Week 02
Equity Valuation
See Berk (2018) Chapters 7 and 10 (Relevant Parts Only)
FINS2615 Intermediate Business Finance
Tim Berger & Dr. Ian Kwan
tim.berger@unsw.edu.au
Introduction Co-Lecturer
Background
►
Founder of KLIND [klaɪnd], a contemporary
strategy studio solely focused on Customer
Experience and Customer Strategy
►
Bayer (Life Science, Pharmaceutical) - Adviser to Bayer on the sale of its
Environmental Science business for $2.6 billion to international private
equity firm Cinven.
►
Former Associate Director in the Strategy &
Transactions practice focusing on large
integration and transformations.
►
►
Experienced in the design and execution of
integration and reorganisation programs.
KKR (Private Equity) – Adviser to KKR on the acquisition of a majority
interest in Colonial First State (CFS) from CBA. Provided Day-1 readiness
and post Day-1 transition support to stand-up CFS as a standalone
business.
►
Lendlease (Construction, Infrastructure) – Adviser to Lendlease on the
sale of Lendlease Engineering to Spanish Group Acciona.
►
Zurich (Insurance) – Lead PMO for Project Swift at Zurich. Setup and
managed a taskforce to prepare the completion net asset statement and
capital reconciliation required under the closing conditions of the
acquisition.
►
Daimler (Automotive) – Lead PMO for Project Future at Daimler. Project
Future has been the biggest reorganisation project in Daimler’s history
involving more than 1,000 team members worldwide and more than 20
external advisory firms. The project affected more than 800 companies
in 65 countries, requiring the transfer of more than 150,000 employees
in Germany and at the same time streamlining operations to reflect new
divisional structure.
Tim Berger
Contact
2
Selected Engagements
Qualifications
►
tim.berger@unsw.edu.au
►
LL.M. in Mergers & Acquisitions
►
linkedin.com/in/bergertim1/
►
MBA
►
klind.cx
►
B.Eng. in Industrial Engineering
►
Lived in GER, UK, US, KSA, UAE and AU, and
worked with C-suite executives for over 10
years
The Story of Corporate Finance
Financial Mathematics
Debt
Cash
Flows
$
𝑁𝑃𝑉 = &
!"#
𝐶𝐹!
− 𝐶𝐹%
1+𝑟 !
Discount
rate
Types
of CF
Week 01
Equity
Week 02
Working capital
management
Week 05
Cash Flow Forecasting
Weeks 03 and 04
Capital Asset Pricing Model
Week 07
Cost of capital
Week 08
(Risk & Return)
Capital structure
3
Week 09
Important: As you take this course, clarify how to connect the different parts of the story!
Week 10:
Payout policy
Contents of lecture
1. Types of Equity
2. Equity valuation models (review)
3. Equity valuation and Excel Functions
4. Estimating dividends in DDM
5. Share repurchases
6. Team Assignment – A deep dive
4
Learning Outcomes – This lesson
To review types of equity and equity models
1. Intrinsic valuation models
To introduce EXCEL FUNCTIONS that aid visualizing equity valuation
• NPV; XNPV; STOCKHISTORY; PV Growing Annuity
This Photo by Unknown Author is
licensed under CC BY-NC-ND
5
To determine the size of equity cash flows
• Estimating the dividends
• Role of Share repurchases
1. Types of equity
Ordinary shares / equity
An ordinary share is equity ownership of the residual cash flows of a business, i.e. remaining cash flows after paying
all operations, financing, investment costs: employees, suppliers, debt interest and principal, taxes, etc.
Ordinary equity / Common shareholders rights include:
• Voting rights – rights to elect members on the Board of Directors; vote on significant matters;
• Profit rights – when the Board declares dividend, shareholders have proportional rights to receive it
• Right to residual value – After bankruptcy, shareholders have rights to residual assets after creditors paid.
• Right to limited liability – If firm goes bankrupt, common shareholders have limited liability.
Dividends
• Dividends are payment to shareholders of the profits of a firm;
• The Board of Directors has no obligation to declare a dividend -- important advantage of corporations
• A firm can go bankrupt for not repaying its debts, but it cannot go bankrupt for not paying dividends.
• Dividends are not liabilities of the firm until declared; once declared, they must be paid.
• Many firms do not pay regular dividends, e.g. start ups
7
Preference shares / equity
Preference (or Preferred) shares:
• Are similar to common shares in that they have equity-like qualities – ownership of profits
• Are similar to debt in that the have debt-like qualities – holders receive regular dividend payments
• Preferred dividends are obligatory, but holder cannot sue the firm if it is not paid, i.e. not a liability.
• If preferred dividend not paid now, its payment may be deferred and interest may be charged.
• But are different from common shares in terms of voting rights: preference shares have NO voting rights
• If firm goes bankrupt
• Debtholders get paid first
• Then preferred shareholders are paid (i.e. debt is senior)
• Then common shareholders are paid (i.e. preferred shares are senior).
Conclusion: From highest to lowest risk:
1. Common shares,
2. Preferred shares, then
3. Debt.
8
Shareholders / Equity holders
are paid by dividends and share repurchases
Discussed in Week 3
Dividends +
Repurchases
Operating
assets
Net cash flow
Reinvested cash flow
(Capex & NWC investments)
9
Shareholders
FCF
Interest
+
Principal
Debtholders
At what level are dividends in the CorpFinStory?
Cash
Flows
$
𝑁𝑃𝑉 = &
!"#
Security level CF
CF that affect a big part of the company
E.g. introducing new product that
changes the company strategy
𝐶𝐹!
− 𝐶𝐹%
1+𝑟 !
Discount
rate
(Risk & Return)
10
Project level CF
CF that affect only a small part of the
company E.g. replacing production
machines in a manufacturing plant
Project level
discount rate
Rate depends on the risk of the
project compared with the average
risk of the company
Portfolio level
discount rate
Rate depends on the risk of the company
compared with other companies that
have the same types of risks,
i.e. portfolio view
Important: As you take this course, be aware of which level of the story you are referring to!
2. Equity Valuation Methods
Valuation is the application of standard formula
Single Cash Flows
Present Value: 𝐶@ = 𝐶A 1 + 𝑟
Perpetuities
𝑃𝑉(𝑂𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝑃𝑉(𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
BA
Future Value: 𝐶A = 𝐶@ 1 + 𝑟
A
Multiple Cash Flows
𝑃𝑀𝑇
𝑟
Annuities
𝑃𝑀𝑇
𝑃𝑉(𝑂𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝐴𝑛𝑛𝑢𝑖𝑡𝑦) =
1− 1+𝑟
𝑟
𝑃𝑀𝑇
𝑟−𝑔
𝑃𝑀𝑇
1+𝑔
𝑃𝑉(𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝐴𝑛𝑛𝑢𝑖𝑡𝑦) =
1−
𝑟−𝑔
1+𝑟
&$
$
Converting Rates
𝐸𝐴𝑅
1+
1
12
#
𝐴𝑃𝑅
= 1+
𝑚
'
m = frequency of compounding per year
APR = Annual Percent Rate (quoted rate compounded m/year)
EAR = Effective Annual Rate (rate with m=1 compound/year)
Stock Valuation Methods
Three basic methods of valuing stock (equity market shares)
1. Intrinsic valuation methods – present value of future expected cash flows
Methods we know from financial mathematics.
2. Comparables methods – comparing similar companies
Uses Equity Multiples and Enterprise Value Multiples
3. Options methods – a stock is valued as a put option on debt
Sophisticated option pricing algorithms… beyond this course.
13
Revise
Intrinsic Valuation
OUR FOCUS
Constant perpetual
DDM
Intrinsic
valuation
models
Dividend discount
models (DDM)
Growing perpetual
DDM
Assume the stock pays regular
dividends. A good model for
large, stable firms in stable
industries. E.g., utilities,
supermarkets, etc.
2-stage DDM
Discounted Cash Flow
(DCF) Models, or
Free Cash Flow Models
These models are useful for stocks that don’t pay
dividends and the DDM is not useful. It is a more
flexible model because it includes the DDM.
14
Multi-stage DDM
The Idea of DDM
For 1 holding period:
𝐷# + 𝑃#
𝑃% =
1 + 𝑟(
For 2 holding periods:
𝐷#
𝐷) + 𝑃)
𝑃% =
+
1 + 𝑟(
1 + 𝑟( )
For 3 holding periods:
𝐷#
𝐷)
𝑃% =
+
1 + 𝑟(
1 + 𝑟(
… For n holding periods:
𝐷#
𝐷)
𝑃% =
+
1 + 𝑟(
1 + 𝑟(
𝐷* + 𝑃*
+
)
1 + 𝑟( *
𝐷$&#
𝐷$
+
⋯
+
+
)
1 + 𝑟( $&#
1 + 𝑟(
= 𝑃𝑉(𝑎𝑙𝑙 𝑓𝑢𝑡𝑢𝑟𝑒 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤𝑠)
15
𝑃$
+
$
1 + 𝑟(
$
0
1
−𝑃%
𝐷#
𝑃#
0
1
2
−𝑃%
𝐷#
𝐷)
𝑃)
0
1
2
3
−𝑃%
𝐷#
𝐷)
𝐷*
𝑃*
0
1
2
…
n
−𝑃%
𝐷#
𝐷)
…
𝐷$
𝑃$
The Idea of DDM
For n holding periods:
𝑃% =
PV of future dividends
𝐷#
𝐷)
+
1 + 𝑟(
1 + 𝑟(
) + ⋯+
𝐷$&#
𝐷$
+
1 + 𝑟( $&#
1 + 𝑟(
PV of selling price
$+
𝑃$
1 + 𝑟(
$
Models’ assumptions about
dividends & selling price
(I) Constant
perpetual DDM
Dividend
discount models
(DDM)
(II) Growing
perpetual DDM
(III) 2-stage DDM
16
𝐷#
𝑃% =
𝑟(
𝐷# = 𝐷) = ⋯ = 𝐷$ , 𝑛 → ∞
𝐷) = 𝐷# (1 + 𝑔)
𝐷* = 𝐷# (1 + 𝑔))
…
𝐷$ = 𝐷# (1 + 𝑔)$&# , 𝑛 → ∞
• Stage 1: Div are anything
• Stage 2: 𝑃$ = PV(perpuity)
Constant
perpetuity
Growing
perpetuity
𝑃% =
𝐷#
𝑟( − 𝑔
𝑃$ =
𝐷$+#
𝑟(
𝐷$+#
𝑃$ =
𝑟( − 𝑔
(I) Constant perpetual DDM
Assumption: 𝐷# = 𝐷) = ⋯ = 𝐷$ (i.e. dividends all the same)
𝐷#
𝐷#
𝑃% =
+
1 + 𝑟(
1 + 𝑟(
=
𝐷#
𝑟(
𝐷#
𝑃% =
𝑟(
17
𝐷#
) + ⋯+ 1 + 𝑟
(
𝐷#
$&# + 1 + 𝑟
(
𝑃$
$+ 1+𝑟
(
As n à ∞
=0
$
As n à ∞, becomes a (ordinary) perpetuity
We have seen this
before as follows…
𝑃𝑉(𝑂𝑟𝑑 𝑃𝑒𝑟𝑝) =
𝑃𝑀𝑇
𝑟
(II) Growing perpetual DDM: Gordon Growth Model
Assumptions: 𝐷) = 𝐷# (1 + 𝑔)
𝑃% =
𝐷#
𝐷)
+
1 + 𝑟(
1 + 𝑟(
𝐷* = 𝐷# (1 + 𝑔))
) + ⋯+
…
𝐷$&#
𝐷$
+
1 + 𝑟( $&#
1 + 𝑟(
𝐷$ = 𝐷# (1 + 𝑔)$&# growing dividends
$+
𝑃$
1 + 𝑟(
$
𝐷#
𝐷# (1 + 𝑔)
𝐷# (1 + 𝑔)$&) 𝐷# (1 + 𝑔)$&#
𝑃$
=
+
+
⋯
+
+
+
1 + 𝑟(
1 + 𝑟( )
1 + 𝑟( $&#
1 + 𝑟( $
1 + 𝑟(
=
$
𝐷#
𝑟( − 𝑔 As n à ∞, converts to a constant-growth perpetuity
This is called the
Gordon Growth Model
𝐷#
𝑃% =
𝑟( − 𝑔
We have seen this
before as follows…
𝑃𝑀𝑇
𝑃𝑉(𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝) =
𝑟−𝑔
IMPORTANT:
• 𝒓𝑬 > g always!
• g < 0 is also possible (diminishing perpetuity)
18
As n à ∞
=0
Example 1: Which model do we use?
Example 1. The CFO of Jake’s Pizza Delivery Franchise business suggested in an interview with equities
analysts that next year’s dividend of $2.50 will be the limit for the foreseeable future. If the analysts’ estimate
for the stock’s cost of equity is 12.5%, what is the estimated stock price?
First question is: Which model should we use? (1) Constant perpetual DDM or (2) Growing perpetual DDM?
ANS: We are told “$2.50 will be the limit” à use constant perpetual DDM model
0
−𝑃%
1
𝐷! = 2.5
2
…
2.50
…
∞
…
2.50
…
𝑟! = 0.125
𝑃% =
Note: Follow the standard pattern: Put a ring around the CF and value in the NW corner!
𝐷! is the cash flow AFTER t=0! Sounds obvious but this is a common error students make!!
19
𝐷#
2.50
=
= 20
𝑟( 0.125
Example 2: Which model do we use?
Example 2. One year later, Jake’s Pizza Delivery Franchise business announces an unexpected growth opportunity that
suggests a long-term growth in annual earnings of 2.1%. If there is no change expected in the dividend payout ratio,
what is the new stock price estimation?
Again, Which model do we use? (1) Constant perpetual DDM or (2) Growing perpetual DDM?
ANS: We are given a “long-term annual growth rate” à use growing perpetual DDM model
NOTE: In using this model, the crucial step is to get 𝑫𝟏 correct. Everything else follows.
If the current dividend (without the growth opportunity) is $2.50 and an annual growth in dividends
of g=2.1% is expected (with the opportunity), then 𝐷# = 2.5(1 + 𝑔). We can now draw the timeline:
0
−𝑃%
1
𝐷! = 2.5(1.021)
2
2.5 1.021
…
"
∞
…
…
2.5 1.021
#
…
Note: Follow the standard pattern: Put a ring around the CF and value in the NW corner!
𝐷! is the cash flow AFTER t=0! Sounds obvious but this is a common error students make!!
20
𝐷#
𝑃% =
𝑟( − 𝑔
2.5(1.021)
=
= 24.54
0.125 − 0.021
(III) 2-Stage DDM
STAGE 1: Within horizon
Find PV of future dividends
within the horizon
STAGE 2: Beyond horizon
Find PV of selling price or terminal
value beyond the horizon
𝐷#
𝐷$
𝑃% =
+ ⋯+
1 + 𝑟(
1 + 𝑟(
= 𝑃./012 #
𝑃$
+
$
1 + 𝑟(
+
0
1
2
𝐷#
𝐷$
… n-1 n n+1 n+2 …
∞
$
𝑃./012 )
𝐷!
𝑃#
𝐷!"# 𝐷!"$
=
Stage 1 Assumption:
Dividends can be anything
EITHER:
OR:
21
=
𝐷#
𝐷$
+ ⋯+
1 + 𝑟(
1 + 𝑟(
$+
𝐷$+# /𝑟(
1 + 𝑟( $
𝐷$+#
𝐷#
𝐷$
𝑟( − 𝑔
=
+ ⋯+
+
1 + 𝑟(
1 + 𝑟( $
1 + 𝑟( $
𝑃$ =
𝐷$+#
𝑟(
𝐷$+#
𝑃$ =
𝑟( − 𝑔
Stage 2 Assumption: Constant perpetuity
𝐷$+# = 𝐷$+) = ⋯
Stage 2 Assumption: Growing perpetuity
𝐷$+) = 𝐷$+# (1 + 𝑔)
𝐷$+* = 𝐷$+# 1 + 𝑔 )
𝐷$+- = 𝐷$+# 1 + 𝑔 * …
Example 3: Which model do we use?
Example 3 (continuing example 2). More information is received on Jake’s Pizza Delivery Franchise announcement and analysts
adjust their estimations. Based on the information, it is projected that dividends will be $2.60, $2.90, $3.35, and $3.60 at the end
of the next four years. Thereafter, they may reach a limit of $3.70 for quite a while. Given these estimations, what is the new
stock price estimation? Cost of capital has increased slightly to 12.9% due to the increased risks.
1
Which model should we use?
ANS 1: As we have annual estimates within horizon and stable estimates of dividends beyond horizon, use a 2-stage DDM.
Which terminal value assumption should we use?
ANS 2: Given limited growth in dividend beyond horizon, use a constant perpetuity dividend assumption.
Draw timeline with the cash flows:
0
2
1
−𝑃%
4
3
2
2.60
2.90
3.35
4
3 28.68 …
3.60
5
6
3.70
3.70
5
22
𝑃% =
2.6
2.9
+
1.129
1.129
&+
3.35
1.129
'+
The new price is estimated at $26.78
3.60 + 28.68 …
= 26.7755
1.129 $
∞
3.70
Note! Use the standard pattern!
Again! Use the standard pattern!
4
….
3
𝐷#
𝑃$ = 𝑃𝑉 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑝𝑒𝑟𝑝 𝑑𝑖𝑣 =
𝑟!
3.70
=
= 28.6821705 …
0.129
The three dots (…) mean there are more digits.
Students should save them all in the calculator’s
memory and use them all in the final calculation!!
(III) 2-Stage DDM à Multi-stage DDM
The 2-stage model is the most important model….
𝑃% =
𝐷#
𝐷$
+ ⋯+
1 + 𝑟(
1 + 𝑟(
+
$
𝑃$
1 + 𝑟(
$
= 𝑃./012 # +
𝑃./012 )
…. because all multi-stage models just repeat the method with different growth rates at each stage:
𝑃% =
𝐷#
𝐷$
+ ⋯+
1 + 𝑟(
1 + 𝑟(
+
$
𝑃$
1 + 𝑟(
$
= 𝑃./012 # +
Firms develop through different stages:
1. Start-up:
Stage 1: no dividends
2. Growth:
Stage 2: low dividends
3. Maturity:
Stage 3: stable dividends
4. Decline:
Stage 4: declining dividends
𝑃./012 ) +
à g=0
à g>0 but small
à g>0 but medium/high
à g<0
To calculate the intrinsic value of the stock price:
1. use an appropriate cash flow model for each stage,
2. calculate the value for each stage based on the dividend growth rate,
3. discount all values to the present moment (i.e. t=0), and then
4. sum the present values to find the overall stock price.
23
… + 𝑃./012 $
Practice 1: 2-stage
Prescott Pharmaceuticals will pay an annual dividend of $1.50 one year from now. Analysts expect this dividend to
grow at 18.6% per year thereafter until the end of year twelve. Afterwards, growth will level off at 4.0% per year.
What is the price of a Prescott share if the firm's equity cost of capital is 11.0%?
0
Stage 1:
1
2
3
…
11 12 13 14 … n ∞
……
……
……
……
……
……
….
….
….
.
….
……
……
……
……
……
……
24
𝐶!)
𝑟−𝑔
……
……
……
……
𝑆ℎ𝑎𝑟𝑒 𝑝𝑟𝑖𝑐𝑒 = 𝑃./012# + 𝑃./012) =
𝑃!" =
……
……
……
𝑃$%&'(!
#
……
……
……
PV(Growing Annuity)
𝐶!
1+𝑔
=
1−
𝑟−𝑔
1+𝑟
0
1.5
0
1.5
0
1.5
0
……
1.5 ………
……
0
1.5
"
86
1.1 !
0
1.5 .186
1
0
1.5
0
1.5
𝐶# = 1.50
𝑟 = 11.0%
𝑔 = 18.6%
𝑛 = 12
Stage 2:
𝐶#) =……………………………………….………
𝐶#* = 𝐶#) ∗………………….…………….
𝑟 = 11.0%
𝑔 = 4.0%
PV(Growing Perpetuity)
Soln
Stock Valuation Methods
Three basic methods of valuing stock (equity market shares)
1. Intrinsic valuation methods – present value of future expected cash flows
Methods we know from financial mathematics.
Revise later
2. Comparables methods – comparing similar companies
Uses Equity Multiples and Enterprise Value Multiples
3. Options methods – a stock is valued as a put option on debt
Sophisticated option pricing algorithms… beyond this course.
25
Learning Outcomes – This lesson
To review types of equity and equity models
1. Intrinsic valuation models
To introduce EXCEL FUNCTIONS that aid visualizing equity valuation
• NPV; XNPV; STOCKHISTORY; PV Growing Annuity
This Photo by Unknown Author is
licensed under CC BY-NC-ND
26
To determine the size of equity cash flows
• Estimating the dividends
• Role of Share repurchases
3. Equity Valuation and Excel functions
Excel function: =NPV(…)
Rate per
period
Video Excel Demo: NPV demo
Cash flow 1
=NPV(r, value1, [value2],…)
The Excel function is solving the formula:
$
𝑁𝑃𝑉 = &
!"#
𝑉𝑎𝑙𝑢𝑒!
𝑉𝑎𝑙𝑢𝑒1
𝑉𝑎𝑙𝑢𝑒2
=
+
+ …
1+𝑟 !
1+𝑟 #
1+𝑟 )
Value1 is always discounted by 1 period
REALLY IMPORTANT. The NPV(…) Excel function is NOT solving:
$
𝑁𝑃𝑉 = &
!"#
𝑉𝑎𝑙𝑢𝑒!
+ 𝑉𝑎𝑙𝑢𝑒%
1+𝑟 !
This formula is what we would usually call NPV because
it accounts for the price paid, the initial (negative) CF.
This can be a source of confusion! So beware!!
28
The NPV(…) function does
not account for CF at t=0!
Important conclusions:
• =NPV(…) will ignore empty cells
• If there is no cash flow, write zero!
Can you contribute to the online forum? Find GOOD online videos that explain how to use these EXCEL functions.
Excel function: =XNPV(…)
Rate per year
(not rate per period)
Video Excel Demo: XNPV demo
Array of cash flows
=XNPV(r, values, dates)
Array of dates
The Excel function is solving :
$
𝑉𝑎𝑙𝑢𝑒!
𝑋𝑁𝑃𝑉 = &
1 + 𝑟 (6"&6#)⁄*89
!"#
NOTE THE MATHEMATICAL DIFFERENCE WITH Excel’s NPV(…).
In XNPV, the first cash flow is assumed to be at t=0 (counted as n=1), but
in NPV(…), the first cash flow is assumed to be at t=1.
Excel’s NPV(…) function is solving:
$
𝑁𝑃𝑉 = &
!"#
𝑉𝑎𝑙𝑢𝑒!
1+𝑟 !
But Excel’s XNPV(…) function is essentially solving:
$
𝑁𝑃𝑉 = &
!"#
𝑉𝑎𝑙𝑢𝑒!
+ 𝑉𝑎𝑙𝑢𝑒%
1+𝑟 !
Important conclusions:
• =XNPV(…) assumes the first CF is at t=0
• =NPV(…) assumes the first CF is at t=1
Note: I write “essentially” solves, not “exactly” solves
as there are things that make this a poor comparison.
29
Can you contribute to the online forum? Find GOOD online videos that explain how to use these EXCEL functions.
Excel function: =PV(…) Growing Annuity
0
1
0
PMT
3
2
PMT
∗ 1+𝑔
4
PMT
∗ 1+𝑔
#
&
n
….
PMT
∗ 1+𝑔
'
n+1
PMT
∗ 1+𝑔
()#
0
….
∞
0
Follow “standard pattern” which “takes all the growing PMT cash flows,
wraps them up into a “ball”, and finds their total present value in NW corner”
Formula for PV(Growing Annuity)
𝑃𝑀𝑇
1+𝑔
𝑃𝑉(𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝐴𝑛𝑛𝑢𝑖𝑡𝑦) =
1−
𝑟−𝑔
1+𝑟
$
𝑃𝑀𝑇 = First cash flow at t=1
𝑟 = discount rate
𝑔 = growth rate of cash flows
𝑛 = number of cash flows
Finding the PV of a growing annuity is very common, but there is no standard Excel
function for this. Instead, we adapt the standard =PV(..) function.
Excel function for PV(Growing Annuity)
𝑃𝑉(𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝐴𝑛𝑛𝑢𝑖𝑡𝑦) = PV(r, nper, pmt, [fv], [type]) / (1 + 𝑔)
First Cash Flow at t=1
Video Excel Demo: PV Growing Annuity
This website gives the full explanation:
http://www.tvmcalcs.com/calculators/apps/excel_
graduated_annuities
“correction” of =PV(..)
Number of cash flows
Net Rate =
30
!"#
−
!"$
1
Warning: This is NOT the same as
!*'
!*+
in the formula above!
Excel function: = STOCKHISTORY(…) & Stock Data Type
The function =STOCKHISTORY(…) retrieves historical data about a financial instrument.
• Stock prices
• Bond prices
• Foreign currency prices
• Bit coin, etc…
=STOCKHISTORY(stock, start_date, [end_date], [interval], [headers], [property1] … [property5])
“Stocks” Data Type à access to historical data: prices, descriptions, volumes, etc
Video Excel Demo: Stockhistory(…)
AFTERPAY
$180.00
$160.00
$140.00
$120.00
$100.00
$80.00
$60.00
$40.00
$20.00
1/
4/
20
21
2/
4/
20
21
3/
4/
20
21
4/
4/
20
21
5/
4/
20
21
6/
4/
20
21
7/
4/
20
21
8/
4/
20
21
9/
4/
20
21
10
/4
/2
02
1
11
/4
/2
02
1
12
/4
/2
02
1
$-
31
Can you contribute to the online forum? Find GOOD online videos that explain how to use these EXCEL functions.
Learning Outcomes – This lesson
To review types of equity and equity models
1. Intrinsic valuation models
To introduce EXCEL FUNCTIONS that aid visualizing equity valuation
• NPV; XNPV; STOCKHISTORY; PV Growing Annuity
This Photo by Unknown Author is
licensed under CC BY-NC-ND
32
To determine the size of equity cash flows
• Estimating the dividends
• Role of Share repurchases
4. Estimating Dividends in the DDM
Estimating Dividends in DDM: the intuition
Dividends + Share
Repurchases
Generation and
distribution of NCF
Real assets
Shareholders
Net Cash
Flow (NCF)
Interest
+
Principal
Debtholders
Reinvested @ ROI%
(part of NCF)
Growth in assets
Intuition by Example:
Real assets
EPS growth 12.8c
Reinvested @
***𝑹𝑶𝑰 = 𝟐𝟔%
*EPS 182c
Retain
(plough back)
27%
Reinvested profit 49.1c
34
Dividend
Payout 73%
**DPS 132.9c
*EPS = Earnings per share
**DPS = Dividends per share
***ROI = Return on investment
Intuition: Reinvestment of earnings generates higher earnings
and growth in real assets in the next operating cycle.
Estimating Dividends in DDM: the trade-off
If stock price is valued
according to the model:
𝐷!
𝑃% =
𝑟& − 𝑔
ANS 1: by increasing the next dividend 𝐷#
…how can company managers
maximize the stock price?
ANS 2: by increasing all future dividends through 𝑔
Ideally, both should be done. But can both be done together? ANS: NO!
There is a fundamental trade-off. If you
increase one, you can’t increase the other.
KEY QUESTION: Why is there a trade-off between the size of the next dividend and future dividends?
Intuitive answer:
• Earnings reinvestment à increase efficiency & capability of the firm’s operations improves à 𝑔 increases
• If the firm pays out a high next dividend (high 𝐷# ), then little will be left to reinvest à low 𝑔
• If the firm pays out a low next dividend (low 𝐷# ), then larger reinvestment can be made à high 𝑔
• Hence the firm must choose to either high 𝐷# /low 𝒈 or low 𝑫𝟏 /high 𝒈, it can’t maximize both simultaneosly.
35
Estimating Dividends in DDM: the formulas
How to increase next dividend 𝐷# (Dividends/share)?
Dividends are a portion of the earnings of the firm.
Berk Equation 7.8
36
𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠/
𝐷/ =
×𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜/
𝑠ℎ𝑎𝑟𝑒𝑠/
Where Dividend Payout Ratio (DPR) is the portion of total
earnings paid as dividends. E.g. If earnings is $100 and $60 is
paid in dividends, then DPR = 60%.
Increasing 𝐷/ , dividends at time t, can be done by:
1. Increasing Earnings
2. Increasing Dividend Payout Ratio
3. Decreasing the number of shares outstanding
Earnings Retention Ratio (ERR) is the portion of total earnings
retained for reinvestment in the firm’s operations. E.g, ERR =
40/100 = 40%.
So, DPR + ERR = 1
Estimating Dividends in DDM: the formulas
How to increase all future dividends through 𝑔?
Let’s assume all increases in earnings are exclusively from new investments,
while all existing investments maintain earnings at the same level:
Change in earnings = New investment ´ Return on investment (1)
New investment = Earnings ´ Earnings Retention Ratio
(2)
Substitute (2) into (1):
Change in earnings = Earnings ´ Earnings Retention Ratio ´ Return on investment
Divide both sides by Earnings:
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠
= 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 ×𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠
=𝑔
Berk Equation 7.12
𝑔 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑅𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 ×𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
37
Why is this a reasonable assumption?
An example:
A firm has assets of $1000.
In Year 1, it earns $200 this year.
Return on Investment = 20% (=200/1000).
If firm does nothing but maintain its assets,
then earnings in Year 2 should also be $200.
However, if firm reinvests $100 of the $200,
then assets increase to $1100. If ROI stays at
20%, then earnings next year will be $220
(=1100*20%).
Hence:
change in earnings = $20
= $100 x 20%
= New investment x Return on investment.
This is equation (1).
Estimating Dividends in DDM: the formulas
Formulas you need to know
“Law of Conservation of Earnings”: 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑎𝑦𝑜𝑢𝑡 𝑅𝑎𝑡𝑖𝑜/ + 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑅𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜/ = 1
𝑃% =
𝐷!
𝑟& − 𝑔
𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠/
𝐷/ =
×𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑎𝑦𝑜𝑢𝑡 𝑅𝑎𝑡𝑖𝑜/
𝑆ℎ𝑎𝑟𝑒𝑠/
𝑔 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑅𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜/ ×𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡/
Thus from the formula we see that the firm’s managers can increase share price by:
Either: increasing the next dividend 𝐷# (Dividends/share),
it needs to:
(a) to increase earnings, or
(b) increase dividend payout, or
(c) reduce the number of shares outstanding
Or: increasing all future dividends by increasing 𝑔,
it needs to:
(a) increase earnings retention, or
(b) increase the return on new investments.
Conclusion: By the “Law of Conservation of Earnings”:
A firm cannot simultaneously increase both Dividend Payout and Earnings Retention: there is always a trade-off.
38
Estimating Dividends in DDM: the choice
Firms face a trade-off, so they must choose between one of two choices, either:
1. To Retain Earnings and invest in the firm’s operations i.e., low 𝑫𝟏 /high 𝒈, or
2. To Payout Dividends to increase share price, but leaving little to invest, i.e., high 𝐷# /low 𝒈 .
ANOTHER KEY QUESTION: How to decide between these two choices?
Intuitively, if the rate of return from investing earnings (ROI) is higher than the firm’s
expected return on equity (𝑟( ), then this is good for shareholders, otherwise earnings
should be paid out as dividends.
In other words:
If 𝑅𝑂𝐼 > 𝑟( then retain earnings and invest to increase future dividends
If 𝑅𝑂𝐼 < 𝑟( then payout dividends by increasing the next dividend
39
An Example of what this mean:
You sell shares worth $1000 and
invest in new a machine. Your
investors expect a return of 10%
(expected return on equity) and
the machine promises a return of
12% (return on investment).
Is this a good or bad deal?
ANS: Since 𝑅𝑂𝐼 > 𝑟, then this is a
good deal!
If 𝑅𝑂𝐼 < 𝑟, then it would be a
bad one.
Example 5: Reinvesting dividends
See Berk (2018), Example 7.3:
Crane Sporting Goods expects earning per share (EPS) of $6 in the coming year. Crane’s current share price is $60,
expects to pay out all the earnings, and has no expectations for share price growth.
Suppose Crane announces a change in policy and cuts its dividend payout rate to 75% for the foreseeable future and
uses the retained earning to open new stores with an expected return on investment of 12%. If the risk of the new investments
is the same as existing investments, then the firm’s equity cost of capital is unchanged. What effect will the change in investment
policy have on Crane’s share price?
4 Under the new policy, what is the next
dividend? Dividend growth rate?
2 With the policy change:
1 Without policy change:
• EPS=$6/share
• EPS=$6/share = 𝐷#
𝐷# = 𝐸𝑃𝑆* ×𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜*
• Payout rate = 75%
• Current price =$60 = 𝑃%
= 6 ∗ 0.75 = $4.50
• 𝑟! = 10%
• Growth = 0
Since risk is the same as before, the
• Payout rate = 100%
𝑔 = 𝑅𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 ×𝑅𝑂𝐼
rate should be the same as before
= 1 − 0.75 ∗ 12% = 3%
• 𝑅𝑂𝐼 = 12%
What is unknown?
𝐷#
4.50
5
• Equity cost of capital
𝑃
=
=
= 64.2857
%
3 Since 𝑅𝑂𝐼 > 𝑟! , the change in policy
𝑟! − 𝑔 0.1 − 0.03
should have a positive affect on the
𝐷#
6
𝑟! =
+𝑔=
+ 0 = 10%
stock price as the new investments
𝑃%
60
ANSWER: As expected (from step 3), the
6
have a higher return than the cost of
new policy will mean a stock price increase
capital.
to $64.29 up from $60.
40
Example 6: Reinvesting dividends
See Berk (2018), Example 7.4:
Suppose Crane actually does cut its dividend payout rate to 75% and invests in new stores (as in the previous
example). However, the return on investment turns out to be 8% rather than 12%. If the expected EPS is still $6 and equity cost
of capital 10%, what will happen to Crane’s share price?
1
Updated policy change:
• EPS=$6/share
• Payout rate = 75%
• 𝑟! = 10%
• 𝑅𝑂𝐼 = 8%
2 Since 𝑅𝑂𝐼 < 𝑟! , any investments in new
stores will not have sufficient return to
cover even the cost of capital.
Hence, we should expect a negative effect
on the share price.
41
3 What is the next dividend? Dividend growth rate?
𝐷# = 𝐸𝑃𝑆* ×𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜*
= 6 ∗ 0.75 = $4.50
𝑔 = 𝑅𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 ×𝑅𝑂𝐼
= 1 − 0.75 ∗ 8% = 2%
𝐷#
4.50
=
= 56.25
𝑟! − 𝑔 0.1 − 0.02
5
𝑃% =
6
ANSWER: The low return on investment will mean a
share price falls to $56.25 down from $64.29, as
expected from step 2.
Practice 2: Estimating dividends
“until” usually means “until and including”
Adapted from Berk (2018), Example 7.5
Small Fry Ltd’s new potato chips have received phenomenal market response, so to accelerate growth it decides to reinvest all its earnings to
expand operations. It has recently announced an EPS of $2 and expects to grow at 20% annually until Year 4. At that point, reinvestment will be
cut to pay 60% of earnings as dividends. It’s long-run growth rate will reach 4% thereafter. If cost of equity is 8%, what is the share price today?
Remember: only cash flows will increase the value of the stock price! So regardless of earnings, stock price will not change
unless dividends are paid. So we need to estimate future dividends.
Earnings Year:
0
EPS growth rate
EPS ($)
2.00
Dividends
Dividend payout rate
Dividend ($)
1
20%
2(1.2)
2
20%
2 1.2
0%
0
0
0%
0
0
&
3
20%
…………
4
…………
…………
5
…………
…………
6
…………
…………
…………
…………
…………
…………
…………
…………
…………
…………
…………
…………
…………
…………
7
…………
APR=8% m=1
𝑃𝑉 =
𝑃𝑉 =
42
𝐷#
=
𝑟−𝑔
Soln
5. Share repurchases
Share repurchases: How and Why
Publicly listed companies with shares traded on the stock market often engage in Share Repurchase.
How?
• Engage in open market purchases in the stock market
• Selective purchases from shareholders
They do this for several reasons:
• To change their capital structure to achieve a certain target (we learn about this in Week 9)
• To increase leverage without issuing debt (we learn about leverage in Week 9)
• To get ready for a strategic move (e.g. window dressing before M&A)
• To return cash to their shareholders, which is form of dividend payment
• To return cash like a dividend, but without changing their dividend policy (Week 10)
44
Share Repurchases vs. Dividends
Firms return cash to shareholders by paying them dividends.
But share repurchases are another almost equivalent* way cash is repaid to shareholders.
For example:
A shareholder owns 200 shares currently priced at $50 a share.
Price/ share
Total value BEFORE dividend/ repurchase
$50
Shares
200
Total Value
$10,000
The firm pays a dividend of $2/share. The share price drops
by $2, but the shareholder receives $2/share. Consequently,
there is no change in wealth from paying a dividend.
Cash received from dividend
Share value after dividend
Total value AFTER dividend
$2
$48
200
200
$400
$9,600
$10,000
Alternatively, the firm buys back 8 shares at the current
market price, or $400 worth of shares. The number of shares
held by the shareholder reduces while receiving cash from
selling the repurchased shares. Again, there is no change in
wealth due to the share repurchase.
Cash received from share repurchase
Share value after share repurchase
Total value AFTER share repurchase
$50
$50
8
192
$400
$9,600
$10,000
The shareholder’s mix of shares & cash is the same regardless
whether they were paid with dividends or with share repurchases.
Conclusion: Share repurchases and Dividends are almost equivalent* ways of returning cash to shareholders.
* Share repurchases and dividends would be exactly equivalent if it were not for differences in income tax rates
charged on income from share repurchases vs. income from dividends. We discuss this in Week 10 (Payout Policy).
45
DDM and Total Payout Model (TPM)
An alternative model to DDM:
The Dividend Discount Model (DDM) values
shares from the perspective of a single shareholder
(i.e. per share model):
Total Payout Model (TPM) avoids the problems of DDM and
includes both dividends and share repurchases.
Total Value = PV(Future payout cash flows)
= PV(Future Dividends)
Total Value = PV(All future payout cash flows)
= PV(Fut. Dividends + Fut. Net Share Purchases)
= PV(Fut. Dividends) + PV(Fut. Net Share Purchases)
Price/share = Total Value / No. of shares
= PV(Fut. Dividends) / No. of shares
= PV(Dividends per share)
Problems with DDM:
1. Ignores share repurchases, i.e. assumes no
share repurchases.
2. Does not handle changes in number of shares.
3. Assumes dividend growth rate, earnings
growth rate, and share price growth rate are all
the same.
46
Price/share = Total Value / No. of shares
Total Payout for 1 period = Total Payout Rate Earnings
Note: When share repurchases are included, earnings growth rate and
dividend growth rate are no longer the same. See next example.
Total Value = PV(Total Future Dividends & Net Share Purchases)
𝑃𝑉 =
𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑦𝑜𝑢𝑡 𝑅𝑎𝑡𝑒 ×𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠
𝑟−𝑔
Total Payout Model: Example
Berk (2018) Example 7.6 Titan Industries has 217 million shares outstanding and expects year-end earnings of $860 million.
Titan plans to payout 50% of earnings, with 30% as dividends and 20% as share repurchases. If annual earnings growth is
expected at 7.5%, payout rates remain constant, and equity cost of capital is 10%, what is the share price?
r = 10%
217 million shares
Earnings = $860 million
30% dividend payout
20% share repurchases
i.e. total payout rate = 50%
𝑔:89(;(<6 = 7.5%
Dividend per share =
'%% × ./%0
&#10
Next total payout = total payout rate earnings
Since payout rates remain constant, total payout will grow at 𝑔:89(;(<6
This means Dividend Yield =
Total Value = PV(Total Future Dividends & Net Share Purchases)
Recall: 𝑟! =
𝑃𝑉 =
𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑦𝑜𝑢𝑡 𝑅𝑎𝑡𝑒 ×𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠
𝑟−𝑔
50%×860
=
= 17.2 𝐵𝑖𝑙𝑙𝑖𝑜𝑛
0.10 − 0.075
Share price =
47
An interesting aside: With share repurchases, is the share
price growth rate the same as the earnings growth rate?
Using DDM (Per share model)
#1.& B;CC;D(
&#1 0;CC;D(
= $79.26/𝑠ℎ𝑎𝑟𝑒
23.
23.
= 1.189 $/𝑆ℎ𝑎𝑟𝑒
#.#.5
= 15.&/ = 1.5%
+ 𝑔6789: ⇒ 𝑔6789: = 10% − 1.5% = 8.5%
In other words, 𝑔6789: > 𝑔:89(;(<6 .
Why? As shares are repurchased, there are fewer shares
outstanding, so growth per share increases.
To check that 𝑔!"#$%$&' = 7.5% and 𝑔'("#! = 8.5% are consistent with each
other, see note 5 at the end of Berk (2018) Chapter 7. Also see Appendix here.
Learning Outcomes – This lesson
To review types of equity and equity models
1. Intrinsic valuation models
To introduce EXCEL FUNCTIONS that aid visualizing equity valuation
• NPV; XNPV; STOCKHISTORY; PV Growing Annuity
This Photo by Unknown Author is
licensed under CC BY-NC-ND
48
To determine the size of equity cash flows
• Estimating the dividends
• Role of Share repurchases
6. Team Assignment
A deeper dive
Read the comprehensive
Team Assignment Assessment Guide in
Moodle >> Team Assignment
Read this first before asking question!
See Moodle >> Team Assignment Folder >> Team Assignment Assessment Guide
The idea of the team assignment
Working in
Teams of 4-5
students from
the same
tutorial group
Choose a
publicly listed
stock
Analyse the stock
• Financials
• Strategy
• Competition
• Industry
• Technology, etc
Inform
ation
in
team
assess the
m
guide
is prev ent
ailing
Value the stock
using all the
financial tools and
models in FINS2615
Recommend to
investors to BUY or
SELL the stock
Make a video of your
recommendation
Integrating FINS2615 with your finance career
Team
Assignment
Finance
Showcase
International
competitions
Connecting students to finance industry / competitive environment
50
Watch the videos of
other teams’
recommendations
and decide
whether you (as an
investor) should
buy or sell their
stock.
Choose your team … becomes official in Week 3
Criteria
Moodle
51
Criteria for team members:
• 4-5 members from the same tutorial group
• All genders must be represented and as much as possible equally represented
• All member to agree on one stock to analyse
Inform
ation
in
team
assess the
m
guide
is prev ent
ailing
Stock Selection Criteria and process
Inform
ation
in
team
assess the
m
guide
is prev ent
ailing
Make a buy/sell recommendation of ANY publicly listed company (e.g., Australia, China, Europe, US,) according to the
following selection criteria:
1. The stock can be found within the FactSet universe [data availability]
2. The company has at least 5 years of complete financial data [data stability]
3. The company has annual sales of US$300 million or more in each of the last 3 years [size]
4. Your team members are all interested in investing in and can explain why [common interest]
5. The stock you choose is different to all other stocks being analyzed in your tutorial group [originality]
6. Do not choose banks or insurance (financial) firms [avoid non-standard financial statements]
7. The stock is not Apple, Google, Microsoft, Amazon, or Tesla [avoid massive hard to value companies]
SIX DELIVERABLES FOR TEAM ASSIGNMENT THROUGHOUT THE TERM
First deliverable
Week 3
Second deliverable
Week 5
Third deliverable
Week 9
Fourth deliverable
Week 9-10
Fifth deliverable
Week 10
Sixth deliverable
Week 11
Choose Team and
Stock
Submit
preliminary
recommendation
Submit final
recommendation
Ask questions
about other
Teams’
recommendations
Answer questions
about your Team’s
recommendation
Submit your Team
feedback
DO NOT WORRY ABOUT THE MANY DELIVERABLES! I WILL BE GUIDING YOU AND REMINDING YOU! J J J
52
Choose your stock from FactSet
What is FactSet?
• FactSet is a financial information portal on companies, markets, instruments, etc.
• You need this for your Team Assignment.
How to get access to FactSet?
• Every student will be given a FactSet account.
• Wait to receive an email from FactSet that tells you how to access it.
• If you enrolled on time, an email be sent to you by end of Week 2.
• If by Week 3 you don’t have a FactSet account, follow instructions in
Moodle …
Moodle >> Team Assignment – Help and Hints >> HELP ON FACTSET
53
Inform
ation
in
team
assess the
m
guide
is prev ent
ailing
Refer to your Team and Stock Code by FactSet Code
•
•
Every team is numbered
Call your stock by FactSet code convention
Examples:
• Team 10 [FMG-AU] – Fortescue Metals Group AU
• Team 66 [WOW-AU] – Woolworths Australia
• Team 72 [MC-FR] – Louis Vuitton Moet Hennessy France
• Team 75 [005930-KRX] – Samsung Korea
54
Inform
ation
in
team
assess the
m
guide
is prev ent
ailing
Notes
Financial Data can be in any currency
Your analysis can be any currency, just be consistent
Aarrgghh…Too complicated??!!!
Don’t worry. Be happy!!!
I will help you with reminders!
Let’s go one week at a time!!
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Tutorial preparation
Tutorial ahead
57
•
Pre-tutorial work
• 20-30 minutes required.
•
Tutorial work
• Valuing stocks using different models
• Estimating dividends
• Some excel exercises
Appendix
Practice 1: 2-stage Soln
Prescott Pharmaceuticals will pay an annual dividend of $1.50 one year from now. Analysts expect this dividend to
grow at 18.6% per year thereafter until the end of year twelve. Afterwards, growth will level off at 4.0% per year.
What is the price of a Prescott share if the firm's equity cost of capital is 11.0%?
0
Stage 1:
2
3
…
11 12 13 14 … n ∞
Stage 2:
𝐶#) = 1.50 1.186 ##
𝐶#* = 𝐶#) ∗ (1.04)
𝑟 = 11.0%
𝑔 = 4.0%
!"
#0
4
1.0
!!
"
86
4
1.1
0
1.0
!!
1.5
)
86
.04
1.1 !! (1
0
1.5
86
1.1 !!
0
1.5 .186
1 !/
0
1.5 .186
1
0
1.5
"
86
1.1 !
0
1.5 .186
1
0
1.5
0
1.5
𝐶# = 1.50
𝑟 = 11.0%
𝑔 = 18.6%
𝑛 = 12
1
PV(Growing Annuity)
𝐶!
1+𝑔
𝑃$%&'(! =
1−
𝑟−𝑔
1+𝑟
1.50
1.186
=
1−
0.11 − 0.186
1.11
= 23.956794
#
𝑃!" =
𝐶!)
𝑟−𝑔
PV(Growing Perpetuity)
!"
1.50 1.186 !! 1.04
=
= 145.531879
0.11 − 0.04
𝑃$%&'(" = 𝑃!" 1.11
0!"
= 41.5989
𝑆ℎ𝑎𝑟𝑒 𝑝𝑟𝑖𝑐𝑒 = 𝑃./012# + 𝑃./012) = 23.9567 + 41.5989 = 65.5556 ≈ $𝟔𝟓. 𝟓𝟔
59
Back
Practice 2: Estimating dividends Soln
Adapted from Berk (2018), Example 7.5
Small Fry Ltd’s new potato chips have received phenomenal market response, so to accelerate growth it decides to reinvest all its earnings to
expand operations. It has recently announced an EPS of $2 and expects to grow at 20% annually until Year 4. At that point, reinvestment will be
cut to pay 60% of earnings as dividends. It’s long-run growth rate will reach 4% thereafter. If cost of equity is 8%, what is the share price today?
Remember: only cash flows will increase the value of the stock price! So regardless of earnings, stock price will not change
unless dividends are paid. So we need to estimate future dividends.
Earnings Year:
0
EPS growth rate
EPS ($)
2.00
Dividends
Dividend payout rate
Dividend ($)
1
20%
2(1.2)
2
20%
2 1.2
0%
0
0
0%
0
0
&
3
20%
2 1.2
0%
0
0
'
4
20%
2 1.2
$
60%
2 1.2 $(0.6)
= 2.4883
5
4%
2 1.2 $(1.04)
6
4%
2 1.2
60%
2 1.2 $(0.6)(1.04)
= 2.5879
60%
2 1.2 $(0.6) 1.04
= 2.6914
$
1.04
7
4%
&
&
APR=8% m=1
𝑃𝑉 =
𝑃𝑉 =
60
'(.(%*+
!.%,E
= 𝟒𝟗. 𝟑𝟖*
𝐷#
2.4883
=
= 62.2075
𝑟 − 𝑔 0.08 − 0.04
*Berk (2018) gives the answer as $49.42. The difference is due to rounding.
Back
Total Payout Model: Example Explained
NOW
800
=860/1.075
Earnings ($M)
Total
payout
model
800 * 20% = 160
800 * 30% = 240
Total Value ($M)
430
= 17.2 𝐵𝑁
0.1 − 0.075
61
860 * 20% = 172
860 * 30% = 258
g=7.5%
#1&
./
217 Given
17.2𝐵
= 79.26
217
Price/Share ($/share)
g=8.5% calculated
Dividends/Share ($/share)
240/217 = 1.106
Earnings/Share ($/share)
800/217 = 3.8688 g=8.5%
Reconciliation:
YEAR-END
860 Given
g=7.5%
Repurchases ($M)
Dividends ($M)
No. of shares (M)
Per
Share
Model
g=7.5% given
g=8.5%
g(total payout model) = 7.5%
g(per share model) = 8.5%
430
= 2 share repurchased è 217 – 2 = 215
79.26 1.085 = 86.00
258/215 = 1.2
860/215 = 4
1.075 ∗
217
= 1.085
215
Conclusion:
Total Payout & Per Share
Models are actually equivalent,
except for the scaling factor for
the shares repurchased.
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