analysing risk and return as basis of portfolio formation

1. Portfolio Management

3. Portfolio Investment
• Portfolio investments are investments in
the form of a group (portfolio) of
financial assets, including transactions in
equity, securities, such as common
stock, and debt securities, such as
bonds, certificate of deposits and
debentures

4. Portfolio Investment
• A portfolio investment is
• a passive investment of securities in a
portfolio
• It is made with the expectation of earning a
return
• The expected return is directly correlated
with the investment's expected risk

5. Portfolio Investment
Portfolio investment is distinct from direct
investment
• Direct investment involves taking a big
enough share ownership in a target
company
• Direct investment possibly involves with
day-to-day management of investment

6. Portfolio Management
Portfolio management is about the
knowledge and skills of making decisions
about investment mix, asset allocation for
individuals and institutions, and balancing
risk against return.

7. Portfolio Risk and Returns
• Investment in bonds and shares have
good returns
• At the same time there is high level of
risk attached to them
• So a good scientific and analytical skill is
needed to manage them

8. Portfolio Risk and Returns
• The classical advice is “never put all
your eggs in one basket”
• To an Investor: “Never put all your
investable funds in one security”.
• Investor should invest in a well
diversified portfolio to optimize the
overall risk-return profile

9. Goal of Portfolio Management
• An investor wants to reduce the overall
risk of his portfolio without diluting the
returns
• So in portfolio management we are
concerned with risk and return.
• We want to achieve a good balance of
risk and return

10. Rate of Return: Single asset
Rate of return
• The rate of return of an asset for a given
period (usually one year) is defined as:
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 + (𝐸𝑛𝑑𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒)
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
𝐂𝐮𝐫𝐫𝐞𝐧𝐭 𝐲𝐢𝐞𝐥𝐝
+
𝐸𝑛𝑑𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
𝐂𝐚𝐩𝐢𝐭𝐚𝐥 𝐠𝐚𝐢𝐧/𝐥𝐨𝐬𝐬 𝐲𝐢𝐞𝐥𝐝

11. Rate of Return: Single asset
Consider the following information about a
certain equity stock
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 =
240 + (6900 − 6000)
6000
= 0.19 = 19%
Price at the beginning of the year Po TZS 6000
Dividend paid at the end of the year D1 TZS 240
Price at the end of the year P1 TZS 6900

12. Rate of Return: Single asset
• The rate of return of 19% in our example
may be broken into current yield (profit)
and capital gain/loss
• 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 =
240
6000
+
6900−6000
6000
= 0.04 + 0.15 = 0.19 = 19%
• 4% is the current yield (profit) and 15%
the capital gain

13. Rate of Return: Single asset
Rate of Return by Probability Distribution
• When you invest in a stock you know that the
return from it can take various possible
values
• Furthermore, the likelihood of those possible
returns can vary
• So we can think in terms of a probability
distribution

14. Rate of Return: Single asset
• Recall: for a probability distribution
1. The possible outcomes must be mutually
exclusive and collectively exhaustive
2. The probability assigned to an outcome may
vary between 0 and 1
3. The sum of probabilities assigned to various
possible outcomes is 1

15. Rate of Return: Single asset
Consider two equity shares: TCC share and
TOL share. TCC share may provide a return of
16%, 11% or 6% with certain probabilities
associated with them based on the state of
the economy. TOL share may earn a return of
40%, 10%, -20% or with the same probabilities
based on the state of the economy.

16. Rate of Return: Single asset
• The probability distributions of the two
share are shown in the following chart:
The expected rate of return is the weighted average
of all possible returns multiplied by their respective
probabilities
State of the
economy
Probability of
occurrence
Rate of Return (%)
TCC TOL
Boom 0.30 16 40
Normal 0.50 11 10
Recession 0.20 6 -20

17. Rate of Return: Single asset
• 𝐸 𝑅 = where
• E(R) = Expected return
• Ri = Return for the ith possible outcome
• Pi = Probability associated with Ri
• n = number of possible outcomes
• So E(R) is the weighted average of possible
outcomes (weight – associated probability)

18. Rate of Return: Single asset
Expected Return TCC stock
State of the Economy 𝑝𝑖 𝑅𝑖 𝑝𝑖𝑅𝑖
1. Boom 0.30 16 4.8
1. Normal 0.50 11 5.5
1. Recession 0.20 6 1.2
E(R) = ΣpiRi =11.5%
Expected Return TOL stock
State of the Economy 𝑝𝑖 𝑅𝑖 𝑝𝑖𝑅𝑖
1. Boom 0.30 40 12.0
1. Normal 0.50 10 5.0
1. Recession 0.20 -20 -4.0
E(R) = ΣpiRi =13.0%

19. Rate of Return on a Portfolio
• The expected return on a portfolio is
simply the weighted average of the
expected returns on the assets
comprising the portfolio
• When a portfolio consists of two
securities
• 𝐸 𝑅𝑝 = 𝑤1𝐸 𝑅1 + 1 − 𝑤1 𝐸(𝑅2)

20. Expected Return on a Portfolio
Where
𝐸 𝑅𝑝 = expected return on a portfolio
𝑤1 = proportion of a portfolio invested in security 1
𝐸 𝑅1 = expected return on security 1
1 − 𝑤1 = Proportion of a portfolio invested in
security 2
𝐸(𝑅2)= expected return on security 2

21. Rate of Return on a Portfolio
Consider a portfolio consisting of two
securities A and B. The expected returns
on these two securities are 10% and 18%
respectively. If the proportion of the
portfolio invested in A and B are 40% and
60% respectively. What is the expected
return on the portfolio?

22. Rate of Return on a Portfolio
Solution
𝐸 𝑅𝑝 = 𝑤1𝐸 𝑅1 + 1 − 𝑤 𝐸(𝑅2)
𝐸 𝑅𝑝 = 0.4 10% + 0.6(18% = 14.8%
• In general when a portfolio consists of n
securities
𝐸 𝑅𝑝 =
𝑖=1
𝑛
𝑤𝑖𝐸(𝑅𝑖)

23. Risk
• Risk is the extent to which actual returns
deviate from expected returns
• It is measured by the variance/standard
deviation
• The variance of a probability distribution is
given by the formula
• 𝜎2 = 𝑝𝑖 {𝑅𝑖 − 𝐸 𝑅 }2,
• Where 𝜎2 = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

24. Risk
𝜎 = (𝜎2
)
1
2 where 𝜎 is standard deviation
• The basic purpose to calculate the
standard deviation is to measure the
extent of variability of possible returns
from the expected return
• Several other measures can be used, but
standard deviation is the most popular

25. Calculation of Risk
Given the following:
• Calculate the risk of the two securities
𝜎2
= 𝑝𝑖 {𝑅𝑖 − 𝐸 𝑅 }2

26. Calculation of Risk

27. Calculation of Risk

28. Types of Risk
• Standard deviation measures the total risk
associated with a security
• The total risk is made up of two components:
• Systematic and Non-systematic risk
• Non-systematic risk is the risk specific to a
company
• Business risk and financial risk

29. Types of Risk
• Non-systematic risk is associated with the
security of a particular company, and can be
eliminated/reduced by combining it with
another security having negative correlation
• This is the process known as the
diversification of non-systematic risk

30. Types of Risk
• Systematic Risk: is the variability in security returns
caused by changes in the economy or the market
• Factors such as interest rates, inflation and state of the
market determine systematic risk
• All securities are affected by such changes to some
extent
• Some to a great extent and others to a less extent
• More sensitive securities have higher systematic risk
• It can be measured by relating that security variability
vis a vis variability in the stock market index

31. Risk
• Through Diversification, by combining many securities
in a portfolio, the non-systematic risk can be
eliminated or substantially mitigated
• However, ultimately when the size of the portfolio
reaches a certain limit it will contain only the
systematic risk of securities included in the portfolio

32. Non-Systematic
Risk
Systematic
risk
1 5 10
No. of Securities
Risk
Relationship between Diversification and Risk

33. Portfolio Risk
• The variance and standard deviation of
return are the statistical measures of
risk in investment
• The variance of a portfolio can be
written as the sum of 2 terms

34. Diversification And Portfolio Risk
Suppose you have TZS 1,000,000 to invest and you want
to invest it equally in two stocks A and B.
• The return on these stocks depends on the state of the
economy.
• Your assessment suggests that probability distribution
of the returns on stocks A and B are shown above.

35. Diversification And Portfolio Risk
• For the sake of simplicity all the five states of the economy
are assumed to be equi-probable.
• We can calculate the return on a portfolio consisting of
stocks A and B in equal proportions.

36. Diversification And Portfolio Risk
15
-5
5
35
25
-5
15
25
5
35
5 5
15
20
30
1 2 3 4 5
Stock A Stock B Portfolio

37. Diversification And Portfolio Risk
𝜎𝑝
2 = 𝑤A
2
𝜎A
2
+ 𝑤B
2
𝜎B
2
+ 2𝑤A𝑤B𝜌AB𝜎A𝜎B
𝑃𝑖 𝑅𝑖 𝑃𝑖𝑅𝑖 𝑅𝑖 − 𝐸 𝑅𝐴 [𝑅𝑖 − 𝐸 𝑅𝐴 ]2
𝑃𝑖[𝑅𝑖 − 𝐸(𝑅)]2
0.2 15 3 0 0 0
0.2 -5 -1 -20 400 80
0.2 5 1 -10 100 20
0.2 35 7 20 400 80
0.2 25 5 10 100 20
𝐸 𝑅𝐴 = 𝑃𝑖𝑅𝑖 = 15 𝑃𝑖[𝑅𝑖 − 𝐸(𝑅)]2
= 𝜎𝐴
2
= 200
𝜎𝐴 = (𝜎𝐴
2
)
1
2 = 200
1
2 = 14.14

38. Diversification And Portfolio Risk
𝜎𝑝
2
= 𝑤A
2
𝜎A
2
+ 𝑤B
2
𝜎B
2
+ 2𝑤A𝑤B𝜌AB𝜎A𝜎B
𝑃𝑖 𝑅𝑖 𝑃𝑖𝑅𝑖 𝑅𝑖 − 𝐸 𝑅𝐵 [𝑅𝑖 − 𝐸 𝑅𝐵 ]2 𝑃𝑖[𝑅𝑖 − 𝐸(𝑅𝐵)]2
0.2 -5 -1 20 400 80
0.2 15 3 0 0 0
0.2 25 5 10 100 20
0.2 5 1 -10 100 20
0.2 35 7 20 400 80
𝐸 𝑅𝐵 = 𝑃𝑖𝑅𝑖 = 15 𝑃𝑖[𝑅𝑖 − 𝐸(𝑅𝐵)]2 = 𝜎𝐵
2
= 200
𝜎𝐵 = (𝜎𝐵
2
)
1
2 = 200
1
2 = 14.14

39. Diversification And Portfolio Risk
Portfolio risk The 2 Security Case
𝜎𝑝
2
= 𝑤A
2
𝜎A
2
+ 𝑤B
2
𝜎B
2
+ 2𝑤A𝑤B𝜌AB𝜎A𝜎B
𝜌AB =
𝐶𝑜𝑣𝐴𝐵
𝜎A𝜎B
=
𝑃𝑖 [𝑅𝐴 − 𝑅𝐴][𝑅𝐵−𝑅𝐵]
𝜎A𝜎B
Security 1 Security 2
Security A 𝑤A
2
𝜎A
2 𝑤A𝑤B𝜎AB
Security B 𝑤B𝑤A𝜎AB 𝑤B
2
𝜎B
2

40. Diversification And Portfolio Risk
𝜌AB =
𝐶𝑜𝑣𝐴𝐵
𝜎A𝜎B
=
𝑃𝑖[𝑅𝐴 − 𝑅𝐴][𝑅𝐵−𝑅𝐵]
𝜎A𝜎B
=
−20
14.14 × 14.14
=
−20
200
= −0.1
𝜎𝑝
2 = 𝑤A
2
𝜎A
2
+ 𝑤B
2
𝜎B
2
+ 2𝑤A𝑤B𝜌AB𝜎A𝜎B
= 0.25 × 200 + 0.25 × 200 + {2 0.5 0.5 −0.1 14.14 14.14 } = 90
𝑃𝑖 𝑅𝑖 𝑃𝑖𝑅𝑖 𝑅𝑖 − 𝐸 𝑅𝐴 𝑃𝑖 𝑅𝑖 𝑃𝑖𝑅𝑖 𝑅𝑖 − 𝐸 𝑅𝐵 [𝑅𝐴−𝑅𝐴][𝑅𝐵−𝑅𝐵
0.2 15 3 0 0.2 -5 -1 20 0
0.2 -5 -1 -20 0.2 15 3 0 0
0.2 5 1 -10 0.2 25 5 10 -100
0.2 35 7 20 0.2 5 1 -10 -200
0.2 25 5 10 0.2 35 7 20 200
𝐸 𝑅𝐴 = 𝑃𝑖𝑅𝑖 = 15 𝐸 𝑅𝐵 = 𝑃𝑖𝑅𝑖 = 15 =-100

41. Diversification And Portfolio Risk
Return and risk of a portfolio depends on the following
two sets of factors
1. Returns and risks of individual securities and the
covariance between the securities forming the
portfolio
2. Proportion of investment in each security

42. Diversification & Reduction or dilution of Portfolio risk
• The process of combining more than one security in a
portfolio is known as diversification
• The main purpose of diversification is to reduce or
dilute the total risk without sacrificing portfolio return
1. If securities returns are perfectly positively correlated,
the correlation coefficient 𝜌AB = +1 and the returns
of the securities move up or down together
𝜎𝑝 = 𝑤𝐴𝜎𝐴 + 𝑤𝐵𝜎𝐵
2. If the securities returns are perfectly negatively
correlated

43. Diversification & Reduction or dilution of Portfolio risk
• The two returns always move in exactly opposite
direction and the correlation coefficient becomes -1
• 𝜎𝑝 = 𝑤𝐴𝜎𝐴 − 𝑤𝐵𝜎𝐵
3. If Securities returns are not correlated i.e. they are
independent, the coefficient of correlation of these
two securities would be zero
𝜎𝑝 = 𝑤𝐴
2𝜎𝐴
2 + 𝑤𝐵
2𝜎𝐵
2

44. Portfolio with More Than Two Securities
• The formula for calculation of expected portfolio
return is the same for a portfolio with two securities
𝐸 𝑅𝑝 =
𝑖=1
𝑛
𝑤𝑖𝐸(𝑅𝑖)

45. Market Risk
• Market risk of a security reflects its sensitivity to the
market movements
• Different securities seem to display differing
sensitivities to the market movement
• The sensitivity of a security to market movement is
called beta (𝛽)

46. Market Risk
• Beta 𝛽 measures the extent to which the return on a
security fluctuates with the returns on the market
portfolio
• The beta for the market is, by definition, 1
• A security which has a beta of, say, 1.5 experiences
greater fluctuation than the market portfolio
• More precisely, if the return on market portfolio is
expected to increase by 10%, the return on the
security with beta of 1.5 is expected to increase by
15% (1.5x10%)

47. Market Risk
• On the other hand, a security which has a beta of, say,
0.8 fluctuates lesser than the market portfolio.
• If the return on the market portfolio is expected to rise
by 10%, the return on the security with a beta of 0.8 is
expected to rise by 8% (0.8x10%).
• Individual security betas generally fall in the range of
0.3 to 2.0 and rarely assumes a negative value

48. Capital Assets Pricing Model
• The Capital Assets Pricing Model is given by the following
equation
• 𝑅𝑖 = 𝑅𝑓 + 𝛽 𝑅𝑚 − 𝑅𝑓
• Ri is the required return of an investment
• Rf is the risk free rate (the rate of return on government
securities)
• Rm is the market return (average returns of all risky assets in
the market)
• 𝛽 is the measure of the sensitivity or responsiveness of the
security returns to the general market returns

49. Security Market Line
• Security market line is the graphical representation of the
CAPM
• The line indicates the rate of return required to compensate
the given level of risk

50. International Diversification
• Experience show that diversification across industries
lead to a lower level of risk for a given level of
expected returns.
• Fully diversified domestic portfolio is about 27% less
risky as a typical individual stock
• However, ultimately the advantages of such
diversification are limited because all companies in a
country are subject to the same business cycles
• Through international diversification, the variability of
returns can be reduced further

51. International Diversification
• The risk that is systematic in the domestic economy
may be unsystematic in the context of the global
economy
• e.g. oil crisis helps oil exporting countries while
hurts non oil countries
• The standard deviation in a fully internationally
diversified portfolio appear to be as little as 11.7% of
that of individual securities

52. International Diversification

53. Benefits of International Investing
1. International focus offers more opportunity than
domestic investment
• This is because investment available within a
country offer only a small percentage of investment
in the global market
2. The expanded available securities suggest the
possibility of achieving better risk-return trade off
than by investing solely in domestic securities
• Higher return for the same level of risk or less risk
for the same level of return

54. Benefits of International Investing
• The broader the internationalization of the
portfolio, the more stable are the returns
and the less is the risk
Optimal International asset Allocation
• International diversification that combines
stock and bond investments is substantially
less risky than international stock
diversification alone

55. Efficient Frontier and Diversification
• Efficient Frontier is the set of portfolios that has the
smallest possible standard of risk
• The graph in the next slide illustrates the effect of
international diversification on the efficient frontier
• International diversification pushes out the efficient
frontier, allowing investors to reduce their risk and
increase their expected return

56. Efficient Frontier and Diversification

57. Barriers to International Diversification
• The following are barriers of investing overseas
1. Legal, informational and economic impediments that
segment national capital markets preventing them to
seek to invest abroad
2. Lack of liquidity
3. Currency controls, specific tax regulations, relatively
less developed capital markets in some countries
4. Lack of readily available comparable information on
potential foreign security acquisition
5. Home bias investors tend to prefer domestic assets

58. Measuring total return from foreign assets
Local currency return =foreign currency return x Currency gain (loss)
1 + 𝑅𝐻 = 1 +
𝑃1 − 𝑃0 + 𝐷1
𝑃0
(1 + 𝑔)
𝑅𝐻 = 1 + 𝑅𝐹 (1 + 𝑔) - 1