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1. Introduction and Basics of
Investments

12/20/2013

1


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


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
Let us Start the session!!!

1. Introduction and Basics of
Investments

12/20/2013

4


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
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

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6
 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

7
Investments Parameters



◦

Return

◦

Risk

◦

Time Horizon

◦

Tax Considerations

◦

Liquidity

◦

Marketability
1. Introduction and Basics of
Investments

12/20/2013

8
•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

9
Next
How to Measure Return and Risk???

1. Introduction and Basics of
Investments

12/20/2013

10
Return



◦



Risk
◦ Historical



HPR

◦ Expected



◦

Historical

HPY

Expected

1. Introduction and Basics of
Investments

12/20/2013

11
12/20/2013

2. Return and Risk

12
What we did in last class…

12/20/2013

2. Return and Risk

13
◦ 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
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

2. Return and Risk

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15
◦ 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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16


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]

2. Return and Risk

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17
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
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
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
12/20/2013

Probability of Occurrence

5%
2. Return and Risk

20
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
12/20/2013

2. Return and Risk

21
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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2. Return and Risk

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2. Return and Risk

24




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

12/20/2013

25




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


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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27


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
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%

12/20/2013

2. Return and Risk

Equity A Equity B

29
• 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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2. Return and Risk

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2. Return and Risk

32
Thank You!!!

12/20/2013

2. Return and Risk

33
12/20/2013

2. Return and Risk

34
What we did in last class…

12/20/2013

2. Return and Risk

35
◦ 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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2. Return and Risk

37




Markowitz Portfolio

Market Model/Index Model

2. Return and Risk

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38
◦ 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
Utility
U2
U1

Wealth
2. Return and Risk

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40
Expected
Return of
Portfolio

S.D. of
Portfolio
2. Return and Risk

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41
Expected
Return of
Portfolio

Risk Averse

Risk Taker

S.D. of
Portfolio
2. Return and Risk

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42







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


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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44
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2. Return and Risk

45


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
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
Weight

Expected Returns of Securities
0.2
0.3
0.3
0.2
Expected Return of Portfolio

12/20/2013

2. Return and Risk

0.1
0.11
0.12
0.13

0.02
0.033
0.036
0.026
0.115

48
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2. Return and Risk

49
Expected Return
Year

Stock A

Stock B

Portfolio AB

2001

-10%

40%

15%

35%

-5%

15%

2004

-5%

35%

15%

2005

12/20/2013

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
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
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
Expected Return
Year

Stock A

Stock B

Portfolio AB

2001

40%

40%

40%

-5%

-5%

-5%

2004

35%

35%

35%

2005

12/20/2013

-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
50%
40%
30%
Series1

20%

Series2
10%

Series3

0%
-10%

2001

2002

2003

2004

2005

-20%

2. Return and Risk

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54
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

12/20/2013

55
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
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
Thank You!!!

12/20/2013

2. Return and Risk

58
12/20/2013

2. Return and Risk

59
12/20/2013

2. Return and Risk

60
What we did in last class…

12/20/2013

2. Return and Risk

61
◦ 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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62
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
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
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
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
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
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

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68
Expected Return

Efficient Sets of Portfolios

Standard Deviations

2. Return and Risk

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69
Thank You!!!

12/20/2013

2. Return and Risk

70
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2. Return and Risk

71


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










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












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.
Return

Indifference
Curve
Linear Efficient
Curve

M

Risky Securities
Efficient Curve

Risk
Free Rate

Risk
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…..
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….
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.
Return

Rm

Rf
σm

Risk

σp
Rp = Rf + (Rm- Rf) X σp

σm
 Slope of line is price of risk
 And Intercept is price of time


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









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)
(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.
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
B
D
Feasible Sets

C

A
Efficient Sets and Feasible Sets
IC 3

IC 2

B

D
Feasible Sets

IC 1

A

C
Stocks
Deviation
A
B

Expected Return Standard
5%

20%

15%

40%
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.
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%
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%
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
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
ri = αiI + βiI rI
βiI = σiI
σI2
σiI = Covariance
σI2 = Variance of Market Index
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%
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%
σi2 =βiI2X σI2 + σεi2
Where

,

σi2 = variance of security i

βiI2X σI2 = Market risk of security i
σεi2 = Unique risk of security i
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
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
σ2p =β2pIσ2I + σ2εp
Where ,
β2pI = [∑Xi βiI] 2 ----- Systematic Risk
σ2εp = ∑Xi2 σ2εi ----- Unique Risk
σp

Total Risk

Unique Risk

Market Risk

N
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






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
Duration, Convexity and Portfolio
Immunization
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
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.
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
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






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








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.
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
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
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
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
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


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)
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
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%
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






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
Price
Pricing error
from convexity

Yield

Duration
(approximates a
line vs a curve)
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 (%)


Definition of convexity:
◦ The rate of change of the slope of the price/yield
curve expressed as a fraction of the bond’s price.
1.
2.
3.

Inverse relationship between convexity and
coupon rate
Direct relationship between maturity and
convexity
Inverse relationship between yield and
convexity




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
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.
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.
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
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
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
Accumulated value of
invested payment







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




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
 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 $)


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


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








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.
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.
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.
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
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.
1. introduction   basics of investments.ppt

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1. introduction basics of investments.ppt

  • 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 7
  • 8. Investments Parameters  ◦ Return ◦ Risk ◦ Time Horizon ◦ Tax Considerations ◦ Liquidity ◦ Marketability 1. Introduction and Basics of Investments 12/20/2013 8
  • 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 9
  • 10. Next How to Measure Return and Risk??? 1. Introduction and Basics of Investments 12/20/2013 10
  • 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 12/20/2013 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 2. Return and Risk 12/20/2013 15
  • 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 12/20/2013 16
  • 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] 2. Return and Risk 12/20/2013 17
  • 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)] 12/20/2013 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)] 12/20/2013 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 12/20/2013 Probability of Occurrence 5% 2. Return and Risk 20
  • 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 12/20/2013 22
  • 23. 2. Return and Risk 12/20/2013 23
  • 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 12/20/2013 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 12/20/2013 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 12/20/2013 27
  • 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 12/20/2013 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% 12/20/2013 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 12/20/2013 30
  • 31. 2. Return and Risk 12/20/2013 31
  • 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 12/20/2013 36
  • 38.   Markowitz Portfolio Market Model/Index Model 2. Return and Risk 12/20/2013 38
  • 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 12/20/2013 39
  • 41. Expected Return of Portfolio S.D. of Portfolio 2. Return and Risk 12/20/2013 41
  • 42. Expected Return of Portfolio Risk Averse Risk Taker S.D. of Portfolio 2. Return and Risk 12/20/2013 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 12/20/2013 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 12/20/2013 44
  • 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 12/20/2013 46
  • 48. Weight Expected Returns of Securities 0.2 0.3 0.3 0.2 Expected Return of Portfolio 12/20/2013 2. Return and Risk 0.1 0.11 0.12 0.13 0.02 0.033 0.036 0.026 0.115 48
  • 50. Expected Return Year Stock A Stock B Portfolio AB 2001 -10% 40% 15% 35% -5% 15% 2004 -5% 35% 15% 2005 12/20/2013 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
  • 53. Expected Return Year Stock A Stock B Portfolio AB 2001 40% 40% 40% -5% -5% -5% 2004 35% 35% 35% 2005 12/20/2013 -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
  • 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 12/20/2013 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 12/20/2013 57
  • 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 12/20/2013 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% 12/20/2013 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% 12/20/2013 2. Return and Risk 65
  • 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 12/20/2013 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 12/20/2013 68
  • 69. Expected Return Efficient Sets of Portfolios Standard Deviations 2. Return and Risk 12/20/2013 69
  • 72.
  • 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.
  • 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.
  • 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.
  • 85.
  • 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
  • 88. Efficient Sets and Feasible Sets IC 3 IC 2 B D Feasible Sets IC 1 A C
  • 89.
  • 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
  • 95.
  • 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
  • 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
  • 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
  • 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.