Lecture 4

Lecture 4
Risk, Return &
Their Evaluations
(Individual assets & portfolios)
Financial Management(N12403)
Lecturer:
Farzad Javidanrad (Autumn 2014-2015)

Some Concepts in Statistics
Some Basic Concepts:
• Variable: A letter (symbol) which represents the elements of a specific set.
• Random Variable: A variable whose values are randomly appear based on
a probability distribution.
• Probability Distribution: A corresponding rule (function) which
corresponds a probability to the values of a random variable.
• Variables (including random variables) are divided into two general
categories:
1) Discrete Variables, and
2) Continuous Variables

• A discrete variable is the variable whose elements (values) can be
corresponded to the values of the natural numbers set or any subset of that.
So, it is possible to put an order and count its elements (values). The
number of elements can be finite or infinite.
• For a discrete variable it is not possible to define any neighbourhood,
whatever small, at any value in its domain. There is a jump from one value
to another value.
• If the elements of the domain of a variable can be corresponded to the
values of the real numbers set or any subset of that, the variable is called
continuous. It is not possible to order and count the elements of a
continuous variable. A variable is continuous if for any value in its domain a
neighbourhood, whatever small, can be defined.

• Probability Distribution: A rule (function) that associates a probability
either to all possible elements of a random variable (RV) individually or a
set of them in an interval.*
• For a discrete RV this rule associates a probability to each possible individual
outcome. For example, the probability distribution for occurrence of a Head
when filliping a fair coin:
𝒙 0 1
𝑃(𝑥) 0.5 0.5
In one trial 𝐻, 𝑇
𝒙 0 1 2
𝑃(𝑥) 0.25 0.5 0.25
In two trials
𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇
𝒙 = 𝑷𝒓𝒊𝒄𝒆 (+1) --- (0) (-1)
𝑃(𝑥) 0.6 0.1 0.3
Change in the price of
a share in one day
• The probability distribution for the price change of a share in stock market

• Expected Value (Probabilistic Mean Value): It is one of the most important
measures which shows the central tendency of the distribution. It is the
weighted average of all possible values of random variable 𝑥 and it is shown
by 𝐸(𝑥).
• For a discreet RV (with n possible outcomes)
𝐸 𝑥 = 𝑥1 𝑃 𝑥1 + 𝑥2 𝑃 𝑥2 + ⋯ + 𝑥 𝑛 𝑃 𝑥 𝑛 =
𝑖=1
𝑛
𝑥𝑖 𝑃(𝑥𝑖)
• For a continuous RV
𝐸 𝑥 =
−∞
+∞
𝑥. 𝑓 𝑥 𝑑𝑥
Where 𝑓 𝑥 is the probability density function (PDF) or simply probability
function and have different forms depending on the distribution.

• Properties of 𝐸(𝑥):
i. If 𝑐 is a constant then 𝐸 𝑐 = 𝑐 .
ii. If 𝑎 𝑎𝑛𝑑 𝑏 are constants then 𝐸 𝑎𝑥 + 𝑏 = 𝑎𝐸 𝑥 + 𝑏 .
iii. If 𝑎1, … , 𝑎 𝑛 are constants then
𝐸 𝑎1 𝑥1 + ⋯ + 𝑎 𝑛 𝑥 𝑛 = 𝑎1 𝐸 𝑥1 + ⋯ + 𝑎 𝑛 𝐸(𝑥 𝑛)
Or
𝐸(
𝑖=1
𝑛
𝑎𝑖 𝑥𝑖) =
𝑖=1
𝑛
𝑎𝑖 𝐸(𝑥𝑖)
iv. If 𝑥 𝑎𝑛𝑑 𝑦 are independent random variables then
𝐸 𝑥𝑦 = 𝐸 𝑥 . 𝐸 𝑦

v. If 𝑔 𝑥 is a function of random variable 𝑥 then
𝐸 𝑔 𝑥 = 𝑔 𝑥 . 𝑃(𝑥)
𝐸 𝑔 𝑥 = 𝑔 𝑥 . 𝑓 𝑥 𝑑𝑥
• Variance: To measure how random variable 𝑥 is dispersed around its expected
value, variance can help. If we show 𝐸 𝑥 = 𝜇 , then
𝑣𝑎𝑟 𝑥 = 𝜎2 = 𝐸[ 𝑥 − 𝐸 𝑥
2
]
= 𝐸[ 𝑥 − 𝜇 2]
= 𝐸[𝑥2 − 2𝑥𝜇 + 𝜇2]
= 𝐸 𝑥2 − 2𝜇𝐸 𝑥 + 𝜇2
= 𝐸 𝑥2 − 𝜇2

𝑣𝑎𝑟 𝑥 =
𝑖=1
𝑛
𝑥𝑖 − 𝜇 2. 𝑃(𝑥)
𝑣𝑎𝑟 𝑥 = −∞
+∞
𝑥𝑖 − 𝜇 2
. 𝑓 𝑥 𝑑𝑥
• Properties of Variance:
i. if 𝑐 is a constant then 𝑣𝑎𝑟 𝑐 = 0 .
ii. If 𝑎 and 𝑏 are constants then 𝑣𝑎𝑟 𝑎𝑥 + 𝑏 = 𝑎2
𝑣𝑎𝑟(𝑥) .
iii. If 𝑥 and 𝑦 are independent random variables then
𝑣𝑎𝑟 𝑥 ± 𝑦 = 𝑣𝑎𝑟 𝑥 + 𝑣𝑎𝑟(𝑦)
For discreet RV
For continuous RV

• Sample Mean and Sample Variance: The formulae for mean and variance in a sample are
different.
• Sample mean for data without frequency is the simple mean value:
𝑋 =
𝑥1+𝑥2+⋯+𝑥 𝑛
𝑛
= 𝑖=1
𝑛
𝑥 𝑖
𝑛
• And for a grouped data with frequency is:
𝑋 =
𝑥1 𝑓1 + 𝑥2 𝑓2 + ⋯ + 𝑥 𝑛 𝑓𝑛
𝑓1 + 𝑓2 + ⋯ + 𝑓𝑛
=
𝑖=1
𝑛
𝑥𝑖 𝑓𝑖
𝑛
• Sample variance, using Bessel’s correction (changing 𝑛 to (𝑛 − 1)) :
𝑆2
=
𝑖=1
𝑛
𝑥𝑖 − 𝑋 2
𝑛 − 1
And for grouped data with frequency:
𝑆2 =
𝑖=1
𝑛
𝑓𝑖 𝑥𝑖 − 𝑋 2
𝑛 − 1
And obviously, the standard
deviation will be:
𝑆 = 𝑆2

Correlation:
Is there any relation between:
 fast food sale and different seasons?
 specific crime and religion?
 smoking cigarette and lung cancer?
 maths score and overall score in exam?
 temperature and earthquake?
risk of one group of bonds with the risk of other group of bonds?
 To answer each question two sets of corresponding data need to be randomly
collected.
Let random variable "𝑥" represents the first group of data and random variable "𝑦"
represents the second.
Question: Is this true that students who have a better overall result are good in
maths?
Some Concepts in StatisticsSome Concepts in Statistics

Our aim is to find out whether there is any linear association between
𝑥 and 𝑦. In statistics, technical term for linear association is “correlation”.
So, we are looking to see if there is any correlation between two scores.
 “Linear association” : variables are in relations at their levels, i.e. 𝑥 with
𝑦 not with 𝑦2
, 𝑦3
,
1
𝑦
or even ∆𝑦.
Imagine we have a random sample of scores in a school as following:

In our example, the correlation between 𝑥 and 𝑦 can be shown in a scatter diagram:
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
Y
X
Correlation between maths score and overall score
The graph shows a positive
correlation between maths
scores and overall scores, i.e.
when x increases y increases
too.

Different scatter diagrams show different types of correlation:
• Is this enough? Are we happy?
Certainly not!! We think we know things better when they are described by
numbers!!!!
Although, scatter diagrams are informative but to find the degree (strength) of a
correlation between two variables we need a numerical measurement.
Adoptedfromwww.pdesas.org

Following the work of Francis Galton on regression line, in 1896 Karl Pearson
introduced a formula for measuring correlation between two variables, called
Correlation Coefficient or Pearson’s Correlation Coefficient.
For a sample of size 𝑛, sample correlation coefficient 𝑟𝑥𝑦 can be calculated by:
𝒓 𝒙𝒚 =
𝟏
𝒏
(𝒙𝒊 − 𝒙)(𝒚𝒊 − 𝒚)
𝟏
𝒏
(𝒙𝒊 − 𝒙) 𝟐 . 𝟏
𝒏
(𝒚𝒊 − 𝒚) 𝟐
=
𝒄𝒐𝒗(𝒙, 𝒚)
𝑺 𝒙 . 𝑺 𝒚
Where 𝑥 and 𝑦 are the mean values of 𝑥 and 𝑦 in the sample and 𝑆 represents the
biased version of “standard deviation”*. The covariance between 𝑥 and 𝑦, (𝑐𝑜𝑣 𝑥, 𝑦 )
shows how much 𝑥 and 𝑦 change together.

Alternatively, if there is an opportunity to observe all available data, the
population correlation coefficient (𝜌 𝑥𝑦) can be obtained by:
𝝆 𝒙𝒚 =
𝑬 𝒙𝒊 − 𝝁 𝒙 . (𝒚𝒊 − 𝝁 𝒚)
𝑬 𝒙𝒊 − 𝝁 𝒙
𝟐. 𝑬(𝒚𝒊 − 𝝁 𝒚) 𝟐
=
𝒄𝒐𝒗(𝒙, 𝒚)
𝝈 𝒙 . 𝝈 𝒚
Where 𝐸, 𝜇 and 𝜎 are expected value, mean and standard deviation of the
random variables, respectively and 𝑁 is the size of the population.
Question: Under what conditions can we use this population correlation
coefficient?

 If 𝒙 = 𝒂𝒚 + 𝒃 𝒓 𝒙𝒚 = 𝟏
Maximum (perfect) positive correlation.
 If 𝒙 = 𝒂𝒚 + 𝒃 𝒓 𝒙𝒚 = −𝟏
Maximum (perfect) negative correlation.
 If there is no linear association between 𝑥 and 𝑦 then 𝑟𝑥𝑦 = 0
Note 1: If there is no linear association between two random variables they might
have non linear association or no association at all.
For all 𝒂 , 𝒃 ∈ 𝑹
And 𝒂 > 𝟎
For all 𝒂 , 𝒃 ∈ 𝑹
And 𝒂 < 𝟎

• In our example, the sample correlation coefficient is:
𝒙𝒊 𝒚𝒊 𝒙𝒊 − 𝒙 𝒚𝒊 − 𝒚 𝒙𝒊 − 𝒙 . (𝒚𝒊 − 𝒚) (𝒙𝒊− 𝒙 ) 𝟐
(𝒚𝒊− 𝒚 ) 𝟐
70 73 12 13.9 166.8 144 193.21
85 90 27 30.9 834.3 729 954.81
22 31 -36 -28.1 1011.6 1296 789.61
66 50 8 -9.1 -72.8 64 82.81
15 31 -43 -28.1 1208.3 1849 789.61
58 50 0 -9.1 0 0 82.81
69 56 11 -3.1 -34.1 121 9.61
49 55 -9 -4.1 36.9 81 16.81
73 80 15 20.9 313.5 225 436.81
61 49 3 -10.1 -30.3 9 102.01
77 79 19 19.9 378.1 361 396.01
44 58 -14 -1.1 15.4 196 1.21
35 40 -23 -19.1 439.3 529 364.81
88 85 30 25.9 777 900 670.81
69 73 11 13.9 152.9 121 193.21
5196.9 6625 5084.15
𝒓 𝒙𝒚 =
𝟏
𝒏
(𝒙𝒊 − 𝒙)(𝒚𝒊 − 𝒚)
𝟏
𝒏
(𝒙𝒊 − 𝒙) 𝟐 . 𝟏
𝒏
(𝒚𝒊 − 𝒚) 𝟐
= 𝟓𝟏𝟗𝟔.𝟗
𝟔𝟔𝟐𝟓×𝟓𝟎𝟖𝟒.𝟏𝟓
=𝟎.𝟖𝟗𝟓
which shows an strong positive correlation between maths score and overall score.

Positive Linear
Association
No Linear
Association
Negative Linear
Association
𝑺 𝒙 > 𝑺 𝒚 𝑺 𝒙 = 𝑺 𝒚 𝑺 𝒙 < 𝑺 𝒚
𝒓 𝒙𝒚 = 𝟏
Adaptedandmodifiedfromwww.tice.agrocampus-ouest.fr
𝒓 𝒙𝒚 ≈ 𝟏
𝟎 < 𝒓 𝒙𝒚 < 𝟏
𝒓 𝒙𝒚 = 𝟎
−𝟏 < 𝒓 𝒙𝒚< 𝟎
𝒓 𝒙𝒚 ≈ −𝟏
𝒓 𝒙𝒚 = −𝟏
Perfect
Weak
No
Correlation
Weak
Strong
Perfect
Strong

Some properties of the correlation coefficient: (Sample or population)
a. It lies between -1 and 1, i.e. −1 ≤ 𝑟𝑥𝑦 ≤ 1.
b. It is symmetrical with respect to 𝑥 and 𝑦, i.e. 𝑟𝑥𝑦 = 𝑟𝑦𝑥 . This means the
direction of calculation is not important.
c. It is just a pure number and independent from the unit of measurement
of 𝑥 and 𝑦.
d. It is independent of the choice of origin and scale of 𝑥 and 𝑦’s
measurements, that is;
𝑟𝑥𝑦 = 𝑟 𝑎𝑥+𝑏 𝑐𝑦+𝑑 (𝑎, 𝑐 > 0)

e. 𝑓 𝑥, 𝑦 = 𝑓 𝑥 . 𝑓(𝑦) 𝑟𝑥𝑦 = 0
Important Note:
Many researchers wrongly construct a theory just based on a simple correlation
test.
 Correlation does not imply causation.
If there is a high correlation between number of smoked cigarettes and the number
of infected lung’s cells it does not necessarily mean that smoking causes lung
cancer. Causality test (sometimes called Granger causality test) is different from
correlation test.
In causality test it is important to know about the direction of causality (e.g. 𝒙 on 𝒚
and not vice versa) but in correlation we are trying to find if two variables moving
together (same or opposite directions).
𝒙 and 𝒚 are statistically independent, where
𝒇(𝒙, 𝒚) is the joint Probability Density
Function (PDF)

How to Bring Risk into Our Calculation?
• In all previous lectures we intentionally avoid talking about risk and taking it
into consideration but in the real world, no investment project can be defined
out of risk.
• Risk does exist because we are surrounded by uncertainty in our everyday life;
so, the question is how to measure it?
• Thousands years ago Babylonians developed a business insurance system for
their shipments. Romans also developed the idea of life insurance to protect
the family of a died person.
• The insurance business did not have much progress until some theoretical
advances happened in the probability theory. This theory allows us to quantify
the outcome of uncertain events based on the number of occurrences of those
events.

• The probability theory does not predict the time of events but it provides a base
to predict the likelihood of occurrence of events based on their relative
frequencies. Where;
𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒂𝒏 𝒆𝒗𝒆𝒏𝒕 =
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒕𝒉𝒂𝒕 𝒕𝒉𝒆 𝒆𝒗𝒆𝒏𝒕 𝒉𝒂𝒑𝒑𝒆𝒏𝒔
𝑻𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒆𝒗𝒆𝒏𝒕𝒔
• The relative frequency itself can be derived either from historical data or from a
theoretical model which assigns the same probability to equally likely events.
• According to the theoretical model the probability of having 6 when rolling a
fair dice is 𝑃 = 1/6; although, this is just an approximation but it is an
accurate approximation specifically when the number of trials increases (law of
large numbers).

• In our example we know all possible events; in fact the set of all possible
outcomes of a random experiment (sample space), which represents the
population is 1, 2, 3, 4, 5, 6 and we assume each one has an equally likely
chance to happen.
• But what happens when we do not know all possible events or the equally likely
assumption is not true? Here we need to focus on historical data.
• Historical data as a sample which comes from a population has enough
information to allow us to estimate and make some claims on parameters of the
population (i.e. mean value 𝜇 and variance 𝜎2).
• If the distribution of the random variable is known the claims on population
parameters can be tested easily using sample information.

• For example, sample mean 𝑋 is a good approximation for the population mean
𝜇 (or expected value of 𝑋; i.e. 𝐸 𝑋 ) and it gets much closer to that if the sample
size increases (theoretically 𝑛 → ∞).
• According to the law of large numbers (weak or strong version) we have:
𝑋 → 𝐸 𝑋 = 𝜇 𝑓𝑜𝑟 𝑛 → ∞
The law of large number is important
as it indicates a stable behaviour of 𝑋
around 𝐸 𝑋 = 𝜇 with increasing the
sample size.
Adopted from http://www.ats.ucla.edu/stat/stata/ado/teach/heads.htm

• All the developments of the probability theory caused a solid foundation to be
made for the analysis of raw data.
• Insurance companies use probability theory to work on historical raw data to
measure the risk involved in some events, such as accidents, natural disasters
and etc. These calculations allow them most of the time to be on a safe side and
make profit.
• In stock and bond markets risk could be source of extra returns but it might be
disastrous too, if the level of risk was not measured properly.

Broad View on Risk
• Risk, for those who do not want to be too much involved in the stock or bond
markets, is defined in terms of stability of returns and safe keeping the initial
investment. So:
Long-term government bonds The safest (very low risk)
corporate bonds & stocks paying
dividends
3rd safest
Non-dividend paying stocks
Short-term government bonds 2nd safest
4th safest

Mean-Variance Framework
• For those who are not risk averse government bonds are not very attractive. To
measure the level of risk involved in investing on non-government bonds, the
standard deviation of returns (or variance of returns) can be used as a natural and
correct measure of risk assuming the returns are distributed normally.
• Suppose we have a historical data for different assets’ returns over a period of 𝑛 years
𝑖 = 1,2,3, … , 𝑛 . For each asset’s return 𝑟𝑖 we need to calculate the mean and standard
deviation as following:
𝐸 𝑟𝑖 = 𝑟 =
𝑖=1
𝑛
𝑟𝑖
𝑛
𝑉𝑎𝑟 𝑟𝑖 = 𝜎2 = 𝐸 𝑟𝑖 − 𝐸(𝑟𝑖) 2 = 𝐸 𝑟𝑖 − 𝑟 2 =
𝑖=1
𝑛
𝑟𝑖 − 𝑟 2
𝑛 − 1
𝑆𝐷 𝑟𝑖 = 𝑣𝑎𝑟(𝑟𝑖) = 𝑖=1
𝑛
𝑟𝑖 − 𝑟 2
𝑛 − 1

• If there are two assets with the same expected returns; the one with the lowest
standard deviation, reflects the lowest volatility in returns.
Adopted from
http://www.pyramis.com/ecompendium/us/archive/2013/q2/articles/2013/q2/investing-strategies/alternative-for-pension-plans/index.shtml
Adopted from
http://seekingalpha.com/article/281569-letting-the-tail-wag-the-dog-transforming-extreme-risk-into-normal-risk

• Suppose you are going to buy 100 toys from China and sell it on-line. If during the sale period,
which is designed for one week, you find costumers for all toys you will gain 70% profit. In
case, you sell half of the toys you will lose 10% and if the sale is less than half, you will lose
50%. Imagine the probability for each scenario is 50%, 30% and 20%, respectively.
a) How much is the expected profit?
b) How much is the risk of this investment?
To answer part a) the expected profit is:
𝐸 𝑥 = 𝑥𝑖. 𝑃 𝑥𝑖 = 70 × 0.5 + −10 × 0.3 + −50 × 0.2 = 22%
And the variance is:
𝑉𝑎𝑟 𝑥 = 𝑥𝑖 − 𝐸 𝑥
2
. 𝑃 𝑥𝑖 = 70 − 22 2 × 0.5 + −10 − 22 2 × 0.3 + −50 − 22 2 × 0.2 = 2496
And the standard deviation is 𝑆𝐷 𝑥 = 2496 ≈ %49.95

• If there is another investment project with the same level of expected profit return but
less volatile it will be rational to invest in the second project.
• Mean-variance framework will be useful if the assumption of normally distributed
returns is true, but in many investment situations, returns are not normally
distributed.
• Even if the returns from different projects do not follow normal distribution but they
follow an identical distribution we can still use this framework.
• In reality, there are many investment projects. We can think of variety of different
investments with different expected returns with different risk levels.

• The mean-variance framework hints at diversification of assets in order to
reduce the level of risk associated to any specific asset because
1. At any given level of standard deviation, a portfolio of assets will almost
provide a higher return than an individual asset.
2. With diversification of assets and increasing number of them in a portfolio,
the unique risk related to an individual
asset moves toward the market risk,
which the latter is affected by state
of the whole economy.
Adopted from http://www.studyblue.com/notes/note/n/portfolio-theory-and-diversification/deck/889088
Standarddeviationas%=risk(percentage)
Number of assets

Portfolio Risk
• To understand how diversification reduce the level of risk we need to find the
relation between portfolio risk and individual financial asset risk.
• Suppose we have two assets 1 & 2 in a portfolio. Now, let’s:
 𝑟1= return from asset 1
 𝑟2= return from asset 2
 𝜎1= standard deviation related to asset 1 (risk of asset 1)
 𝜎2= standard deviation related to asset 2 (risk of asset 2)
𝜎𝑖 = 𝐸 𝑟𝑖 − 𝐸(𝑟𝑖) 2
 𝜎12= covariance between asset 1 and asset 2
𝜎12 = 𝑐𝑜𝑣(𝑟1, 𝑟2) = 𝐸 [𝑟1−𝐸 𝑟1 . [𝑟2 − 𝐸 𝑟2 ])
 𝜌= correlation coefficient between two assets (𝜌 =
𝜎12
𝜎1. 𝜎2
)

Portfolio Risk
• If 𝜔1and 𝜔2 are the proportions of assets (stock) 1 and 2 in the portfolio then
the expected return and the variance of the returns in the portfolio are:
𝐸 𝝎 𝟏 𝑟1 + 𝝎 𝟐 𝑟2 = 𝝎 𝟏 𝐸 𝑟1 + 𝝎 𝟐 𝐸(𝑟2)
𝑉𝑎𝑟 𝝎 𝟏 𝑟1 + 𝝎 𝟐 𝑟2 = 𝝎 𝟏
𝟐
. 𝜎1
2
+ 𝝎 𝟐
𝟐
. 𝜎2
2
+ 2𝝎 𝟏 𝝎 𝟐. 𝜎12
= 𝝎 𝟏
𝟐
. 𝜎1
2
+ 𝝎 𝟐
𝟐
. 𝜎2
2
+ 2𝝎 𝟏 𝝎 𝟐. 𝜌. 𝜎1. 𝜎2
• If two assets (stocks) are not correlated at all; 𝑐𝑜𝑣 𝑟1, 𝑟2 = 0, which means 𝜌 =
0. This rarely happens because, for example, in stock market a considerable
change in the price of one asset has an impact (weak or strong) on price of
other assets; therefore, each asset is assumed to be a perfect substitution for
another asset).
𝑐𝑜𝑣(𝑟1, 𝑟2)

Portfolio Risk
• For any value of 𝜌 (knowing that: −1 < 𝜌 < 1), variance of each individual
asset is bigger than the variance of the portfolio of assets.
• To find out why, assume 𝜔1 = 𝜔2 = 0.5 and also 𝜎1
2
= 𝜎2
2
= 𝜎∗
2
, so;
𝑉𝑎𝑟 𝝎 𝟏 𝑟1 + 𝝎 𝟐 𝑟2 = 𝝎 𝟏
𝟐
. 𝜎1
2
+ 𝝎 𝟐
𝟐
. 𝜎2
2
+ 2𝝎 𝟏 𝝎 𝟐. 𝜌. 𝜎1. 𝜎2
= 0.5𝜎∗
2 + 0.5𝜎∗
2. 𝜌
= 0.5𝜎∗
2
(1 + 𝜌)
• For all values of 𝜌, the value of 0.5(1 + 𝜌) is less than 1 or in an extreme case
when 𝜌 = 1 (perfect linear association between two assets) it is equal to 1.
• Therefore,
𝑉𝑎𝑟 𝑟1 > 𝑉𝑎𝑟 𝜔1 𝑟1 + 𝜔2 𝑟2
𝑉𝑎𝑟 𝑟2 > 𝑉𝑎𝑟 𝜔1 𝑟1 + 𝜔2 𝑟2

Portfolio Risk
• The story is the same when there are more than two assets in the portfolio. For
example, with three assets we have:
𝑉𝑎𝑟 𝝎 𝟏 𝑟1 + 𝝎 𝟐 𝑟2 + 𝝎 𝟑 𝑟3 = 𝝎 𝟏
𝟐
𝜎1
2
+ 𝝎 𝟐
𝟐
𝜎2
2
+ 𝝎 𝟑
𝟐
𝜎3
2
+ 2𝝎 𝟏 𝝎 𝟐 𝜎12 + 2𝝎 𝟏 𝝎 𝟑 𝜎13 + 2𝝎 𝟐 𝝎 𝟑 𝜎23
=
𝑖=1
3
𝝎𝒊
𝟐
𝜎𝑖
2
+ 2
𝑖=1
3
𝑗=2
3
𝝎𝒊. 𝝎𝒋 . 𝜎𝑖𝑗 (𝑖 < 𝑗)
=
𝑖=1
3
𝑗=1
3
𝝎𝒊. 𝝎𝒋 . 𝜌. 𝜎𝑖 . 𝜎𝑗 (𝑖 = 𝑗 → 𝜌 = 1)
• In case, we have 𝑛 assets, the portfolio variance (𝜎 𝑃
2
)will be:
𝜎 𝑃
2
=
𝑖=1
𝑛
𝑗=1
𝑛
𝝎𝒊. 𝝎𝒋 . 𝜌. 𝜎𝑖. 𝜎𝑗 (𝑖 = 𝑗 → 𝜌 = 1)
𝝆. 𝝈𝒊. 𝝈𝒋

Portfolio Risk
• If the proportion of each asset’s return in the portfolio is equal (𝜔𝑖 =
1
𝑛
) and
the variance related to each asset can be substitute with an average variance
(𝜎∗
2
) and for more simplification imagine that the values of covariance between
any two assets are the same (for example, equal to an average 𝜎𝑖𝑗); we have:
𝜎 𝑃
2
= 𝑛
1
𝑛
2
𝜎∗
2 +
𝑛 − 1
𝑛
. 𝜎𝑖𝑗
• It is obvious that 𝜎 𝑃
2
has an inverse relationship with the number of assets (𝑛)
in the portfolio.
• As the number of assets in the portfolio increases, the variance of the portfolio
will be more dependent on the covariance between assets’ returns and less
dependent on their individual variances.
𝜎 𝑃
2
→ 𝜎𝑖𝑗 𝑤ℎ𝑒𝑛 𝑛 → +∞
2
𝑖=1
3
𝑗=2
3
𝝎𝒊. 𝝎𝒋 . 𝜎𝑖𝑗 =
2
𝑛2
× 𝐶 𝑛
2
× 𝜎𝑖𝑗
=
𝑛 − 1
𝑛
. 𝜎𝑖𝑗

• The risk of a well-diversified portfolio depends on the [overall] market risk of the
[all] securities included in the portfolio. [Brealeyet al., p178]
• It is more important to know how an individual security contributes to the overall
portfolio’s risk rather than knowing how risky it is in isolation. This means that we
need to measure the market risk of a security, that is, how sensitive the security is to
market movements. The measure for this sensitivity is beta(𝜷):
𝛽 =
𝜎𝑖𝑚
𝜎 𝑚
2
Where 𝜎𝑖𝑚is the covariance between the security (stock) returns and the market returns
𝜎 𝑚
2 and is the variance of the returns in the market.
• Securities with 𝜷>𝟏 amplify the overall movements of the market, with 0<𝜷<𝟏 move
in the same direction as the market but slower than the market. 𝜷=𝟏 or close to that
represents the market portfolio.
Market Risk & Security (Asset) Beta

Market Risk & Security Beta
• This sensitivity measure also shows the security’s contribution to portfolio risk.
• To show this in a simple way imagine investors hold a combination of two assets in
their portfolio; one is the market portfolio (which represents the average risk) and the
second could be any asset added to the previous set.
• The risk of the first asset will be the risk it adds on to the market portfolio. If
 𝜎 𝑚
2
is the variance of the returns in the market portfolio (before adding a new asset)
and
 𝜎𝑖
2
is the variance of the individual asset being added to the portfolio (with proportion
𝜔)then
𝜎 𝑛𝑒𝑤
2 = 𝝎 𝟐 𝜎𝑖
2
+ 𝟏 − 𝝎 𝟐 𝜎 𝑚
2
+ 𝟐𝝎 𝟏 − 𝝎 𝜎𝑖𝑚
If the proportion of the new asset 𝜔 is really small (for e.g 0.01), 𝜎𝑖
2
can be ignored
(why?) and there is only 2𝜔 1 − 𝜔 proportion of covariance will be added to the new
portfolio variance.

Lecture 4

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