risk+and+return part+1

7/30/2019 Risk+and+Return Part+1

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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.

risk+and+return part+1

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