4-risks-variable-income.pptx

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MANAGING CONTINUOUS RISKS – VARIABLE INCOME ASSETS

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Contents

1. Return & Risk measures for 2-asset portfolio2. Return & Risk measures for 3-asset portfolio3. Recognising the efficient & inefficient portfolios4. Generalisation: The N-security portfolio5. Markowitz Portfolio Theory6. Sharpe Single Index Model7. CAPM

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1. 2-Security Portfolio: Eg.1

• Security 1: Expected return: 16%, SD = 10%• Security 2: Expected return: 14%, SD = 16%• Correlation between 1 & 2: 0.5• Total funds available = 200000• Funds to be allocated to 1: 120000• Remaining to be allocated to 2• Calculate expected return & risk of the

portfolio

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1A: 2-Security Portfolio: Eg.2

• Security 1: SD = 2%, Weight: 2/3• Security 2: SD = 4%, Weight: 1/3• Calculate portfolio SD for the cases: r = -0.5, r = 0, r = 0.5, r = +1• What are your conclusions?

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1A: 2-Security Portfolio: Ans.2• Portfolio SD are:

• The smaller the correlation the between the securities the lower is the portfolio risk.

• When r = +1 the portfolio SD is the weighted average of the individual SDs

• Favourable effects of diversification occur only when securities are perfectly positively correlated.

r PORTFOLIO SD

-0.5 1.34

0 1.9

+0.5 2.3

+1.0 2.658

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1B: 2-Security Portfolio: Illustration

• Fischer/Jordan/p579: Diagram

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2. 3-Security Portfolio: Eg.3

• Security 1: Expected return: 16%, SD = 10%• Security 2: Expected return: 14%, SD = 15%• Security 3: Expected return: 20%, SD = 20%• Correlation between 1 & 2: 0.3; between 1 & 3 = 0.5;

between 2 & 3 = 0.6• Total funds available = 200000• Funds to be allocated to 1: 100000• Funds to be allocated to 2: 60000• Remaining in 3• Calculate expected return & risk of the portfolio

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2. 3-Security Portfolio: Illustration

• Fischer/Jordan/p580: Diagram• Fischer/Jordan/p581: DiagramObservations:• The locus of 2-security portfolio is a curve whereas

the locus of a 3-security portfolio is a region in the risk-return space.

• The no. of 3-security portfolios is enormous – much more than the no. of 2-security portfolios

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3. Recognising the Efficient & Inefficient Portfolios

• By referring to the portfolios one can identify the optimal (efficient) & non-optimal (inefficient) portfolios

• An efficient portfolio is one that has either (1) More return than any other portfolio with the same risk, OR (2) Less risk than any other portfolio with the same return

• Conversely a portfolio is inefficient (or dominated) if some other portfolio lies above it or to the left of it in the risk-return space

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4. N-Security Portfolio

• N-security formula for return• N-security formula for risk

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5. Markowitz Portfolio Theory

• Aka: Modern Portfolio Theory• Markowitz devised a computational model to

identify the efficiency locus – the portion on the risk-return space on which the efficient portfolios lie

• This locus is called efficient frontier• Portfolios lying below this frontier in the risk-

return space are feasible but not efficient• Portfolios lying above are not feasible

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• No. of inputs required in the MPT for a N-security portfolio: 1. N expected returns, 2. N variances of returns, & 3. (N2 – N)/2 Covariances

• Eg.4: Calculate the no. of inputs required for portfolios of following numbers of securities: 10, 50, 100 & 1000

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• Ans-4:• No. of securities: No. of inputs • 10 65• 50 1325• 100 5150• 1000 501500

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• The massive requirement of data is a limitation of the theory

• This happens because for each pair of securities in the portfolio correlation / covariance was required

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6. Sharpe Single Index Model

• In order to reduce the massive data requirements of the MPT, Sharpe proposed the Single Index model according to which return from every security was related to the market

• Hence instead taking the covariance between every pair of security, the covariance of each security with the market can be taken

• This reduced the data inputs to 3N + 2 for a N-security portfolio

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7. CAPM

• Sharpe Single Index model reduced the data requirements for identifying the efficient frontier

• However MPT & SSI had considered only risky assets in market – they did not consider portfolios that could be made by combining risky assets with risk-free assets

• Further both did not explain what the relationship between return & risk would be

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7. CAPM

• When a risk-free investment opportunity is available in the market, investors can create portfolios by combining risky assets with the risk-free asset

• What are the consequences?

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CAPM

• Explains the behaviour of the Efficient Frontier in the MPT when a riskless asset is introduced

• Riskless asset represents the opportunity to lend/borrow at the risk-free rate

• Proved that all investors depending on their risk appetite will combine the risk-free asset with only one specific efficient portfolio in the Efficient Frontier

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CAPM• This particular efficient portfolio was called

the market portfolio• Investors with more/less risk appetite will

make different combinations of the risk-free asset & the market portfolio thus changing the shape of the Efficient Frontier• Unlevered (lending) portfolios & Levered

(borrowing) portfolios• Shape of the efficient frontier changes from a

curve to a straight line

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Numerical Problems on Lending / Borrowing Portfolios

Eg.5: An investor has Rs. 200000. A risky portfolio has expected return of 20% & SD of 16%. Calculate the return & risk of the following portfolios:

(a) He invests Rs. 120000 in the risky portfolio & lends the remaining amount at risk-free rate of 10%.

(b) He borrows Rs. 100000 at the risk-free rate of 10% & invests the total amount in the risky portfolio.

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Modified Efficient Frontier With Lending & Borrowing Opportunities

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Modified Efficient Frontier…

• With the change in the shape of the efficient frontier it could now be represented mathematically

• With the efficient frontier now turned into a straight line on the risk-return space, it could be represented mathematically as a relationship between risk & return with risk as an independent variable

• But the issue is what should be the measure of risk

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Modified Efficient Frontier…

Two relationships between risk & return emerge from the modified efficient frontier on the basis of how risk is defined:

A. Capital Market Line: Represents the relationship between risk & return for efficient portfolios / assets

B. Security Market Line: Represents the relationship between risk & return for any portfolio / asset whether efficient or not (CAPM)

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A: Capital Market Line

• The modified efficient frontier which includes the market portfolio as one of the points on it is a straight line which originates from Rf & extends indefinitely beyond the market portfolio

• Equation: Where:

jfj RR

m

fm RR

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• This straight line representing the modified efficient frontier is on the risk-return space is called Capital Market Line (CML)

• The equation for CML represents the relationship between risk & return for efficient portfolios

• The slope λ represents the price of risk in the market

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Eg.6: The risk-free rate of return is 10%. The expected rate of return on the market index is 15% and the variance of market returns is 25%2 .

(a) Formulate the CML(b) Use the CML to estimate the expected return

for a portfolio that has a variance of 81%2 .

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B: Security Market Line

• The CML represents the relationship between risk & return for efficient portfolios

• It does not represent the relationship between risk & return for inefficient portfolios & individual securities

• The expected return & SD for inefficient portfolios & individual securities would be below the CML

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• Such portfolios would be found all throughout the feasible region below the CML because except the efficient frontier (CML) the points lying on the remaining portion of the feasible region represent the inefficient portfolios – an individual security is an inefficient portfolio

• Rational investors would not hold individual securities & inefficient portfolios because their risk-return combination is not optimal

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• So any rational investor would not hold individual securities instead of holding a diversified portfolio consisting of many securities

• Moreover rational investors would also not hold inefficient portfolios because they have the opportunity to include more securities in the portfolio & make it efficient

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• By ignoring the opportunity to create efficient portfolios & deliberately holding inefficient portfolios & individual securities, investors would be taking more risk than necessary

• Hence the capital market will not compensate for the greater risk they have taken

• The capital market will compensate only for the risk of an efficient portfolio – which is a well-diversified portfolio

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• A well-diversified portfolio is supposed to have only systematic risk – which arises out of exposure to market risk

• All the unsystematic risk of a well-diversified portfolio is eliminated by diversification effect

• So the appropriate measure of risk for any security is its systematic risk

• And the expected return for any security or an inefficient portfolio should be a function of its systematic risk

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• The systematic risk of a security is captured by its covariance of returns with the market portfolio

• Hence there can be a relationship between the expected return of an individual security & its covariance with market portfolio

),(),(

miCovmmCovRR

RR fmfi

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• The CAPM, given below, derives out of the above relationship

• The graphical representation of the CAPM

relationship is called Security Market Line (SML)

ifmfi RRRR )(

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Eg.7: The risk-free rate of return is 8%. The market index has an expected return of 16% and a variance of 144%2. What should be the expected return on a security which has a covariance with market of 216%2 ?

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• The CAPM can be generalised across all assets & portfolios, whether efficient or inefficient

• Because all assets are to be priced in such a manner that the expected return compensates only for the systematic risk, the CAPM can be considered to be the general pricing model for all assets whether they are efficient or not

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Assumptions of CAPM• No transaction costs• No taxes• Investors have identical information• Borrowing & lending is possible at risk free rate• Borrowing & lending rates are equal• Including all assumptions of MPT

4-risks-variable-income.pptx

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