Slides - Session 3+4.pdf111111111111111111111111111111

finance

1. Session 3 & 4:
Risk, Return, and
Portfolio Theory
1

2. Returns, Risk, and Portfolio
Optimisation
2

3. Session Reading
•Bodie/Kane/Marcus: Chapter
5 for a revision of return and risk
calculations. Chapter 7 for in-
depth reading (sections 7.1 –
7.4). See questions at end of
Chapter 7 for possible exam
style questions.
3

4. Rates of
Return: Single
Period
• Holding-period return (HPR) measures
the rate at which the investor’s funds have
grown during the investment period
4

5. Rates of
Return:
Multiple
Periods
•Often you will be interested in average
investment returnsover longer periods
of time
•i.e. you are interested in knowing the
performance of your investment(s)
over ‘n’ periods
•Two methods:
➢Arithmetic average
➢Geometric average
5

6. Performance of a sample mutual fund
6

7. Returns using arithmetic averaging
Arithmetic Averaging
R
a= (r
1+ r
2+ r
3+ r
4) / 4
R
a= (.10 + 0.25 –0.20 + 0.25) / 4
R
a= 0.10 or 10%
7n
r
r
n
1t
t
A

=
=

8. Returns using geometric averaging
•Arithmetic average ignores the effects of compounding.
Geometric average is calculated by compounding the actual
period-by-period returns and then finding the equivalent single
per-period return
8()()() 1r1...r1r1r
1/n
n21G −+++=
r
g= [(1+0.10) x (1+0.25) x (1-0.20) x (1+0.25)]
1/4
-1
r
g= [(1.1) x (1.25) x (0.8) x (1.25)]
1/4
–1
r
g= (1.5150)
1/4
–1 = 0.0829 = 8.29%

9. Estimating future
returns and
risks: Scenario
Analysis
•Scenario analysis is the process of
devising a list of possible economic
scenariosand specifying the
likelihoodof each one as well as the
HPRthat will be realised in each case
•The list of possible outcomes with
associated probabilities is a
probability distribution
9

10. An example of scenario
analysis
Consider an investment in a broad portfolio of stocks:
10
State of
the
Economy
Scenario
“s”
Probability
“p(s)”
Return
“r(s)”
Boom 1 0.25 44%
Normal 2 0.50 14%
Recession 3 0.25 -16%

11. Expected Return
•The expected return is the weighted average of returns in all
possible scenarios s = 1, … Swith weights equal to the
probability of that particular scenario
11() ()()
=
=
S
1s
srsprE
p(s) = probability of a state
r(s) = return if a state occurs

12. Expected Return: Example
12() ()()
=
=
S
1s
srsprE
State of the
Economy
Scenario
“s”
Probability
“p(s)”
Return
“r(s)”
Boom 1 0.25 44%
Normal 2 0.50 14%
Recession 3 0.25 -16%
Expected return “E(r)” =
Probability x Return
“p(s) x r(s)”
0.25 x 44% +
0.50 x 14% +
0.25 x -16% =
= 14%

13. Expected Variance
•Variance is the dispersion of returns
•Variance tells us about the potential for deviation of the return
from its expected value
•The bigger the variance, the more the propensity for rates of
return to deviate from their expected value
13() ()()() 
=
−==
S
1s
22
rEsrspσrVar

14. Variance: Example
14() ()()() 
=
−==
S
1s
22
rEsrspσrVar
State of the
Economy
Scenario
“s”
Probability
“p(s)”
Return
“r(s)”
Boom 1 0.25 44%
Normal 2 0.50 14%
Recession 3 0.25 -16%
Expected Return E(r) =14%
Variance (r) “σ
2
” =
Var(r)
“σ
2
”
0.25[44%-14%]
2
+
0.50[14%-14%]
2
+
0.25[-16%-14%]
2
=
450

15. Standard Deviation
•Standard deviation is simply the square root of variance
•In the current example, the standard deviation (“σ”) = 450
1/2
= 21.21%
15() ()()()  ()rVarrEsrspσrSD
S
1s
2
=−==
=

16. 16

17. Exercise 1An investor has the following returns over a period of 5 years: 10%, 5%, -2%, 8%, and 12%.
Calculate the arithmetic rate of return (ARR) for this investment.
17
1-
2-Calculate the geometric rate of return (GRR) for the same investment. The Arithmetic Rate of Return (ARR) is:

18. EXERCISE 2A stock has the following possible returns under different economic conditions:
Economic Condition Probability Return (%)
Boom 0.3 25%
Normal 0.5 10%
Recession 0.2 -5%
Questions:
1. Calculate the expected return of the stock.
2. Compute the variance of the stock returns.
3. Determine the standard deviation of the stock returns.
18

19. Normal Distribution with mean return 12.5% and
stddev20.3%
19

20. Real return distribution in practice
are not normal
•Normal distribution is a theoretical distribution
•-Return distribution exhibit more features like skewness
and leptokurtic features
20

21. Skewness (asymmetry) Kurtosis
=0 if normal dist. =3 if normal dist.
21
Note: No need to memorize these 2 formulas by heart for the exam!

22. In excel
•Functions to calculate the first four
moments of return distribution from a
series of returns are available
•mean or (average) for the mean
return.
•std (for standard deviation)
•skew (for the skewness)
•kurt ( for the Kurtosis)
22

23. 23Example 1: Stock Returns (Skewness = -0.5, Kurtosis = 1.5)
Asset: XYZ Company Stock
Data: Daily return data for 1 year
Values:
Mean (Average) = 0.000371724 (Small positive average return)
Skewness = -0.516679437 (Negative skew)
Kurtosis = 1.422484973 (Platykurtic, less extreme events than normal distribution)

24. Explanation:
Skewness (-0.52): The negative skew indicates that the distribution of returns has a longer
left tail. This means that the stock is more likely to have extreme negative returns (large
losses) than large positive ones. You could see frequent small gains but the potential for
occasional, larger-than-usual losses.
Kurtosis (1.42): The kurtosis of 1.42 is lower than that of a normal distribution (which is 3).
This indicates a platykurtic distribution, meaning the stock’s returns are less likely to
produce extreme values (both large gains or large losses) than a normal distribution would
predict.

Comment:
This type of distribution can be seen in stocks that have relatively stable but volatile periods
where large negative movements are more likely. The risk of a big loss is present, but
extreme gains are rarer. This makes it a stock where investors may need to be cautious
about the potential for large losses while the returns seem generally stable. 24

25. 25Example 2: Cryptocurrency (Skewness = 0.8, Kurtosis = 4.0)
Asset: ABC Cryptocurrency (e.g., Bitcoin)

Data: Daily return data for 1 year
Values:
Mean (Average) = 0.0025 (Small positive average return)
Skewness = 0.8 (Positive skew)
Kurtosis = 4.0 (Leptokurtic, higher chance of extreme events)

26. Explanation:
Skewness (0.8): The positive skew indicates that the distribution of returns has a longer
right tail. This means that the cryptocurrency is more likely to experience large positive
returns (big gains) than large negative returns (losses). In other words, while you might
have small losses or stable returns frequently, large upside potential exists.

Kurtosis (4.0): A kurtosis of 4.0 is higher than the normal distribution value of 3, indicating
that the distribution is leptokurtic. This means there are fatter tails and a higher probability
of extreme events—both large gains and large losses—than a normal distribution would
predict.

Comment:
This is a classic example of a volatile asset like cryptocurrency, where you can have
relatively frequent small losses or gains, but the possibility of significant price swings
(either huge gains or steep losses) is much more likely than in a normal distribution.
Investors need to be aware that the risk is higher, but the potential for large rewards
(extreme gains) also exists. 26

27. 27Example 3: Corporate Bond (Skewness = -0.2, Kurtosis = 3.1)
Asset: A corporate bond (e.g., high-rated bond)
Data: Monthly return data for 5 years
Values:
Mean (Average) = 0.005 (Positive average return)
Skewness = -0.2 (Slight negative skew)
Kurtosis = 3.1 (Leptokurtic)

28. Explanation:
Skewness (-0.2): The slight negative skew indicates that the return distribution is slightly
tilted toward negative outcomes, but not excessively so. While the bond may provide stable
returns over time, there is a small probability of experiencing a larger-than-usual loss.
However, the skew is not as extreme as in the first example of the stock.

Kurtosis (3.1): The kurtosis value of 3.1 is slightly higher than the normal distribution’s
value of 3, meaning the bond’s return distribution is slightly leptokurtic. This suggests that
while extreme outcomes (large gains or large losses) are not highly probable, they are more
likely than in a normal distribution. Investors in corporate bonds might expect stable
returns most of the time, but there’s a small chance of an extreme negative event (a default
or market crash) affecting returns more significantly.
Comment:
For a corporate bond, the distribution of returns suggests a relatively safe investment, but
with some risk. The slight negative skew and slightly leptokurtic distribution show that
while the bond will likely provide steady returns, investors should still be aware of the
possibility of extreme (though rare) outcomes—such as a credit downgrade or market
shock that could lead to a significant loss. 28

29. Summery:
•The return of the portfolio is measured
by the expected return (usually
geometric average)
•The total risk of the portfolio is
measured by the standard deviation of
returns
•Higher moments like skewness and
kurtosis can give significant
information about return distributions.
•In case of different scenarios, returns
and variances are to be multiplied by
their probabilities to get the expected
return and the risk.
29

30. Constructing
Portfolios
30

31. Constructing Portfolios
•Learning objectives:
-Understand risk decompositionbetween systematicand non-
systematicrisk
-Show how covarianceand correlationaffect the risk of diversified
portfolios
•Construct efficientportfolios
•Calculate the compositionof the optimal risky portfolio
31

32. systematic
and non-
systematic
risk
32

33. Sources of Risk Affecting Portfolios
33
Macroeconomic factors:
-Inflation rate
-interest rate
-Business cycle
Micro-factors:
-Firm specific
-Firm’s management style
-Firm’s R&D

34. Macro-economic factors
affect bothA and B
Micro-factors are
differentfor A and B
Stock A
Stock B
34

35. Components of Risk
•Market or systematic risk: risk related to the macro economic factor or
market index
•Unsystematic risk: firm-specific risk, not related to the macro economy
35
Total risk = Systematic + Non-Systematic
Total risk = Non-diversifiable + Diversifiable
•Securities have different proportions of systematic and non-
systematic risk

36. Measure of systematic risk
•The relative sensitivity or volatility to
market moves (i,e, The systematic non-
diversifiable risk) is represented by a
measure (beta)
36

37. Measure of systematic risk
•Bêta = 1 : Asset moves in the same direction and in the same
amount as the market
•0<Bêta < 1 : Asset moves in the same direction, but in a lesser
amount than the market
•Bêta > 1 : Asset moves in the same direction, but in a greater
amount than the market
•Bêta =0 : Asset movements are independent of those of the
market
•Bêta =-1 : Asset moves in the opposite direction and in the
same amount as the market
37

38. Measure of
systematic
risk
-Assets with lower Beta
are less exposed to
systematic risk.
38

39. Measure of systematic risk
•Consider the following information
• Standard deviation Bêta
•–Asset A 20 % 1,25
•–Asset B 30 % 0,95
••Which asset has the highest total risk?
••Which asset has the highest systematic risk?
••Which assets will have higher return?
39

40. Beta of a portfolio
Consider the following information about 4 stocks (A, B, C and
D)
stock weight Bêta
A 0,133 3,69
B 0,2 0,64
C 0,267 1,64
D 0,4 1,79
1
•What is the beta of the portfolio ?
&#3627408437;????????????&#3627408436;??????= 0,133 (3,69) + 0,2 (0,64) + 0,267 (1,64) + 0,4 (1,79) =
1,773
40

41. The diversification
41

42. Example of Diversification
•Suppose you owned shares in an airline and an oil company
•We would expect the oil company stock price to rise when
oil prices rise
•We would expect the airline’s stock price to fall when oil
prices rise
•Therefore, the two effects are offsetting, thereby stabilising
portfolio returns
42

43. Diversification to
Reduce Risk !
•Suppose the market contains a
large number of securities …
•Adding more securities into an
existing portfolio would continue to
reduce the risk exposure
•Portfolio volatility should continue
to fall
•However, even with a large number
of securities in a portfolio, you may
not eliminate all risk
•Returns of all securities are
affected by the risk involved in the
macro-economy
43

44. Portfolio risk as a function of the number of
stocks
44

45. Portfolio risk decreasesas
diversification increases[NYSE study]
45

46. Covariance and
correlation
46

47. Covariance
and
Correlation
• Correlation : The degree to which
economic variables are observed to
move together
Correlation is the most used method, but
we need to calculate covariance before
we can calculate correlation
Covariance: The degree to which
economic variable deviate with the same
direction from their average.
47

48. Let’s Assume we have Two Risky Assets
48

49. 1. Covariance
49()
( )( ) 
1
,
1
,,
−
−−
=

=
N
rrrr
rrCov
N
t
BtBstS
BS
E.g.If Covis > 0, then when stocks produces a return
above their average returns, the bonds tends to follow a
similar pattern

50. Correlation
•Correlation coefficient falls between -1 (perfect
negative) and +1 (perfect positive) while -∞<COV<+∞
•Zero correlation means that the returns on the two
assets are unrelated to each other
50( , )
SB
SB
SB
Cov r r


=

51. Returns on two assets (A and B) over a 10-
year period
0
2
4
6
8
10
12
0
2
4
6
8
10
12
14
1989199019911992199319941995199619971998
-4
-2
0
2
4
6
8
10
12
14
16
1989199019911992199319941995199619971998
Perfect positive: rho=1
Perfect negative: rho=-1
No correlation: rho=0
51

52. Three rules for two-
asset portfolios
52

53. Rule 1: Returns
The rate of return on the portfolio is a weighted average of the
returns on the component securities with the investment
proportions as weights
Remember that the sum of the weights equals to 1 or 100%
53P B B S S
r w r w r=+

54. Example: Rule 1 -Returns
•You have €1,000 to invest and you put €400 in
security A and €600 in security B
•Next month the return for A is 10% and for B is 6%
•What is the % return?
54P B B S S
r w r w r=+ ( )( )%6.7%66.0%104.0 =+=
P
r

55. Rule 2: Expected Returns
The expected rate of return on the portfolio is a
weighted average of the expected returns on the
component securities
55( ) ( ) ( )
P B B S S
E r w E r w E r=+
If there are nsecurities in the portfolio:() ()
=
=
n
i
iiP rEwrE
1

56. Example: Rule 2 –Expected Returns
•If security S has a 20% expected return and B has 10% expected
return
•What is the expected return for the portfolio if you allocate
equal money to A and B?
56( ) ( ) ( )
P B B S S
E r w E r w E r=+ ()( )( )%15%105.0%205.0 =+=
P
rE

57. Rule 3: Variance and Standard Deviation
The variance of the portfolio return is a sum of the
contributions of the component security variances
plus
A term that involves the correlation coefficient between
the returns of the component securities
572 2 2
( ) ( ) 2( )( )
P B B S S B B S S BS
w w w w     = + +
where ρ
BSis the correlation coefficient between
the returns on the stock and bond funds
Note: standard deviation is the square root of variance

58. Example: Rule 3 –Variance and Standard
DEviation
•Suppose σ
bonds= 12% and σ
stocks= 25% and
assume that ρ
bonds,stocks= 0.2. What is the total
variance of the portfolio if we invest 75% in bonds
and 25% in stocks
58( )( ) ( )( )20.02525.01275.022525.01275.0
222
++=
var= 142.56
Note: the lower the correlation the lower is portfolio variance
SD = 11.94

59. ExercicesYou have invested in two assets:
- Asset A: $5,000 with an expected return of 8%
- Asset B: $3,000 with an expected return of 10%
Calculate the expected return of the portfolio.
59Given the following information:
- Asset X: Expected return = 12%, Standard deviation = 18%
- Asset Y: Expected return = 9%, Standard deviation = 15%
- Portfolio Weights: 60% in X and 40% in Y
- Correlation coefficient (ρ) between X and Y = 0.3
Calculate the portfolio standard deviation.
Exercice 1
Exercice 2

60. The Efficient
Frontier
60

61. The
efficient
frontier
61
Construct a risk/return
plot of all possible
portfolios (with different
weights)
For each return, more
than one portfolio can be
found
Those portfolios that are
not dominated (in terms
of risk-return) constitute
the efficient frontier

62. Investment opportunity set for stocks and bonds
62
No point plot
above the line
Points below the efficient
frontier (like Z) are dominated
All portfolios
on the (pink)
line are
efficient
Y

63. Diversification
and Correlation
•There are benefits to diversification
whenever asset returns are less than
perfectly correlated (+ & -correlation)
•Assets are almost always less than
perfectly correlated.
63

64. Investment
opportunity
set for stocks
and bonds
with various
correlations
64

65. Efficient
frontier
with many
risky
assets
65

66. Example
Portfolio Standard Deviation = 28.1%
Portfolio Expected Return = 17.4%
66
Correlation Coefficient = 0.40
Assets σ % of
portfolio
Avg
Return
Bonds 28 60% 15%
Stocks 42 40% 21%
Let’s add another stock ‘New Corp’ to the portfolio
There are benefits to diversification whenever asset returns are less than
perfectly correlated

67. Example
NEW Portfolio Standard Deviation = 23.43%
NEW Portfolio Expected Return = 18.20%
67
Correlation Coefficient = 0.30
Assets σ % of
portfolio
Avg
Return
Portfolio 28.1 50% 17.4%
New Corp 30 50% 19%
NOTE: Higher Return & Lower Risk due to benefitsof
Diversification
If opposite case, Bad choice of New Corp!

68. Including a
risk-free
asset and
the CaL
68

69. Extending
to include
risk-free
asset
•We can expand the asset
allocation problem to include a risk-
free asset in the risky portfolio
•The optimal combination
becomes linear
•A single combinationof risky and
risk-free assets will dominate
69

70. Capital Allocation Line (CAL)
•A risky investment portfolio (i.e. a risky asset) can be characterised
by its reward-to-variability ratio (return divided by risk)
•This ratio is the slope of the CAL, the line connecting the risk-free
asset to the risky asset
•All combinations of the risky and risk-free asset lie on this line
•Investors will prefer a steeper sloping CAL because that means
higher expected returns for any level of risk
•CAL shows the risk-return trade-offs available by mixing risk-free
assets with the investor’s risky portfolio
70

71. •S (slope) = reward-to-variability ratio
71

72. Opportunity set using stocks and bonds and two Capital
Allocation Lines
72

73. The best CAL?
•We want the CAL that gives the best portfolio
combination
•For this we compare reward-to-variability ratios.
Which is steepest?
•What will give us the best CAL?
•Usually considered to be a combination of all stocks and
bonds (and, indeed, all possible assets)
73()
Af
A
E r r

−

74. Optimal Capital Allocation Line for bonds, stocks and T-Bills
74

75. Optimal Portfolio
•What is the optimal risky portfolio?
•In this case, it is the blend of stocks and bonds that gives ‘O’ portfolio. In reality, it
will be all possible assets.
•What is the optimal choice between risky assets and risk-free assets?
•Depends on what investors want? Risk? Return?
•Remember that the stock, bond combination does not change
•Only the choice between how much to invest in the risky assets and how much to
invest in the risk-free assets
75

76. Optimal Portfolio: Tobin [1958]
“separation property”
•Portfolio choice can be separated into two independent
tasks:
1.Determination of the optimal risky portfolio
“the best risky portfolio is the same for all investors
regardless of risk aversion”
2.Construct a portfolio choosing between risk-free and
the risky assets dependent on investor risk preferences
76