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Capital market history + risk and
return
Chapter 12 and 13
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Learning objectives
How to calculate return on an Investment
The Historic returns of various types of investments
Risks of such Investments
Lessons from the study of capital markets History
How to calculate expected returns and variance of risky assets
Impact of diversification on portfolio risk and return
The systematic risk principle
How to measure systematic risk
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TheImportance of FinancialMarkets
Financial markets allow companies, governments and
individuals to increase their utility
Savers have the ability to invest in financial assets so
that they can defer consumption and earn a return to
compensate them for doing so
Borrowers have better access to the capital that is
available so that they can invest in productive assets
Financial markets also provide us with information about
the returns that are required for various levels of risk
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Returns
What makes up the total return?
- Return usually has two components,
- First, the cash that you receive directly while you ownthe investment(the income component).
- Second, the capital gain/loss gain/loss due to change
in the price of the asset
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Example
An investor bought 100 Anglo American shares at thebeginning of the year at R40 per share. At the end of the
year management decided to pay a dividend of R2 per
share. Also, the value of the share rises to R45 per share
by the end of the year.
Calculate the income component and capital gain/loss
component separately in money terms.
Calculate the dividend yield and capital gains yield
What is the total return in both money and percentage
terms?
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Solution The income component
Dividend per share multiplied by the number of
shares purchased
Income component = R2 * 100
= R200
Capital gain/(loss) = the change in share price multipliedby the number of shares purchased, i.e. (P1-
P0)*number of shares purchased
Where P1 is the price at the end of the year and P0 is the
price at the beginning of the year
Capital gain/(loss) = R(45 40)* 100
= R5 * 100
= R500
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Solution continued
Dividend yield = dividend per share/ price per share at
the beginning of the year multiplied by 100/1
Dividend yield = (2/40)* 100
= 5%
Capital gains yield = ((P1 P0)/P0)*100
Similar to slide no. 6, P1 is the price at the end of the year
and P0 is the price at the beginning of the year
Capital gain/(loss) yield =((45-40)/40)*100
= 12.5%
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Solution continuedTotal return in money terms = total dividend income received plus
total capital gains/(loss)
=R( 200 + 500)
= R700
Total return in % terms = (total dividend income received plus totalcapital gains/(loss))/amount invested)*100
= (700/4000)*100
= 17.5%
or dividend yield plus capital gains yield
= 5% + 12.5%
= 17.5%
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Average returns
Average return is the return that you expect to get from an
Asset per year on average.
Calculating Average returns.
-Add up the returns for the Asset over the period underconsideration
-Divide the total return over the period by the number of years in
the period.
To calculate real returns, you need to remove the effect of
inflation, thus
Real return = Nominal return the inflation rate
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Risk premium
The
extra
return earned for taking on risk
Treasury bills are considered to be risk-free, because they are
virtually free of any default risk over their short life.
The risk premium is the return over and above the risk-free rate
Or
it is the excess return or additional return earned by moving
from a relatively risk free investment to a risky one.
Risk premium can as well be interpreted as a reward for bearing
risk(hence, the name risk premium)
Risk premium = Expected return risk free return
Or Average return risk free return
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Variance and Standard Deviation
Variance and standard deviation measure the volatility of
asset returns
The greater the volatility the greater the uncertainty
Historical variance = sum of squared deviations from the
mean / (number of observations 1)
Standard deviation = square root of the variance
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Variance and Standard Deviation:example
Year Actual
return
Average
return
Deviation
from the
mean
Squared
deviation
1 0.15 0.105 0.045 0.002025
2 0.09 0.105 -0.015 0.000225
3 0.06 0.105 -0.045 0.002025
4 0.12 0.105 0.015 0.000225
Totals 0.42 0.00 0.0045
Variance = 0,0045 / (4-1) = 0,0015
Standard Deviation = 0,03873 or 3.87%
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Expected returns and variances
In the previous chapter we calculated returns and variances
based on historical data.
In this chapter, we analyse returns and variance when the
information we have concerns future possible returns and their
probabilities.
How do we calculate expected returns and variances given
future returns and their probabilities?
Lets assume that there are two states in the economy, i.e. the
boom and a recession.
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Calculation of expected return
Lets further assume that the boom and recession are equally
likely(i.e. a 50- 50 chance of each)
Refer to the table below for the information about Asset A and B.
State of the
economy
Probability of state of
economy
Share returns if state occurs
A B
Recession 0.5 -0.20 0.30
Boom 0.5 0.70 0.10
1.0
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Calculation of expected return
illustrated (share A)
State of the
economy
Probability of
state of
economy
Share returns
if state occurs
Calculation Result
Share A
Recession 0.5 -0.20 0.5*-0.20 -0.10
Boom 0.5 0.70 0.5*0.70 0.35
Expected
return0.25
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Risk premium
We defined risk premium as the difference between the return
on a risky investment and the risk free rate investment
Using projected returns, the expected risk premium is the
difference between expected return on a risky investment and
a return on a risk free investment.
Example
Assume the risk free rate is 8%Calculate the projected risk premium for share A and B based
on the expected returns for share A and B.
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Solution to example
Risk premium for share A.
Risk premium = expected return(A) Risk
free rate(Rf)
= E(RA) - Rf
= 0.25- 0.08
=0.17 or 17%
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End of todays lecture
Tomorrow we continue with the rest of the
content from chapter 13
Thank you
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Risk and return_part 2
Chapter 13
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Calculating variance an dstandard
deviation
To calculate Variance you need to do the following:
Determine the squared differences from the expected return
Multiply each possible squared deviation by its probability
Add up the results from the second step and what you get is
the variance.
The standard deviation, like before, is the square root of the
variance.
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An example
Use information provided for share A and B .
Note: expected returns on share A and B are 25% and 20%
respectively.
Also, for a given year, share A will return either -20% or 70%
while share B will return either 30% or 10%
Assume a 50-50 chance for each of the two states of the
economy(boom and recession)
Required: Calculate the variance and standard deviation for
share A and B
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Solution
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Solution continued
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An example with unequal probabilities
State of economy Probability of state
of economy
Return deviation
from expected
return
Squared deviation
from expected
return
Product of 2 & 4
Share A
Recession 0.8 -0.2-(-0.02) 0.0324 0.02592
Boom 0.2 0.7-(-0.02) 0.5184 0.10368
A2 = 0.12960
Share B
Recession 0.8 0.3-0.26 0.0016 0.00128
Boom 0.2 0.1-0.26 0.0256 0.00512B
2 =0.00640
Thus, the standard deviations for A and B
are 36% and 8% respectively
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Portfolios
Previous discussions focused on individual assets
Investors actually hold a portfolio of Assets
Hence, investors tend to own more than a single asset
Thus, a portfolio is a group of assets such as shares
and bonds held by an investor.
Given that Investors hold a portfolio rather than asingle asset, it becomes necessary to be able to
calculate portfolio returns and variances.
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Portfolio weights
Portfolio weights- percentage of a portfolios total
value in a particular asset.
Example: Assume we have R1000 in asset A and
4000 in Asset B, our total portfolio is worth R5000
Weight of Asset A = 1000/5000 = 20%
Weight for Asset B = 4000/5000 = 80%
Therefore, the portfolio weights are 0.2 and 0.8
Observation: weights should add up to 1.
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Portfolio expected returns
Assume that the portfolio is made up of Asset A and B.
Further assume that the weights are 0.2 and 0.8 for Asset A and
B respectively. Use the information in the table below to
calculate portfolio returns.
State of economy Probability of the
state of economy
Returns-Asset A Returns- Asset B
Recession 0.5 -0.2 0.3
Boom 0.5 0.7 0.1
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Steps for calculating portfolio returns
Calculate the expected return on each asset
Multiply the expected return on each asset by the
weight of that asset in a portfolio
Add up the results from the second step and the
result is the portfolio return.
Now, lets use the information in the table above and
the weights of asset A and B as assumed above.
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Example solution
Expected return on Asset A
E(RA) = 0.5*(-0.2) + 0.5*(0.7)= 0.25
Expected return on Asset B
E(RB) = 0.5*(0.3) + 0.5*(0.1)= 0.20
Portfolio weights are 0.2 and 0.8 for asset A and B
respectively
Portfolio return = WA*(E(RA) + WB*(E(RB)
Rp = 0.2*(0.25) + 0.8*(0.2) = 0.21 or 21%
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Portfolio variance and standard
deviation
Calculating portfolio Variance and standard deviation under
given economic conditions and their probabilities.
State of theeconomy
Probability of stateof the economy
Share A rate ofreturn if state
occurs
Share B rate ofreturn if state
occurs
Recession 0.10 -0.20 0.30
Normal 0.60 0.10 0.20
Boom 0.30 0.70 0.50
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Heres a comprehensive illustration
Suppose you have R20 000 in total.
If you put R6000 in share A and the remainder in B what is the
expected return, variance and standard deviation on:
(i) individual assets and (ii) on your portfolio?
Hint: for the portfolio, first of all calculate the portfolio weights, e.g.
Weight for share A = 6 000/20 000
WA = 0.3 or 30%
Therefore WB = 1- WA (note: your weights must add up to one)
WB = 1- 0.3
WB = 0.7 or 70%
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Example solution
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Solution continued
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Solution continued
Alternatively, calculate the portfoliosreturns in each of the states as below:State of the economy Probability of the state
of the economy
Portfolio returns if state
occurs
Recession 0.10 0.3*(-0.2)+0.7*(0.3) =
0.15
Normal 0.60 0.3*(0.1)+0.7*(0.2) =
0.17
Boom 0.30 0.3*(0.7)+0.7*(0.5) =
0.56
The Portfolios expected return = E(Rp)
E(Rp) = 0.1*(0.15) + 0.6*(0.17) + 0.3*(0.56) = 28.5%
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Solution continued
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Diversification
Diversification- when you have two or more assets ofdifferent classes in your portfolio.
The reason behind diversification is that it reduces portfoliorisk as measured by the portfolios standard deviation.
The extent of the reduction of risk depends on thecorrelation between the assets in the portfolio.
Correlation is the measure of the extent to which thereturns on two assets move together.
Correlation can be positive, negative or zero.
It ranges between -1 and 1
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The principle of diversification
The principle of diversification tells us that spreading aninvestment across assets( forming a portfolio) will eliminate
some of the risk.
However, note that there is a minimum level of risk that
can not be eliminated simply by diversification.
That risk which can not be eliminated by diversification is
called non diversifiable risk.
Note; diversification reduces risk but to a certain point.
Some risk is diversifiable while some is not.
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Systematic and unsystematic risk
Systematic risk , is the risk that influences a large number
of assets.
Because systematic risks are marketwide effects, they aresometimes known as market risks.
Unsystematic risk is the risk that affects at most a smaller
number of assets. This risk is sometimes called unique or
asset specific
Note the distinction between the above two types of risks
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Diversification and unsystematic risk
Unsystematic risk is that risk that is particular to a single
asset or at most a smaller group.
Value of the assets being affected by company specific
events
By holding a large portfolio, the value of the portfolio will go
up due to positive company specific events and some will
go down due to negative company specific events
Thus net effect will be relatively small as the effects tend to
cancel out each other.
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Hence, this is why some variability
associated with individual companies can
be eliminated due to diversificationBy combining assets into a portfolio, the
unique or unsystematic events both
positive and negative effects tend to cancel
out each other.
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Diversification and unsystematic risk
Note: Unsystematic risk is essentially eliminated by
diversification, thus a relatively large portfolio has almost
no unsystematic risk.
Unsystematic risk is also known as diversifiable risk.
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Systematic risk
This is the risk that can not be eliminated through
diversification
Why?
Because by definition, it affects all assets to somedegree.
Thus, it does not matter how many assets we have in a
portfolio, the systematic risk does not go away.
Systematic risk is also known as non-diversifiable risk.
Note: Total risk = systematic risk + unsystematic risk.