Cost of capital

Weighted Average Cost of Capital (WACC):

$$WACC=w_d{r}_d(1-t)+w_p{r}_p+w_e{r}_e$$

where $t$ is tax rate.

• $r_d=r_f+\text{spread}$
• $r_e=r_f+(ERP+IRP)$
  • ERP: equity risk premium
  • IRP: idiosyncratic risk premium

Cost of capital factors

Top-down, External	Bottom-up, Company specific
Capital availability	Revenue, earnings and cash flow volatility
Market conditions	Asset nature and liquidity
Legal and regulatory considerations, country risk	Financial strength, profitability, and leverage
Tax jurisdiction	Security features

• Capital avaliability: Lower

• Market condition

  • Inflation: Higher
  • Receession: Higher
  • Expansion: Lower
  • Lower interest rante or exchange volatility: Lower

• Legal and regulatory: Greater, Lower

• Higher marginal income tax rate: Lower (look at formula, greater tax shield effect)

• Revenue, earning, and cash flow volatility:

Factor	Cost of Capital
Higher stability in sales	Lower
Higher revenue concentration	Higher
Higher earnings predictability	Lower
Higher operating and financial leverage	Higher
Higher ESG risk	Higher

• Asset nature and liquidity

Factor	Cost of Capital
Higher % of fungible, tangible assets	Lower
Higher % of liquid assets	Lower

• Financial strength

Factor	Cost of Capital
Higher profitability	Lower
Higher cash flow generation	Lower
Higher interest coverage, liquidity	Lower
Higher leverage ratios	Higher

• Security features

	Feature	Cost of Capital
Debt	Callability	Higher
	Putability	Lower
	Convertibility	Lower
Preferred stock	Cumulative	Lower
Common stock	Inferior cash flow/voting rights	Higher

Cost of Debt

• Traded debt: YTM
• Non-traded debt: Simiilar bond yield or maturity with credit rating. Without credit rating: synthetic credit ratings.

Synthetic credit rating example

Rating class	Interest Coverage Ratio	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	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%

• Bank loans: The latest interest rate.

NOTE

Amortizing loans typically have lower cost of debt than non-amortizing

• Leases

  • Rate implicit in the lease (RIIL)

  With the following equation (considering financing lease)

  $$\text{PV of lease payment}+\text{PV of residual value to lessor}=\text{Fair value of leasedn asset}+\text{Lessor’s direct initial cost}$$
  • Incremental borrowing rate (IBR): The rate of a collateralized loan over the same term

Equity Risk Premium (ERP)

Historical Approach

$$ERP=r_m-r_f$$

We can take multiple average from historical data to estimate future ERP.

**Key assumptions **:

• Returns are stationary
• Market are relatively efficient
• Average return should be an unbiased estimate of expected returns in the long run

Key decisions:

• Selection of equity index
  • Typically broad-based, market-value-weighted indexes
• Selection fo time period

Time Period	Advantages	Disadvantages
Longer time period	Fluctuating volatility has less effect	Not representative of the current market
Shorter time period	Representative of the current market environment	Increases the likelihood of greater noise (covering only a portion of a business cycle or a period of disruption)

• Selection of mean type

	Advantages	Disadvantages
Arithmetic Mean	• Easy to calculate • Considers all observations	• Sensitive to extreme values • Overestimates expected terminal value of wealth
Geometric Mean	• Considers all observations • Gives outliers less weight • Estimates expected terminal value of wealth

• Selection of risk-free rate proxy

	Advantages	Disadvantages
Government bond YTM	More closely matches infinite equity duration	Not a completely risk-free return
Government bill rate	Exact estimate of the risk-free rate	Not closely matches infinite equity duration

Limitation:

• ERP can vary overtime
• Survivorship bias tend to inflate historical estimates of the ERP

Forward-looking Approach

• Survey-based estimates
• Dividend discount models
• Macroeconomic modeling

Survey-based

• Assess expectation by asking people what they expect
• Estimates tend to be sensitive to recent market returns

Dividend Discount Models (DDM)

From the Gordon growth model, assuming constant P/E, earnings, dividends, and price will grow at the same rate.

$$ERP=E(\frac{D_1}{V_0})+E(g)-r_f$$

Macroeconomic Model

$$ERP=[DY+\mathrm{\Delta }(\frac{P}{E})+i+g-\mathrm{\Delta }S]-E(r_f)$$

NOTE

Similar to the growth function in Economics , where $\mathrm{\Delta }S$ is the net change of Buybacks + Relative Dynamism. The idea is identical to that. Simply different notations.

Factors			Symbol
Dividend yield			$DY$
Expected capital gain	Expected repricing		$\mathrm{\Delta }(P/E)$
	Earnings growth/share	Expected inflation	$i$
		Real economic growth	$g$
		Change in shares outstanding	$\mathrm{\Delta }S$

NOTE

Expected inflation $i$ can be estimated from $YT{M}_{treasurybond}-YT{M}_{TIPS}$

Limitations

• Survey
  • Samping and response bias
  • Behavioral biases
• DDM
  • Assumption too strict
• Macroeconomic models
  • Modelling errors from macro data
  • Behavioral biases

Cost of Equity

Models

• DDMs
• Bond yield plus risk premium approach (BYPRP)
• CAPM
• Fama-French

DDM

• Requirements: Publically traded, stable and predictable dividend
• Gordon growth model:
  • $r_e=\frac{D_1}{P_0}+g$
• Multiyear financial forecast:
  • $P_0=(\sum _{t=1}^n\frac{D_1}{(1+r_e)})+\frac{P_n}{(1+r_e{)}^n}$, using IRR to calculate $r_e$.

BYPRP

$$r_e=r_d+RP$$

where $RP$ can be exstimated using historical mean difference in returns between euity market index and corporate bond index.

• Estimating cost of debt provides a starting point estimate of the return demanded by debt investor
• Determination of RP is relatively arbitrary
• Requires company to have traded debt
• If a company has multiple traded debt with different features, it is hard to select.

CAPM

$$r_e=r_f+\beta (ERP)$$

Fama-French Models

$$r_e=r_f+{\beta }_{1}ERP+{\beta }_{2}SMB+{\beta }_{3}HML+{\beta }_{4}RMW+{\beta }_{5}CMA$$

• SMB: Small minus big, size premium
• HML: High(book-to-market) minus low, value premium
• RMW: Robust minus weak, profitability premium
• CMA: Conservative minus aggressive, investment premium

Extended CAPM

$$r_e=r_f+{\beta }_{\text{peer}}(ERP)+SP+SCRP$$

• ${\beta }_{\text{peer}}$: Industry $\beta$, from a peer group opf publicly traded companies in the same industry
• SP: Size premium
• SCRP: Specific company risk premium

Build-up Approach

$$r_e=r_f+ERP+SP+IP+SCRP$$

NOTE

When industry $\beta$ cannot found, we assume $\beta =1$.

• IP: Industry risk premium

Country Risk Premium (CRP)

CRP is required by investor for the added risk of investing in emerging market

$$CRP=\text{Sovereign yield spread}\times \frac{{\sigma }_{\text{equity}}}{{\sigma }_{\text{bond}}}$$

• Soverign yield spread: The yield on emerging market bonds minus the yield on developed market government bonds.
• ${\sigma }_{\text{equity}}$: Volatility of the local country's equity market
• ${\sigma }_{\text{bond}}$: Volatility of the local country's bond market

Global CAPM

• Single factor: A global market index as $r_m$
• Assuming no significant risk differences across countries

International CAPM

$$E(r_e)=r_f+{\beta }_G(E(r_{gm})-r_f)+{\beta }_c(E(r_c)-r_f)$$

• $r_{gm}$: Global index
• $r_c$: foreign currency index