The country risk premium represents the anticipated additional risk of the market relative to a benchmark developed market. One way to calculate it is the sovereign yield spread: the yield on the debt of the local country, denominated in the currency of the benchmark developed country, less the yield on a similar maturity sovereign bond in that developed country. Analysts generally hope that this spread is an adequate approximation.

The weakness of the method is stated plainly: a bond yield spread is being used to estimate an equity risk premium. Legal and market environments differ across countries, and the spread on sovereign bonds may not be appropriate for a cost of equity at all.

Exhibit 14: Country risk premiums based on the sovereign yield spread

Country	Sovereign risk rating (10 = most risk)	CRP
A	6	3.90%
B	2	0.50%
C	5	2.75%
D	7	5.40%
E	4	1.75%
F	10	19.50%
G	9	14.50%
H	1	0.0%
I	3	1.0%
J	8	9.20%

The premium rises monotonically with the rating across all ten countries, but not linearly. Moving from rating 1 to rating 5 costs 2.75 percentage points; moving from rating 5 to rating 10 costs a further 16.75 points. Country risk is convex in the rating.

Damodaran (2021) refined the estimate by scaling the sovereign yield spread by the relative volatility of equity and bond returns in the local market.

The two sigmas are the volatility of the local equity market and of the local bond market.

The adjustment fixes the conceptual objection above, since it converts a bond-based spread into an equity-scaled one. Its cost is a data requirement: the local country must have both historical equity and bond return series long enough to measure volatility.

Extended models for international operations

Where a company is exposed to the risk of a country, an adjustment to the required return on equity is needed. Three routes are available: the global CAPM, the international CAPM, and the country spread and risk rating models already described.

In the global CAPM, a global market index is the single factor, and no significant risk differences across countries are assumed. The likely outcome is a low, or even negative, slope coefficient, because the correlation between emerging and developed market returns is generally quite low. Adding a second factor such as domestic market index returns mitigates the problem to a degree, but only where reliable financial data exist in the emerging market.

The international CAPM regresses the returns on an emerging market stock against the risk premium on a global index and, in addition, against that of a wealth-weighted foreign currency index.

Proxies for the global index include the MSCI All Country World Index and the FTSE All-World Index. The currency index aggregates the return from investing in foreign currency relative to the domestic currency of the company, weighted by country relative wealth rather than by market capitalisation. Each currency contributes its expected exchange rate move plus whatever riskless return is available in that country.

The two betas have distinct interpretations. The sensitivity to the global index depends on how connected the company is to the global economy against its local one, so a lower global beta marks a company that is largely a local business. The sensitivity to the currency index depends on whether the cash flows of the company respond to exchange rates through imports, exports and investments.

Choosing among the international methods

There is no generally accepted methodology for estimating a country risk premium for companies operating in a developing country, and that is the honest summary of the position. Two guidelines follow:

If the operations of the company are global but confined to developed countries, the global CAPM and the international CAPM are reasonable methods to apply.
If operations extend into developing countries, the position is less clear. Estimating a country risk premium from the sovereign yield approach may be appropriate, but those estimates are based on historical rates and may not reflect the premium going forward.

Example 12 · Worked

An analyst is estimating the country risk premium for the Makinassi Company, headquartered in Country X, which makes 40% of its sales in Country Y.

Sovereign spreads and return volatilities

Country	Sovereign country yield spread	Standard deviation of equity returns	Standard deviation of bond returns
X (headquarters)	1.5%	2.0%	1.0%
Y (local)	3.2%	4.0%	2.5%

Using the Damodaran model, calculate the country risk premium the analyst should use for Makinassi.

Solution. Three steps, and the first is the one most often missed.

Step 1. Take the spread relative to the home country, not the raw local spread. Seen from a company based in Country X, the spread that matters is 3.2% − 1.5% = 1.7%.

Step 2. Scale by the relative volatility of the local market. Use the equity and bond volatilities of Country Y, the local market, not Country X:
CRP = 0.017 × (0.04 ÷ 0.025) = 0.017 × 1.6 = 0.0272.

Step 3. Adjust for the exposure of the company. Makinassi has 40% of sales in Country Y, so lambda is 0.40:
Premium = 0.40 × 0.0272 = 0.01088.

The analyst should add a country risk premium of 1.088% to the cost of equity for Makinassi.

KM is a junior analyst at Atla Investments. Her manager has asked her to estimate the cost of debt and the cost of equity for Gretna Engines as a starting point for a WACC and a valuation. This section works the case through end to end, because the sequence of decisions is the point, not any single number.

Gretna Engines: company profile

Item	Description
Company	Small capitalisation, publicly traded
Business model	Manufacturer of small engines for boats and recreational all-terrain vehicles; operates with a relatively high proportion of fixed costs
Industry	Industrial equipment (cyclical)
Revenues, earnings, cash flows	Trending upward in recent years but varying considerably over the business cycle
Nature of assets	Primarily inventory and the property, plant and equipment of its engine production facilities

Gretna has been trading well recently. Several years ago, however, a sharp decline in boat and ATV sales pushed the company into a liquidity crisis, and it issued redeemable preferred stock to shore up its position, at a rather high cost. Management has now signalled in recent filings that, given favourable market conditions, it intends to issue new unsecured debt and use the proceeds to retire the preferred shares at par.

Exhibit 15: Gretna capital structure and capital type information

Capital type	Current capital structure	Selected information
Debt	20%	One issue only: 7% coupon, seven years left to run, paying semiannually; straight unsecured paper rated BBB; thinly traded, so no dependable YTM
Preferred equity	15%	7% dividend rate, redeemable now at a par value of 1,000 per share; traded frequently, at a current price of 980
Common equity	65%	Actively traded

Exhibit 16: Liquid BBB-rated comparable bonds

Bond	Coupon rate	Remaining maturity	Current price (per 100 of par)
Bond 1	5%	4 years	99.50
Bond 2	7%	4 years	106.46
Bond 3	6%	8 years	100
Bond 4	8%	8 years	112.42

Exhibit 17: CAPM and five-factor betas and risk premiums

Model	Factor	Factor beta	Risk premium
CAPM	Market (ERP)	0.91	5.5%
FF5	Market (ERP)	0.95	5.5%
FF5	Size (SMB)	0.45	1.8%
FF5	Value (HML)	0.14	3.9%
FF5	Profitability (RMW)	−0.19	3.1%
FF5	Investment (CMA)	0.30	3.7%

Betas were estimated by regressing Gretna excess returns on the relevant risk factors over the most recent 60 months. KM uses the 20-year government benchmark rate of 2.1% as the risk-free proxy. For the BYPRP figure she takes 6.2% as the historical excess of equity returns over long-dated corporate bond yields. She tells her manager that she estimated the ERP of 5.5% using the historical approach, choosing the short-term government bill rate and an arithmetic mean.

Example 13 · Worked

1. Identify two characteristics of the Gretna business model that might raise its cost of capital.

Solution. Three are available and any two suffice.
First, high volatility in revenues and earnings, which follows from the cyclical industry Gretna operates in. Companies with volatile revenues and earnings face a higher cost of capital than companies with stable ones.
Second, the relative illiquidity of the asset base, which consists of inventory and specialised production facilities. Asset bases with a low proportion of liquid assets carry a higher cost of capital.
Third, high operating leverage, since the company runs a high proportion of fixed costs, which amplifies any revenue swing into a larger earnings swing.

2. How might KM estimate a current cost of debt, and what is it?

Solution. The Gretna issue is thinly traded, so no reliable YTM is available and matrix pricing on the four similarly rated liquid bonds is the appropriate route. Three steps.

Step 1. Compute the YTM on each comparable from its market price. All four are semiannual pay, so N is in half-years, PMT is the half-year coupon, and the periodic rate is doubled to annualise.
Bond 1: N = 8, PV = −99.5, PMT = 2.5, FV = 100, giving 2.570% × 2 = 5.140%.
Bond 2: N = 8, PV = −106.46, PMT = 3.5, FV = 100, giving 2.595% × 2 = 5.191%.
Bond 3: N = 16, PV = −100, PMT = 3, FV = 100, giving 3.000% × 2 = 6.000%. This one can be read without a calculator: the bond trades at par, so its YTM equals its coupon rate.
Bond 4: N = 16, PV = −112.42, PMT = 4, FV = 100, giving 3.010% × 2 = 6.021%.

Step 2. Average the YTMs at each maturity.
Four-year average = (5.140% + 5.191%) ÷ 2 = 5.166%.
Eight-year average = (6.000% + 6.021%) ÷ 2 = 6.011%.

Step 3. Interpolate linearly out to seven years. Linear interpolation assumes the yields between the two known points are evenly spaced.
Difference: 6.011% − 5.166% = 0.845% across 8 − 4 = 4 years.
Annual increment: 0.845% ÷ 4 = 0.211%.
Five years: 5.166% + 0.211% = 5.377%.
Six years: 5.377% + 0.211% = 5.588%.
Seven years: 5.588% + 0.211% = 5.799%, or about 5.8%.

Gretna debt has seven years remaining, so the matrix estimate is 5.8%. One qualification: this figure was derived from bonds that are more liquid than the thinly traded Gretna issue, so the actual Gretna yield would likely be somewhat higher, to compensate investors for liquidity risk.

3. What is the current cost of preferred equity?

Solution. The preferred is actively traded at 980. With a 7% dividend rate on a par value of 1,000, the annual dividend is 70. Preferred with no growth is a perpetuity, which is the DDM solved for the required return with a growth rate of zero:
Cost of preferred equity = 70 ÷ 980 = 7.14%.
Note that the redemption price is par, 1,000, while the market price is 980, so the preferred trades at a slight discount to the price at which it would be retired.

4. How might KM estimate the cost of debt if management executes the plan to redeem the preferred?

Solution. Debt and preferred equity currently make up 20% + 15% = 35% of the capital structure. Replacing the preferred entirely with debt would produce a structure of 35% debt and 65% common equity. The two anchors are the current estimated cost of debt of 5.8% and the current cost of preferred of 7.14%. Because debtholders rank ahead of preferred shareholders in a claim on assets, the new debt should cost more than 5.8% but less than 7.14%. The increment over 5.8% reflects the additional leverage that comes with taking debt from 20% to 35% of the structure.

5. What market conditions would lead management to conclude that now is a good time to issue?

Solution. A relatively low risk-free rate, relatively tight credit spreads, or both, since the cost of debt is the sum of those two components. The evidence is in the prices. The preferred trades at 980 but must be redeemed at par of 1,000, a slight premium to the market price, so management is willing to pay up to retire it. That management still regards the trade as attractive implies that the debt it would issue is cheap, which points to a low risk-free rate and tight spreads.

6. What could management do to lower the cost of the new debt at issuance?

Solution. Three options, each of which hands something of value to the investor.
First, issue secured debt collateralised on some of the property, plant and equipment. Secured debt is normally cheaper than unsecured, because collateral reduces loss given default.
Second, attach a put option, which lets investors sell the bond back before maturity and is therefore worth paying for in the form of a lower yield.
Third, attach a conversion feature, which gives investors an equity option and again lowers the initial yield. The cost of the second and third options is a possible future refinancing at higher rates or future equity dilution respectively.

7. What is the Gretna cost of common equity under the CAPM, the five-factor model and the BYPRP approach?

Solution.
CAPM. r_e = 0.021 + 0.91(0.055) = 0.021 + 0.05005 = 0.0711, or 7.11%.

Five-factor model. Watch the sign on the profitability beta, which is negative:
r_e = 0.021 + 0.95(0.055) + 0.45(0.018) + 0.14(0.0390) − 0.19(0.031) + 0.30(0.037)
r_e = 0.021 + 0.05225 + 0.0081 + 0.00546 − 0.00589 + 0.0111 = 0.0920, or 9.20%.

BYPRP. Add the estimated premium of 6.2% to the matrix-priced cost of debt of 5.8%:
r_e = 0.058 + 0.062 = 0.12, or 12%.

8. Given Panel A of Exhibit 17, why might the CAPM estimate not be a reasonable one?

Solution. Line the three estimates up against the cost of preferred equity.
CAPM: 7.11%. Five-factor model: 9.20%. BYPRP: 12.00%. Cost of preferred: 7.14%.
Common shareholders hold a residual claim ranking below preferred shareholders, so they must demand a higher return than preferred holders. The CAPM estimate of 7.11% is below the 7.14% cost of preferred equity, which inverts the required ordering by three basis points. On that ground alone the CAPM estimate is not a realistic figure for the cost of common equity.

9. Why might the KM estimate of the ERP be relatively high or low, given her two choices?

Solution. Both of her choices push the estimate up.
She used the short-term government bill rate as the risk-free proxy. Assuming a normal upward-sloping yield curve over most of the estimation period, short-term yields sat below long-term yields, so subtracting the smaller number leaves a larger premium.
She also used an arithmetic rather than a geometric mean, which very likely raises the estimate again, since the arithmetic mean of a volatile return series exceeds its geometric mean.
Her ERP of 5.5% is therefore likely to be high relative to an estimate built on the long-term government bond YTM and a geometric mean.

Figure 5: Interpolating a seven-year yield from a four-year and an eight-year point

Only the two large points are observed. Linear interpolation is the straight line between them, so the intermediate yields are read straight off it in equal steps of 0.211 percentage points per year. The method says nothing about the true shape of the credit curve between the two anchors.

What a three basis point violation actually tells you

The gap between the CAPM estimate of 7.11% and the preferred cost of 7.14% is tiny, and it would be a mistake to read it as a precise measurement of anything. What it does is expose an inconsistency in the inputs. A beta of 0.91 for a small, cyclical, operationally leveraged manufacturer is implausibly low, and the ERP of 5.5% was itself built on assumptions that bias it upward, so the CAPM answer is low despite a generous premium. When a model output violates the seniority ordering of the capital structure, the correct response is to re-examine the inputs, not to accept the number because the arithmetic is right. Here the five-factor estimate of 9.20%, which picks up the size and investment exposures the single-factor model misses, is the more defensible figure.

LM works in corporate development at Hydrocrop Ltd, a manufacturer of irrigation equipment. Management is considering acquiring Precision Irrigation, a private business whose software raises irrigation efficiency. It sits in an emerging market where sovereign risk runs higher. LM has been asked to estimate the WACC of Precision. This case combines almost everything in the reading: peer benchmarking, a synthetic credit rating, a country risk premium, an unlevered and re-levered beta, two cost of equity methods, and a WACC.

Exhibit 18: Selected information for Precision and its peer group

Item	Precision Irrigation	Software industry average
Cash and near-cash items	9%	14%
Trade receivables	10%	12%
Inventories	4%	3%
Remaining current assets	5%	4%
Net property, plant and equipment	21%	30%
Goodwill and other intangibles	47%	32%
All other assets	4%	5%
Debt outstanding (millions)	18.4	296.4
Assets in total (millions)	105.2	1,276.2
EBITDA (millions)	12.2	177.4
Interest charge (millions)	1.6	23.5
Beta	Not available	1.25
Marginal tax rate	20%	25%

Two further facts about Precision. The founder and chief executive remains deeply involved in every part of the operation, with no succession plan in place. Approximately 60% of revenues come from software subscriptions, and 70% come from five major customers located close to one another.

Exhibit 19: Synthetic credit rating schedule

Credit rating	Interest coverage	D/E	Credit spread
AAA	IC > 11 times	D/E < 15%	0.82%
AA	9 < IC < 11	15% < D/E < 20%	1.09%
A	7 < IC < 9	20% < D/E < 25%	1.46%
BBB	5 < IC < 7	25% < D/E < 30%	2.15%
BB	1.4 < IC < 3	30% < D/E < 40%	2.88%

The coverage band on the BB row is printed in the source as 3 < IC < 1.4, which cannot be read literally since the lower bound exceeds the upper. It is shown here in the order the surrounding rows imply. Nothing in the case turns on it, because Precision lands in the A band on both ratios.

Exhibit 20: Factor risk premiums assigned by LM

Factor	Risk premium
Market (ERP)	6%
Size (SP)	5%
Industry (IP)	1%
Specific-company (SCRP)	6%

The yield to maturity on the 10-year benchmark government bond of the emerging country is 5.41%, and interest expense is fully deductible. The corporate development team normally assigns a size premium of 3% to 6% and a specific-company premium of 4% to 8% for private companies. LM judges that a country risk premium of 2% is warranted for the higher sovereign risk, and assumes the current capital structure is the long-term target.

Example 14 · Worked

1. Estimate the after-tax cost of debt for Precision.

Solution. The bonds are not traded, so a synthetic rating is required. Compute the two ratios first.

Interest coverage = 12.2 ÷ 1.6 = 7.63.
D/E = 18.4 ÷ (105.2 − 18.4) = 18.4 ÷ 86.8 = 0.2120, or 21.20%. Equity is total assets less total debt, since no other information is given.

Against Exhibit 19, a coverage ratio of 7.63 falls in the A band (7 < IC < 9) and a D/E of 21.20% also falls in the A band (20% < D/E < 25%). Unlike the Gamma case in Section 4, the two ratios agree, so the synthetic rating is A and the implied credit spread is 1.46%.

Build the yield from three components: the benchmark government yield, the credit spread, and the country risk premium.
r_d = 5.41% + 1.46% + 2.00% = 8.87%.

Interest is fully deductible at a marginal rate of 20%, so:
After-tax cost of debt = 8.87% × (1 − 0.20) = 7.096%.

2. Why is a size premium near the top of the 3% to 6% range justified?

Solution. Because Precision is very small relative to its peer group. Measured by total assets, Precision at 105.2 million is less than one tenth the size of the average listed company in the industry at 1,276.2 million, a ratio of about 12 to 1. Since the size premium is assumed to be inversely related to company size, a premium of 5% near the high end of the range is reasonable.

3. Which characteristics of Precision justify a higher specific-company premium, and which justify a lower one?

Solution. Four arguments push the premium up and two pull it down.

Higher. The asset base is far more intangible than the peer group, 47% against 32%, and holds less cash, 9% against 14%; asset bases weighted toward intangible and illiquid assets carry a higher cost of equity. Customer concentration is severe, with roughly 70% of revenues from five customers. Those customers sit close to one another geographically, adding geographic concentration risk. And the founder and chief executive is involved in every part of the business with no succession plan, which is significant key-person risk.

Lower. Roughly 60% of revenues come from software subscriptions, so a high proportion of revenue is recurring, which implies steadier earnings and cash flow. And Precision is less leveraged than the average software company: its D/E of 0.2120 compares with an industry figure of 296.4 ÷ (1,276.2 − 296.4) = 0.3025, and its interest coverage of 7.63 is slightly above the industry figure of 177.4 ÷ 23.5 = 7.55.

4. Estimate the cost of equity for Precision using the extended CAPM and the build-up approach.

Solution.
Extended CAPM. Precision has no beta of its own, so borrow the industry beta and adjust it for leverage. Two steps.

Unlever the industry beta at the industry tax rate of 25% and industry D/E of 0.3025:
β_Asset = 1.25 ÷ [1 + (1 − 0.25)(0.3025)] = 1.25 ÷ 1.226875 = 1.019.

Re-lever at the Precision tax rate and leverage, 20% and 0.2120:
β_Precision = 1.019 × [1 + (1 − 0.20)(0.2120)] = 1.019 × 1.1696 = 1.19.

Note the direction: Precision is less levered than the industry, so its re-levered beta of 1.19 sits below the industry beta of 1.25, and above the asset beta of 1.019.

Now build the required return, adding the country risk premium of 2%:
r_e = 5.41% + 1.19(6%) + 5% + 1% + 6% + 2% = 5.41% + 7.14% + 14% = 26.55%.

Build-up approach. No beta at all, so the ERP enters unadjusted:
r_e = 5.41% + 6% + 5% + 6% + 2% = 24.41%.

The two answers differ by 2.14 percentage points. Part of the gap is the beta of 1.19 scaling up the 6% ERP, worth 1.14 points, and part is the 1% industry premium, which the build-up calculation here does not include.

A note on the source: the CFA Institute text introduces these two answers with the figures 26.56% and 25.41% and then sets out equations that produce 26.55% and 24.41%. The inputs are reproduced here exactly, and they give 26.55% and 24.41%.

5. Estimate the WACC of Precision using the build-up estimate of the cost of equity.

Solution. Take the weights from the balance sheet, using book values since no market value of equity exists for a private company.
w_d = 18.4 ÷ 105.2 = 0.1749.
w_e = (105.2 − 18.4) ÷ 105.2 = 0.8251.

Apply the two-term WACC, using the pre-tax cost of debt of 8.87% with the tax shield applied inside the formula:
WACC = (0.1749)(0.0887)(1 − 0.20) + (0.8251)(0.2441)
WACC = 0.012409 + 0.201408 = 0.2138, or 21.38%.

Equity dominates the answer. It supplies 82.51% of the funding at 24.41%, contributing 20.14 percentage points of the 21.38% total, while debt contributes only 1.24 points. That is the arithmetic reason why the estimate of the cost of equity, and therefore the size and specific-company premiums, matters far more to this valuation than the synthetic credit rating does.

A note on the source: the CFA Institute text introduces this answer with the figure 21.13% and then sets out the equation that produces 21.38%. The inputs are reproduced here exactly, and they give 21.38%.

Reading the whole estimate back

A WACC of 21.38% is a very high discount rate by developed-market standards, and it is worth seeing where it came from before accepting it. The government benchmark contributes 5.41 percentage points, the country risk premium 2 points to both components, and the private-company premiums for size and specific risk contribute 11 points to the equity side alone. In other words, roughly half of the required return on equity is made up of judgemental premiums that no market price validates. That is the honest position with a private emerging-market target, and it is why the range guidance the corporate development team applies, 3% to 6% for size and 4% to 8% for specific risk, exists at all: it constrains how far judgement can move the answer. An analyst who wanted to challenge this valuation would attack those two premiums, not the synthetic rating or the beta.

Check yourself

1. A jurisdiction raises its corporate income tax rate. Holding everything else constant, what happens to the after-tax cost of debt for issuers there, and what single condition must hold for that answer to be right?

The after-tax cost of debt falls, because each unit of deductible interest now shelters more tax. The condition is that interest is in fact deductible and the company has taxable income before interest to absorb the deduction. Where interest is not deductible, or where the company has already hit a statutory cap on the deductible amount, no tax adjustment is made at all and WACC is unaffected.

2. Two issuers are identical except that one has issued callable bonds and the other putable bonds. Which pays the higher yield at issuance, and does that ordering necessarily persist?

The callable issuer pays more at issuance, because the call benefits the issuer and disadvantages the investor, who demands compensation. The putable issuer pays less, because the put benefits the investor. The ordering does not necessarily persist. If rates fall, the callable issuer refinances and its cost drops. If rates rise, the putable issuer has bonds put back to it and must refinance at the higher prevailing rate, so its cost rises.

3. A company has a synthetic rating model that maps interest coverage to a B rating and D/E to a BBB rating for the same borrower. What should the analyst do?

Neither rating can simply be adopted, because the model gives a conflicting signal. The analyst should examine both ratios historically for that borrower to see whether either is trending, and use judgement informed by the trend to settle the rating. Improving coverage might justify moving up a notch from the rating the current coverage level implies.

4. An analyst estimates a historical ERP using a short-term bill rate and an arithmetic mean, in a market whose yield curve has normally been upward sloping. Is the estimate likely to be high or low relative to one using a long bond yield and a geometric mean, and why does each choice work in that direction?

High, on both counts. A short-term bill rate has historically sat below the long bond yield in an upward-sloping curve, so subtracting the smaller number leaves a larger premium. And the arithmetic mean of a volatile return series exceeds its geometric mean, so the measured average excess return is larger. The two choices compound rather than offset.

5. Under the Grinold–Kroner model, the dividend yield is 2.5%, expected repricing is −0.5%, expected inflation is 2.0%, real growth is 1.5%, the expected change in shares outstanding is +0.4%, and the expected risk-free rate is 3.0%. What is the ERP?

Earnings growth per share is i + g − ΔS = 2.0 + 1.5 − 0.4 = 3.1%. Then ERP = [2.5 + (−0.5) + 3.1] − 3.0 = 5.1 − 3.0 = 2.1%. Note that share issuance dilutes, so a positive change in shares outstanding subtracts. Note also that raising expected inflation would not change the answer, because it would lift the expected risk-free rate by the same amount.

6. Why is illiquidity of a private company normally handled as a valuation discount rather than as a premium added to the required return on equity?

Because the convention treats it as a reduction in the estimated value of the equity interest, called a discount for lack of marketability, rather than as an element of the discount rate. The size, industry and specific-company premiums already loaded into the required return cover the other risks of private ownership. Adding an illiquidity premium to the discount rate as well as applying a marketability discount to the value would count the same problem twice.

7. A private target has a D/E ratio of 0.50 and a marginal tax rate of 30%. Its listed peer group has a beta of 1.10, a D/E ratio of 0.25 and a tax rate of 25%. Estimate the beta to use for the target, and say in which direction the adjustment moves.

Unlever the peer beta first: 1.10 ÷ [1 + (1 − 0.25)(0.25)] = 1.10 ÷ 1.1875 = 0.926. Then re-lever at the target leverage and tax rate: 0.926 × [1 + (1 − 0.30)(0.50)] = 0.926 × 1.35 = 1.25. The adjustment moves the beta up, because the target carries twice the leverage of the peer group. The asset beta of 0.926 is the business risk alone, stripped of both capital structures.

8. A cost of equity model returns a figure below the cost of the preferred equity of the same issuer. Is that possible, and what should the analyst conclude?

It is arithmetically possible but economically inconsistent. Common shareholders hold a residual claim ranking below preferred shareholders, so they must require a higher return. A model output that inverts that ordering signals a problem in the inputs, most often an implausibly low beta or a mis-specified risk-free proxy, and the correct response is to re-examine the inputs and prefer an estimate from a model that captures the missing exposures.

End of lesson.

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