Financial Management (Lecture 4)

Financial Management(N12403)

Lecture 4 Risk, Return & Lecturer: Their Evaluations Farzad Javidanrad (Individual assets & portfolios)

(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

Some Concepts in Statistics

• 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.

Some Concepts in Statistics • 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

• The probability distribution for the price change of a share in stock market 𝒙 = 𝑷𝒓𝒊𝒄𝒆

(+1)

--- (0)

(-1)

𝑃(𝑥)

0.6

0.1

0.3

Change in the price of a share in one day

Some Concepts in Statistics

• 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)

𝑛

𝐸 𝑥 = 𝑥1 𝑃 𝑥1 + 𝑥2 𝑃 𝑥2 + ⋯ + 𝑥𝑛 𝑃 𝑥𝑛 =

𝑥𝑖 𝑃(𝑥𝑖 ) 𝑖=1

• 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.

Some Concepts in Statistics • 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 , … , 𝑎𝑛 are constants then 𝐸 𝑎1 𝑥1 + ⋯ + 𝑎𝑛 𝑥𝑛 = 𝑎1 𝐸 𝑥1 + ⋯ + 𝑎𝑛 𝐸(𝑥𝑛 ) Or

𝑛

𝐸(

𝑛

𝑎𝑖 𝑥𝑖 ) = 𝑖=1

𝑎𝑖 𝐸(𝑥𝑖 ) 𝑖=1

iv. If 𝑥 𝑎𝑛𝑑 𝑦 are independent random variables then 𝐸 𝑥𝑦 = 𝐸 𝑥 . 𝐸 𝑦

Some Concepts in Statistics 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

2

= 𝐸[ 𝑥 − 𝐸 𝑥 ] = 𝐸[ 𝑥 − 𝜇 2 ] = 𝐸[𝑥 2 − 2𝑥𝜇 + 𝜇2 ] = 𝐸 𝑥 2 − 2𝜇𝐸 𝑥 + 𝜇2 = 𝐸 𝑥 2 − 𝜇2

Some Concepts in Statistics 𝑛

For discreet RV

𝑥𝑖 − 𝜇 2 . 𝑃(𝑥)

𝑣𝑎𝑟 𝑥 = 𝑖=1

For continuous RV

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)

Some Concepts in Statistics • 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 +⋯+𝑥𝑛 𝑛

=

𝑛 𝑖=1 𝑥𝑖

𝑛

• And for a grouped data with frequency is:

𝑥1 𝑓1 + 𝑥2 𝑓2 + ⋯ + 𝑥𝑛 𝑓𝑛 𝑋= = 𝑓1 + 𝑓2 + ⋯ + 𝑓𝑛

𝑛 𝑖=1 𝑥𝑖 𝑓𝑖

𝑛

• Sample variance, using Bessel’s correction (changing 𝑛 to (𝑛 − 1)) :

𝑆2 = And for grouped data with frequency: 𝑆2 =

𝑛 𝑖=1

𝑥𝑖 − 𝑋 𝑛−1

𝑛 𝑖=1 𝑓𝑖

𝑥𝑖 − 𝑋 𝑛−1

2

2

And obviously, the standard deviation will be: 𝑆 = 𝑆2

Some Concepts in in Statistics Some Concepts Statistics 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?

SomeConcepts Conceptsin inStatistics Statistics Some Imagine we have a random sample of scores in a school as following:

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 1 2 3 𝑦 not with 𝑦 , 𝑦 , or even ∆𝑦. 𝑦

SomeConcepts ConceptsininStatistics Statistics Some In our example, the correlation between 𝑥 and 𝑦 can be shown in a scatter diagram:

Correlation between maths score and overall score 100

The graph shows a positive correlation between maths scores and overall scores, i.e. when x increases y increases too.

90 80 70

Y

60 50 40 30 20 10

0 0

10

20

30

40

50 X

60

70

80

90

100

SomeConcepts Conceptsin inStatistics Statistics Some Different scatter diagrams show different types of correlation:

Adopted from www.pdesas.org

• 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.

Some Concepts Concepts in in Statistics Statistics Some 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:

𝒓𝒙𝒚 =

𝒏 𝟏 (𝒙𝒊 𝒏 𝟏 (𝒙𝒊

− 𝒙)(𝒚𝒊 − 𝒚)

− 𝒙)𝟐 .

𝒄𝒐𝒗(𝒙, 𝒚) = 𝒏 𝟐 𝑺𝒙 . 𝑺𝒚 𝟏 (𝒚𝒊 − 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.

Some Concepts Concepts in in Statistics Statistics Some Alternatively, if there is an opportunity to observe all available data, the population correlation coefficient (𝜌𝑥𝑦 ) can be obtained by:

𝝆𝒙𝒚 =

𝑬 𝒙𝒊 − 𝝁𝒙 . (𝒚𝒊 − 𝝁𝒚 ) 𝑬 𝒙𝒊 − 𝝁𝒙 𝟐 . 𝑬(𝒚𝒊 − 𝝁𝒚 )𝟐

𝒄𝒐𝒗(𝒙, 𝒚) = 𝝈𝒙 . 𝝈𝒚

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?

Some Concepts in Statistics  If 𝒙 = 𝒂𝒚 + 𝒃

For all 𝒂 , 𝒃 ∈ 𝑹 And 𝒂 > 𝟎

𝒓𝒙𝒚 = 𝟏

Maximum (perfect) positive correlation.  If 𝒙 = 𝒂𝒚 + 𝒃

For all 𝒂 , 𝒃 ∈ 𝑹 And 𝒂 < 𝟎

𝒓𝒙𝒚 = −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.

Some SomeConcepts Conceptsin inStatistics Statistics • 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 85 22 66 15 58 69 49 73 61 77 44 35 88 69

73 90 31 50 31 50 56 55 80 49 79 58 40 85 73

12 27 -36 8 -43 0 11 -9 15 3 19 -14 -23 30 11

13.9 30.9 -28.1 -9.1 -28.1 -9.1 -3.1 -4.1 20.9 -10.1 19.9 -1.1 -19.1 25.9 13.9

166.8 834.3 1011.6 -72.8 1208.3 0 -34.1 36.9 313.5 -30.3 378.1 15.4 439.3 777 152.9

144 729 1296 64 1849 0 121 81 225 9 361 196 529 900 121

193.21 954.81 789.61 82.81 789.61 82.81 9.61 16.81 436.81 102.01 396.01 1.21 364.81 670.81 193.21

5196.9

6625

5084.15

𝒓𝒙𝒚 =

𝒏 𝟏 (𝒙𝒊 𝒏 𝟏 (𝒙𝒊

− 𝒙)(𝒚𝒊 − 𝒚)

− 𝒙)𝟐 .

𝒏 𝟏 (𝒚𝒊

− 𝒚)𝟐

=

𝟓𝟏𝟗𝟔.𝟗 =𝟎.𝟖𝟗𝟓 𝟔𝟔𝟐𝟓×𝟓𝟎𝟖𝟒.𝟏𝟓

which shows an strong positive correlation between maths score and overall score.

Some SomeConcepts Conceptsin inStatistics Statistics 𝑺𝒙 > 𝑺𝒚 𝒓𝒙𝒚 = 𝟏

𝑺𝒙 = 𝑺𝒚

𝑺𝒙 < 𝑺𝒚 Perfect

Adapted and modified from www.tice.agrocampus-ouest.fr

𝒓𝒙𝒚 ≈ 𝟏

Strong

𝟎 < 𝒓𝒙𝒚 < 𝟏

Weak

𝒓𝒙𝒚 = 𝟎 −𝟏 < 𝒓𝒙𝒚 < 𝟎

No Correlation

Positive Linear Association

No Linear Association

Weak

𝒓𝒙𝒚 ≈ −𝟏

Strong

𝒓𝒙𝒚 = −𝟏

Perfect

Negative Linear Association

Some Some Concepts Concepts in in Statistics Statistics 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)

Some SomeConcepts Conceptsin inStatistics Statistics e. 𝑓 𝑥, 𝑦 = 𝑓 𝑥 . 𝑓(𝑦)

𝑟𝑥𝑦 = 0 𝒙 and 𝒚 are statistically independent, where 𝒇(𝒙, 𝒚) is the joint Probability Density Function (PDF)

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).

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.

How to Bring Risk into Our Calculation? • 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; 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒕𝒉𝒂𝒕 𝒕𝒉𝒆 𝒆𝒗𝒆𝒏𝒕 𝒉𝒂𝒑𝒑𝒆𝒏𝒔 𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒂𝒏 𝒆𝒗𝒆𝒏𝒕 = 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).

How to Bring Risk into Our Calculation? • 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 𝜎 2 ). • If the distribution of the random variable is known the claims on population parameters can be tested easily using sample information.

How to Bring Risk into Our Calculation? • 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

How to Bring Risk into Our Calculation? • 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)

Short-term government bonds

2nd safest

corporate bonds & stocks paying dividends

3rd safest

Non-dividend paying stocks

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: 𝑛 𝑖=1 𝑟𝑖 𝐸 𝑟𝑖 = 𝑟 = 𝑛 𝑛 2 𝑟 − 𝑟 𝑖 𝑖=1 𝑉𝑎𝑟 𝑟𝑖 = 𝜎 2 = 𝐸 𝑟𝑖 − 𝐸(𝑟𝑖 ) 2 = 𝐸 𝑟𝑖 − 𝑟 2 = 𝑛−1 𝑆𝐷 𝑟𝑖 =

𝑣𝑎𝑟(𝑟𝑖 ) =

𝑛 𝑖=1

𝑟𝑖 − 𝑟 𝑛−1

2

Mean-Variance Framework • 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

Mean-Variance Framework • 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

And the standard deviation is 𝑆𝐷 𝑥 = 2496 ≈ %49.95

2

× 0.3 + −50 − 22

2

× 0.2 = 2496

Mean-Variance Framework • 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.

Mean-Variance Framework • 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.

Standard deviation as % = risk (percentage)

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.

Number of assets Adopted from http://www.studyblue.com/notes/note/n/portfolio-theory-and-diversification/deck/889088

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 )  𝜎12 = covariance between asset 1 and asset 2 𝜎12 = 𝑐𝑜𝑣(𝑟1 , 𝑟2 ) = 𝐸 [𝑟1 −𝐸 𝑟1

2

. [𝑟2 − 𝐸 𝑟2 ])

 𝜌= correlation coefficient between two assets (𝜌 =

𝜎12 ) 𝜎1 . 𝜎2

Portfolio Risk • If 𝜔1 and 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 𝝎𝟏 𝑟1 + 𝝎𝟐 𝑟2 = 𝝎𝟏 𝐸 𝑟1 + 𝝎𝟐 𝐸(𝑟2 ) 𝑉𝑎𝑟

𝝎𝟏 𝑟1 + 𝝎𝟐 𝑟2 = 𝝎𝟐𝟏 . 𝜎12 + 𝝎𝟐𝟐 . 𝜎22 + 2𝝎𝟏 𝝎𝟐 . 𝜎12 = 𝝎𝟐𝟏 . 𝜎12 + 𝝎𝟐𝟐 . 𝜎22 + 2𝝎𝟏 𝝎𝟐 . 𝜌. 𝜎1 . 𝜎2

𝑐𝑜𝑣(𝑟1 , 𝑟2 )

• 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).

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 𝜎12 = 𝜎22 = 𝜎∗2 , so;

𝑉𝑎𝑟 𝝎𝟏 𝑟1 + 𝝎𝟐 𝑟2 = 𝝎𝟐𝟏 . 𝜎12 + 𝝎𝟐𝟐 . 𝜎22 + 2𝝎𝟏 𝝎𝟐 . 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 𝝎𝟏 𝑟1 + 𝝎𝟐 𝑟2 + 𝝎𝟑 𝑟3 = 𝝎𝟐𝟏 𝜎12 + 𝝎𝟐𝟐 𝜎22 + 𝝎𝟐𝟑 𝜎32 + 2𝝎𝟏 𝝎𝟐 𝜎12 + 2𝝎𝟏 𝝎𝟑 𝜎13 + 2𝝎𝟐 𝝎𝟑 𝜎23 3

3

3

𝝎𝟐𝒊 𝜎𝑖2 + 2

= 𝑖=1

3

𝝎𝒊 . 𝝎𝒋 . 𝜎𝑖𝑗

(𝑖 < 𝑗)

𝑖=1 𝑗=2

𝝆. 𝝈𝒊 . 𝝈𝒋

3

=

𝝎𝒊 . 𝝎𝒋 . 𝜌. 𝜎𝑖 . 𝜎𝑗

(𝑖 = 𝑗 → 𝜌 = 1)

𝑖=1 𝑗=1

• In case, we have 𝑛 assets, the portfolio variance (𝜎𝑃2 )will be: 𝑛

𝑛

𝜎𝑃2 =

𝝎𝒊 . 𝝎𝒋 . 𝜌. 𝜎𝑖 . 𝜎𝑗 𝑖=1 𝑗=1

(𝑖 = 𝑗 → 𝜌 = 1)

Portfolio Risk 1 ) 𝑛

• If the proportion of each asset’s return in the portfolio is equal (𝜔𝑖 = 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: 1 2 𝜎𝑃 = 𝑛 𝑛

2

𝑛−1 2 𝜎∗ + . 𝜎𝑖𝑗 𝑛

3

3

2

𝝎𝒊 . 𝝎𝒋 . 𝜎𝑖𝑗 = 𝑖=1 𝑗=2

=

2 × 𝐶𝑛2 × 𝜎𝑖𝑗 𝑛2 𝑛−1 . 𝜎𝑖𝑗 𝑛

• It is obvious that 𝜎𝑃2 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. 𝜎𝑃2 → 𝜎𝑖𝑗 𝑤ℎ𝑒𝑛 𝑛 → +∞

Market Risk & Security (Asset) Beta • 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): 𝜎𝑖𝑚 𝛽= 2 𝜎𝑚 Where 𝜎𝑖𝑚 is the covariance between the security (stock) returns and the market returns 2 and is the variance of the returns in the market. 𝜎𝑚 • Securities with 𝜷>𝟏 amplify the overall movements of the market, with 0