Topic 3:
Discounted cash flow applications to
security valuation
Purpose:
This lecture covers the basics of
the DCF approach to security
valuation
-1-
Time and Money are Inseparable
-2-
Money Grows with Compound
Interest
-3-
… And Shrinks with Inflation
We have to
find out what
happens after
the
combined
effects of
both interest
and inflation- 4 -
Discounted Cash Flow (DCF)
Money in
hand today is
worth more
than money
you have to
wait for
-5-
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)
-6-
The DCF approach to coupon bonds:
•  Computing price,
with a known
required rate of
return:
• Computing yield
-to-maturity
P0 =
Face Value n Coupon Pmt
+∑
(1+ R )n
(1 + R)i
i=1
Market Price =
 equals the rate
implied by the
market price
 search by trial
-and-error for
unknown R
n Coupon Pmt
Face Value
+ ∑
n
i
(1+ R)
i = 1 (1 + R)
-7-
Example 1: Computing Price
•  Face Value is $1,000
•  Coupon rate is 7%
•  Then FV is 1000
•  PMT is 35
(semi-annual payments)
•  Market rate is 8%
(semi-annual compounding)
•  Maturity is 20 years
(7% of $1000, divided by 2)
•
•
•
•
•
Interest is 8
P/YR is 2
N is 40
Compute PV
= $901.04
•  Negative sign in display
reflects sign convention
-8-
Example 2: Computing Yield
•  Face Value is $1,000
•  Coupon rate is 7%
•
•
•
•
•
(semi-annual payments)
•  Maturity is 20 years
•  Price is $815.98
Then FV is 1000
PMT is 35
P/YR is 2
N is 40
PV is 815.98
–  Make it negative to reflect
sign convention
•  Compute interest
= 9.00%
-9-
Table Illustrating Coupon Bias and Convexity
old
rate
new
rate
old price
new price
capital gain
(loss)
relative
change
20-year, 10% bonds
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$849.54
$849.54
$1,197.93
$1,197.93
$603.99
$603.99
$685.14
$1,092.01
$1,000.00
$1,462.30
$569.71
$642.26
($164.40)
$242.47
($197.93)
$264.37
($34.28)
$38.27
-19.35%
+28.54%
-16.52%
+22.07%
-5.68%
+6.34%
10-year, 10% bonds
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$885.30
$885.30
$1,135.90
$1,135.90
$668.78
$668.78
$745.14
$1,065.04
$1,000.00
$1,297.55
$634.86
$705.46
($140.16)
$179.74
($135.90)
$161.65
($33.92)
$36.68
-15.83%
+20.30%
-11.96%
+14.23%
-5.07%
+5.48%
20-year, 5% bonds
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$473.38
$473.38
$703.11
$703.11
$321.13
$321.13
$370.28
$631.97
$571.02
$884.43
$300.77
$344.15
($103.10)
$158.59
($132.09)
$181.32
($20.36)
$23.02
-21.78%
+33.50%
-18.79%
+25.79%
-6.34%
+7.17%
10-year, 5% bonds
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$598.55
$598.55
$796.15
$796.15
$432.20
$432.20
$490.28
$739.84
$688.44
$925.61
$406.64
$460.00
($108.27)
$141.29
($107.71)
$129.46
($25.56)
$27.80
-18.09%
+23.61%
-13.53%
+16.26%
-5.91%
+6.43%
- 10 -
Picture of Convexity
Illustration
Illustration
of
Convexity
Illustrationof
ofConvexity
Convexity
$1,200.00
$1,200.00
$1,200.00
$1,000.00
$1,000.00
$1,000.00
Price
Price
Price
$800.00
$800.00
$800.00
$600.00
$600.00
$600.00
$400.00
$400.00
$400.00
$200.00
$200.00
$200.00
1-year
5-year10-year
1-year
30-year20-year20-year
10-year
10-year
5-year
5-year 5-year
1-year1-year 1-year
$0.00
$0.00
$0.00
111
333
555
777
999 11
11
13
15
17
19
21
23
25
27
29
11 13
13 15
15 17
17 19
19 21
21 23
23 25
25 27
27 29
29
Rate
Rate
(%)
Rate(%)
(%)
- 11 -
Picture of 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 (%)
- 12 -
McCauley’s Duration
0
1
2
3
$300
$110
$121
$133.10
$100
$100
$100
Discounted @ 10%
Duration is the weighted average maturity
( 100
300 )
Duration = 1
+ 2
( 100
300)
+ 3
( 100
300)
Duration = 2
- 13 -
Risk factors for bondholders:
•
•
•
•
Purchasing power risk
Interest rate risk
Reinvestment risk
Default risk
- 14 -
Money Shrinks with Inflation
Let’s find out
what
happens after
the
combined
effects of
both interest
and inflation
- 15 -
Inflation and the price level:
Let i = the rate of inflation
Future Price Level = Present Price Level * (1 + i)
n
Illustration:
•
•
•
•
Let i = 10% and present price level = 100
1 year in the future, price level will be 110
2 years in the future, price level will be 121
3 years in the future, price level will be 133.10
- 16 -
Illustration of Fisher Effect
(annual compounding)
Today:
•  Invest $100
•  Desired r is 3%
One year from now:
•  Collect $107.12
•  Spend $3 * 1.04 = $3.12
–  real profit of $3
•  Expected inflation is 4% •  Reinvest $104
–  Keeps real principal
Desired nominal return is
intact
7.12%
R = i + r (1+i)
R = i + r + ri
R = r + i + ri
- 17 -
Fisher Effect
R = r + i + ri
Illustrations:
•  r = 3%, i = 4%, R = 7.12%
•  r = 4%, i = 5%, R = 9.20%
•  r = 5%, i = 6%, R = 11.30%
- 18 -
Illustration of Real Return from Investing
(annual compounding)
Year zero: Invest $100
•  R is 15%
•  i is 12%
One year later:
•  Collect $115
•  Reinvest $112
–  Keeps real principal intact
•  Spend $3
•  Expected real profit is
$3/1.12 = $2.68
Expected real return is 2.68%
r = (R – i)/(1 + i)
- 19 -
Fisher Effect
r = (R - i) / (1+i)
Illustrations:
•  R = 15%, i = 12%, r = 2.68%
•  R = 13%, i = 10%, r = 2.73%
•  R = 10%, i = 7%, r = 2.80%
- 20 -
Practice Set 2, Problem 1
U.S.A.
R = 3%
i = 2%
r = (3% – 2%) / 1.02
= 0.98%
Germany
R = 4%
i = 1%
r = (4% – 1%) / 1.01
= 2.97%
Real rate is higher in Germany, so money would flow from
U.S. to Germany
- 21 -
Practice Set 2, Problem 2
UK
R = 4%
i = 3%
r = (4% – 3%) / 1.03
= 0.97%
Japan
R = 5%
i = 1%
r = (5% – 1%) / 1.01
= 3.96%
Real rate is higher in Japan, so money would flow from
UK to Japan
- 22 -
Practice Set 2, Problem 3
U.S.A
r = 3%
i = 4%
R = r + i +ri
= 0.03 + 0.04 + (0.03 * 0.04)
= 7.12%
Real rate is higher in Japan, so money would flow from
UK to Japan
- 23 -
What about tax?
There is less
purchasing
power after
inflation
…and even
less after you
have paid
income tax
on the
interest you
earn
- 24 -
After-Tax Real Rate
•  If marginal tax rate is 25%
For every $100 extra you earn, you keep $75
For every $100 of deductible spending, you pay $75
After-Tax = Pre-Tax (1 - t)
•  After-tax real rate
r=
R (1 – t) – i
1+i
- 25 -
Illustration of Real Return after tax
(Example 3)
Year zero: Invest $100
•  R is 9%
•  i is 4%
•  Marginal tax rate is
25%
One year later:
•  Collect $109
•  Pay tax of $2.25
–  Tax is 25% of $9
–  $106.75 left over
•  Reinvest $104
–  Keeps real principal intact
•  Spend $2.75
•  Expected real profit is
$2.75/1.04 = $2.64
R (1 – t) – i
Expected real return after tax is
r=
1+i
2.64%
- 26 -
Once upon a time when we had
high inflation, high tax
Let R = 20%, i = 12%, t = 70%
r=
r=
R (1 – t) – i
1+i
20% (1 – .7) – 12%
1.12
r = – 5.36%
- 27 -
Recent Numbers (Problem 27)
Let R = 4%, i = 3%, t = 35%
r=
R (1 – t) – i
1+i
r=
4% (1 – .35) – 3%
1.03
r = – 0.39%
- 28 -
Problem 28
Let R = 6.9%, i = 5%, t = 35%
r=
r=
R (1 – t) – i
1+i
6.9% (1 – .35) – 5%
1.05
r = – 0.49%
- 29 -
Illustration of Fisher Effect
(monthly compounding)
Month zero:
•  Invest $10,000
•  Nominal APR = 15%
–  R/m is 1.25%
One month later:
•  Collect $10,125
•  Reinvest $10,100
•  Annual inflation = 12%
–  i/m is 1%
–  Keeps real principal
intact
•  Expected real profit is •  Spend $25
$25/1.01 = $24.75
Expected real return is .2475% per month (or
approx 2.97% annually, with P/YR = 12)
r/m = (R/m – i/m)/(1 + i/m)
- 30 -
More About Coping with Inflation
- 31 -
Purchasing Power:
•  If you stuffed a $100 bill into your mattress and
left it there during a year of 10% inflation, its
purchasing power would shrink
Purchasing Power = $100 *
1
1.10
Purchasing Power = $90.91
- 32 -
Purchasing Power:
•  After two years, the money in the
mattress would shrink more:
Purchasing Power = $100 *
Purchasing Power = $100 *
1
(1.10)
2
1
1.21
Purchasing Power = $82.64
- 33 -
Purchasing Power:
•  After three years, the money in the
mattress would shrink still more:
Purchasing Power = $100 *
Purchasing Power = $100 *
1
(1.10)
3
1
1.331
Purchasing Power = $ 75.13
- 34 -
Future Purchasing Power
•  To compute the amount of purchasing power you will
have in the future as a result of saving today
–  first calculate how many dollars you will have
–  then adjust for inflation, as follows:
Future Purchasing Power =
Future Purchasing Power =
Future Amount
(1 + i)
PV (1 +R)
(1 + i)
n
n
n
= PV
(1 +R)
n
(1 + i)
n
- 35 -
Illustration
•
•
•
•
•
Invest $1000
Earn 15%
Inflation is 12%
Wait 5 years
How much purchasing power will you have?
  P/YR is 1, N is 5, PV is -1000, I/YR is 15, PMT is 0
  Calculate FV = 2011.36
•  Then deflate:
  I/YR is 12
  Everything else stays the same
  Calculate PV = 1141.30
•  Real return is 2.68% compounded annually
- 36 -
How to incorporate inflation into a series:
Illustration (cash flows adjusted for inflation):
PV = 0
+ 100
(1.03)
1
+ 100
(1.03)
2
+ 100
(1.03)
3
= 282.86
Illustration (nominal cash flows):
PV = 0
+ 110
(1.133)
1
+ 121
(1.133)
2
+ 133.103 = 282.86
(1.133)
r = 3%, i = 10%, therefore R = 13.3%
- 37 -
Basic techniques for inflation
adjustment for investment analysis
•  Make cash flow estimates in terms of
tomorrow’s dollars and evaluate using the
nominal discount rate
•  Or, make cash flow estimates in terms of
today’s dollars and evaluate using the real
discount rate
- 38 -
Example 4: Inflation Adjustment
Nominal cash flows
•  C(0) = -$250
•  C(1) = $110
•  C(2) = $121
•  C(3) = $133.10
•  Nominal Required
Return = 13.3%
•  NPV = $32.86
Inflation =10%
Real cash flows
•  C(0) = -$250
•  C(1) = $100
•  C(2) = $100
•  C(3) = $100
•  Real Required Return =
3%
•  NPV = $32.86
- 39 -
Deflation
•  Deflation is just like inflation, but with a negative i
•  If you stuffed a $100 bill into your mattress and left it
there during a year of 10% deflation, its purchasing
power would grow
Purchasing Power = $100 *
Purchasing Power = $100 *
1
1 - .10
1
.90
Purchasing Power = $111.11
- 40 -
Deflation
•  After two years, the money in the mattress would buy
even more than before:
1
Purchasing Power = $100 *
2
(1 -.10)
Purchasing Power = $100 *
1
.81
Purchasing Power = $ 123.46
- 41 -
Fisher Effect with Deflation
r = (R – i) / (1 + i)
In this case, i is negative, so be careful with the signs
r = (R + d) / (1 – d)
Illustrations:
•  R = 6%, i = –4%, r = 10% / .96 = 10.42%
•  R = 8%, i = –5%, r = 13% / .95 = 13.68%
•  R = 8%, i = –10%, r = 18% / .9 = 20.00%
- 42 -
Understanding the Yield Curve
- 43 -
Practice Set 2, Problem 4
Two-year bond
R = 6.5%
P/YR = 1
N=2
Let’s say PV = $10,000
Then FV = $11,342.25
Rollover Strategy
Start with $10,000
1st year add 6%
$10,600
2nd year add 7.5%
$11,395
2-year average return is
6.75%
Expected return is higher with rollover strategy
What risks are involved?
- 44 -
Practice Set 2, Problem 5
Two-year bond
R = 7%
P/YR = 1
N=2
Let’s say PV = $10,000
Then FV = $11,449.00
Rollover Strategy
Start with $10,000
1st year add 6%
$10,600
2nd year add 7.5%
$11,395
2-year average return is
6.75%
Expected return is higher with two-year bond
What pressures would result?
Given the expectations, equilibrium 2-year rate would be
6.75% (Problem 6)
- 45 -
Practice Set 2, Problem 7
Start with $100
1st year add 5%
$105
2nd year add 6%
$111.30
3rd year add 7%
$119.09
Average return:
PV is –100
FV is 119.09
P/YR is 1
N is 3
Calculate interest
Result is 6.00 %
- 46 -
Practice Set 2, Problem 8
Start with $100
1st year add 6%
$106
2nd year add 6.5%
$112.89
3rd year add 7%
$120.79
4th year add 8%
$130.46
Average return:
PV is –100
FV is 130.46
P/YR is 1
N is 4
Calculate interest
Result is 6.87 %
- 47 -
The yield curve:
•  R = r + inflation adjustment + risk
adjustment
•  Three 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”
–  Preferred Habitat Theory
- 48 -
Let's see how different theories explain what
we observe:
•  Upward sloping yield curve
R
Maturity
R
•  Flat yield curve
Maturity
R
•  Downward sloping yield
curve
Maturity
- 49 -
Picture of 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
17
19
21
23
25
27
29
Rate (%)
- 50 -
Equity
- 51 -
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 =
R=
Dividend
R
Dividend
P0
- 52 -
Example 5: Computing Price
•  Par Value is $100
•  Dividend rate is 7%
•  Market rate is 8%
•  Then Price = $7/.08
•  = $87.50
- 53 -
Example 6: Computing Yield
•  Par Value is $100
•  Dividend rate is 7%
•  Market Price is
$73.68
•  Then Yield = 7/73.68
= 9.50%
- 54 -
Common Equity
Discounted Cash Flow Approach to
Measuring the Firm Foundation
- 55 -
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
- 56 -
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
- 57 -
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
- 58 -
Example 7: 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
=
- 59 -
Example 8: 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%
- 60 -
Example 9: 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%
- 61 -
The dividend approach to
evaluating common stock:
∞
•  The general form:
•  The Gordon “constant
dividend growth”
model:
•  Which reduces by
means of math
wizardry to a simple
form
P0 = ∑
i=1
DIVi
(1 + Ri )i
DIV 0 (1 + g)i
(1 + R)i
i=1
∞
P0 = ∑
P0 =
DIV1
DIV 0 (1 + g)
=
R− g
R− g
- 62 -
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
- 63 -
Example 10: Computing Price
•  Current dividend is
$2
•  Growth rate is 5%
•  Required return is
12%
•  Then Price
= (2*1.05)/(.12-.05)
= $30.00
- 64 -
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
- 65 -
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
- 66 -
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?
•  The company would have to be well-positioned
in a growing market, with strong competitive
advantages, in order be attractive at this price
- 67 -
Another illustration, 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?
- 68 -
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
- 69 -
The Efficent Markets approach:
•  Best prediction of the price
tomorrow is the price today,
adjusted for drift
Pˆ t+1 Φ t = Pt × 1 + Rˆ t
(
)
•  We can estimate drift using
the Capital Asset Pricing
Model (CAPM)
–  expected reward is proportional
to risk-bearing
Rasset j = RSafe + Relative Risk Index
asset j
(Raverage − RSafe )
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The CAPM
•  This can be stated
more compactly:
•  CAPM tells its story
better in another
form:
(
R j = Rf + β j Rm − Rf
(
R j − Rf = β j Rm − Rf
)
)
Risk Premium j = β j × Risk Premium
Average
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Example 11: Required Return
•  TCS stock has half
the average risk
•  Average risk
investment returns
12%
•  T-Bills return 3%
•  Then Required Return
= 3% + .5(12% - 3%)
= 3% + 4.5%
= 7.5%
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Example 11: Price Forecast
•
•
•
•
Suppose TCS stock price is $100 today
TCS pays no dividends
Required ROR is 7.5%
What is the best forecast of the stock
price a year from now?
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Example 12: Required Return
•  ACU stock has twice
the average risk
•  Average risk
investment returns
12%
•  T-Bills return 3%
•  Then Required Return
= 3% + 2(12% - 3%)
= 3% + 18%
= 21%
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Example 12: Price Forecast
•
•
•
•
Suppose ACU stock price is $100 today
ACU pays no dividends
Required ROR is 21%
What is the best forecast of the stock
price a year from now?
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