Indifference Curve

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Portfolio Theory

Transcript of Indifference Curve

Portfolio Theory

Indifference CurveIndifference Curve

Expected Return E(r)

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Represents individuals willingness to trade-off return and risk Assumptions:

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1) 5 Axioms 2) Prefer more to less (Greedy) 3) Risk aversion 4) Assets jointly normally distributed

Increasing UtilityStandard Deviation (r)

DominanceExpected Return 4 2 1 Standard Deviation 3

2 dominates 1; has a higher return 2 dominates 3; has a lower risk 4 dominates 3; has a higher return

Jointly normally distributed?

Individual stock return may not be normally distributed, but a portfolio consists of more and more stocks would have its return increasingly close to being normally distributed.

Jointly normally distributed?1st moment: Mean = Expected return of portfolio 2nd moment: Variance = Variance of the return of portfolio (RISKNESS) Mean and Var as sole choice variables => Distribution of return can be adequately described by mean and variance only That means, distribution has to be normally distributed.

2 reasons to support mean-variance criteria 1) As the table shows, a portfolio with large number of risky assets tend to be close to normally distributed. 2) The fact that investors rebalance their own portfolios frequently will act so as to make higher moments (3rd , 4th , etc) unimportant (Samuelson 1970)

Math Review I Asset js return in State s: rjs = (Ws W0) / W0 Expected return on asset j: E(rj) = ssrjs Asset js variance: 2j = ss[rjs - E(rj)]2 Asset js standard deviation: j = 2j

Math Review I Covariance of asset is return & js return: Cov(ri, rj)= E[(ris - E(ri)) (rjs - E(rj))] =ss[ris - E(ri)] [rjs - E(rj)] Correlation of asset is return & js return: ij = Cov(ri, rj) / (ij) -1 ij 1 When ij = 1 => i and j are perfectly positively correlated. They move together all the time. When ij = -1 => i and j are perfectly negatively correlated. They move opposite to each other all the time.

A simple example: Asset j60% $10,000 40% $80,000 Bad State: $100,000 = -20% rbad = ($80,000 $100,000) / $150,000 Good State: rgood = ($150,000 $100,000) / $100,000 = 50%

Expected Return:E(rj) = ssrjs = 60%(50%) + 40%(-20%) = 22%

Variance:2j = ss[rjs - E(rj)]2 = 60%(50%-22%)2 + 40%(-20%-22%)2 = 11.76%

Standard Deviation:j = 2j = 11.76% = 34.293%

Math Review II 4 properties concerning Mean and Var Let be random variable, a be a constant 1) E(+a) = a + E() 2) E(a) = aE() 3) Var(+a) = Var() 4) Var(a) = a2Var()

Portfolio Theory a bit of history Modern portfolio theory (MPT)or portfolio theorywas introduced by Harry Markowitz with his paper "Portfolio Selection," which appeared in the 1952 Journal of Finance. 38 years later, he shared a Nobel Prize with Merton Miller and William Sharpe for what has become a broad theory for portfolio selection. Prior to Markowitz's work, investors focused on assessing the risks and rewards of individual securities in constructing their portfolios. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. Intuitively, this would be foolish. Markowitz formalized this intuition. Detailing a mathematics of diversification, he proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics. In a nutshell, inventors should select portfolios not individual securities. (Source: riskglossary.com)

Illustration: 2 risky assets Assume you have 2 risky assets (x & y) to choose from, both are normally distributed. rx ~ N(E(rx), 2x) & ry ~ N(E(ry), 2y) You put a of your money in x, b in y. a+b=1 Portfolio Expected Return: E(rp) = E[arx + bry]=aE(rx)+ bE(ry)

Illustration: 2 risky assets rx ~ N(E(rx), 2x) & ry ~ N(E(ry), 2y) Portfolio Variance: 2p = E[rp - E(rp)]2 = E[(arx + bry)-E[arx + bry]]2 = E[(arx - aE[rx])+(bry - bE[bry])]2 = E[a2(rx - E[rx])2 + b2(ry - E[ry])2 + 2ab(rx (ry - E[ry])] = a2 2x + b2 2y + 2abCov(rx, ry) = a2 2x + b2 2y + 2abCov(rx, ry) 2p = a2 2x + b2 2y + 2abxyxy p = (a2 2x + b2 2y + 2abxyxy )

- E[rx])

Illustration: 2 risky assetsp = (a2 2x + b2 2y + 2abxyxy) p increases as xy increase. Implication: given a (and thus b), if xy is smaller, variance of portfolio is smaller. Diversification: you want to maintain the expected return at a definite level but lower the risk you expose. Ideally, you hedge by including another asset of similar expected return but highly negatively correlated with your original asset.

DiversificationProposition: portfolio of less than perfectly correlated assets always offer better riskreturn opportunities than the individual component assets on their own. Proof: If xy = 1 (perfectly positively correlated) then, p = a x + b y If < 1 (less than perfectly correlated) then, p < a x + b y

Varying the portion on X & YSuppose:

E(rp)13%

rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2) E(rp) = E[arx + bry]=aE(rx)+ bE(ry)

%80% 100% a

Varying the portion on X & YSuppose: p rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2) p = (a2 2x + b2 2y + 2abxyxy )20%

xy =1 xy =-112%

xy =0.3

0%

100%

a

Min-Variance opportunity set with the 2 risky assetsE(rp)13%

= -1 =. 3

= -1

=1p

%8

12%

20%

Min-Variance opportunity set with the Many risky assetsE(rp)Efficient frontier

Individual risky assets Min-variance opp. set

p

Min-Variance opportunity setE(rp)Min-Variance Opportunity set the locus of risk & return combinations offered by portfolios of risky assets that yields the minimum variance for a given rate of return

p

Efficient setE(rp)Efficient set the set of mean-variance choices from the investment opportunity set where for a given variance (or standard deviation) no other investment opportunity offers a higher mean return.

p

Individuals decision making with 2 risky assets, no risk-free assetE(rp)U U U

Efficient set S P Q More risk-averse investor Less risk-averse investor

p

Introducing risk-free assets Assume borrowing rate = lending rate Then the investment opp. set will involve any straight line from the point of risk-free assets to any risky portfolio on the min-variance opp. set However, only one line will be chosen because it dominates all the other possible lines. The dominating line = linear efficient set Which is the line through risk-free asset point tangent to the min-variance opp. set. The tangency point = portfolio M (the market)

Capital market line = the linear efficient setE(rp)

E(Rm)

M

5%=Rf mp

Individuals decision making with 2 risky assets, with risk-free assetE(rp)B Q M A rfp

CML

Implication All an investor needs to know is the combination of assets that makes up portfolio M as well as risk-free asset. This is true for any investor, regardless of his degree of risk aversion.