1. Valuing Stocks 2
Helen Evans

2. Reminder
P0 = Div1 + P1
1+r
Last week we explained todays stock price in terms of the expected dividend, Div1 and the expected stock price P1
To use this equation, we need to know P1
But - Future stock prices are difficult to forecast
However, we can use the Dividend Discount Model

3. Worked Example
P0 = Div1 + P1
1+r
ABC stock has just paid a dividend of $10 and dividends are expected to grow by 5% p.a.
The share price in 1 years time is expected to be $110
Stocks which show a similar level of risk have an expected return of 20% p.a.
Calculate the current share price

4. Consider the 2 year investor
An investor who intends to sell the stock in 1 year would value the stock as:
P0 = Div1 + P1
1+r
An investor who intends to sell the stock in 2 years would value the stock as:
P0 = Div1 + Div2 + P2
1+r (1+r)^2
Where Div 2 is the dividend at the end of year 2, and P2 is the price at the end of year2.
An investor who intends to sell the stock in 3 years would value the stock as:
P0 = Div1 + Div Div3 + P3
1+r (1+r)^ (1+r)^3

5. Worked Example – ABC continued
P0 = Div1 + Div2 + P2
1+r (1+r)^2
ABC stock has just paid a dividend of $10 and dividends are expected to grow by 5% p.a.
The share price in 2 years time is expected to be $120
Stocks which show a similar level of risk have an expected return of 20% p.a.
Calculate the current share price

6. Dividend Discount Model
The current value of a stock,P0, is equal to the present value of the future cashflows which an investor receives from ownership
The current value of a stock is the present value of the dividends it will pay over the investment horizon plus the present value of the stock price at the time, t, that the investor intends to sell the stock
P0 = Div1 + Div2 + Div3 + …+Divt + Pt
1+r (1+r)^2 (1+r)^ (1+r)^t

7. What is the stock price at time t?
The value of the stock at time t, Pt, is equal to the present value of the future cashflows which an investor receives from ownership from time t onwards
The stock price at the horizon date, Pt, is determined by expectations of dividends from that that date forward
Pt = Div(t+1) + Div(t+2) + Div(t+3) + …
1+r (1+r)^ (1+r)^3

8. Add the two together P0 = Div1 + Div2 + Div3 + …+Divt + Pt
1+r (1+r)^2 (1+r)^ (1+r)^t
Pt = Div(t+1) + Div(t+2) + Div(t+3) + …
1+r (1+r)^ (1+r)^3
P0 = Div1 + Div2 + Div3 + …
1+r (1+r)^2 (1+r)^3

9. Dividend Discount Model
The value of a stock is the present value of all the future dividends
Assume dividend payments remain constant at the level Div1
Reminder
Value of a perpetuity paying £L p.a. = L/r
P0 = Div1 r

10. Gordon Growth Model P0 = Div1 r – g Constant growth formula
In reality, companies are expected to grow and be profitable, so dividends will increase
Assume that dividends grow at a constant rate,g
P0 = Div1 + Div1(1+g) + Div2(1+g)^2 + …
1+r (1+r)^ (1+r)^3
P0 = Div1
r – g
Constant growth formula
It can be shown that this simplifies to

11. Worked Example – ABC stock
ABC stock has just paid a dividend of £10
Annual dividend growth rate is 5%
Discount rate is 15%
P0 = 10 x (1.05)/(15% - 5%) = £105
Observations
Dividend is assumed to come at the end of the first period
Formula only valid if g<r

12. Task DEF stock has just paid a dividend of £25
Annual dividend growth rate is 10%
Discount rate is 20%
What is the fair price for DEF?
P0 = 25 x (1.1)/(20% - 10%) = £275

13. Expected Rate of Return
Constant growth formula can be rearranged to deduce expected rate of return
E(r) = Div1 + g P0
= dividend yield + growth rate
= expected return for all stocks with this level of risk

14. Task DEF stock has just paid a dividend of £25
Annual dividend growth rate is 10%
DEF is trading at £275
What is the discount rate for DEF?
E(r) = Div1 + g = 25x % = 20% P

15. Expected Rate of Return
E(r) = Div1 + g P0
= dividend yield + growth rate
= expected return for all stocks with this level of risk
ABC stock has just paid a dividend of £10
Annual dividend growth rate is 5%
Discount rate is 15%
Suppose ABC develops a new product which increases expected growth in dividends from 5% to 10%
Will the expected return change?

16. Will the expected return change?
Because the market is efficient, investors will assimilate this information immediately and the price will jump to
P0 = 10 x 1.1/(15% - 10%) = £220
E(r) = Div1/P0 + g
= 11/ %
= 15%
Risk has not changed, so expected return remains the same

17. Mature Stocks
In reality, stocks often exhibit high levels of growth for the first few years then reach a plateau with more stable levels of earnings
Constant growth formula is unsuitable
Technique
Value dividends over rapid growth period
Estimate stock price at end of rapid growth period
P0 = PV dividends in rapid growth period + PV stock price at end rapid growth period

18. Worked Example – XYZ stock
Investors require a 12% rate of return from XYZ stock
XYZ has just paid a dividend of £20
Over the next 5 years, dividends are expected to grow at a rate of 10%, after that the company is expected to provide a growth rate of 5%
What is the current price?

19. Solution
Technique
PV dividends (Y1 –Y5) = 20x x 1.1^ x 1.1^5
^ ^5
= 94.77
Using constant growth formula
P5(price at end y5) = Div6/(r-g)
= 1.05 x 20 x 1.1^5/(12% - 5%)
= 33.82/(12% - 5%) =
P0 = PV dividends (1-5) + PV (P5)
= /1.12^5
= £368.92

20. Task Investors require a 20% rate of return from HIJ stock
HIJ has just paid a dividend of £10
Over the next 3 years, dividends are expected to grow at a rate of 5%, after that the company is expected to provide a growth rate of 2%
What is the current price?

21. Solution
Technique
PV dividends (Y1 –Y3) = 10x x 1.05^ x 1.05^3
^ ^3
= 23.1
Using constant growth formula
P3(price at end y3) = Div4/(r-g)
= 1.02 x 10 x 1.05^3/(20% - 2%)
= 65.59
P0 = PV dividends (1-3) + PV (P3)
= /1.2^3
= £61.06

22. Summary Gordon Growth Model
Initial high level of growth often followed followed by long term mature, lower rate rate