1. introduction basics of investments.ppt

04/11/20231. Introduction and Basics of Investments

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1. Introduction and Basics of Investments


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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?


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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


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Let us Start the session!!!


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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


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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


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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


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Investments Parameters

◦ Return

◦ Risk

◦ Time Horizon

◦ Tax Considerations

◦ Liquidity

◦ Marketability

Investment


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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


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Next How to Measure Return and Risk???


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Return

◦ Historical

HPR

HPY

◦ Expected

Risk

◦Historical

◦Expected

Investment

04/11/2023 2. Return and Risk 12

2. Return and Risk


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?


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


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%


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]


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%


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%


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%


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%


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%


Probability Distribution of Return


So there is a risk of earning more than one return or

uncertainty in return


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?


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.


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


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


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%


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


Skewness and Kurtosis


So How Return and Risk should be related…..next class


End of Lecture 2Thank You!!!


3. Markowitz Portfolio Theory


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


Portfolio Theories


Portfolio Theories

Markowitz Portfolio

Market Model/Index Model


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.


Utility TheoryUtility

Wealth

U1

U2


Indifference CurveExpectedReturn of Portfolio

S.D. of Portfolio


Risk Averse - Taker

Risk Averse

Risk Taker

ExpectedReturn of Portfolio

S.D. of Portfolio


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”.


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.


Measuring Portfolio Return


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.


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%


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


Measuring Portfolio Risk


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%


Portfolio Risk

-20%

-10%

0%

10%

20%

30%

40%

50%

2001 2002 2003 2004 2005

Series1

Series2

Series3


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%


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


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%


S.D. 22.64% 22.64% 22.64%


Portfolio Risk

-20%

-10%

0%

10%

20%

30%

40%

50%

2001 2002 2003 2004 2005

Series1

Series2

Series3


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%


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



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



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.


End of LectureThank You!!!


So How a Multiple Choice of Portfolio can be compared using Markowitz

Portfolio Theory …..next class


4. Markowitz Portfolio Theory – Efficient Frontier


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


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


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%


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%


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


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


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


Efficient Sets of Portfolios

Efficient Sets of Portfolios

Standard Deviations

Expe

cted

Ret

urn


End of LectureThank You!!!


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)


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


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.


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


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.


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