1. A journey of discovery Involving Risk
The Cost of Equity
A journey of discovery
Involving Risk
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2. Definition of Risk The chance that the outcome will not be as expected
Question
Is risk a bad thing?
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3. Measures used Expected Return Cash Flow Probability Expected Return3, 3,
4, ,600 4, 5,
,000
(Mean)
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4. Measures Used Variance The average of the mean squared error terms
or in other words
The difference between the outcome as expected and the mean, then squared, then times the probability and then added up.
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5. Measures Used Example using numbers Prob CF ER CF-ER (CF-ER)2 x Prob
, , ,000, ,000
, , ,000
,000 1,
, , ,000
, , ,000, ,000
4, ,000
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6. Measures Used Is this a useful number?
Not to me but we need it to find the:-
Standard Deviation which is the square root of the variance
And this is a number that can be used
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7. Measures Used Following through the current example, with a
Variance of 300,000 then the
Standard Deviation (sd) is 547.7
We may use this to work out the chance of an
event happening. Assuming a normal (bell shaped)
distribution then we know that 68.46% of outcomes will be within one sd of the mean, 95.44% within two sds and 99.74% within 3 sds
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8. Measures used
Question. What probability is there that we will make a cash flow of 3,753 or more?
1) 3,753 is 247 away from the mean
2) 247 represents 247/547.7 = 45.0% of one standard deviation
3) Look in the normal probability distribution table
4) .45 of an sd = .3264, or area under the curve to the left of this point is 32.64% so area to the right must be 67.36
5) so there is a 67.36% chance we will make 3,753 or more
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9. Portfolio Theory
So far we have looked at the risk of one asset on its own
But normally assets are held as part of a portfolio - two or more assets
What happens to our risk measurements when there is more than one asset?
Question?
What would you do with £5,000,000 and
why?
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10. Portfolio Considerations
We have two questions about a portfolio
1)In a portfolio, what is the expected return of the portfolio?
2)In a portfolio of two (or more) assets, will the risk of variability be greater or smaller?
We had better find out
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11. Portfolio Expected Return
Luckily it is easy to work out as it is simply the weighted average of the returns of the assets in the portfolio.
So, two assets A and B
Expected Return on A = 5%
Expected Return on B = 14%
Portfolio made up of ¾ A and ¼ B
Return is .75 (5) (14) = 7.25%
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12. Variance of a Portfolio
But is it that simple for the variance?
Clearly not ER
Umbrellas ER ER
Cider
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13. Variance of a Portfolio
The riskiness of an asset held in a portfolio is different from that of an asset held on its own
Variance can be found using the following formula
Var Rp = w2Var(RA) + 2w(1-w)Cov(RARB)+(1-w)2VarRB
Cov stands for Covariance
Covariance is a measure of how random variables, A & B move away from their means at the same time
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14. Variance of a Portfolio continued
With regard to the formula, we know
* The weights (w) and (1-w) of the assets,
because we decide what they will be
*How to work out the variance of A and B
because we have just done it.
But
We just need the covariance and that is
easy to work out
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15. Variance of A and B and Covariance
Work out variance of each asset
A is a Steel company
Col:
Prob Return Expected col 2- ER (col4)2 x col 1
on steel Return
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16. Variance of A and B and Covariance
B is a Building Company
Prob Return Expected col 2- ER (col 4)2 x col 1
on Build Return
SD 14.75
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17. Variance of A and B and Covariance
To find the covariance we simply multiply column 4 from steel by column 4 from building and multiply by the probability and add them all
Prob Col 4 Steel Col 4 Build
x x =
x x =
x x =
x x =
x x =
Covariance
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18. Variance of the Portfolio
So A. ER = 5% Var = SD = 7.37
B. ER = 14% Var = SD = 14.75
Covar =
Create portfolio of 75% A and 25% B
ER = .75 x x 14 = 7.25
Now insert the figures into the formula
Var Rp = w2Var(RA) + 2w(1-w)Cov(RARB)+(1-w)2VarRB
= (.75)2 (54.4)+2(.75X.25)( )+(.25)2 (217.6)
= (-40.8)
Var = 3.4
SD =
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19. Now Try Portfolio ER% SD 100% A 5 7.37 100% B 14 14.75
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20. Answer Portfolio ER% SD 100% A 5 7.37 100% B 14 14.75
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21. Generalise from 2 Asset ModelC
E Rp A
Portfolio Opportunity Set
sd Rp A C
= Efficient Set
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22. Portfolio Opportunity Set Sd Rp
Capital Market Line
ERp
* Market Portfolio Rf
Portfolio Opportunity Set
Sd Rp
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24. Capital Asset Pricing Model CAPM
It was realised that total RISK could be split into two parts
Diversifiable or unsystematic risk and
Undiversifiable or systematic risk
In addition
It was recognised that if risk could be diversified away cheaply and easily then there should be no reward for taking it on
Now look at Table What do you notice
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25. CAPM
However even if you had a well diversified portfolio there is a risk, market risk, you could not diversify away because certain risks affect everything e.g. the state of the economy, the price of oil etc
However these factors do not affect everything to the same degree
Therefore a new measure has to be used which does not measure the total risk of an asset or a portfolio but which measures its risk relative to a well diversified portfolio
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26. CAPM This measure is called BETA
Beta = Covariance of Asset and Portfolio
Variance of the Market
Beta enables us to estimate the un-diversifiable risk of an asset and compare it with the un-diversifiable risk of a well diversified portfolio
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27. CAPM
Example
First we need the covariance between the asset and the market
We could work it out as we did for the covariance of assets A and B
We may also use the Correlation Coefficient, pa,m, and the SDs of the market and asset as follows
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28. CAPM Cov asset and market =pa,m sda sdm SD Stock A = 28.1
SD Market = 12
P = .6
Covariance = 202
Variance of the Market = 144
So Beta = 202 = 1.4
144
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29. CAPM
To work out what the return should be on any asset all we need do is work out what return we should be getting on a well diversified portfolio, work out the extra risk (beta) involved in the asset under consideration and stick the result into an equation
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30. CAPM The equation is ERA = RF + (ERM –RF)B Where RF = Risk Free Rate
ERM- RF = Premium expected for holding risky assets
Historically has been 6 to 7 % now considered closer to 3 or 4
B =Beta
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31. Security Market Line Rm Market Portfolio Rf 0 1.0 2.0 Beta
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32. CAPM Example Risk Free Rate = 7% Market Premium = 3%
Beta of asset = 1.4 (i.e. riskier than the market)
Then expected/required (note it is expected!) return on the asset is
ERA = 7 + (10-7)1.4 = 11.2
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33. CAPM So, nice and easy But Any Problems? Well, does it work?
Yes and No
What is the evidence?
Is there anything else?
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34. CAPM Empirical evidence shows higher risk/higher return BUT
not as high as predicted, the slope of the SML is flatter
Small company effect
Book value effect
Assumptions
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35. Cost of Equity Alternatives
The Arbitrage Pricing Model/Theory
ERA = RF + [S1 – RF]bj1 + ………[SK – RF]bjk
The Gordon Dividend Growth Model (B&M&A p65)
R= D1 + G Po
PO = today’s share price (£4.50), D1= the next dividend
( 30 pence)
G = estimate of growth in dividend (7%)
R = = 14%
450
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