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04/11/20231. Introduction and Basics of Investments
1
1. Introduction and Basics of Investments
04/11/20231. Introduction and Basics of Investments
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
What this Paper is All About?
04/11/20231. Introduction and Basics of Investments
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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.
Topics and References
04/11/20231. Introduction and Basics of Investments
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Let us Start the session!!!
04/11/20231. Introduction and Basics of Investments
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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.
Investment
04/11/20231. Introduction and Basics of Investments
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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
Investment
04/11/20231. Introduction and Basics of Investments
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Investment Avenues
Shares
Bonds
Mutual Funds
Debentures
PF
Gold Silver Real Estate
Indira Vikas Patra Post Office Deposits Bank Deposits NSC
04/11/20231. Introduction and Basics of Investments
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Investments Parameters
◦ Return
◦ Risk
◦ Time Horizon
◦ Tax Considerations
◦ Liquidity
◦ Marketability
Investment
04/11/20231. Introduction and Basics of Investments
9
Risk-Return Trade off
Risk
Return
•Gold
•Real Estate
•Shares•MFs Equity Fund
•Bonds•PF
•Debentures•MFs Debt Funds
•Bank Deposit
• NSC, Post-Office DepositKisan Vikas Patra
•Derivatives
04/11/20231. Introduction and Basics of Investments
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Next How to Measure Return and Risk???
04/11/20231. Introduction and Basics of Investments
11
Return
◦ Historical
HPR
HPY
◦ Expected
Risk
◦Historical
◦Expected
Investment
04/11/2023 2. Return and Risk 12
2. Return and Risk
04/11/2023 2. Return and Risk 13
What we did in last class…
04/11/20232. Return and Risk 14
We covered in last class
◦ Why people invest?
◦ What they want from their investment?
◦ Where all they can invest and what parameters they adopt to invest?
04/11/20232. Return and Risk 15
Investment
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
04/11/20232. Return and Risk 16
How do we measure return?
◦ 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%
04/11/20232. Return and Risk 17
Computing Mean Historical Return
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]
04/11/2023 2. Return and Risk 18
How AM is different to GM
YearBeginning
ValueEnding
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)] 1/3 – 1 = 3.35%
04/11/2023 2. Return and Risk 19
How AM is different to GM
YearBeginning
ValueEnding
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)] 1/2 – 1 = 0.00%
04/11/2023 2. Return and Risk 20
How do we Calculate Expected ReturnExpected 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 Probability of Occurrence
Deep Recession 5%
Mild Recession 20%
Average Economy 50%
Mild Boom 20%
Strong Boom 5%
04/11/2023 2. Return and Risk 21
How do we Calculate Expected Return
Economy Growth
Probability of Occurrence T-Bills
Corporate Bonds
Equity A
Equity B
Deep Recession 5% 8% 12% -3% -2%
Mild Recession 20% 8% 10% 6% 9%
Average Economy 50% 8% 9% 11% 12%
Mild Boom 20% 8% 8.50% 14% 15%
Strong Boom 5% 8% 8% 19% 26%
100%
Expected Rate
of Return 8.00% 9.20% 10.30% 12.00%
04/11/20232. Return and Risk 22
Probability Distribution of Return
Probability Distribution of Equity "A"
0%
10%
20%
30%
40%
50%
60%
Dispersion from Expected Return
Prob
abili
ty
Series1
Series1 5% 20% 50% 20% 5%
-13.300% -4.300% 0.700% 3.700% 8.700%
04/11/20232. Return and Risk 23
Probability Distribution of Return
04/11/2023 2. Return and Risk 24
So there is a risk of earning more than one return or
uncertainty in return
04/11/20232. Return and Risk 25
What is Risk
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?
04/11/20232. Return and Risk 26
What is Risk 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.
04/11/20232. Return and Risk 27
Risk Measures for Historical Returns
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
04/11/20232. Return and Risk 28
Risk Measurement for Expected Return 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
04/11/2023 2. Return and Risk 29
Return and Risk MeasurementExpected Return or Risk
Measure T-BillsCorporate
Bonds Equity A Equity B
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%
04/11/20232. Return and Risk 30
Things to look Measuring Risk
• Variance and Standard DeviationThe spread of the actual returns around the expected return; The greater thedeviation of the actual returns from expected returns, the greater the variance
• SkewnessThe biasness towards positive or negative returns;
• KurtosisThe shape of the tails of the distribution ; fatter tails lead to higher kurtosis
04/11/20232. Return and Risk 31
Skewness and Kurtosis
04/11/2023 2. Return and Risk 32
So How Return and Risk should be related…..next class
04/11/2023 2. Return and Risk 33
End of Lecture 2Thank You!!!
04/11/2023 2. Return and Risk 34
3. Markowitz Portfolio Theory
04/11/2023 2. Return and Risk 35
What we did in last class…
04/11/20232. Return and Risk 36
We covered in last classes
◦ 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
04/11/2023 2. Return and Risk 37
Portfolio Theories
04/11/20232. Return and Risk 38
Portfolio Theories
Markowitz Portfolio
Market Model/Index Model
04/11/20232. Return and Risk 39
Markowitz Portfolio Theory
◦ 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.
04/11/20232. Return and Risk 40
Utility TheoryUtility
Wealth
U1
U2
04/11/20232. Return and Risk 41
Indifference CurveExpectedReturn of Portfolio
S.D. of Portfolio
04/11/20232. Return and Risk 42
Risk Averse - Taker
Risk Averse
Risk Taker
ExpectedReturn of Portfolio
S.D. of Portfolio
04/11/20232. Return and Risk 43
Rules of Indifference Curve
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”.
04/11/20232. Return and Risk 44
Indifference Curve
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.
04/11/2023 2. Return and Risk 45
Measuring Portfolio Return
04/11/20232. Return and Risk 46
Risk - Return on a Portfolio
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.
04/11/2023 2. Return and Risk 47
Expected Return of Single Security
Probability Possible Returns
0.35 0.08 0.028
0.3 0.1 0.03
0.2 0.12 0.024
0.15 0.14 0.021
Expected Return 10.30%
04/11/2023 2. Return and Risk 48
Expected Return of Portfolio
Weight Expected Returns of Securities
0.2 0.1 0.02
0.3 0.11 0.033
0.3 0.12 0.036
0.2 0.13 0.026
Expected Return of Portfolio 0.115
04/11/2023 2. Return and Risk 49
Measuring Portfolio Risk
04/11/2023 2. Return and Risk 50
Portfolio Risk
Expected Return
Year Stock A Stock B Portfolio AB
2001 40% -10% 15%
2002 -10% 40% 15%
2003 35% -5% 15%
2004 -5% 35% 15%
2005 15% 15% 15%
Avg Return 15% 15% 15%
S.D. 22.64% 22.64% 0.00%
04/11/20232. Return and Risk 51
Portfolio Risk
-20%
-10%
0%
10%
20%
30%
40%
50%
2001 2002 2003 2004 2005
Series1
Series2
Series3
04/11/20232. Return and Risk 52
Portfolio Risk
Expected Return
YearStock
AStock
BPortfolio
AB
2001 40% -10% 15%
2002 -10% 40% 15%
2003 35% -5% 15%
2004 -5% 35% 15%
2005 15% 15% 15%
Avg Return 15% 15% 15%
S.D.22.64
%22.64
% 0.00%
-20%
-10%
0%
10%
20%
30%
40%
50%
2001 2002 2003 2004 2005
Correlation Coefficient = -1.0
04/11/2023 2. Return and Risk 53
Portfolio Risk
Expected Return
Year Stock A Stock B Portfolio AB
2001 -10% -10% -10%
2002 40% 40% 40%
2003 -5% -5% -5%
2004 35% 35% 35%
2005 15% 15% 15%
Avg Return 15% 15% 15%
S.D. 22.64% 22.64% 22.64%
04/11/20232. Return and Risk 54
Portfolio Risk
-20%
-10%
0%
10%
20%
30%
40%
50%
2001 2002 2003 2004 2005
Series1
Series2
Series3
04/11/20232. Return and Risk 55
Portfolio Risk
Expected Return
YearStock
AStock
BPortfolio
AB
2001 -10% -10% -10%
2002 40% 40% 40%
2003 -5% -5% -5%
2004 35% 35% 35%
2005 15% 15% 15%
Avg Return 15% 15% 15%
S.D.22.64
%22.64
% 22.64%
-20%
-10%
0%
10%
20%
30%
40%
50%
2001 2002 2003 2004 2005
Correlation Coefficient = +1.0
04/11/20232. Return and Risk 56
Measuring Portfolio Risk
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
04/11/20232. Return and Risk 57
Measuring Portfolio Risk
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.
04/11/2023 2. Return and Risk 58
End of LectureThank You!!!
04/11/2023 2. Return and Risk 59
So How a Multiple Choice of Portfolio can be compared using Markowitz
Portfolio Theory …..next class
04/11/2023 2. Return and Risk 60
4. Markowitz Portfolio Theory – Efficient Frontier
04/11/2023 2. Return and Risk 61
What we did in last class…
04/11/20232. Return and Risk 62
We covered in last classes
◦ 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
04/11/2023 2. Return and Risk 63
Expected data for the Two Securities
ER 0.103 0.12
Variance 0.0019310 0.00232
SD 0.04394315 0.048166
Coefficient of Variation 0.42663248 0.401386
Covariance 0.00202
Correlation Coefficient 0.95436882 -0.75
Risk Tolerance 0.5
04/11/2023 2. Return and Risk 64
Expected Return for PortfoliosPortfolios 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%
04/11/2023 2. Return and Risk 65
Expected Risk for Portfolios
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%
04/11/2023 2. Return and Risk 66
Summary Sheet for the Portfolios
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 1 0.1200 0.00232 0.04816638 0.11536
04/11/20232. Return and Risk 67
Feasible Sets of Portfolios
Feasible Sets of Portfolios
0.10000.10500.11000.11500.12000.1250
0 0.01 0.02 0.03 0.04 0.05 0.06
Standard Deviations
Exp
ecte
d R
etur
n
04/11/20232. Return and Risk 68
Eliminating Inferior PortfoliosTwo 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
04/11/20232. Return and Risk 69
Efficient Sets of Portfolios
Efficient Sets of Portfolios
Standard Deviations
Expe
cted
Ret
urn
04/11/2023 2. Return and Risk 70
End of LectureThank You!!!
04/11/2023 2. Return and Risk 71
So How a Portfolio can be optimised using Markowitz Portfolio Theory
…..next class
CAPM – An Equilibrium Model
We have done till now in Portfolio Management…
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
Assumptions
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 risk-free 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
Implication of Assumptions
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.
Linear Efficient Frontier
Risk
Return
RiskFree Rate
Risky SecuritiesEfficient Curve
Linear EfficientCurve
IndifferenceCurve
M
CAPMSo 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…..
CAPM
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….
CAPMThe 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.
The Efficient Set
Risk
Return
Rm
σpσm
Rf
The Efficient Set
Rp = Rf + (Rm- Rf) X σp
Slope of line is price of risk
And Intercept is price of time
σm
CAPM 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
CAPM
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)
Inputs required to use the CAPM -
(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.
Portfolio Analysis – Efficient Frontier
Efficient Frontier
Two Conditions
1) Offer Maximum Return for varying levels of Risk, and
2) Offer Minimum Risk for varying levels of expected returnAll the feasible sets are not efficient unless it passes through this test
Efficient Sets and Feasible Sets
Feasible Sets
A
D
C
B
Efficient Sets and Feasible Sets
Feasible Sets
A
D
C
B
IC 2
IC 1
IC 3
How to form Efficient Frontier ?
2 Stock Case
Stocks Expected Return Standard Deviation
A 5% 20%
B 15% 40%
Formula
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.
Expected Return for Portfolios
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%
Standard Deviation of Portfolio
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%
Efficient Frontier
Upper and Lower Bounds to Portfolios
0.00%2.00%
4.00%6.00%
8.00%10.00%
12.00%14.00%
16.00%
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00%
Standard Deviations
Expe
cted
Retu
rn
Market Models
Market Model
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
Graphical Presentation of Market Model
ri = αiI + βiI rI
Beta
βiI = σiI
σI2
σiI = CovarianceσI2 = Variance of Market Index
Random Error
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%) = -5%
11% - (-1% +8%) = 4%
Graphical Presentation of Market Model
Infotech versus S&P 500: 1992-1996
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%
Security’s Total Risk
σi2 =βiI
2X σI2 + σεi
2
Where ,σi
2 = variance of security i
βiI2X σI
2 = Market risk of security i
σεi2 = Unique risk of security i
Portfolios Return
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
.....continued
rp = ∑Xi (αiI + βiI rI + εiI)
= ∑Xi αiI + (∑Xi βiI ) rI + ∑XiεiI
= αpI + βpI rI + εpI
Where i range from o to n.
Intercept Slope X independent Variable
Random Error
Portfolio Risk
σ2p =β2
pIσ2I + σ2
εp
Where ,β2
pI = [∑Xi βiI] 2 ----- Systematic Risk
σ2εp = ∑Xi
2 σ2εi ----- Unique Risk
Risk and Diversification
Unique RiskMarket Risk
Total Riskσp
N
Calculations
StockPortfolio
Weight BetaExpected Return of
StockVariance 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
Questions
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) = 0.020Covariance (A,C) = 0.035Covariance (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
Universal Principles for bonds
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.
Price
YTM
Bond Pricing Relationships
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)
Interest Rate Sensitivity
Bond Coupon Maturity YTM
ABCD
Change in yield to maturity (%)
Perc
en
tag
e c
han
ge in
bon
d p
rice
0
A 12% 5 years 10%
B 12% 30 years 10%
C 3% 30 years 10%
D 3% 30 years 6%
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 short-term bonds
Price sensitivity is inversely related to the yield to maturity at which the bond is selling
Interest Rate Sensitivity (contd.)
Duration 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.
Example: 8-year, 9% annual coupon bond
0
200
400
600
800
1000
1200
1 2 3 4 5 6 7 8
Year
Cas
h flo
w Bond Duration = 5.97 years
PV of cash flows
Actual cash flows
Area where PV of CF before and after balance out
icerP)y1(CF
wt
tt
twtDT
1t
tCF
Macaulay Duration: Calculation
PV of cash flowsas a % of bond
price
Cash Flow for Period t
Duration and Price Volatility An adjusted measure of duration can be used to approximate the price
volatility of a bond
mYTM
1
DurationMacaulay Duration Modified
Where:m = number of payments a yearYTM = nominal YTM
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
2.0
40
1040
sum
34.553
855.611
964.540
.0358
.8871
1.000
.0537
1.7742
1.8853
Duration Calculation Example
Eg. Coupon = 8%, yield = 10%, years to maturity = 2
DURATION
Why is duration a key concept?
1. It’s a simple summary statistic of the effective average maturity of the portfolio;
2. It is an essential tool in immunizing portfolios from interest rate risk;
3. It is a measure of interest rate risk of a portfolio
4. Equal duration assets are equally sensitive to changes in interest rates
Price change is proportional to duration and not to maturity
Duration/Price Relationship
y
yD
P
P
1
)(
y
DD
1* yD
PP *
• Where D = duration
D* is the 1st derivative of bond’s price with respect to yield ie. D* = (-1/P)(dP/dY)
Duration/Price Relationship
y
yD
P
P
1
)(
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
Duration/Price Relationship
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%)?
= (-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* = = 10/1.04 =9.615
Therefore , % change in price = (-9.615)(.5%) = -4.81%
y
D
1
ΔyDP
ΔP *
yDPP *
Rules for Duration
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 lower
Convexity 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
Duration and Convexity
Price
Duration
(approximates a line vs a curve)
Pricing error from convexity
Yield
Convexity of Two Bonds0
Change in yield to maturity (%)
Perc
en
tag
e c
han
ge in
bon
d p
rice
Bond A
Bond B
A is more convex than B:If rates inc A’s price falls less than B’sIf 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)
Convexity Definition of convexity:
◦ The rate of change of the slope of the price/yield curve expressed as a fraction of the bond’s price.
Properties of Convexity
1. Inverse relationship between convexity and coupon rate
2. Direct relationship between maturity and convexity
3. Inverse relationship between yield and convexity
Immunization - Defined 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
Immunization 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 horizonis 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 bond is shorter than his investment horizonD= H If Holding Period (H) matches Duration (D), the two risks will
exactly offset each other – Bond is said to be immunized.
Immunization explained with example 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 numberYrs. Remaining
until obligationAccumulated value of
invested payment
If rates remain at 8% 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
Payment numberYrs. Remaininguntil obligation
Accumulated value of invested payment
if rates fall to 7% Formula used Value of formula
1 4 800*(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 0 800*(1.07)^0 800
sale of bond 0 10800/1.07 10093.45794
14694.04915
Payment numberYrs. Remaining
until obligationAccumulated value of
invested payment
if rates fall to 9% formula used value of formula
1 4 800*(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 0 800*(1.09)^0 800
sale of bond 0 10800/1.09 9908.256881
14696.02537
Accumulated value of invested payment
So, at 7%, the fund is Rs.14,694.05/- having surplus of Rs. 0.77 and at 9%, surplus as compared to the amount at 8% is Rs. 2.74.So, we can see that by matching the duration of assets and liabilities we are able to protect the funds from interest rate fluctuation. When coupon rate decreases, increase in resale value of bond balances the reduction in the coupon payments. Increase in interest rate decreases the resale value of bond but balanced by the increased coupon payment , thus, shields the portfolio from interest rate fluctuation.
Difficulties in Maintaining Immunization Strategy
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
Immunization and duration of bond portfolios
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
Application of Immunization Concepts Risk Immunization: elimination of interest rate risk by
matching duration of financial assets and liabilities Financial Institutions: Banks especially utilize these
techniques
Assets of BankLoans to customersAutoMortgageStudent
(Bank is Owed this $)
Liabilities of BankDeposits from Customers
CDsBank accounts
(Bank Owes this $)
Risk Immunization
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=5 yr
Risk Immunization
For a bank to not be speculating on interest rates
Duration of Assets = Duration of Liabilities
Assets of Bank- Duration=15 yr
• Liabilities of Bank- Duration=15 yr
Bank Immunization Case 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.
Table below illustrates the impact of interest rate changes for a bank with no immunization.
Balance Sheet of Simple National Bank
Original Position
Assets
Liabilities Loan Portfolio Value $1,000 Portfolio Duration 5 years Interest Rate 10%
Deposit Portfolio Value $1,000 Portfolio Duration 1 year Owners' Equity $0 Interest Rate 10%
Following Rise in Rates to 12 Percent Assets
Liabilities
Loan Portfolio Value $909 Deposit Portfolio Value $982 Owners' Equity - $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:
P )r + (1
)r + (1 d D - = dP i
i
i i i
Deposit PortfoliodP = -1 (.02/1.10) $1,000 = -$18.18
Loan PortfoliodP = -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
Liabilities
Loan Portfolio Value $1,000 Portfolio Duration 3 years Interest Rate 10%
Deposit Portfolio Value $1,000 Portfolio Duration 3 years Owners' Equity $0 Interest Rate 10%
Following Rise in Rates to 12 Percent Assets
Liabilities
Loan Portfolio Value $945 Deposit Portfolio Value $945 Owners' Equity $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.
Bank Immunization Case (contd.)