# A journey of discovery Involving Risk

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A journey of discovery Involving Risk
The Cost of Equity A journey of discovery Involving Risk copyright anbirts

Definition of Risk The chance that the outcome will not be as expected

Measures used Expected Return Cash Flow Probability Expected Return
3, 3, 4, ,600 4, 5, ,000 (Mean) copyright anbirts

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. copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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? copyright anbirts

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 copyright anbirts

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% copyright anbirts

Variance of a Portfolio
But is it that simple for the variance? Clearly not ER Umbrellas ER ER Cider copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 = copyright anbirts

Now Try Portfolio ER% SD 100% A 5 7.37 100% B 14 14.75

Answer Portfolio ER% SD 100% A 5 7.37 100% B 14 14.75

Generalise from 2 Asset Model
C E Rp A Portfolio Opportunity Set sd Rp A C = Efficient Set copyright anbirts

Portfolio Opportunity Set Sd Rp
Capital Market Line ERp * Market Portfolio Rf Portfolio Opportunity Set Sd Rp copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

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 copyright anbirts

CAPM So, nice and easy But Any Problems? Well, does it work?
Yes and No What is the evidence? Is there anything else? copyright anbirts

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 copyright anbirts

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 copyright anbirts

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