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We will use an example to illustrate how private and public companies estimate the required return on equity and cost of debt.

John Marcus, an analyst at Glendale Investments, has been tasked with estimating the cost of equity and debt for DMC, a company they are looking to invest in.

Marcus notes the following about DMC:

$$ \begin{array}{l|l} \text{Business Model} & {\text{DMC manufactures automobiles with a} \\ \text{large proportion of their costs being fixed} }\\ \hline \text{Company} & \text{Publicly traded small-cap company} \\ \hline \text{Industry} & \text{Industrial equipment} \\ \hline \text{Nature of assets} & {\text{Majority of assets are in the form of PPE} \\ \text{and inventory of finished vehicles} } \\ \hline {\text{Revenues, earnings,} \\ \text{and cashflows}} & {\text{Have been on an upward trend but vary} \\ \text{over the business cycle} } \end{array} $$

DMC has had high sales and profits recently. However, five years back, the sale of their flagship vehicle, Icarus, declined due to new European emissions regulations. This put the company in a liquidity crisis, prompting it to raise cash by issuing redeemable preferred shares, which came at a high cost.

DMC’s management has noted that the market conditions have improved and will issue unsecured debt to retire the preferred stock. Exhibit 1 below shows DMC’s current capital structure.

$$ \textbf{Exhibit 1: DMCs Capital Structure} \\ \begin{array}{c|c|c} \textbf{Type of Capital} & \textbf{Current Capital} & \textbf{Selected Capital} \\ & \textbf{Structure} & \textbf{Type Information} \\ \hline \text{Common shares} & 70\% & \text{Actively traded} \\ \hline \text{Debt} & 15\% & { \text{Single debt issue: 8% coupon} \\ \text{rate, 7 years remaining maturity} \\ \text{with semiannual payments.} \\ \text{Straight unsecured debt; BBB} \\ \text{rating with no YTM} \\ \text{information.}} \\ \hline \text{Preferred shares} & 15\% & { \text{Frequently traded} \\ \text{The current share price is 890} \\ \text{The dividend rate is 8%,} \\ \text{redeemable at a par value of} \\ \text{1000 per share} }\\ \end{array} $$

Marcus gathers information on four corporate bonds, each of which has a BBB rating, just like DMC’s. Refer to Exhibit 2 below.

$$ \textbf{Exhibit 2: Information on Liquid BBB- rated Bonds} \\ \begin{array}{c|c|c|c} & \textbf{Coupon} & \textbf{Remaining} & \textbf{Current} \\ & \textbf{Rate} & \textbf{Maturity} & \textbf{price*} \\ \hline \text{Bond A} & 8\% & 8 \text{ years} & 113 \\ \hline \text{Bond B} & 5\% & 4 \text{ years} & 99 \\ \hline \text{Bond C} & 7\% & 4 \text{ years} & 107 \\ \hline \text{Bond D} & 6\% & 8 \text{ years} & 101 \end{array} $$

*Current price per 100 of per value

Marcus uses the Fama-French five-factor model and CAPM to estimate DMC’s cost of equity. He does this by regressing DMC’s excess returns on risk factors for the most recent 60 months. His risk-free proxy is the 20-year government benchmark of 2%. Information on the CAPM and Fama-French Five factor-beta and risk premiums are illustrated in Exhibit 3 below.

$$ \textbf{Exhibit 3: CAPM and 5-factor Beta and Risk Premiums} \\ \begin{array}{c|c|c|c} & \textbf{Factor} & \textbf{Risk Premium} & \textbf{Beta} \\ \hline {\text{FFM risk} \\ \text{premium and} \\ \text{5-factor beta} } & & & \\ \hline & \text{Market (ERP)} & 6\% & 0.91 \\ \hline & \text{Size (SMB)} & 2\% & 0.45 \\ \hline & \text{Value (HML)} & 3.9\% & 0.2 \\ \hline & \text{Profitability (RMW)} & 3.2\% & -0.18 \\ \hline & \text{Investment (CMA)} & 2.6\% & 0.4 \\ \hline {\text{CAPM risk} \\ \text{premium and} \\ \text{Factor beta}} & & & \\ \hline & \text{Market (ERP)} & 6\% & 0.95 \end{array} $$

Marcus uses the BYPRP approach to estimate the cost of equity for DMC. He uses a historical risk premium for the estimate. The historical approach informs his arrival at the ERP of 6%. He opts to use an arithmetic mean and the short-term government bill in the estimate.

- Estimate DMC’s cost of debt.
- Calculate the current cost of preferred equity.

The first step will be to calculate the YTM of each bond.

**Bond A YTM:** N= 16; PV=-113; PMT = 4; FV = 100; CPT I/Y = 2.968% x 2 = 5.936%

**Bond B YTM:** N= 8; PV= -99: PMT =2.5; FV = 100; CPT I/Y = 2.640% x 2 = 5.28%

**Bond C YTM:** N= 8; PV= -107: PMT =3.5; FV = 100; CPT I/Y =2.523% x 2 = 5.046%

**Bond D YTM:** N= 16; PV=-101: PMT = 3; FV = 100; CPT I/Y =2.921% x 2 = 5.842%

The second step will be to calculate the average YTM for 4-year and 8-year maturity.

$$ \text{Average YTM (4-year Maturity)}=\frac{(5.28\%+5.046\%)}{2}=5.163\% $$

$$ \text{Average YTM (8-year Maturity)}=\frac{(5.968\%+5.842\%)}{2}=5.905\% $$

We then use linear interpolation to estimate the average YTMs for 5-, 6-and 7-year maturities. We do this by first computing the difference between the 8-year average YTM and the 4-year average YTM (in this case, 5.905% − 5.163% = 0.742%). Then, we divide the difference by the difference between the years of the known yields (8 − 4 = 4; 0.742%/4 = 0.1855%).

The 0.1855% is the estimated annual incremental average yield after year 4 as the term to maturity increases.

Estimated average YTM in year 5 = Year 4 average YTM + 0.1855% = 5.163% + 0.1855% = 5.3485%

Estimated average YTM in year 6 = Year 5 average YTM + 0.1855% = 5.3485% + 0.1855% = 5.534%

Estimated average YTM in year 7 = Year 6 average YTM + 0.1855% = 5.534% + 0.1855% = 5.7195%

From the matrix pricing, DMC’s debt is likely to have a YTM of 5.72%

**Solution to 2: The Current Cost of Preferred Equity**

Since DMC’s preferred stock has an annual dividend rate of 8% and a par value of 1000, the annual dividend will be 80. Using the DDM formula with a growth rate of 0:

$$ \begin{align*} r_e & =\frac{D_1}{P_0}+g \\ & =\frac{70}{890}+0=0.07865=7.87\% \end{align*} $$

## Question

From the case above, the cost of common equity, using the Fama-French five-factor model, is

most likelyto be:

- 7.7%.
- 9.6%.
- 11.72%.
## Solution

The correct answer is

B.It is calculated using the following formula:

$$ \begin{align*} r_e & =r_f+\beta_1ERP+\beta_2SMB+\beta_3HML+\beta_4RMW+\beta_5CMA \\ & =0.02+0.91\left(0.06\right)+0.45\left(0.02\right)+0.2\left(0.039\right) \\ & -0.18(0.032)+0.4(0.026) \\ & =0.09604=9.6\% \end{align*} $$

A is incorrect. This is the cost of common equity calculated using the CAPM model.$$ \begin{align*} r_e & =r_f+\beta_1ERP \\ & =0.02+0.95\left(0.06\right)=0.077=7.7\% \end{align*} $$

C is incorrect. The cost of common equity was calculated using the BYPRP, as shown below.$$ \begin{align*} r_e & =r_d+RP \\ &= 0.0572+0.06=0.1172=11.72\% \end{align*} $$

Reading 20: Cost of Capital: Advanced Topics

*LOS 20 (e) Estimate the cost of debt or required return on equity for a public and a private company.*