modern portfolio theory-markowitz

8/9/2019 Modern Portfolio Theory-Markowitz

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Modern Portfolio Theory

Prof Mahesh KumarAmity Business School


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Modern Portfolio Theory

Harry Markowitz William F. Sharpe Merton Miller

Modern portfolio theory was introduced by HarryMarkowitz in 1952. Markowitz, Sharpe & Miller were co-recipients of the Nobel Prize in Economics in 1990 fortheir pioneering work in portfolio theory


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Expected ReturnThe expected return on a portfolio is the weightedaverage of the returns of each asset within theportfolio

Example: A portfolio is comprised of three securitieswith the following returns:

Security Return % of Portfolio

A 5% 30%B 10% 45%

C 15% 25%


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Expected ReturnThe expected return of the portfolio is theweighted average:

!

!

!

!

n

j j j 1

r r

. 1 0 % . 1 5 % . 5

9 . 7 5 %

r j = retur n at time period j= expected retur n

w j = proportio n of the portfolio comprised of asset jr


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Portfolio Risk: Two Risky AssetsStandard deviation of a two-asset portfolio iscalculated as follows:

A A A A, A = w + w + w w

A

A,

= standard deviation

w = the proportion of the portfolio comprised of A = the correlation coefficient between A &


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Portfolio Risk: Two Risky Assets

2 2 2 2A A B B A B A,B A B= w + w + 2w w

Portfolio risk is driven mainly by thecorrelation between the assets!!


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CorrelationCorrelation is a measure of the linear relationshipbetween two assets

Correlation varies between perfect negative (-1) toperfect positive ( 1)Perfect negative correlation: when the return onasset A rises, the return on Asset B falls and viceversaPerfect positive correlation: the returns on asset Aand Asset B move in perfect unison


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Correlation & Risk Reduction B

ARA

A B

RBPerfectNegative Correlation

PerfectPositive Correlation

Less thanPerfect Correlation

To minimize portfolio risk, choose assets that havevery low correlations with each other.


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Moving Toward Many Risky AssetsWhen the portfolio consists of many risky assets,they form a plot similar to a broken egg shellshapeEach dot within the broken egg shell shaperepresents the risk/return profile for a single riskyasset or portfolio of risky assetsTo maximize return per unit of risk assumed, aninvestor would always choose an asset or portfoliothat plots along the efficient frontier


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Portfolios: Many Risky Assets

YouYou wouldwould nevernever choosechoose AssetAsset A,A, asas youyou cancan earnearn a a higherhigher returnreturnwithwith similarsimilar riskrisk byby choosingchoosing the the assetasset thatthat plotsplots alongalong thethe EfficientEfficientFrontierFrontier. .

RA

A

Return

Standard Deviation

A


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Choosing a Portfolio: So FarThe investor first decides how much risk toassume

The investor then chooses the portfolio thatplots along the efficient frontier with thatamount of risk


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Introducing the Risk Free Security

When a risk-free asset (Treasury Bill) isintroduced into the set of risky assets, a new

efficient frontier emergesThis new efficient frontier is known as theCapital Market Line (CML)The CML represents all possible portfolioscomprised of Treasury Bills and the MarketPortfolio


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Adding the Risk-Free Asset

RM

M

Return

Standard Deviation

A

Rf

Capital Market Line


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Capital Market Line (CML)

To maximize return for an amount of risk, investors shouldhold a portion of their assets in T-bills and a portion in themarket portfolio.

Linear relationship between risk and returnTo earn an expected return greater than the return on themarket portfolio, invest more than 100% of ones own wealthin the market portfolio.

PortfolioW ell-d iversified f M arket f Portfolio Market

R = r r - r


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What are we Missing?We know:

I nvestors should split their assets betweenTreasury bills and the market portfolioTo reduce risk, invest a greater proportion of assets in Treasury billsTo enhance expected return, invest a greaterproportion of assets in the market portfolio

modern portfolio theory-markowitz

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