Business Finance
Michael Dimond
Risk and the costs of capital
• When considering common equity, preferred equity and debt,
the owners of these securities each bear a different level of
risk. The higher the risk, the higher the rate of return will be.
• Usually, this is what you expect:
Rf < Kd < Kpfd < Ke
and…
inflation < Rf
• What does this tell you about the return demanded by each
type of investor?
• What does that imply about the risk borne by each?
• Real Rates, Risk Premium, etc.
• Market efficiency (the EMH)
Michael Dimond
School of Business Administration
Determining Kd with YTM
• Remember from valuing bonds that the “i” in the TVM
relationship is the yield on a bond.
• Yield is the rate of return demanded by investors in debt.
• To find the Cost of Debt (Kd), solve for the yield of a bond.
• Remember: Costs of capital are forward-looking numbers. It
is what an investor will demand from future performance.
• What does this mean for the cost of debt (Kd)?
Michael Dimond
School of Business Administration
The “Approximation Formula”
• The “approximation formula” (PMT+((FV-PV)/n))/((FV+PV)/2)
works only when bonds are selling close to par
• I have tried to eliminate it from the homework, but a couple of
multi-part problems require it as one of the parts
• Because it is not accurate, please do not use the
approximation formula unless you are specifically told to. I
will not use it on any exams.
Michael Dimond
School of Business Administration
Determining Ke with the Dividend Growth Model
• Ke is the required rate of return for equity investors
“Cost of Equity”
• We can derive this from the dividend discount model:
P0 = D1/(r-g)
:. (r-g) = D1/P0
:. r = D1/P0 + g
• Since r is the required rate of return, Ke = D1/P0 + g
• How do you test the answer you get?
• What if you wanted to solve for the expected growth rate?
• There is another way to find the cost of equity: CAPM
Michael Dimond
School of Business Administration
Determining Ke with CAPM
• Ke is the required rate of return for equity investors
“Cost of Equity”
• The CAPM (Capital Asset Pricing Model) is a formula used to
compute Ke
Ke = Rf + β(Rm – Rf)
or
Ke = Rf + β(MRP)
• MRP is the Market Risk Premium
MRP = Rm – Rf
• Remember: Costs of capital are forward-looking numbers. It
is what an investor will demand from future performance.
• What does this mean for the cost of equity (Ke)?
Michael Dimond
School of Business Administration
About beta
• Beta (β) represents how well an asset’s return correlates with
the return on the market.
• Correlation, not Causality
• Beta measures sensitivity to economic inputs
• Beta is the slope of the line showing the relationship of the data
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
Mkt Return
16%
13%
7%
14%
12%
-5%
-9%
-12%
4%
8%
A Return
28%
22%
14%
18%
20%
-2%
2%
-8%
9%
13%
30%
25%
20%
A Return
15%
Linear (A Return)
y = 1.0893x + 0.0637
10%
5%
0%
-20%
-10%
0%
10%
20%
-5%
-10%
• What is the difference between Systematic and Unsystematic risk?
• Can beta be zero? Can beta be negative?
Michael Dimond
School of Business Administration
Can beta really be negative?
What does a negative beta
imply about the stock
performance?
What does a negative beta
imply about Ke?
Michael Dimond
School of Business Administration
Determining Kpfd with the Dividend Discount Model
• The value of a perpetuity is how we price preferred equity:
PVperp = CF/r
:. Price = Dividend/Kpfd
:. Price x Kpfd = Dividend
:. Kpfd = Dividend / Price
• So… If XYZ Company has 8% preferred stock with a $20.00
par value which is selling for $28.00, what would be the Cost
of Preferred Equity?
• 8% x 20.00 = 1.60
• 1.60 / 28.00 = 0.0571 = 5.71%
Michael Dimond
School of Business Administration
Market Value vs Book Value
• Investors have an opinion about the value of a company. A
stockholder believes the price of stock is appropriate value,
and this is different than what the balance sheet shows for
the value of equity.
• The balance sheet shows the book value. The market price
determines the market value.
• Example: Book Value
• Example: Market Value (“Market Cap” or Market Capitalization)
• P/B ratio
Michael Dimond
School of Business Administration
WACC
• Investors care about market value more than book value.
• Costs of capital are used in making investing decisions.
• :. If we wanted to know the overall cost of capital for a
company (the cost of equity and the cost of debt combined),
we would use a weighted average of the percentages, and
weight them based on the market values.
• The Weighted Average Cost of Capital is called the WACC.
WACC = we x Ke + wd x Kd (1-t)
• If there is preferred stock, we just expand the formula:
WACC = wpfd x Kpfd + we x Ke + wd x Kd (1-t)
Michael Dimond
School of Business Administration
Weights of Equity & Debt
• The weight is the proportion of that type of capital compared
with the total capital in the firm.
• What is the weight of each type of capital below?
• Equity = E = $600MM
• Debt = D = $300MM
• Preferred Stock = Pfd = $100MM
•
•
•
•
Total Capital = TC = 600+300+100 = 1,000
We = E/TC = 600/1,000 = 0.60
Wd = D/TC = 300/1,000 = 0.30
Wpfd = Pfd/TC = 100/1,000 = 0.10
• Note: the sum of the weights always equals 1.00
Michael Dimond
School of Business Administration
Determining the WACC
• You may need to compute the WACC from information
presented like this:
• ABC Company needs to know their WACC. They have $10MM in 8.5% bonds
payable, which sell for $1,125, have semiannual payments and mature in ten
years. Their tax rate is 34%. They have 1 million shares of stock which paid
$1.80 in dividends last year. The dividends are expected to grow 7% per year
forever and the stock currently sells for $27.50. They also have 10,000 shares
of 9% preferred stock with a $100 par value which sells for $125.
• What is the best approach to solving this problem?
Michael Dimond
School of Business Administration
Break a complicated problem into smaller pieces
•
ABC Company needs to know their WACC. They have $10MM in 8.5% bonds
payable, which sell for $1,125, have semiannual payments and mature in ten
years. Their tax rate is 34%. They have 1 million shares of stock which paid
$1.80 in dividends last year. The dividends are expected to grow 7% per year
forever and the stock currently sells for $27.50. They also have 10,000
shares of 9% preferred stock with a $100 par value which sells for $125.
• Find the market values
•
•
•
•
MVe = $27.50 x 1MM = $27.5MM
MVd = 10MM ÷ 1,000 (assumed FV) x $1,125 = $11.25MM
MVpfd = $125 x 10,000 = $1.25MM
TC = $27.5MM + $11.25MM + $1.25MM = $40.0MM
• Find the weights & tax shield
•
•
•
•
We = 27.5 ÷ 40.0 = 0.6875
Wd = 11.25 ÷ 40.0 = 0.2813
Wpfd = 1.25 ÷ 40.0 = 0.0313
(1-t) = (1 - 0.34) = 0.66
Michael Dimond
School of Business Administration
Break a complicated problem into smaller pieces
•
ABC Company needs to know their WACC. They have $10MM in 8.5% bonds
payable, which sell for $1,125, have semiannual payments and mature in ten
years. Their tax rate is 34%. They have 1 million shares of stock which paid
$1.80 in dividends last year. The dividends are expected to grow 7% per year
forever and the stock currently sells for $27.50. They also have 10,000
shares of 9% preferred stock with a $100 par value which sells for $125.
• Find the different costs of capital
• Kd : Solve for the yield
find i, where: n=10x2, PV=-1125, FV=1000, PMT=0.085x1000÷2 [i=3.3800]
:. Kd = 2x3.38% = 6.76%
• Ke : Use the Dividend Discount Model or the CAPM (based on data available)
Ke = r = D1/P0 + g
Ke = [(1.80 x 1.07) / 27.50] + 0.07 = 0.1400
:. Ke = 14.00%
• Kpfd : Find the return on a non-growing perpetuity
Kpfd = r = D1/P0 = (0.09 x 100)/125 = 0.0720
:. Kpfd = 7.20%
Michael Dimond
School of Business Administration
Determining the WACC
• From there, put the pieces of the formula in place, then
work through the formula in steps
•
•
•
•
WACC = wpfd x Kpfd +
we x Ke + wd x Kd x (1-t)
WACC = 0.0313 x 0.072 + 0.6875 x 0.14 + 0.2813 x 0.0676 x 0.66
WACC =
0.0023
+
0.0963
+ 0.2813 x 0.0446
WACC =
0.0023
+
0.0963
+
0.0126
WACC = 0.1112 = 11.12%
0.0446 is the
After-Tax
Cost of Debt
• By parsing the problem, you avoid errors
• Try another example:
• XYZ Company needs to know their WACC. They have $9.9MM in 8.25%
bonds payable, which sell for $1,100, have semiannual payments and mature
in twelve years. Their tax rate is 35%. They have 1 million shares of stock
which currently sell for $28.75. They also have 8,000 shares of 9.25%
preferred stock with a $100 par value which sells for $110. U.S. government
bonds currently yield 3.15% and the expected return on the market is 10.38%.
XYZ has a beta of 1.25
Michael Dimond
School of Business Administration
Determining the WACC
• Try another example:
• XYZ Company needs to know their WACC. They have $9.9MM in 8.25%
bonds payable, which sell for $1,100, have semiannual payments and mature
in twelve years. Their tax rate is 35%. They have 1 million shares of stock
which currently sell for $28.75. They also have 8,000 shares of 9.25%
preferred stock with a $100 par value which sells for $110. U.S. government
bonds currently yield 3.15% and the expected return on the market is 10.38%.
XYZ has a beta of 1.25
• Market Values & Weights
•
•
•
•
MVe = $28,750,000
MVpfd = $ 880,000
MVd = $10,890,000
TC = $40,520,000
we = 0.7095
wpfd = 0.0217
wd = 0.2688
1.0000
a
• Tax Shield = (1 – t) = (1 - 0.35) = 0.65
Michael Dimond
School of Business Administration
Determining the WACC
• Try another example:
• XYZ Company needs to know their WACC. They have $9.9MM in 8.25%
bonds payable, which sell for $1,100, have semiannual payments and mature
in twelve years. Their tax rate is 35%. They have 1 million shares of stock
which currently sell for $28.75. They also have 8,000 shares of 9.25%
preferred stock with a $100 par value which sells for $110. U.S. government
bonds currently yield 3.15% and the expected return on the market is 10.38%.
XYZ has a beta of 1.25
• Costs of Capital
• Ke = Rf + β(Rm – Rf) = 0.0315 +1.25(0.1038-0.0315) = 0.1219 = 12.19%
• Kpfd = Dividend / Price = 9.25/110 = 0.0841 = 8.41%
• Kd = Find YTM where i is semiannual and…
n = 24 semiannual, PV = -1,100, FV = 1,000, PMT = 41.25 semiannual
i = 3.5021 :. YTM = 7.0043% or 0.0700
Michael Dimond
School of Business Administration
Determining the WACC
• From there, put the pieces of the formula in place, then
work through the formula in steps
•
•
•
•
WACC = wpfd x Kpfd +
we x Ke + wd x Kd x (1-t)
WACC = 0.0217 x 0.0841 + 0.7095 x 0.1219 + 0.2688 x 0.0700 x 0.65
WACC =
0.0018
+
0.0865
+ 0.2688 x 0.0455
WACC =
0.0018
+
0.0865
+
0.0122
WACC = 0.1005 = 10.05%
Michael Dimond
School of Business Administration
Real life is more complicated
• Equity can come from retained earnings or from new
investment.
• New issues of stocks and bonds come with flotation costs,
price adjustments, etc. which must be factored in.
• Growth rates are rarely stated, so they must be computed.
Michael Dimond
School of Business Administration
Kd with flotation cost
• ABC Company is in the 40% tax bracket and can sell 15-year
bonds ($1,000 par) paying annual interest of 12%. Market
rates are slightly below that for bonds of this rating, so the
bonds will sell for $1,100. To issue the bonds, ABC will have
flotation costs of $30 per bond. Find Kd.
•
•
•
•
•
n = 15 years (annual)
i = ?? Solve for YTM (annual)
PMT = 12% x 1,000 (annual) outflow
PV = 1,100 – 30 (expected proceeds less flotation costs) inflow
FV = 1,000 outflow
•
•
•
•
i = 11.0252
YTM = 11.03%
Kd = 11.03%
Kd (After-tax) = 11.03% x (1-0.40) = 6.62%
Michael Dimond
School of Business Administration
Kpfd with flotation cost
• XYZ Company is issuing preferred stock with an 8% dividend
and a $120 par value. The shares will sell for $129.60 and
have flotation costs of $7.20 per share. What is the cost of
preferred equity?
•
•
•
•
•
•
Par = 120.00
Dividend = 8% x 120 = 9.60
Selling price = 129.60
Flotation costs = 7.20
Net proceeds = 129.60 – 7.20 = 122.40
Kpfd = 9.60 / 122.40 = 0.0784 = 7.84%
Michael Dimond
School of Business Administration
Ke with flotation cost
• ABC Company is considering a SEO (Seasoned Equity
Offering). Their stock currently sells for $48.22, but the new
shares will be underpriced by $1.00 and have $2.86 per
share in flotation costs. The planned dividend per share is
$1.45 for the coming year, and they expect the growth of
dividends to follow the same average growth it has for the
past 5 years. Historic annual DPS are:
•
•
•
•
•
2008
2009
2010
2011
2012
2.12
2.30
2.60
2.92
3.10
• What is their Cost of Equity for the SEO?
• Ke = D1 / P0 + g… but what is g?
Michael Dimond
School of Business Administration
Ke with flotation cost – finding the growth rate
• Historic annual DPS for the last 5 years are:
•
•
•
•
•
2008
2009
2010
2011
2012
2.12
2.30
2.60
2.92
3.10
• Growth can be found by solving for the CAGR (Compound
Annual Growth Rate)
•
•
•
•
(3.10/2.12) = 1.46
1.46 ^ (1/4) = 1.0997 (a 5-year sample means we see 4 years of compounding)
CAGR = 1.0997 – 1 = 0.0997 = 9.97%
g = 9.97%
Michael Dimond
School of Business Administration
Ke with flotation cost
• ABC Company is considering a SEO (Seasoned Equity
Offering). Their stock currently sells for $48.22, but the new
shares will be underpriced by $1.00 and have $2.86 per
share in flotation costs. The planned dividend per share is
$1.45 for the coming year, and they expect the growth of
dividends to follow the same average growth it has for the
past 5 years. Historic annual DPS are:
•
•
•
•
•
2008
2009
2010
2011
2012
2.12
2.30
2.60
2.92
3.10
• What is their Cost of Equity for the SEO?
• Ke = D1 / P0 + g
• Ke = 1.45 / P0 + 0.0997 … what should we use for P0?
Michael Dimond
School of Business Administration
Ke with flotation cost
• Their stock currently sells for $48.22, but the new shares will
be underpriced by $1.00 and have $2.86 per share in
flotation costs.
• Net Proceeds will be 48.22 – 1.00 – 2.86 = 44.36
• What is their Cost of Equity for the SEO?
• Ke = D1 / P0 + g
• Ke = 1.45 / 44.36 + 0.0997 = 0.1324 = 13.24%
• What if they used Retained Earnings instead of issuing new
stock?
• Ke = 1.45 / 48.22 + 0.0997 = 0.1298 = 12.98%
• Why is the cost of equity higher for new stock than for
retained earnings?
• What would happen if a company always issued new stock
instead of funding growth from retained earnings?
Michael Dimond
School of Business Administration