Investment

1. 1. Introduction and Basics of
Investments
12/20/2013
1

2. 
The purpose of this paper is to help you learn how to manage your Money so
that you will derive the maximum benefit from what you earn.

To accomplish you need
1) to learn about investment alternatives that are available today,
2) to develop a way of analyzing and thinking about investments that will
remain with you in years to come when new and different opportunities
become available.

The paper mixes theory, practical, and application of the theories using
modern/contemporary tool Microsoft Excel.

Evaluation – (Internal -100) and BREAKUP WILL BE TOLD AT LATER
STAGE.

Classes – 30 classes
1. Introduction and Basics of
Investments
12/20/2013
2

3. 
The detailed topics are given separately as a file, but in brief we shall be discussing over
following topics
a) Investments Basics – Risk and Return Measurement
b) Modern Portfolio Theories
c) Equity Analysis and Debt Analysis
d) Portfolio Optimization
e) Portfolio Evaluation

References:
a) Investment Analysis and Portfolio management by Frank K. Reilly and
Keith C. Brown. – Thomson Publication
b) Investments by William F. Sharpe, Gordon J. Alexander, and Jeffery V.
Bailey. – Prentice Hall Publication
c) Class Notes and Handouts.
1. Introduction and Basics of
Investments
12/20/2013
3

4. Let us Start the session!!!
1. Introduction and Basics of
Investments
12/20/2013
4

5. 
Is the current commitment of rupees for a period of time in order to derive
future payments that will compensate the investor for
a) the time the funds are committed (Pure time value of
money or rate of interest)
b) the expected rate of inflation, and
c) the uncertainty of the future of payments
(investment risk so there has to be risk premium)

So in short individual does trade a rupee today for some expected future
stream of payments that will be greater than the current outlay.

Investor invest to earn a return from savings due to their deferred
consumption so they require a rate of return that compensates them.
1. Introduction and Basics of
Investments
12/20/2013
5

6. So we answered following Questions?

◦
Why people invest?
◦
What they want from their investment?
And now we will discuss

◦
Where all they can invest and what parameters they adopt to
invest?
◦
How they measure risk and return and how they
1. Introduction and Basics of
Investments
12/20/2013
6

7.  Gold
 Shares
 Silver
 Real Estate
 Bonds
 Indira Vikas Patra
 Post Office Deposits
 Bank Deposits
 Mutual Funds
 Debentures
 PF
 NSC
1. Introduction and Basics of
Investments
12/20/2013
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8. Investments Parameters

◦
Return
◦
Risk
◦
Time Horizon
◦
Tax Considerations
◦
Liquidity
◦
Marketability
1. Introduction and Basics of
Investments
12/20/2013
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9. •Derivatives
Return
•Shares
•MFs Equity Fund
•Real Estate
•MFs Debt Funds
•Debentures
•NSC, Post-Office Deposit
Kisan Vikas Patra
•PF
•Bonds
•Bank Deposit
•Gold
Risk
1. Introduction and Basics of
Investments
12/20/2013
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10. Next
How to Measure Return and Risk???
1. Introduction and Basics of
Investments
12/20/2013
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11. Return

◦

Risk
◦ Historical

HPR
◦ Expected

◦
Historical
HPY
Expected
1. Introduction and Basics of
Investments
12/20/2013
11

12. 12/20/2013
2. Return and Risk
12

13. What we did in last class…
12/20/2013
2. Return and Risk
13

14. ◦ Why people invest?
◦ What they want from their investment?
◦ Where all they can invest and what parameters they adopt to
invest?
2. Return and Risk
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14

15. Return

◦
Historical

HPR
(Holding Period Return)

HPY
(Holding Period Yield)
◦
Expected

Risk
◦ Historical
 Variance and Standard
Deviation
 Coefficient of Variance
◦ Expected
 Variance and Standard
Deviation
 Coefficient of Variance
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16. ◦ HPR - When we invest, we defer current consumption in order to add our wealth so
that we can consume more in future, hence return is change in wealth resulting from
investment. If you commit Rs 1000 at the beginning of the period and you get back Rs
1200 at the end of the period, return is Holding Period Return (HPR) calculated as
follows
 HPR = (Ending Value of Investment)/(beginning value of Investment) = 1200/1000 = 1.20
◦ HPY – conversion to percentage return, we calculate this as follows,
 HPY = HPR-1 = 1.20-1.00 = 0.20 = 20%
◦ Annual HPR = (HPR)1/n = (1.2) ½, = 1.0954, if n is 2 years.
◦ Annual HPY = Annual HPR – 1 = 1.0954 – 1 = 0.0954 = 9.54%
2. Return and Risk
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17. 
Over a number of years, a single investments will likely to give
high rates of return during some years and low rates of return, or
possibly negative rates of return, during others. We can
summarised the returns by computing the mean annual rate of
return for this investment over some period of time.

There are two measures of mean, Arithmetic Mean and Geometric
Mean.

Arithmetic Mean = ∑HPY/n

Geometric Mean = [{(HPR1) X (HPR2) X (HPR3)}1/n -1]
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18. Year
Beginning
Value
Ending
Value
HPR
HPY
1
1000
1150
1.15
0.15
2
1150
1380
1.2
0.2
3
1380
1104
0.8
-0.2
AM = [(0.15) + (0.20) + (-0.20)]/3 = 5%
GM = [(1.15) X (1.20) X (0.80)]
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1/3
– 1 = 3.35%
2. Return and Risk
18

19. Year
Beginning
Value
Ending
Value
HPR
HPY
1
100
200
2.0
1.0
2
200
100
0.5
-0.5
AM = [(1.0) + (-0.50)]/2 = 0.50/2 = 0.25 = 25%
GM = [(2.0) X (0.50)]
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1/2
– 1 = 0.00%
2. Return and Risk
19

20. Expected Return = ∑RiPi,
• where i varies from 0 to n
• R denotes return from the security in i
outcome
• P denotes probability of occurrence of i
outcome
Economy Growth
Deep Recession
5%
Mild Recession
20%
Average Economy
50%
Mild Boom
20%
Strong Boom
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Probability of Occurrence
5%
2. Return and Risk
20

21. Economy
Growth
T-Bills
Corporate
Bonds
Equity
A
Equity
B
5%
8%
12%
-3%
-2%
20%
8%
10%
6%
9%
50%
8%
9%
11%
12%
Mild Boom
20%
8%
8.50%
14%
15%
Strong Boom
5%
8%
8%
19%
26%
8.00%
9.20%
10.30%
12.00%
Deep
Recession
Mild Recession
Average
Economy
Probability of
Occurrence
100%
Expected Rate
of Return
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2. Return and Risk
21

22. Probability Distribution of Equity "A"
60%
Probability
50%
40%
30%
Series1
20%
10%
0%
Series1
-13.300%
-4.300%
0.700%
3.700%
8.700%
5%
20%
50%
20%
5%
Dispersion from Expected Return
2. Return and Risk
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23. 2. Return and Risk
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24. 12/20/2013
2. Return and Risk
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25. 

Webster define it as a hazard; as a peril ; as a
exposure to loss or injury.
Chinese definition –
Means its a threat but at the same time its an
opportunity
So what is in practice risk means to us?
2. Return and Risk
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25

26. 

Actual return can vary from our expected return,
i.e. we can earn either more than our expected
return or less than our expected return or no
deviation from our expected return.
Risk relates to the probability of earning a return
less than the expected return, and probability
distribution provide the foundation for risk
measurement.
2. Return and Risk
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26

27. 
Variance – is a measure of the dispersion of actual outcomes
around the mean, larger the variance, the greater the dispersion.
Variance = ∑(HPYi – AM)2 / (n)
where i varies from 1 to n.
Variance is measured in the same units as the outcomes.

Standard Deviation – larger the S.D, the greater the dispersion
and hence greater the risk.

Coefficient of Variation – risk per unit of return,
= S.D/Mean Return
2. Return and Risk
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28. 
Variance – is a measure of the dispersion of possible
outcomes around the expected value, larger the variance, the
greater the dispersion.
Variance = ∑(ki – k)2 (Pi)
where i varies from 1 to n.
Variance is measured in the same units as the outcomes.


Standard Deviation – larger the S.D, the greater the dispersion
and hence greater stand alone risk.
Coefficient of Variation – risk per unit of return,
= S.D/Expected Return
2. Return and Risk
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28

29. Expected Return or Risk
Measure
T-Bills
Corporate
Bonds
Expected return
8%
9.20%
10.30%
12.00%
Variance
0%
0.71%
19.31%
23.20%
Standard Deviation
0%
0.84%
4.39%
4.82%
Coefficient of Variation
0%
0.09%
0.43%
0.40%
Semi variance
0.00%
0.19%
12.54%
11.60%
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2. Return and Risk
Equity A Equity B
29

30. • Variance and Standard Deviation
The spread of the actual returns around the expected return; The greater the
deviation of the actual returns from expected returns, the greater the varian
• Skewness
The biasness towards positive or negative returns;
• Kurtosis
The shape of the tails of the distribution ; fatter tails lead to higher kurtosis
2. Return and Risk
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31. 2. Return and Risk
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32. 12/20/2013
2. Return and Risk
32

33. Thank You!!!
12/20/2013
2. Return and Risk
33

34. 12/20/2013
2. Return and Risk
34

35. What we did in last class…
12/20/2013
2. Return and Risk
35

36. ◦ How do we calculate Risk and Return of a single Security?
◦ Historical and Expected Risk and Return
◦ Concept of Price Adjustments - Bonus, Stock Split, and Demerger
2. Return and Risk
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37. 12/20/2013
2. Return and Risk
37

38. 

Markowitz Portfolio
Market Model/Index Model
2. Return and Risk
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39. ◦ Measure of Return – Probability Distribution and its Weighted Average
Mean.
◦ Measure Risk – Standard Deviation (Variability) of Expected Return of a
Portfolio?
◦ Investors do not like risk and like return.
◦ Nonsatiation – always prefer higher levels of terminal wealth to lower levels
of terminal wealth.
◦ Risk Aversion – investor choose the portfolio with smaller S.D. ( not like Fair
Gamble).
◦ Investors get positive utility with return as they help them in maximising
wealth and vice-versa with Risk.
2. Return and Risk
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39

40. Utility
U2
U1
Wealth
2. Return and Risk
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40

41. Expected
Return of
Portfolio
S.D. of
Portfolio
2. Return and Risk
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41

42. Expected
Return of
Portfolio
Risk Averse
Risk Taker
S.D. of
Portfolio
2. Return and Risk
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42

43. 



All the portfolios on a given indifference curve provide same
level of utility.
They Never Intersect Each Other otherwise they will violate
law of transitivity.
An investor has an infinite number Indifference Curves.
A risk-averse investor will find any portfolio that is lying on
an indifference curve that is “farther north-west” to be more
desirable than any portfolio lying on an indifference curve
that is “not as far northwest”.
2. Return and Risk
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43

44. 
Every investor has an indifference map representing his/her
preferences for expected returns and standard deviations.

An investor should determine the expected return and standard
deviation for each potential portfolio.

The two assumptions of Nonsatiation and risk aversion cause
indifference curves to be positively sloped and convex.

The degree of risk aversion will decide the extent of positiveness in
slope of indifference curves.

More Flat is the indifference curves of an individual – higher risk
aversion and vice-versa.
2. Return and Risk
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45. 12/20/2013
2. Return and Risk
45

46. 
Expected return of Portfolio
= ∑Xiki
Xi is the fraction of the portfolio in the ith asset, n is
the number of assets in the portfolio. Here i range from
0 to n.
2. Return and Risk
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46

47. Probability
Possible Returns
0.35
0.3
0.2
0.08
0.1
0.12
0.15
Expected Return
12/20/2013
0.028
0.03
0.024
0.14
0.021
10.30%
2. Return and Risk
47

48. Weight
Expected Returns of Securities
0.2
0.3
0.3
0.2
Expected Return of Portfolio
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2. Return and Risk
0.1
0.11
0.12
0.13
0.02
0.033
0.036
0.026
0.115
48

49. 12/20/2013
2. Return and Risk
49

50. Expected Return
Year
Stock A
Stock B
Portfolio AB
2001
-10%
40%
15%
35%
-5%
15%
2004
-5%
35%
15%
2005
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15%
2003
S.D.
-10%
2002
Avg Return
40%
15%
15%
15%
15%
15%
15%
22.64%
22.64%
0.00%
2. Return and Risk
50

51. 50%
40%
30%
Series1
20%
Series2
10%
Series3
0%
-10%
2001
2002
2003
2004
2005
-20%
2. Return and Risk
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51

52. Expected Return
Year
Stock
A
Stock
B
Portfolio
AB
50%
40%
2001
40%
-10%
15%
30%
2002
-10%
40%
15%
2003
35%
-5%
15%
2004
-5%
35%
15%
0%
2005
15%
15%
15%
-10%
Avg
Retur
n
S.D.
20%
10%
2001
2002
2003
2004
2005
-20%
15%
15%
15%
22.64
%
22.64
%
0.00%
Correlation Coefficient = -1.0
2. Return and Risk
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52

53. Expected Return
Year
Stock A
Stock B
Portfolio AB
2001
40%
40%
40%
-5%
-5%
-5%
2004
35%
35%
35%
2005
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-10%
2003
S.D.
-10%
2002
Avg Return
-10%
15%
15%
15%
15%
15%
15%
22.64%
22.64%
22.64%
2. Return and Risk
53

54. 50%
40%
30%
Series1
20%
Series2
10%
Series3
0%
-10%
2001
2002
2003
2004
2005
-20%
2. Return and Risk
12/20/2013
54

55. Expected Return
Stock
A
Year
Stock
B
50%
Portfolio
AB
40%
30%
2001
-10%
-10%
-10%
2002
40%
40%
40%
2003
-5%
-5%
-5%
2004
35%
35%
35%
0%
15%
-10%
2005
Avg
Retur
n
S.D.
15%
15%
20%
10%
2001
2002
2003
2004
2005
-20%
15%
15%
22.64
%
22.64
%
15%
Correlation Coefficient = +1.0
22.64%
2. Return and Risk
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55

56. So Risk is not a simple weighted average of risk with
securities like we did in measuring Expected
Return………..we need to know following things to
measure risk of a Portfolio.

Covariance between two securities

Correlation Coefficient between two securities

Variance of securities

Standard Deviation of Securities
2. Return and Risk
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56

57. Standard deviation of Portfolio =( ∑ ∑Xi Xj σij)1/2
where i and j vary from 0 to n, and σij is covariance
between i and j securities.
σij = ρijσi σj, where σi & σj is standard deviation of i
and j respectively.
2. Return and Risk
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57

58. Thank You!!!
12/20/2013
2. Return and Risk
58

59. 12/20/2013
2. Return and Risk
59

60. 12/20/2013
2. Return and Risk
60

61. What we did in last class…
12/20/2013
2. Return and Risk
61

62. ◦ How do we calculate Risk and Return of a single Security?
◦ Historical and Expected Risk and Return
◦ Concept of Price Adjustments - Bonus, Stock Split, and Demerger
2. Return and Risk
12/20/2013
62

63. ER
0.103
0.12
Variance
SD
Coefficient of Variation
0.0019310
0.04394315
0.42663248
0.00232
0.048166
0.401386
Covariance
Correlation Coefficient
Risk Tolerance
0.00202
0.95436882
0.5
-0.75
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2. Return and Risk
63

64. Portfolios Proportion in X
Proportion in Y
Return
A
1
0
5.00%
B
0.8
0.2
7.00%
C
0.75
0.25
7.50%
D
0.5
0.5
10.00%
E
0.25
0.75
12.50%
F
0.2
0.8
13.00%
G
0
1
15.00%
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2. Return and Risk
64

65. Portfolios
Lower Bound
Upper Bound
No relationship
A
20.00%
20.00%
20.00%
B
10.00%
23.33%
17.94%
C
0.00%
26.67%
18.81%
D
10.00%
30.00%
22.36%
E
20.00%
33.33%
27.60%
F
30.00%
36.67%
33.37%
G
40.00%
40.00%
40.00%
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2. Return and Risk
65

66. Weights
A
B
ER
Variance
SD
Utility
1
1
0
0.1030
0.001931
0.043943145
0.099138
2
0.75
0.25 0.1073
0.0019887
0.044594703
0.103273
3
0.5
0.5
0.1115
0.0020728
0.045527464
0.107355
4
0.25
0.75 0.1158
0.0021832
0.046724592
0.111384
5
0
0.00232
0.04816638
0.11536
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1
0.1200
2. Return and Risk
66

67. Expected Return
Feasible Sets of Portfolios
0.1250
0.1200
0.1150
0.1100
0.1050
0.1000
0
0.01
0.02
0.03
0.04
0.05
0.06
Standard Deviations
2. Return and Risk
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67

68. Two Conditions
1)
Offer Maximum Return for varying levels of Risk,
and
2)
Offer Minimum Risk for varying levels of expected
return
All the feasible sets are not efficient unless it passes
through this test
2. Return and Risk
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68

69. Expected Return
Efficient Sets of Portfolios
Standard Deviations
2. Return and Risk
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69

70. Thank You!!!
12/20/2013
2. Return and Risk
70

71. 12/20/2013
2. Return and Risk
71

73. 
To Identify Investor’s Optimal Portfolio

Investor’s needs to estimate
◦ Expected returns
◦ Variances
◦ Covariances
◦ Riskfree Return


Investor’s need to identify tangency portfolio
The Optimal Portfolio involves an investment in the tangency
portfolio along with either riskfree borrowing or lending to
get linear efficient portfolio

74. 






Investors think in terms of single period and choose portfolios
on the basis of each portfolio’s expected return and standard
deviation over that period.
Investors can borrow/lend unlimited amount at a given riskfree rate.
No restrictions on short sale.
Homogenous Expectations.
Assets are perfectly divisible and marketable at a going price.
Perfect market.
Investors are price takers i.e. their buy/sell activity will not
affect stock price

75. 





Allows us to change our focus from how an individual should invest to
what would happen to securities prices if everyone invested in same
manner.
Enables us to develop the resulting equilibrium relationship between
each security’s risk and return.
Everyone would obtain in equilibrium the same tangency portfolio
(Homogenous Expectation)
Also the linear efficient frontier same for all investors as they face same
risk free rate.
So only reason investors to have dissimilar portfolios is their different
preferences towards risk and return (Indifference Curve).
However they will chose the same combination of risky securities.

76. Return
Indifference
Curve
Linear Efficient
Curve
M
Risky Securities
Efficient Curve
Risk
Free Rate
Risk

77. So we are saying in brief
Separation theorem
The Optimal combination of risky assets for an investor can be
determined without any knowledge of the investor’s preferences
toward risk and return.
Now…..

78. Second Point of CAPM is



Each investor will hold a certain positive amount of each risky
security.
Current market price of each security will be at a level where total
no. of shares demanded equals the no. of shares outstanding.
Risk free rate will be at a level where the total no. of money
borrowed equals the total amount of money lent.
Hence there is an equilibrium or we can say that tangency portfolio
which fulfilled above criteria is also termed as market portfolio. And
we define market portfolio as given in next slides….

79. The Market Portfolio
is a portfolio consisting of all securities I which the proportions
invested in each security corresponds to its relative market value.
The relative market value of a security is simply equal to the
aggregate market value of the security divided by the sum the
aggregate market values of all the securities.

80. Return
Rm
Rf
σm
Risk
σp

81. Rp = Rf + (Rm- Rf) X σp
σm
 Slope of line is price of risk
 And Intercept is price of time

82. 
Uses variance as a measure of risk

Specifies that a portion of variance can be diversified away,
and that is only the non-diversifiable portion that is
rewarded.

Measures the non-diversifiable risk with beta, which is
standardized around one.

Translates beta into expected return Expected Return = Riskfree rate + Beta * Risk Premium

83. 




The risk of any asset is the risk that it adds to the market
portfolio
Statistically, this risk can be measured by how much an asset
moves with the market (called the covariance)
Beta is a standardized measure of this covariance
Beta is a measure of the non-diversifiable risk for any asset can
be measured by the covariance of its returns with returns on a
market index, which is defined to be the asset's beta.
The cost of equity will be the required return,
Cost of Equity = Riskfree Rate + Equity Beta * (Expected Mkt Return
– Riskfree Rate)

84. (A) Risk-free Rate
(B) The Expected Market Risk Premium (The Premium
Expected For Investing In Risky Assets Over The
Riskless Asset)
(C) The Beta Of The Asset Being Analyzed.

86. Two Conditions
1)
2)
Offer Maximum Return for varying levels of
Risk, and
Offer Minimum Risk for varying levels of
expected return
All the feasible sets are not efficient unless it
passes through this test

87. B
D
Feasible Sets
C
A

88. Efficient Sets and Feasible Sets
IC 3
IC 2
B
D
Feasible Sets
IC 1
A
C

90. Stocks
Deviation
A
B
Expected Return Standard
5%
20%
15%
40%

91. Expected Return of Portfolio = ∑Xiri, where i
range from 0 to n.
and X is Proportion of total investment in ith
security and ri is expected return of the security.
Standard deviation of Portfolio =( ∑ ∑Xi Xj σij)1/2
where i and j vary from 0 to n, and σij is
covariance of i and j securities.
σij = ρijσi σj, where σi & σj is standard deviation
of i and j respectively.

92. Portfolios
Proportion in X
Proportion in Y Return
A
1
0
5.00%
B
0.83
0.17
6.70%
C
0.67
0.33
8.30%
D
0.5
0.5
10.00%
E
0.33
0.67
11.71%
F
0.17
0.83
13.30%
G
0
1
15.00%

93. Portfolios Lower Bound
Upper Bound
No relationship
A
20.00%
20.00%
20.00%
B
10.00%
23.33%
17.94%
C
0.00%
26.67%
18.81%
D
10.00%
30.00%
22.36%
E
20.00%
33.33%
27.60%
F
30.00%
36.67%
33.37%
G
40.00%
40.00%
40.00%

94. Expected Return
Upper and Lower Bounds to Portfolios
16.00%
14.00%
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
0.00%
5.00%
10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00%
Standard Deviations

96. ri = αiI + βiI rI + εiI
Where,
ri = return on security i for given period
αiI = intercept form
βiI = slope form
rI = return on market index I for the same
period
εiI =random error

97. ri = αiI + βiI rI

98. βiI = σiI
σI2
σiI = Covariance
σI2 = Variance of Market Index

99. Security A
Security B
Intercept
2%
-1%
Actual Return
on the Market
index X beta
10% X 2% =
12%
10% X 8% = 8%
Actual Return
on Security
9%
11%
Random Error 9% - (2% + 12%) 11% - (-1% +8%) =
= -5%
4%

100. Infotech versus S&P 500: 1992-1996
8.00%
6.00%
4.00%
2.00%
-15.00%
-10.00%
0.00%
-5.00% -2.00%
0.00%
-4.00%
-6.00%
5.00%
10.00%
15.00%
20.00%

101. σi2 =βiI2X σI2 + σεi2
Where
,
σi2 = variance of security i
βiI2X σI2 = Market risk of security i
σεi2 = Unique risk of security i

102. rp = ∑Xi ri
Where i range from o to n. and
Xi = proportion of investment in security i.
ri = expected return of security i.
Also,
ri = αiI + βiI rI + εiI
Hence rp
= ∑Xi (αiI + βiI rI + εiI)
.....continued

103. rp =
∑Xi (αiI + βiI rI + εiI)
= ∑Xi αiI + (∑Xi βiI ) rI + ∑XiεiI
= αpI +
βpI rI
+
Intercept
Slope X independent
Variable
Where i range from o to n.
εpI
Random Error

104. σ2p =β2pIσ2I + σ2εp
Where ,
β2pI = [∑Xi βiI] 2 ----- Systematic Risk
σ2εp = ∑Xi2 σ2εi ----- Unique Risk

105. σp
Total Risk
Unique Risk
Market Risk
N

106. Stock
Portfolio
Weight
Beta
Expected Return of
Stock
Variance of
Stock
A
0.25
0.5
0.4
0.07
B
0.25
0.5
0.25
0.05
C
0.5
1
0.21
0.07
Variance of
Market
0.06

107. 




Residual Variance of each of the stocks?
Beta of the portfolio?
Variance of the Portfolio?
Expected Return on the portfolio?
Portfolio Variance on teh basis of Markowitz
Variance – Covariance formula.
Covariance (A,B) = 0.020
Covariance (A,C) = 0.035
Covariance (B,C) = 0.035

108. Duration, Convexity and Portfolio
Immunization

109. Bondholders have interest rate risk even if
coupons are guaranteed - Why?
Unless the bondholders hold the bond to maturity, the
price of the bond will change as interest rates in the
economy change

110. The following basic principles are universal for bonds :



Changes in the value of a bond are inversely related to changes in
the rate of return. The higher the rate of return (i.e., yield to
maturity (YTM)), the lower the bond value.
Long-term bonds have greater interest rate There is a greater
probability that interest rates will rise (increase YTM) and thus
negatively affect a bond’s market price, within a longer time period
than within a shorter period
Low coupon bonds have greater interest rate sensitivity than high
coupon bonds In other words, the more cash flow received in the
short-term (because of a higher coupon), the faster the cost of the
bond will be recovered. The same is true of higher yields. Again, the
more a bond yields in today’s dollars, the faster the investor will
recover its cost.

111. Bond Pricing
Relationships
Price
Inverse relationship between price and yield
An increase in a bond’s yield to maturity
results in a smaller price decline than the gain
associated with a decrease in yield (convexity)
YTM

112. Bond
Coupon Maturity
YTM
A
B
C
12%
12%
3%
5 years
30 years
30 years
10%
10%
10%
D
3%
30 years
6%
0
Change in yield to maturity (%)
A
B
C
D

113. 


There are three factors that affect the way the price of a bond
reacts to changes in interest rates. These three factors are:
◦ The coupon rate.
◦ Term to maturity.
◦ Yield to maturity.
Long-term bonds tend to be more price sensitive than shortterm bonds
Price sensitivity is inversely related to the yield to maturity at
which the bond is selling

114. 




Duration measures the combined effect of all the factors that
affect bond’s price sensitivity to changes in interest rates.
Duration is a weighted average of the present values of the
bond's cash flows, where the weighting factor is the time at
which the cash flow is to be received.
The weighted average of the times until each payment is
received, with the weights proportional to the present value
of the payment
Duration is shorter than maturity for all bonds except zero
coupon bonds
Duration is equal to maturity for zero coupon bonds
Note: Each time the discount rate changes, the duration
must be recomputed to identify the effect of the
change.
Duration tells us the sensitivity of the bond price to one
percent change in interest rates.

115. 1200
Cash flow
1000
800
Bond Duration = 5.97 years
600
400
200
0
1
2
Actual cash flows
PV of cash flows
3
4
5
6
7
8
Year
Area where PV of CF before and after balance out

116. CF t (1
wt
t
y)
Price
T
D
t wt
t 1
CFt
Cash Flow for Period
t
PV of cash flows
as a % of bond
price

117. An adjusted measure of duration can be used to approximate the price
volatility of a bond
Modified Duration
Macaulay Duration
1
YTM
m
Where:
m = number of payments a year
YTM = nominal YTM

118. Eg. Coupon = 8%, yield = 10%, years to maturity = 2
Time
(years)
C1
Payment
PV of CF
(10%)
C4
Weight
C1 XC4
.5
40
38.095
.0395
.0198
1
40
36.281
.0376
.0376
1.5
40
34.553
.0358
.0537
2.0
1040
855.611
.8871
1.7742
sum
964.540
1.000
1.8853
DURATION

119. 1.
2.
3.
4.
It’s a simple summary statistic of the
effective average maturity of the portfolio;
It is an essential tool in immunizing
portfolios from interest rate risk;
It is a measure of interest rate risk of a
portfolio
Equal duration assets are equally sensitive
to changes in interest rates

120. 
Price change is proportional to duration and
not to maturity
P
P
( y)
D
1
y
• Where D = duration
D
P
D
*
1
y
P
*
D
y
D* is the 1st derivative of bond’s price with respect to yield ie. D* = (-1/P)(dP/dY)

121. Duration/Price Relationship
P
P
( y)
D
1
y
The relative change in the price of the bond
is proportional to the
absolute change in yield [dY ] where the factor of proportionality [D/(1+Y)]
is a function of the bond’s duration.
For a given change in yield, longer duration bonds have greater relative
price volatility. This implies that anything that causes an increase in a
bond's duration serves to raise its interest rate sensitivity, and vice-versa.
 Therefore, if interest rates are expected to fall, bonds with lower
coupons can be expected to appreciate faster than higher coupon
bonds of the same maturity

122. E.g. 1. What would be the percentage change in the price of a bond with a
modified duration of 9, given that interest rates fall 50 basis points (i.e.. 0.5%)?
P
*
D
P
y
= (-9)(-.05%) = 4.5%
E.g. 2. What would be the % change in price of a bond with a Macaulay
Duration of 10 if interest rates rise by 50 basis points (i.e.. 0.5%) The current
YTM is 4%.
D
D* =
1
=
10/1.04 =9.615
y
Therefore , % change in price
ΔP
P
D
*
Δy = (-9.615)(.5%) = -4.81%

123. Rule 1 The duration of a zero-coupon bond
equals its time to maturity
Rule 2 Holding maturity constant, a bond’s
duration is higher when the coupon
rate is lower
Rule 3 Holding the coupon rate constant, a
bond’s duration generally increases
with its time to maturity
Rule 4 Holding other factors constant, the
duration of a coupon bond is higher
when the bond’s yield to maturity is

124. 


Duration approximates price change but
isn’t exact
For small changes in yields, duration is
close but for larger changes in yields, there
can be a large error
Duration always underestimates the value
of bond price increases when yields fall and
overestimates declines in price when yields
rise

125. Price
Pricing error
from convexity
Yield
Duration
(approximates a
line vs a curve)

126. A is more convex than B:
If rates inc  A’s price falls less than B’s
If rates dec  A’s price rises more than B’s
Convexity is desirable for investors so they will pay for it
(ie. A’s yield is probably less than B’s)
Bond A
0
Bond B
Change in yield to maturity (%)

127. 
Definition of convexity:
◦ The rate of change of the slope of the price/yield
curve expressed as a fraction of the bond’s price.

128. 1.
2.
3.
Inverse relationship between convexity and
coupon rate
Direct relationship between maturity and
convexity
Inverse relationship between yield and
convexity

129. 

Classical immunization is a passive bond portfolio
strategy to shield fixed-income assets from
interest rate risk. It is done by setting the duration
of a bond portfolio equal to its time horizon.
In an immunized bond portfolio the effects of
rising rates reducing the capital value of the bonds,
and increasing the return on reinvestment of
coupon payments, exactly offset each other, and
vice-versa.
Immunization techniques thus
- Reduces interest rate risk to zero
- Shields portfolio from interest rate fluctuations

130. Type of Risks to Bondholders

Price risk / Market risk :
An investor who buys a bond with maturity more than his investment horizon is
exposed to market risk : if interest rates go up (down) the investor is worse off
(better off).
D >H The bond exposes the investor to market risk if the duration of the bond
exceeds his investment horizon

Reinvestment risk:
An investor who buys a bond with maturity less than (or equal to) her investment
horizon
is exposed to reinvestment risk. So, if interest rates go up (down) the investor is
better off
(worse off).
D < H The bond exposes the investor to reinvestment risk if the duration of the
D=bond is shorter than his(H) matches Duration (D), the two risks will
H If Holding Period investment horizon
exactly offset each other – Bond is said to be immunized.

131. Banks are concerned with the protection of the current net
worth or net market value of the firm ,whereas, pension fund
and insurance companies are concerned with protecting the
future value of their portfolio. Here I’ll take the example of
pension fund which has to pay back pension fund of Rs.
10,000/- to one of its investor, with guaranteed rate of 8%
after 5 years. So, it is obligated to pay Rs. 10,000
*(1.08)^=Rs. Rs.14,693.28 in years. So, suppose, pension
fund company chooses to fund its obligation with Rs. 10,000
, of 8% annual coupon bond selling at par value with 6 years
maturity. So, if interest rate remains at 8% the amount
accrued will exactly be equal to the obligation of
Rs.14,693.28 in 5 years. Now we consider two scenarios,
where interest rate goes down to 7% and in second case it
reaches 9%. In 7% scenario, amount accrued will be equal to
Rs. 14,694.05 in years and in 9% scenario it will be Rs.
14,696.02 in years. The three scenarios with their
accumulated value of invested payments.

132. Payment number
Yrs. Remaining
until obligation
If rates remain at 8%
Accumulated value of
invested payment
Formula used
Value of formula
1
4
800*(1.08)^4
1088.391168
2
3
800*(1.08)^3
1007.7696
3
2
800*(1.08)^2
933.12
4
1
800*(1.08)^1
864
5
0
800*(1.08)^0
800
sale of bond
0
10800/1.08
10000
14693.28077

133. Yrs. Remaining
until obligation
Payment number
Accumulated value of
invested payment
if rates fall to 7%
Formula used
Value of formula
1
800*(1.07)^4
1048.636808
2
3
800*(1.07)^3
980.0344
3
2
800*(1.07)^2
915.92
4
1
800*(1.07)^1
856
5
sale of bond
4
0
800*(1.07)^0
800
0
10800/1.07
10093.45794
14694.04915

134. Yrs. Remaining
until obligation
Payment number
Accumulated value of
invested payment
if rates fall to 9%
formula used
value of formula
1
800*(1.09)^4
1129.265288
2
3
800*(1.09)^3
1036.0232
3
2
800*(1.09)^2
950.48
4
1
800*(1.09)^1
872
5
sale of bond
4
0
800*(1.09)^0
800
0
10800/1.09
9908.256881
14696.02537

135. Accumulated value of
invested payment

136. 



Rebalancing required as duration declines
more slowly than term to maturity
Modified duration changes with a change in
market interest rates
Yield curves shift
In practice, we can’t rebalance the portfolio
constantly because of transaction costs

137. 

The duration of a bond portfolio is equal to the
weighted average of the durations of the bonds in
the portfolio
The portfolio duration, however, does not change
linearly with time. The portfolio needs, therefore,
to be rebalanced periodically to maintain target
date immunization

138.  Risk Immunization: elimination of interest rate risk by
matching duration of financial assets and liabilities
 Financial Institutions: Banks especially utilize these
techniques

Assets of Bank
Loans to customers
 Liabilities of Bank
Deposits from Customers
Auto
CDs
Mortgage
Bank
accounts
Student
(Bank is Owed this $)
(Bank Owes this $)

139. 
Assets of Bank
◦ Duration=15 yr
• Liabilities of Bank
– Duration=5 yr
 If interest rates drop, the value of assets increases more than the
value of liabilities decreases.
- Bank Value Increases.
 If interest rates increase, the value of the assets decrease more than
the value of liabilities increases.
- Bank Value Drops.
 Bank is speculating on interest rates

140. 
Assets of Bank
- Duration=15 yr
• Liabilities of Bank
- Duration=15 yr
 For a bank to not be speculating on interest rates
 Duration of Assets = Duration of Liabilities

141. 



Commercial banks borrow money by accepting deposits and
use those funds to make loans. The portfolio of deposits and
the portfolio of loans may both be viewed as bond portfolios,
with the deposit portfolio constituting the liability portfolio and
the loan portfolio constituting the asset portfolio.
If a bank’s deposits and loans have different maturities, the
bank may lose money in the event of an overall change in
interest rate levels.
To eliminate this risk, banks may wish to immunize their
portfolio. A portfolio is immunized if the value of the portfolio
is not affected by a change in interest rates.
Immunization is accomplished by managing the duration of the
portfolio.

142. Bank Immunization Case
(contd.)
Balance Sheet of Simple National Bank
Original Position
Assets
Loan Portfolio Value
Portfolio Duration
Interest Rate
Liabilities
$1,000
5 years
10%
Deposit Portfolio Value
Portfolio Duration
Owners' Equity
Interest Rate
$1,000
1 year
$0
10%
Following Rise in Rates to 12 Percent
Assets
Loan Portfolio Value
Liabilities
$909
Deposit Portfolio Value
Owners' Equity
$982
- $72
Notice that the duration of the assets is 5 years and the duration of the
liabilities is 1 year.

143. Bank Immunization Case
(contd.)


Assume that interest rates rise from 10% to 12% on both deposit
and loan portfolios.
What is the change in value of the deposit and loan portfolios?
Applying the following duration formula:
dP i = - D



d (1 + r
i
(1 + r
i
i
)
)
P
i
Deposit Portfolio
dP = -1 (.02/1.10) $1,000 = -$18.18
Loan Portfolio
dP = -5 (.02/1.10) $1,000 = - $90.91
So the deposits (liabilities) have decreased in value by $18.18 and
the assets have decreased in value by $90.91. The combined effect
is equal to a $72 reduction in equity.

144. Bank Immunization Case
(contd.)
Immunized Balance Sheet of Simple National Bank
Original Position
Assets
Loan Portfolio Value
Portfolio Duration
Interest Rate
Liabilities
$1,000
3 years
10%
Deposit Portfolio Value
Portfolio Duration
Owners' Equity
Interest Rate
$1,000
3 years
$0
10%
Following Rise in Rates to 12 Percent
Assets
Loan Portfolio Value
Liabilities
$945
Deposit Portfolio Value
Owners' Equity
$945
$0

145. Bank Immunization Case
(contd.)
The previous table illustrates the impact of interest
rates changes for a bank with immunization. Both the
liabilities and assets have a duration of 3 years.
Estimate the price change using the duration formula:
dP = -3 (.02/1.10) $1,000 = - $54.55
Because the bank is immunized against a change in
interest rates, the change in rates have an equal and
offsetting effect on the liabilities and assets of the
bank
leaving the equity position of the bank unchanged.