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24/08/2020
Foundations of Finance
Lecture 5
Diversification: Defining Risk,
Understanding its Relationship
with Return & Calculating Returns
1. Lecture Overview
In today’s lecture, we will introduce the concepts of
risky assets and diversification. During the course
of the lecture we will examine:
– How the risk of an asset is measured;
– The attitude of investors towards risk;
– The relationship between the risk of an asset and the return
required by investors on the asset;
– What diversification is;
– How to construct a diversified portfolio in practice; and,
– How to calculate realized and expected rates of return and risk
and how they differ.
1
24/08/2020
2. What is a Random Variable?
A random variable is one that can take on
any number of different values. Each value
has an associated probability of occurring.
An example is the return on a share over the
next year. Many different outcomes are
possible, and each possibility has an
associated
probability
of
occurring.
Therefore, the return on a share is a random
variable.
2. What is a Random Variable?
75
65
55
45
35
25
5
15
-5
-15
-25
-35
-45
-55
-65
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
-75
Probability density
The uncertainty associated with the outcome of a
random variable is described by a probability
distribution, which illustrates the relative likelihood of
each possible outcome occurring. The most commonly
used distribution is the normal distribution, an example
of which is provided below.
Stock market return
2
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2. What is a Random Variable?
There are 4 “moments” commonly used to describe the
shape or characteristics of a distribution:
– Mean: The “average” or “expected “value” of the distribution. For a
normal distribution, the mean is equal to the mode (value with the
highest probability of occurrence) and the median (the middle of all
possible values after ordering);
– Variance: A measure of how closely or widely individual values are
spread around the mean value. A small variance means that the
data doesn’t vary a lot from the mean. A large variance means that
the data is more spread out. The standard deviation, , is simply the
square root of the variance. Given its relative ease of interpretation,
standard deviation is often discussed instead of variance;
– Skewness: A measure of the lack of symmetry of a distribution about
its mean. By definition, the normal distribution has zero skewness
(ie the normal distribution is perfectly symmetrical); and,
– Kurtosis: A measure of the “tallness” or “flatness” of the distribution.
By definition, the normal distribution has zero excess kurtosis.
2. What is a Random Variable?
If we assume that the possible values future
values an asset could take are normally
distributed, we don’t need to worry about
describing the skewness or kurtosis of the
distribution, as we already know their values by
definition. Therefore, assuming returns are
normally distributed simplifies the process of
describing possible outcomes as we only need
to consider the mean (expected value) and
standard deviation or variance of these
outcomes.
3
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2. What is a Random Variable?
75
65
55
45
35
25
5
15
-5
-15
-25
-35
-45
-55
-65
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
-75
Probability density
Suppose the return on the stock market is normally distributed with mean
10% and standard deviation 20%. This distribution is illustrated in the
diagram below:
Stock market return
This distribution demonstrates that it is extremely likely that the return on
the market will be between –30% and +50% over the next year. Moreover,
there is high probability that the return will be between –10% and +30%.
That is, most of the mass (or area under the curve) is concentrated in this
region.
3. How Do We Calculate the Expected
Value of a Random Variable?
Recall that the value we expect a random variable
(X) to take is known as its expected value or mean.
The expected value of a random variable, E(r), is
calculated as:
N
E (r )   ri P(ri )
i 1
Where:
ri
= the ith possible outcome; and,
P(ri) = the probability the ith outcome will occur.
4
24/08/2020
3. How Do We Calculate the Expected
Value of a Random Variable?
Example: Expected
Return of a Random
Variable
Possible Outcomes
Probabilities of Outcomes
2
0.028
3
0.056
4
0.083
Consider
an
example
involving the roll of two
dice.
The
possible
outcomes (ie sum of
values obtained on each
roll of the dices) and
associated probabilities are
tabulated below:
5
0.110
6
0.139
7
0.168
8
0.139
9
0.110
10
0.083
11
0.056
12
0.028
EN JE Vi Phi
2
0.028
3
0.056
3. How Do We Calculate the Expected
Value of a Random Variable?
The expected return for the roll of the dice is
calculated as:
E (r )  (2 x0.028)  (3 x0.056)  (4 x0.083)  (5 x0.110)  (6 x0.139)  (7 x0.168) 
(8 x0.139)  (9 x0.110)  (10 x0.083  (11x0.056)  (12 x0.028)
 7.00
5
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3. How Do We Calculate the Expected
Value of a Random Variable?
• Example: Calculating the Expected Rate of
Return for an Investment in Ordinary Shares
3. How Do We Measure the Risk of a
Random Variable?
Standard deviation, , is commonly used as the measure of risk
of a random variable. Basically, the higher the standard
deviation, the further the future value of the random variable can
deviate from the expected value. The standard deviation of a
random variable is calculated as the square root of variance, or:
  2

N
 (r  E (r ))
i 1
i
2
P (ri )
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3. How Do We Measure the Risk of a
Random Variable?
Example: Standard Deviation of a Random Variable
Using the information tabulated in Slide 9, we can calculate the
variance and standard deviation of the value we expect to roll on
the two dice:
 2  (2  7)2 (0.028)  (3  7) 2 (0.056)  (4  7) 2 (0.083)  (5  7) 2 (0.110)  (6  7) 2 (0.139)  (7  7) 2 (0.168) 
(8  7) 2 (0.139)  (9  7) 2 (0.110)  (10  7) 2 (0.083)  (11  7) 2 (0.056)  (12  7) 2 (0.028)
 5.858
  2
 5.858
 2.42
4. What Attitude Do Investors Have
Towards Risk?
One of the main principles of microeconomics is that
economic agents can be modeled as risk averse utility
maximisers.
In finance, we continue to employ this
principle when determining how investors will evaluate risky
investment proposals. More specifically, we make two key
assumptions regarding investor preferences:
1. Investors prefer more wealth to less; and,
2. Investors are risk averse (ie they prefer less risk).
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24/08/2020
4. What Attitude Do Investors Have
Towards Risk?
A standard utility function that exhibits these two traits is included
below. The fact that the utility function is upward sloping indicates
that the investor prefers more to less, no matter how wealthy they
might become. Additionally, the fact that the utility function
increases at a decreasing rate indicates that the investor is
risk averse.
Utility
iii
t
I ii
Wealth
Itt
4. What Attitude Do Investors Have
Towards Risk?
The diagram below illustrates in more detail how a risk-averse investor
who prefers more to less may evaluate an investment proposal. In
particular, we can define an individual to be risk averse if they choose
not to undertake an investment that provides a 50/50 chance of an
increase or decrease in wealth of $x.
Utility
U (W0 + x)
U (W0 )
u
d
U (W0 – x)
W0 – x
W0
W0 + x
Wealth
8
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4. What Attitude Do Investors Have
Towards Risk?
For example, suppose that an investor is faced with a
gamble whereby he/she bets $100 on the toss of a coin. If
it’s heads he/she wins $200, if it’s tails he/she gets nothing.
The gamble creates a 50/50 chance of increasing or
decreasing his/her wealth by $100, so his/her expected
wealth is unchanged. This sort of a gamble is known as a
fair bet. However, if the investor is risk averse, he/she will
actually pay to avoid being subjected to this risk.
4. What Attitude Do Investors Have
Towards Risk?
The bet in the figure on Slide 15 can be summarized as:
Proposal
Outcome

  x


 – x
Prob.
50%
50%
The investor would be pleased if the outcome turns out to
be +x, which is illustrated by the increase in utility on the
graph (u). However, the investor would be extremely
displeased if the outcome turns out to be –x. The decrease
in the utility (d) when the outcome is –x is far greater than
the increase in the utility when the outcome is +x. This
discussion illustrates that in general a risk averse individual
will never take a fair bet.
9
24/08/2020
4. What Attitude Do Investors Have
Towards Risk?
The implication of this for investor behaviour is
that risk averse investors need to be paid to take
on risk. In the presence of risk and uncertainty
investors will demand a risk premium.
4. What Attitude Do Investors Have
Towards Risk?
Example: Demanding a Risk Premium
Imagine a risk averse investor only had a choice of investing in either or
both of the two assets described below:
Asset
E(R)

A
15%
12%
B
15%
15%
If the investor were risk averse, they would never invest in Asset B as it
has a higher level of risk than Asset A but the same return.
10
24/08/2020
5. Diversification
• Diversification allows an individual to reduce the risk of
their investment without sacrificing any expected return
simply by spreading their wealth over a portfolio
comprising a number of assets in an appropriate way.
• How does this relate to investor preferences?
– We know investors are risk averse and therefore prefer less risk
to more.
– Diversification provides a means of reducing risk faced by
investors without sacrificing expected return by combining assets
that don’t move perfectly together in a portfolio.
– Note that the ideas we are about to discuss can be extended to
consider more than 2 assets. You will consider these extensions
in FINM2003 Investments.
5.1 Measuring How Assets Move Together
Correlation and Covariance:
The covariance between variables X and Y, XY, is a
measure of association between the two variables. For
example, we may be interested in whether there is an
association between the return on a company’s stock
and the return on the stock market in general:
– If the market always went up at the same time company’s
stock went up, the covariance would be positive;
– If the return on the stock and the return on the market were
not associated in any way, the covariance would be near
zero; and,
– If when the market went up, the stock went down, the
covariance would be negative.
11
24/08/2020
5.1 Measuring How Assets Move Together
Correlation and Covariance (Continued):
However, covariance is sensitive to the scale of
measurement of X and Y and therefore the degree of
association (as opposed to the sign) is difficult to interpret.
Conversely, the correlation coefficient is a standardised
measure of association between two variables. It is
standardized as correlation measures must lie between
negative one and one. This makes it easy to gauge the
extent to which two variables are associated.
The
correlation coefficient, xy, is calculated as:
 X ,Y
X ,Y 
 X Y
left
Oxy
Oxy
Pxy
7 4087
O O
I
y
4
4
0
2
0.07
5.1 Measuring How Assets Move Together
Correlation:
We now consider the following exhaustive list of
correlation values between two assets, Asset 1 and
Asset 2, 12:
– Case 1: Perfect positive correlation (12=+1);
– Case 2: Perfect negative correlation (12=-1); and.
– Case 3: Non-perfect correlation (-1< 12<+1).
12
24/08/2020
5.1 Measuring How Assets Move Together
Case 1: Perfect positive correlation (12=+1)
Perfect Positive Correlation Between the Returns on Assets 1 and 2
0.18
0.16
Return on Asset 2
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Return on Asset 1
5.1 Measuring How Assets Move Together
Case 2: Perfect negative correlation (12=-1)
Perfect Negative Correlation Between Returns on Assets 1 and 2
0
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Return on Asset 2
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
Return on Asset 1
13
24/08/2020
5.1 Measuring How Assets Move Together
Case 3: Non-perfect correlation (-1< 12<+1)
Correlation of 0.46 Between Returns on Assets 1 and 2
0.16
0.14
Return on Asset 2
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Return on Asset 1
6. Diversification
The optimal investment strategy in terms of
constructing a portfolio to reduce risk depends
on the properties of the two assets and how
they are related to one another. Before we
discuss the “ideal” diversification properties, we
will go through how to calculate the expected
return, standard deviation and variance on a 2asset portfolio.
We will then use these
concepts to prove that diversification allows us
to reduce risk without sacrificing expected
return.
14
24/08/2020
6. Diversification
Consider the two assets below:
E[r]
Asset 1
E[r1]
Asset 2
E[r2]
σ2
σ1
σ
Two of the important characteristics of any asset are the:
1. Expected return; and,
2. Standard deviation around that expected return.
We know that investors prefer higher expected returns and lower
standard deviations (lower risk). Neither of the two assets in
the diagram is superior in both measures.
6.1 Portfolio Expected Return
Now consider forming a portfolio including both
Asset 1 and Asset 2 in some proportion. From
the properties of random variables reviewed
earlier in the lecture, the expected return of this
portfolio, E(Rp), is given by:
E ( R p )  w1 E ( R1 )  w2 E ( R2 )
Where:
w1
=Proportion invested in Asset 1;
w2
=Proportion invested in Asset 2;
E(R1) =Expected return on Asset 1; and,
E(R2) =Expected return on Asset 2.
y
IW.tw
Equally weighted0
W Wz
5
15
24/08/2020
6.1 Portfolio Expected Return
Example: Expected Return of a Portfolio
If a portfolio comprises 50% of Asset A and 50% of Asset B, and
the expected returns on these assets are 10% each, the expected
return on the portfolio is calculated as:
E ( R p )  (0.50 x0.10)  (0.50 x0.10)
 0.10
 10.0%
The fact that the expected return on the portfolio is 10% is
unsurprising given the return on both assets included in the
portfolio is also 10%.
4.2 Portfolio Variance and Standard Deviation
The variance of the expected return on a portfolio is the
weighted sum of the variances of the individual assets plus
the weighted sum of the covariances of the individual assets,
or:
 p 2  w12 12  w2 2 2 2  2w1w2 12
Substituting in the formula for covariance, we can also write
the portfolio variance equation as:
 p 2  w12 12  w2 2 2 2  2w1w2 12 1 2
16
24/08/2020
6.2 Portfolio Variance and Standard Deviation
The standard deviation of the portfolio, p which
provides us with a measure of its risk, is simply the
square root of its variance, or:
 p   p2
6.2 Portfolio Variance and Standard Deviation
Example: Variance and Standard Deviation of a
Portfolio
Return to the earlier example of a portfolio comprising 50% of
Asset A and 50% of Asset B, where the standard deviations of
these assets are both 12%. If the correlation between these assets
is 0.5, the variance and standard deviation of the portfolio are
calculated as:
 p 2  (0.5) 2 (0.12) 2  (0.5) 2 (0.12) 2  (2)(0.5)(0.5)(0.5)(0.12)(0.12)
 0.0108
 p  0.0108
 0.103923
 10.39%
17
24/08/2020
6.2 Portfolio Variance and Standard Deviation
The previous example illustrates a very
important idea: Combining assets that do
not move perfectly together in a portfolio
reduces variance and, therefore, standard
deviation without lowering expected return.
The question is how do we reduce risk by
the greatest amount?
6.2 Portfolio Variance and Standard Deviation
The ideal situation would be to combine assets
that move in opposite directions to one another (ie
have perfectly negative correlation). However,
diversification benefits are still available for assets
with less than perfect positive correlation, though
these benefits will decrease as the coefficient of
correlation between the assets increases.
18
24/08/2020
6.2 Portfolio Variance and Standard Deviation
We will now go on to illustrate this by considering
the (exhaustive) list of possible values of 1,2
discussed earlier and the resulting effects upon
diversification:
– Case 1: Perfect positive correlation (12=+1);
– Case 2: Perfect negative correlation (12=-1); and,
– Case 3: Non-perfect correlation (-1<12 <+1).
6.2 Portfolio Variance and Standard Deviation
We will continue with the example of the 2 asset
portfolio comprising 50% of Asset A and 50% of Asset
B, where the standard deviations of these assets are
both 12%. As the weighting of the assets remains
constant, so too will the portfolio’s expected return.
However, we will show that the risk of the portfolio
decreases as the correlation between the assets
approaches –1.
19
24/08/2020
6.2 Portfolio Variance and Standard Deviation
Case 1: Perfect Positive Correlation
The variance and standard deviation of the portfolio if the
assets are perfectly positively correlated is calculated as:
 p 2  (0.5) 2 (0.12) 2  (0.5) 2 (0.12)2  (2)(0.5)(0.5)(1.0)(0.12)(0.12)
 0.0144
 p  0.0144
 0.12
 0.12
6.2 Portfolio Variance and Standard Deviation
Case 2: Perfect Negative Correlation
The variance and standard deviation of the portfolio if the
assets are perfectly negatively correlated is calculated as:
 p 2  (0.5) 2 (0.12) 2  (0.5) 2 (0.12) 2  (2)(0.5)(0.5)( 1.0)(0.12)(0.12)
 0.0000
 p  0.0000
 0.00
 0.00%
20
24/08/2020
6.2 Portfolio Variance and Standard Deviation
Case 3: Non-Perfect Correlation
The variance and standard deviation of the portfolio if the
assets are non-perfectly correlated with AB=0.46 is
calculated as:
 p 2  (0.5) 2 (0.12) 2  (0.5) 2 (0.12) 2  (2)(0.5)(0.5)(0.46)(0.12)(0.12)
 0.010512
 p  0.010512
 0.1025
 10.25%
6.2 Portfolio Variance and Standard Deviation
42
21
24/08/2020
6.2 Portfolio Variance and Standard Deviation
The weights of each asset in a 2-asset portfolio
resulting in the LEAST possible risk can be
calculated using the minimum variance portfolio
equation:
  
    2  
2
w1 
1,2 1
2
2
2
1
2
1,2
2
1
2
and
w2  1  w1
6.3 Constructing a Diversified Portfolio
A number of commonsense procedures can be useful in
constructing a diversified portfolio. These procedures are detailed
below and involve selecting assets that are relatively unrelated.
1. Diversify across industries: Investing in a number of different
stocks within the same industry does not generate a diversified
portfolio since the returns of firms within an industry tend to be
highly correlated. Diversification benefits can be increased by
selecting stocks from different industries.
2. Diversify across industry groups: Some industries themselves
can be highly correlated with other industries and hence
diversification benefits can be maximized by selecting stocks from
those industries that tend to move in opposite directions or have
very little correlation with each other.
22
24/08/2020
6.3 Constructing a Diversified Portfolio
3. Diversify across geographical regions: Companies whose operations
are in the same geographical region are subject to the same risks in
terms of natural disasters and state or local tax changes. These risks
can be diversified by investing in companies whose operations are not
in the same geographical region.
4. Diversify across economies: Stocks in the same country tend to be
more highly correlated than stocks across different countries. This is
because many taxation and regulatory issues apply to all stocks in a
particular country. International diversification provides a means for
diversifying these risks.
5. Diversify across asset classes: Investing across asset classes such
as stocks, bonds, and real property also produces diversification
benefits. The returns of two stocks tend to be more highly correlated, on
average, than the returns of a stock and a bond or a stock and an
investment in real estate.
6.3 Constructing a Diversified Portfolio
46
23
24/08/2020
7. Realised Return
• Up until now in this lecture, we have only
considered the Expected Return of a
share (or asset).
• We are also interested with the Realised
Return of an investment:
– Realised Return (or Cash Return) measures
the gain or loss on an investment.
47
7. Realised Return
• Example: You invested in 1 share of BHP
for $14.63 a year ago and sold the share
today for $26.54. Assuming no dividends
were paid during the year, calculate the
following for your investment:
– Cash Return; and,
– Realised Return.
48
24
24/08/2020
7. Realised Return
• To calculate Cash Return:
• Therefore, given the information provided on BHP,
the Cash Return:
= $26.54 + 0 – $14.63
= $11.91
49
7. Realised Return
• We can also calculate the realised return (or rate of
return) as a percentage. Simply, it is the cash return
divided by the beginning stock price:
• Therefore, the rate of return of BHP over the year is:
50
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7. Realised Return
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7. Realised Return
• It can be seen that returns from investing in
stocks (or other investments) can be positive or
negative.
• This past performance is not generally an
indicator of future performance. If we want to
examine future performance, we consider
expected returns (as explored earlier in the
lecture) which takes into account the risk and
return relationship.
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8. Which return when?
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9. Arithmetic Vs. Geometric Average
Rates of Return
• We are all familiar with the arithmetic average rate of
return:
– It essentially answers the question, “what was the average of the
yearly rates of return?”
• However, we also need to consider the geometric
average:
– Answers the question: “what was the growth rate of your
investment?”
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9. Arithmetic Vs. Geometric Average
Rates of Return
• Simple Example (pp. 207, textbook):
Year
Annual rate of return
0
Value of the shares
$100
1
50%
$150
2
-50%
$75
• Arithmetic average
• Geometric average
= (50-50) ÷ 2 = 0%
= [(1+Ryear1) × (1+Ryear 2)]1/2 - 1
= [(1+0.5) x (1-0.5)] 1/2 - 1
= [(1.5) × (.5)] 1/2 - 1
= -13.4%
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9. Arithmetic Vs. Geometric Average
Rates of Return
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Next Week….
An important application of the material
discussed in this lecture is in the derivation
of the Capital Asset Pricing Model (CAPM).
We will discuss the CAPM, which provides
the basis for determining an appropriate
discount rate to use in valuing investment
proposals, in next week’s lecture.
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