Today’s agenda
Review what we have learned in the last lecture
Stock and its valuation• Some terminology about a stock
• Value a stock
• Simple dividend discount model
• Dividend growth model
What have we learned in the last lecture
Bond? How to value a bond? Yield to maturity and spot rates? Term structure of interest rates and yield
curve?
Some questions
A bond that pays annual coupon is issued with a coupon rate of 4%, maturity of 30 years, and a yield to maturity of 7%, what will be the rate of return if you buy it now and hold it for one year and the yield to maturity in the next year will be 8%?
What is a stock?
A (common) stock is a financial claim that has the following properties:• A right to receive dividends after creditors
have been paid
• A right to vote at the annual meeting
• A limited liability security
Dividends are periodic cash flows to share holders
Stocks & Stock Market
Primary Market - Place where the sale of new stock first occurs.
Secondary market - market in which already issued securities are traded by investors.
P/E ratio - Price per share divided by earnings per share.
Dividend yield- Dividends per share divided by the stock price
Values of stocks
Book Value of a stock- the value according to the balance sheet in the accounting.
Market Value of a stock – the value according to the traded stock prices in the market.
Stock valuation
When you want to invest in a stock, you are very interested in whether the stock is under-priced or over-priced. To find out, you need to value the stock
Two simple approaches to price a stock• Simple dividend discount model
• Dividend growth model
We will apply these two approaches to real stocks, for example, IBM
Simple dividend discount model: valuing IBM We will first use the dividend discount model to value
the International Business Machine.• What does the company do?
http://finance.yahoo.com/• Symbol “IBM”
• Trades on the NYSE We see price is recently $180
• hit “detailed”
• we see the company is paying $3 dividend per share (we will do an annualized problem for simplicity, here we assume that all the earnings are paid out as dividends)
Valuing IBM (continue)
Let’s suppose IBM is going to continue paying $3 dividend per share, forever
We are planning to buy the stock and hold it forever Of course, we must be able to draw the cash flow
diagram
PV???
$3 $3$3 $3 $3 $3
Yr1 Yr2 Yr3 Yr4 Yr5 Time=infinity
Valuing IBM (continue)
How much is IBM worth?• Suppose the required rate of return by the
investor is 10%.
The present value of future dividend cash flows should equal the price of IBM.
30$1.0
31 r
CPV
Valuing IBM (3)
Clearly, the price calculated using this simple model is below the current market price
Why?• we have undervalued the stock
• the market has overvalued the stock Let’s be humble and assume the former
• where did we go wrong?
Valuing IBM (4)
Sensitivity of our answer to discount rate:
Clearly, this is still not the answer
Discount rate Price
7%
8%
9%
11%
$42.9
$37.5
$33.3
$27.27
Valuing IBM (5)
What if the dividend is not constant ? Suppose the dividend were to grow at 4% per
year:• the next dividend will be $3
• in two years we will receive $3.12
• and so on … Can we derive the formula for a growing
perpetuity?• define g ≡ 4% the growth rate
• define C ≡ 3 the dividend received in year one
Dividend growth model
When dividends grow at a rate of g=4%, the cash flow diagram looks like as follows:
PV???
Yr1 Yr2 Yr3 Yr4 Yr5 Time=infinity
$3.0 $3*(1.04)∞$3.12 $3.37$3.24
Dividend growth model (2)
Based on the diagram, we have the math equation:
)1(
)1(
)1(
)1(
)1(
)1(1 3
2
2 r
gC
r
gC
r
gCrC
PV
Dividend growth (continue)
To calculate the PV of dividend flows with a growth, we can have some math exercise as follows:
grr
rg
S
SSS
S
rgrg
i
i
1
11
1
11
1
1;
1
,111
20
2
because
Dividend growth (continue)
How to calculate dividend perpetuity with a growth:
grC
grr
rC
SrC
rC
rg
rC
r
grg
rC
r
gC
r
gC
r
CPV
i
i
i
i
11
1111
1
)1(
)1(11
11
)1(
)1(
)1(
)1(
)1(
00
21
Dividend growth model (5) Do you think that this formula makes sense ?
When g increases, what will happen to the stock price?
When r increases, what will happen to the stock price?
When g =0, what happens? When g>r, what will happen to the stock price?
• In order to use the formula, r must be greater than g.
gr
CPV
Back to the valuation of IBM Sensitivity of our answer to growth rate of dividends Next year’s dividend is still $3.0 Discount rate is constant at 10%
Certainly, we are close, but g=5% is reasonable?
Growth rate Stock price
1%
2%
3%
4%
5%
$33.3
$37.5
$50.0
$60.0
$75.0
Sensitivity analysis with respect to discount and growth rates
Discount rate
Dividend Growth rate
1%
2%
3%
4%
5%
5.5%
6% 7% 8% 9% 10% 11% 12%
60
75
75100
60 50 43 38 33 30
273033384350
300
150
600
100
200
150
60 50
75
100
120
43
50
43
43
38
55
3338
46
60
6786
75
60 50
Stock price
Another way of looking at stock valuation
Suppose stock A pays dividend of $3 every year, with a discount rate of 10%. What is the stock price now in the following three cases• (a) hold it for ever
• (b) hold for five years
• (c) hold it for twenty years
Another example
Suppose stock A pays dividend of $3 next year, with a constant dividend growth rate of 5% and a discount rate of 10%. What is the stock price now in the following three cases• (a) hold it for ever
• (b) hold for one year
• (c) hold it for two years
More on the dividend discount model
So far, we have used the dividend cash flows to calculate the stock price.
In the real world, can we apply this formula to figure out the stock prices for all the stocks? How?
How to decide on the growth rate
If a firm chooses to pay a lower dividend, and reinvest the funds, the stock price may increase because future dividends may be higher.
Payout Ratio - Fraction of earnings paid out as dividends
Plowback Ratio - Fraction of earnings retained by the firm.
More on the dividend growth
Growth can be calculated by the return on equity times the plowback ratio
Let g= the dividend growth rate
g = return on equity X plowback ratio
Example
Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?
Solution
Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?
P0
5
1267
.$41.
No Growth With Growth
Solution
Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?
P0
5
1267
.$41.
No GrowthWith Growth
g
P
. . .
. .$75.
20 40 08
3
12 08000
The present value of growth opportunities
If the company did not plowback some earnings, the stock price would remain at $41.67. With the plowback, the price rose to $75.00.
The difference between these two numbers (75.00-41.67=33.33) is called the Present Value of Growth Opportunities (PVGO).
PVGO again
Present Value of Growth Opportunities (PVGO) - Net present value of a firm’s future investments.
Valuing Common Stocks
Expected rate of return - The percentage yield that an investor forecasts from a specific investment over a set period of time. Sometimes called the holding period return (HPR).
Expected return
Expected Return – the ratio of the profit over the initial cost
Here, P1 is the expected price in period 1, P0 is the current price and Div1 is the expected dividend payment in period 1.
Expected Return
rDiv P P
P1 1 0
0
An example
Example: A stock pays dividend of $3 every year. The current stock price is
$100. The expected price is $110 for the next year. If you hold the stock this year,
what is the expected rate of return?
Another example
Imagine Corporation has just paid a dividend of $0.40 per share. The dividends are expected to grow at 30% per year for the next two years and at 5% per year thereafter. If the required rate of return in the stock is 15%, calculate the current stock price.