managerial finance exam
Dr. Ekaterina Chernobai
p. 1
FRL 3000
Instructor: Dr. Ekaterina Chernobai
Chapter 8 “Stocks”
1
Dr. Ekaterina Chernobai
p. 2
In Chapter 8
Important examples of financial securities that use Annuity or Perpetuity calculations:
Bonds (Ch. 7) – sold by companies or Governments to borrow money from investors
Stocks (Ch. 8) – sold by companies. Represent company ownership, or “equity”
An important example of perpetuity in real life
Dr. Ekaterina Chernobai
p. 3
Stocks vs. bonds (1 of 4)
Both provide long-term funding for the company
Both provide future income that an investor must consider
Both make future payments periodically
Both can be purchased in a marketplace at a price “today”
Stocks vs. bonds
Dr. Ekaterina Chernobai
p. 4
Stocks vs. bonds (2 of 4)
From the firm’s perspective:
BONDS are long-term debt
STOCK shares are equity
Balance Sheet
Assets
Liabilities & Equity
Long-term debt
Common stock
Income Statement
Sales Revenue
– Costs of goods sold
– Depreciation
= Earnings before interest & taxes
– Interest
= Earnings before taxes
– Taxes
= Net Income (or Profit)
Dividends
Addition to retained earnings
Firm’s snapshot at a point of time
Firm’s performance over a period of time
Short-term liabilities
Retained earnings
Dr. Ekaterina Chernobai
p. 5
Stocks vs. bonds (3 of 4)
Payments on BONDS vs. on STOCK shares
| Bonds | Stocks |
| Pay last on maturity date Pay coupons every year, & a lump-sum payment (Face Value) at maturity Coupons are (usually) fixed every year | Pay indefinitely Pay Dividends every year Dividends can change every year |
0 1 2 3 4 5 6 …
$C $C $C $C
$FV
0 1 2 3 4 5 6 …
$D1 $D2 $D3 $D4 $D5 $D6 …
Dr. Ekaterina Chernobai
p. 6
Stocks vs. bonds (4 of 4)
Stock types
Common stock
Preferred stock
Most common stock type
Dividends are not guaranteed, may change or even equal $0
Less common stock type
Dividends are fixed and usually guaranteed
Has preference over common stock in receiving dividend & in distribution of firm’s assets in bankruptcy
Each year, Firm makes payments to holders of its bonds first, then to holders of preferred stock, & finally to holders of common stock
We’ll mainly look at this type of stock in this Chapter!
Dr. Ekaterina Chernobai
p. 7
Stock value – in general (1 of 5)
Common stock example:
You buy one share of common stock today
You expect to receive a $5 dividend over the next year
You expect to be able to sell that share for $80 in 1 year
Return on investment is 15%
Question: How much would you pay for the share today?
Answer: Stock value today = PV of all future payments
=
=
Dividend next year + Price next year
1 + discount rate
$5 + $80
1 + 0.15
= $73.91
Dr. Ekaterina Chernobai
p. 8
Stock value – in general (2 of 5)
In general:
Dividend in 1 year + Price in 1 year
1 + discount rate
Stock value today =
D1 + P1
1 + R
=
D1 P1
1 + R 1 + R
= +
mathematically, this can be separated into 2 terms:
= +
Dr. Ekaterina Chernobai
p. 9
Stock value – in general (3 of 5)
D1 P1
1 + R 1 + R
= +
Stock value today
Now, we can do the same with “P1”, then with “P2”, and so on:
D2 P2
1 + R 1 + R
D1 D2 P2
1 + R (1 + R)2 (1 + R)2
= + +
D1 D2 D3 P3
1 + R (1 + R)2 (1 + R)3 (1 + R)3
= + + +
D3 P3
1 + R 1 + R
= +
…
D1 D2 D3 D4 D5 …
1 + R (1 + R)2 (1 + R)3 (1 + R)4 (1 + R)5
= + + + + +
and so on forever
D4 P4
1 + R 1 + R
= +
Today’s stock price = Present Value of all future dividends
Stock value today
Stock value today
Stock value today
Dr. Ekaterina Chernobai
p. 10
Stock value – in general (4 of 5)
Since we can’t possibly know all future dividends, we will now look at some special cases with simplified assumptions that make calculations of today’s stock price possible…
Today’s stock price = Present Value of all future dividends
Dr. Ekaterina Chernobai
p. 11
Stock value – in general (5 of 5)
Let’s assume these few cases for future dividends:
1.) Constant dividend (or zero-growth dividend)
The firm will pay a constant dividend forever
This is like preferred stock
Use the perpetuity formula to calculate price
2.) Constant dividend growth
The firm will increase the dividend by a constant percent every period
Use the growing perpetuity formula to calculate price
3.) Non-constant dividend growth
Dividend growth is not consistent initially, but settles down to constant growth eventually
Use a multistage model to calculate price
(e.g., two-stage growth model in “supernormal growth” examples)
Dr. Ekaterina Chernobai
p. 12
1.) Zero-growth dividend (1 of 2)
1.) Constant dividend (or zero-growth dividend)
Same dividend (D) every year forever
PV =
C
R
PV of perpetuity!
0 1 2 3 …
D
D
D
Stock price today, P0
Dividend, D
Dr. Ekaterina Chernobai
p. 13
1.) Zero-growth dividend (2 of 2)
EXAMPLE
Suppose stock of ABC Co. sells for $18 per share. The company is expected to pay a $0.50 dividend every quarter and the required return is 10%.
Are shares of stock priced correctly?
Solution:
Correct price today (P0) =
$0.50
0.025
= $20
quarterly “R” = 0.10 / 4
D
R
quarterly dividend “D”
=
Since stock shares cost $18, this price is ______ than it should be by $____.
lower
2
Dr. Ekaterina Chernobai
p. 14
2.) Constant growth dividend (1 of 7)
D0
Company has just paid
D1 = D0 x(1+g)
D2 = D0 x(1+g)2
D3 = D0 x(1+g)3 …
0 1 2 3 …
2.) Constant growth dividends
Stock price today, P0
Dividend in 1 year, D1
PV =
C
R – g
PV of growing perpetuity!
Rate at which dividends grow each year, g
Dr. Ekaterina Chernobai
p. 15
2.) Constant growth dividend (2 of 7)
D0 x(1+g)
R – g
To find current value, or price, of stock share (P0):
P0 =
D1
R – g
OR
P0 =
because D1 = D0 x(1+g)
P0 = stock value, or price, today
D0 = dividend that’s just been paid
D1 = dividend that will be paid next time (e.g., in 1 year)
g = growth rate of dividends
R = discount rate
If we are given “dividend next year (D1)”
If we are given “dividend today (D0)”
Dr. Ekaterina Chernobai
p. 16
2.) Constant growth dividend (3 of 7)
… price at time t equals:
Dt = D0 x(1+g)t
Dt+1
R – g
Pt =
PV = D0
I/Y = g
N = t
CPT FV
This is again just PV of a growing perpetuity of cash flows after time t
0 1 2 3 … … t t+1 …
Also, at any time t in the future…
… dividend at time t equals:
Dr. Ekaterina Chernobai
p. 17
2.) Constant growth dividend (4 of 7)
EXAMPLE
Suppose XYZ Inc. yesterday paid a dividend of $0.50 per share to each of its stockholders. It is expected to increase its dividend by 2% per year. The market requires a return of 15% on assets of this risk.
1.) How much would new investors be paying for XYZ’s stock today?
2.) What will the stock be worth in 5 years?
Dr. Ekaterina Chernobai
p. 18
2.) Constant growth dividend (5 of 7)
1.) How much would new investors be paying for XYZ’s stock today?
Price today (P0) =
$0.51
0.15 – 0.02
= $3.92
Dividend paid yesterday (D0) = $0.50
D1
R – g
=
= D0 x (1 + g)
= $0.50 x(1 + 0.02)
= $0.51
2.) What will the stock be worth in 5 years?
Dividend in 6 years (D6) =
Price in 5 years (P5) =
= D0 x (1 + g)6
= $0.50 x(1 + 0.02)6
= $0.563
D6
R – g
$0.563
0.15 – 0.02
Dividend in 1 year (D1) =
=
= $4.33
PV = 0.50
I/Y = 2
N = 6
CPT FV
PV = 0.50
I/Y = 2
N = 1
CPT FV
Dr. Ekaterina Chernobai
p. 19
2.) Constant growth dividend (6 of 7)
Turns out that every year price grows at the same rate as dividends :
Price in 5 years (P5) =
D6
R – g
=
D1 x(1+g)5
R – g
=
Price today x (1+g)5
D0 x(1+g)5
Dividend in 5 years (D5) =
same as “price today”, P0
= Dividend today x (1+g)5
Dr. Ekaterina Chernobai
p. 20
2.) Constant growth dividend (7 of 7)
A note about the constant-growth formula:
P0 =
D1
R – g
!!! If R > g………. The only case when the formula works!
!!! If R < g………..Mathematically, Price will be negative. Doesn’t make sense!
Instead, if Dividends grow too fast, Price will be infinitely large
Dr. Ekaterina Chernobai
p. 21
3.) Nonconstant growth dividend (1 of 9)
This covers all other cases!
3.) Non-constant dividend growth
Dr. Ekaterina Chernobai
p. 22
3.) Nonconstant growth dividend (2 of 9)
EXAMPLE
(“No dividend” example)
A company currently pays no dividends. You predict that after 4 years it will start paying dividends for the first time, and the dividend per share will equal $0.80 per quarter, paid out at the end of each quarter. You expect the company to grow at 5% per year indefinitely and adjust its quarterly dividends accordingly in order to reflect its earnings. The required rate of return is 12%.
What is this company’s stock worth today?
g = 5%/4
= 1.25%
per quarter
R = 12%/4
= 3%
per quarter
Dr. Ekaterina Chernobai
p. 23
3.) Nonconstant growth dividend (3 of 9)
Price in 4 years (or 16 quarters)
today
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5 … … …
$.8, then grow by 1.25% every quarter forever
Din 17 quarters
Rquarterly – gquarterly
= $45.71
$0.80
0.03 – 0.0125
growing perpetuity
=
=
$45.71
Price0 =
$45.71
(1+0.03)16
= $28.48
CF0 0
C01 0 F01 15
C02 45.71 F02 1
Find NPV @ I = 3
So, it’s as if we have:
today
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5 … … …
Dr. Ekaterina Chernobai
p. 24
3.) Nonconstant growth dividend (4 of 9)
EXAMPLE
(Uneven growth example)
You want to buy a company’s stock. You predict that the dividend on each share will be:
$1 per share in 1 year,
$2 per share in 2 years,
$3 per share in 3 years,
$4 per share in 4 years,
after which you expect the company to grow at 7% per year indefinitely and adjust its dividends accordingly in order to reflect its earnings.
The required rate of return is 12%.
What is this company’s stock worth today?
$1 $2 $3+$80
Dr. Ekaterina Chernobai
p. 25
3.) Nonconstant growth dividend (5 of 9)
0 1 2 3 4 5 6 …
$1 $2 $3 $4
Price0 =
$1
1+0.12
$2
(1+0.12)2
$83
(1+0.12)3
+
+
= $61.57
Non-constant growth
constant growth @ 7%
PV of growing perpetuity in Yr 3 =
$4
0.12 – 0.07
= $80
$4x(1+0.07)
$4x(1+0.07)2 …
CF0 0
C01 1 F01 1
C02 2 F02 1
C03 83 F03 1
Find NPV @ I=12
0 1 2 3 4 5 6 …
So, it’s as if we have:
Dr. Ekaterina Chernobai
p. 26
3.) Nonconstant growth dividend (6 of 9)
EXAMPLE
(“Supernormal growth” example)
A company has been growing at a phenomenal rate of 40% per year because of unexpected high sales. You expect this trend to continue for another 5 years, after which the company’s growth will fall to 10% per year. The required return is 12%. The company has just paid $5 million in dividends to its shareholders.
What is the total value of its stock?
First growth rate g1=40%
Second growth rate g2=10%
40%
10%
0.12–0.1
Dr. Ekaterina Chernobai
p. 27
3.) Nonconstant growth dividend (7 of 9)
D0=5
0 1 2 3 4 5 6 7 …
growing perpetuity
5 +/- PV
1 N
40 I/Y
CPT FV
D1=7
5 +/- PV
2 N
40 I/Y
CPT FV
D2=9.8
5 +/- PV
3 N
40 I/Y
CPT FV
D3=13.7
5 +/- PV
4 N
40 I/Y
CPT FV
D4=19.2
5 +/- PV
5 N
40 I/Y
CPT FV
D5=26.9
… … … … … …
PV of growing perpetuity in Yr 4
$26.9
= $1,344.56
CF0 0
C01 7 F01 1
C02 9.8 F02 1
C03 13.7 F03 1
C04 1363.76 F04 1
Find NPV @ I=12
Price0 = 890.52
g = 40%
g = 10%
1st constant growth
2nd constant growth
=
7 9.8 13.7 19.2+1,344.56
0 1 2 3 4
So, it’s as if we have:
my solutions
Dr. Ekaterina Chernobai
p. 28
3.) Nonconstant growth dividend (8 of 9)
D0= 5
0 1 2 3 4 5 6 7 …
g = 40%
g = 10%
D1 =
P0 =
1 1+0.4
0.12–0.4 1+0.12
5
= 51.29
growing annuity
5 x (1+0.4) = 7
PV first 5 dividends =
7 x
x 1 –
51.29 + 1,479.02/(1 + 0.12)5 = 890.52
=
D6=
growing perpetuity
5x(1+0.4)5x(1+0.1)
=29.6
P5=
= 1,479.02
D6
R – g
29.6 0.12–0.1
=
7
29.6
textbook’s solutions
Dr. Ekaterina Chernobai
p. 29
3.) Nonconstant growth dividend (9 of 9)
This last example follows two-stage growth model :
P0 = D1
1
R – g1
1 + g1
1 + R
1 – +
Pt
(1 + R)t
Pt = =
Dt+1
R – g2
D0(1 + g1)t (1 + g2)
R – g2
t
where
g1 = the 1st growth rate
g2 = the 2nd growth rate
in the book!
Dr. Ekaterina Chernobai
p. 30
Required return components (1 of 11)
Now, let’s see what we can say about
annual rate of return (“R”)
on a stock
Dr. Ekaterina Chernobai
p. 31
t t+1
Paid price = $500
( Pt )
Dividend paid at year end = $10
(Dt+1)
Total cash received if stock is sold = $560 (Pt+1 + Dt+1)
time line
Required return components (2 of 11)
New price at year end = $550
(Pt+1)
Dollar return over one year = $10 + ($550 – $500) = $60
Percentage return over one year = $60 / $500 = 12%
Today……..Buy a share for $500.
Next year...(1) Receive a $10 dividend;
(2) Share appreciates to $550, can sell it for more $
stock
Dr. Ekaterina Chernobai
p. 32
in $ in %
Dividend income $10 Dividend yield $10/$500 = 2%
Dt+1 Dt+1 / Pt
Capital gain $550 – $500 = $50 Capital gains yield $50/ $500 = 10%
Pt+1 – Pt (Pt+1 – Pt) / Pt
ending – beginning ending – beginning beginning
price price price price price
Total dollar return $550+$10–$500= $60 Total percentage return $60/$500 = 12%
Dt+1 + Pt+1 – Pt (Dt+1 + Pt+1 – Pt )/Pt
Notice that 12% is also simply 2%+10%!
Required return components (3 of 11)
Dr. Ekaterina Chernobai
p. 33
Total PERCENTAGE Return =
= Dt+1+ Pt+1 – Pt
Pt
= Dt+1 Pt+1 – Pt
Pt Pt
= dividend capital
yield gains
yield
= R
Total DOLLAR Return =
= dividend capital
income gain (or loss)
Required return components (4 of 11)
Dr. Ekaterina Chernobai
p. 34
You bought a stock for $35 and you received a dividend of $1.00 after half a year, and another dividend of $0.25 at the end of the year. The stock is now selling for $40.
What is your percentage return?
EXAMPLE
Solution:
Dividend yield =
Capital gains yield =
Total % return (R)=
$1.25 / $35
= 3.57%
($40 – $35) / $35
= 14.29%
= 3.57% + 14.29%
= 17.86%
dividend yield + capital gains yield
Required return components (5 of 11)
Dr. Ekaterina Chernobai
p. 35
Let’s check:
We got R = 17.86%
This means:
Future Value of the investment = initial investment x (1 + R)
= $35 x (1 + 0.1786)
= $41.25
$41.25 is exactly equal to the total dividend $1.25 plus the new stock price $40
Required return components (6 of 11)
Dr. Ekaterina Chernobai
p. 36
Required return components (7 of 11)
Back to “Dividend Growth Model”:
Rearrange:
P0 =
D1
R – g
R – g =
D1
P0
R = + g
D1
P0
Solve for return “R”:
Dr. Ekaterina Chernobai
p. 37
Required return components (8 of 11)
R = + g
D1
P0
As defined earlier, this is “dividend yield”
And so this must be “capital gains yield”
% rate at which dividends grow each year
% rate at which stock price grows each year (see earlier slides)
Both dividends & stock price grow at the same rate each year
where D1 also equals D0 x(1+g)
Dr. Ekaterina Chernobai
p. 38
Required return components (9 of 11)
EXAMPLE
Suppose a firm’s stock is selling for $10.50. It just paid a $1 dividend, and dividends are expected to grow at 5% per year.
What is the dividend yield?
What is the capital gains yield?
D1
P0
=
$1x(1+0.05)
$10.50
=
= 10%
= 5% (given!)
= g
What is the required return?
R = + g
D1
P0
= 10% + 5%
= 15%
Dr. Ekaterina Chernobai
p. 39
Required return components (10 of 11)
EXAMPLE
The current dividend yield on a company’s common stock is 3.2 percent. The company just paid a $1.48 annual dividend and announced plans to pay $1.54 next year. The dividend growth rate is expected to remain constant at the current level.
What is the required rate of return on this stock?
0.032 + 0.04054
= 0.0725, or 7.25%
R = + g
D1
P0
g = % change in dividends
=
D1 – D0
D0
= 0.04054, or 4.054%
1.54 – 1.48
1.48
=
=
Solution:
3.2%
Dr. Ekaterina Chernobai
p. 40
Required return components (11 of 11)
EXAMPLE
A company’s common stock sells for $38 a share and pays an annual dividend that increases by 3 percent annually. The market rate of return on this stock is 8.20 percent.
What is the amount of the last dividend paid?
Solution:
R = + g
D1
P0
D1 = D0 x(1 + g)
0.082 = + 0.03
D0 x(1 + 0.03)
38
D0 =
(0.082 – 0.03) x 38
1 + 0.03
= 1.92
Dr. Ekaterina Chernobai
p. 41
Common vs. preferred stock (1 of 3)
Income Statement for some year
Sales revenue
Cost of good sold
Depreciation
= EBIT
- Interest paid
= Taxable income
Taxes (34%)
= Net income, or profits
Dividends paid
Retained earnings
Corporate bonds, bank loans
1st priority: Preferred stock
2nd priority: Common stock
Dr. Ekaterina Chernobai
p. 42
Common vs. preferred stock (2 of 3)
Common stock
Shareholder rights
- they elect directors, who hire managers to run the company
- subscription (or preemptive) rights = existing stockholders are offered any newly issues shares first before the general public
Proxy voting
- grant the authority to vote to someone else. 1 share = 1 vote
Classes of stock
- company may have more than 1 class of stock. E.g., “class B”, “class C”
Example: Google has Class A (held by the public, 1 stock = 1 vote) and
Class B (held by company insiders, 1 stock = 10 votes).
Dividends
- =share of co’s profits; paying becomes co’s liability once announced; taxed at 15% in year received (in contrast, capital gains taxed only when realized)
Dr. Ekaterina Chernobai
p. 43
Common vs. preferred stock (3 of 3)
Preferred stock
Dividends paid out first
- preferred stock 1st, common stock 2nd
Stated value
- usually $100. So, a phrase “5% preferred stock” means Dividend = $5
Cumulative vs. noncumulative dividends
- Cumulative…….. If not paid this year, will be carried over to next
- Noncumulative….If not paid this year, will NOT be carried over to next
Unpaid dividends are NOT debts of the firm
Is it debt or equity?
- preferred stockholders receive a stated dividend = Debt?
!
Dr. Ekaterina Chernobai
p. 44
Summarize
Stocks = investors’ claims on Firm’s equity
Stock price today = PV of all future dividends
Stock price calculations:
(1) Constant dividend model
(2) Constant dividend growth model
(3) Non-constant dividend growth
no dividends temporarily
uneven growth
supernormal growth
Return
- $ return vs % return
- 2 components of return: dividend yield & capital gains yield
- Dividend yield & capital gains yield in “constant dividend growth” model
Common & preferred stock features