Topic 6: Discounted cash flow
applications to security
valuation
Purpose: This lecture covers the basics
of the DCF approach to
security valuation
-1-
Learning Objectives Satisfied:
5. Bond Valuation
Objectives: Understand the following concepts
3How to read and understand market quotations from
Wall Street Journal
3Basic process used to value bonds, find their yield to
maturity, yield to call
3Important relationships that exist in bond valuation and
implications for investors
-2-
Learning Objectives Satisfied:
6. Stock Valuation
Objectives: Understand the following concepts:
3Basic characteristics of stocks
3How to evaluate preferred and common stock
3How to calculate expected and required rate of return
for stocks
3Assumptions behind models and their limitations
3How to read and interpret stock market quotations.
-3-
The DCF approach in general form
• Given an efficient market, NPV is zero
for a securities transaction
• Therefore, today’s price equals PV of all
future cash flows
n
Price = C0 + C1 /(1+R) + C2 /(1+R)2 + … + Cn /(1+R)
-4-
The DCF approach to coupon bonds:
• Computing price, with
a known required rate
of return:
• Computing yield-tomaturity
– equals the rate
implied by the
market price
– search by trial-anderror
P0 =
Face Value n Coupon Pmt
+Â
(1+ R )n
(1 + R)i
i=1
Market Price =
n Coupon Pmt
Face Value
+ Â
n
i
(1+ R)
i = 1 (1 + R)
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Example 1: Computing Price
• Face Value is $1,000
• Coupon rate is 7%
• Market rate is 8%
(semi-annual)
• Maturity is 20 years
•
•
•
•
•
•
•
Then FV is 1000
PMT is 35
Interest is 8
P/YR is 2
N is 40
Compute PV
= $901.04
• Negative sign in display
reflects sign convention
-6-
Example 2: Computing Yield
• Face Value is $1,000
• Coupon rate is 7%
• Maturity is 20 years
(semi-annual)
• Price is $815.98
•
•
•
•
•
•
•
Then FV is 1000
PMT is 35
N is 40
P/YR is 2
PV is -815.98
Compute interest
= 9.00%
-7-
Table Illustrating Coupon Bias and Convexity
20-year, 10% bonds
10-year, 10% bonds
20-year, 5% bonds
10-year, 5% bonds
old
rate
new
rate
old price
new price
capital gain
(loss)
12%
15%
12%
9%
8%
8%
relative
change
$849.54
$685.14
($164.40)
-19.35%
$849.54
$1,092.01
$242.47
+28.54%
10%
$1,197.93
$1,000.00
($197.93)
-16.52%
6%
$1,197.93
$1,462.30
$264.37
+22.07%
17%
18%
$603.99
$569.71
($34.28)
-5.68%
17%
16%
$603.99
$642.26
$38.27
+6.34%
12%
15%
$885.30
$745.14
($140.16)
-15.83%
12%
9%
$885.30
$1,065.04
$179.74
+20.30%
8%
10%
$1,135.90
$1,000.00
($135.90)
-11.96%
8%
6%
$1,135.90
$1,297.55
$161.65
+14.23%
17%
18%
$668.78
$634.86
($33.92)
-5.07%
17%
16%
$668.78
$705.46
$36.68
+5.48%
12%
15%
$473.38
$370.28
($103.10)
-21.78%
12%
9%
$473.38
$631.97
$158.59
+33.50%
8%
10%
$703.11
$571.02
($132.09)
-18.79%
8%
6%
$703.11
$884.43
$181.32
+25.79%
17%
18%
$321.13
$300.77
($20.36)
-6.34%
17%
16%
$321.13
$344.15
$23.02
+7.17%
12%
15%
$598.55
$490.28
($108.27)
-18.09%
12%
9%
$598.55
$739.84
$141.29
+23.61%
8%
10%
$796.15
$688.44
($107.71)
-13.53%
8%
6%
$796.15
$925.61
$129.46
+16.26%
17%
18%
$432.20
$406.64
($25.56)
-5.91%
17%
16%
$432.20
$460.00
$27.80
+6.43%
-8-
Convexity
Illustration of Convexity
$1,200.00
$1,000.00
Price
$800.00
$600.00
$400.00
20-year
$200.00
10-year
5-year
1-year
$0.00
1
3
5
7
9
11
13
15
Rate
17
19
21
23
25
27
29
(%)
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Coupon Bias
Illustration of Coupon Bias
$1,200.00
10% Coupon
5% Coupon
$1,000.00
Price
$800.00
$600.00
$400.00
$200.00
$0.00
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Rate (%)
- 10 -
Risk factors for bondholders:
•
•
•
•
Purchasing power risk
Interest rate risk
Reinvestment risk
Default risk
- 11 -
The yield curve:
• R = r + inflation adjustment + risk adjustment.
• Inflation adjustment:
R = r + i + ri
r = (R–i)/(1+i)
• Two theories to explain the yield curve
– Liquidity Premium Theory
– Pure Expectations Theory (PET)
• also known as the Rational Expectations Theory
• easily remembered as the “Pet Rat”
- 12 -
Let's see how different theories
explain what we observe:
•
R
Upward sloping yield curve
Maturity
•
R
Flat yield curve
Maturity
•
Downward sloping yield curve
R
Maturity
- 13 -
Convexity
Illustration of Convexity
$1,200.00
$1,000.00
Price
$800.00
$600.00
$400.00
20-year
$200.00
10-year
5-year
1-year
$0.00
1
3
5
7
9
11
13
15
Rate
17
19
21
23
25
27
29
(%)
- 14 -
The DCF approach to preferred stock
• Computing price, with a
known required rate of
return:
• Computing yield, which is
the required rate of return
implied by the market
price:
P0 =
Dividend
R
R=
Dividend
P0
- 15 -
Example 3: Computing Price
• Par Value is $100
• Dividend rate is 7%
• Market rate is 8%
• Then Price = $7/.08
• = $87.50
- 16 -
Example 4: Computing Yield
• Par Value is $100
• Dividend rate is 7%
• Price is $73.68
• Then Yield = 7/73.68
• = 9.50%
- 17 -
Common Equity
Discounted Cash Flow Approach to
Measuring the Firm Foundation
- 18 -
The Debate Over What to Value
•
•
•
•
Earnings
Dividends
Cash Flow
Something More?
– Woolridge (1995) shows that over half the value of a
company’s stock is based on something more than a
simple multiple of earnings
- 19 -
What are the Value Drivers?
• Market value of
physical assets
– Consider change in net worth
when new assets and liabilities
are included in the balance sheet
• When would impact on net
worth be neutral? …
Negative? … Positive?
– You may be able to stop here
if neutral or positive
• Added earning power
derived from new assets
• Option approaches continue
from here
– Value of new opportunities
– Enhanced value of human capital
• Stronger organizational capital
via enhanced flexibility
• New incentives offered to key
decision makers
– Enhanced technology
– Enhanced competitive advantage
– DCF methods focus on these
earnings
– You may be able to stop here, too
- 20 -
The Debate Over How to Forecast
• Multiple of Current Earnings?
• Multiple of Current Cash Flow?
– Considers all that could be taken from the company
• What Should Be the Multiplier?
– Choosing the “comparables” is the part of valuation that is art,
not science
• More Complex Forecast of Future Dividends?
– Constant growth
– Super-normal growth
- 21 -
Discounting the Forecast Cash Flow
• Established family business for sale to employees
• Employees can borrow with terms of eight years
and 15%
– Cash flow stable at $10 mm per year
• Value =
10mm/1.15 + 10mm/1.152 + 10mm/1.153 + 10mm /1.154 +10mm/1.155
+10mm/1.156 + 10mm /1.157 + 10mm /1.158
$44.9 million
Or about 4.5 times cash flow
=
- 22 -
Discounting the Forecast Cash Flow
• Venture capital example: Art Grunnion Boatbuilder
• Suppose forecast cash flows are
–
–
–
–
–
–
-$1 mm now
-$1 mm the first year
-$1 mm the second year
-$1 mm the third year
-$1 mm the fourth year
$10 mm to sell the company in year five
• Find internal rate of return
= 24.07%
- 23 -
Discounting the Forecast Cash Flow
• Another venture capital example
• Suppose forecast cash flows are
–
–
–
–
–
–
-$5 mm now
-$10 mm the first year
-$20 mm the second year
-$50 mm the third year
-$100 mm the fourth year
$1 billion to sell the company in year five
• Opportunity cost of capital 15%
• Value =
-5mm -10mm/1.15 - 20mm/1.152 - 50mm/1.153 - 100mm/1.154 + 1000mm/1.155
Value = $378.3 million
IRR = 109%
- 24 -
Dividend Valuation Model
•
• The general form:
P0 = Â
i=1
• The Gordon “constant
dividend growth” model:
• Which reduces by means of
math wizardry to a simple
form
DIVi
(1 + Ri )i
DIV 0 (1 + g)i
(1 + R)i
i=1
•
P0 = Â
P0 =
DIV1
DIV 0 (1 + g)
=
R- g
R- g
- 25 -
This model can be rearranged
• to find the required
rate of return
implied by the
market price, as
follows:
R =
=
DIV1
+g
Market Price
DIV0 (1 + g)
+g
Market Price
• That is, R = dividend
yield + growth rate
- 26 -
Example 5: Computing Price
• Current dividend is
$2
• Growth rate is 5%
• Required return is
12%
• Then Price =
(2*1.05)/(.12-.05)
• = $30.00
- 27 -
The risk factors of common stock
• Uncertainty about predicting future cash flows
from ongoing operations
• Uncertainty about predicting competitors'
future actions, and their results
• Uncertainty about predicting the future
economic, political, and technological
environments
• Uncertainty about predicting the firm's future
growth opportunities, which depend in large
part on the future environment
- 28 -
Exercise in speculation:
• A company will pay dividends of $1 per share for the
coming year, and the stock is selling for $25 per share.
– You require a 20% rate of return on stock in small
companies like this
– Calculate the growth rate that would be required in
order to make this stock look attractive (plug the
numbers into the formula)
.20 = (1/25) + g
g = .16
- 29 -
Exercise in speculation:
• So, you would have to be confident that the
company could sustained dividend growth of
16% annually into the foreseeable future.
• What stories would you want to be able to tell
about this company in order to make you
reasonably comfortable with buying the stock
at its current price?
- 30 -
Exercise in speculation:
• The company would have to be well-positioned
in a growing market, with strong competitive
advantages, in order be attractive at this price
- 31 -
Another example, The Case of the Crazy P/E Ratio:
•
•
•
•
•
Mousetek corporation owns one asset, a Lear Jet valued at $2.5
million.
There are 20,000 shares of stock outstanding, and no other claims
against assets. Shares are selling at $100 each.
The company operates the jet for charter, and this year earned
only $2,000 after tax. Thus EPS this year was 10¢, making the P/E
ratio astronomical at 1,000 to 1.
All of the earnings were paid out in dividends. The accountant
used the normal growth model and found that the current stock
price reflects growth expectations of 19.88% per year in
perpetuity, assuming a cost of capital of 20%.
Question: Is this a super growth company, or is the market price
of the stock crazy? For that matter, is the accountant crazy?
- 32 -
Shortcomings of the DCF approach
for valuing equity:
•
•
•
•
Depends upon accurate estimates of future cash flows
Fails to consider liquidation value
Fails to consider the value of control
Doesn't adequately deal with growth opportunities
- 33 -
The Efficent Markets approach:
•
•
Best prediction of the price
tomorrow is the price today,
adjusted for drift.
Pˆ t+1 Ft = Pt ¥ 1 + Rˆ t
(
)
We can estimate drift using
the Capital Asset Pricing
Model (CAPM)
– expected reward is
proportional to riskbearing
Rasset j = RSafe + Relative Risk Index asset j ( Raverage - RSafe )
- 34 -
The CAPM
• This can be stated
more compactly:
• CAPM tells its story
better in another
form:
(
R j = Rf + b j Rm - Rf
(
R j - Rf = b j Rm - Rf
)
)
Risk Premium j = b j ¥ Risk Premium Average
- 35 -
Example 6: Required Return
• TCS stock has half
the average risk
• Average risk
investment returns
12%
• T-Bills return 5%
• Then Required
Return = 5% +
.5(12% -5%)
• = 5% + 3.5%
• = 8.5%
- 36 -
Example 6: Price Forecast
•
•
•
•
Suppose TCS stock price is $100 today
TCS pays no dividends
Required ROR is 8.5%
What is the best forecast of the stock
price a year from now?
- 37 -
Example 6: Required Return
• ACU stock has twice
the average risk
• Average risk
investment returns
12%
• T-Bills return 5%
• Then Required
Return = 5% +
2(12% -5%)
• = 5% + 14%
• =19%
- 38 -
Example 6: Price Forecast
•
•
•
•
Suppose ACU stock price is $100 today
ACU pays no dividends
Required ROR is 19%
What is the best forecast of the stock
price a year from now?
- 39 -