copyright anbirts1 the cost of equity a journey of discovery involving risk

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The Cost of Equity

A journey of discovery

Involving Risk

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Definition of Risk

• The chance that the outcome will not be as expected

Question

Is risk a bad thing?

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Measures used

• Expected Return Cash Flow Probability Expected Return 3,000 .10 300 3,500 .20 700 4,000 .40 1,600 4,500 .20 900 5,000 .10 500 1.00 4,000 (Mean)

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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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Measures Used

• Example using numbers

Prob CF ER CF-ER (CF-ER)2 x Prob.10 3,000 300 -1,000 1,000,000 100,000

.20 3,500 700 - 500 250,000 50,000

.40 4,000 1,600 0 0 0

.20 4,500 900 500 250,000 50,000

.10 5,000 500 1,000 1,000,000 100,000

4,000 300,000

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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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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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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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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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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


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) + .25 (14) = 7.25%


Variance of a Portfolio• But is it that simple for the variance?

• Clearly not

Umbrellas

Cider

ER

ER

ER


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


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


Variance of A and B and Covariance

• Work out variance of each asset• A is a Steel companyCol:1 2 3 4 5 Prob Return Expected col 2- ER (col4)2 x col 1

on steel Return .2 -5.5 -1.1 -10.5 22.05 .2 .5 .1 - 4.5 4.05 .2 4.5 .9 - .5 .05 .2 9.5 1.9 4.5 4.05 .2 16.0 3.2 11.0 24.20 5.0 54.4



• B is a Building CompanyProb Return Expected col 2- ER (col 4)2 x col 1 on Build Return.2 35 7.0 21 88.2.2 23 4.6 9 16.2.2 15 3.0 1 .2.2 5 1.0 -9 16.2.2 -8 -1.6 -22 96.8

14.0 217.6

SD 14.75



• 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 .2 x -10.5 x 21 = -44.1 .2 x -4.5 x 9 = - 8.1 .2 x - .5 x 1 = - .1 .2 x 4.5 x -9 = - 8.1 .2 x 11.0 x -22 = -48.4 Covariance -108.8


Variance of the Portfolio

• So A. ER = 5% Var = 54.4 SD = 7.37• B. ER = 14% Var = 217.6 SD = 14.75• Covar = - 108.8• Create portfolio of 75% A and 25% B• ER = .75 x 5 + .25 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)(-108.8 )+(.25)2 (217.6)• = 30.6 + (-40.8) + 13.6• Var = 3.4• SD = 1.8439


Now Try

• Portfolio ER% SD 100% A 5 7.37

100% B 14 14.75

75% A 25% B 7.25 1.84

60% A 40% B

50% A 50% B

40% A 60% B


Answer

• Portfolio ER% SD 100% A 5 7.37

100% B 14 14.75

75% A / 25% B 7.25 1.84

60% A / 40% B 8.6 1.48

50% A / 50% B 9.5 3.7

40% A / 60% B 10.4 5.9


E Rp

Portfolio Opportunity Set o Rp

A

C

Generalise from 2 Asset Model

A C = Efficient Set

E Rp

Portfolio Opportunity Set sd Rp

A

C


A C = Efficient Set

E Rp


A

C


A C = Efficient Set

E Rp


A

C


A C = Efficient Set

E Rp


A

C


A C = Efficient Set

E Rp


A

C


A C = Efficient Set


* Market Portfolio

Capital Market Line

Rf

ERp

Sd RpPortfolio Opportunity Set


E Rp


A

C


A C = Efficient Set

Market Portfolio

Rf

CML

E Rp


A

C


A C = Efficient Set

Market Portfolio

Rf

CML


Capital Asset Pricing ModelCAPM

• 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 16.7. What do you notice


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


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


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


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


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


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


Security Market Line

Rm Market Portfolio

Rf

0 1.0 2.0 Beta


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


CAPM

• So, nice and easy• But Any Problems? Well, does it work? Yes and No

• What is the evidence?• Is there anything else?


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


Cost of EquityAlternatives

• 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 = 30 + .07 = 14% 450

copyright anbirts1 the cost of equity a journey of discovery involving risk

Documents

b slide

b return

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risk slide

mean slide

covariance b

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