Portfolio Theory

Indifference Curve Increasing Utility Expected Return E(r) Represents individual’s willingness to trade-off return and risk Assumptions: 5 Axioms Prefer more to less (Greedy) Risk aversion Assets jointly normally distributed Expected Return E(r) Increasing Utility Standard Deviation σ(r)

Dominance • 2 dominates 1; has a higher return Expected Return 4 2 3 1 Standard Deviation • 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 As the table shows, a portfolio with large number of risky assets tend to be close to normally distributed. 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 j’s return in State s: rjs = (Ws – W0) / W0 Expected return on asset j: E(rj) = ∑sαsrjs Asset j’s variance: σ2j = ∑sαs[rjs- E(rj)]2 Asset j’s standard deviation: σj = √σ2j

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

A simple example: Asset j $150,000 Good State: rgood = ($150,000 – $100,000) / $100,000 = 50% 60% $10,000 40% $80,000 Bad State: rbad = ($80,000 – $100,000) / $100,000 = -20% Expected Return: E(rj) = ∑sαsrjs = 60%(50%) + 40%(-20%) = 22% Variance: σ2j = ∑sαs[rjs- E(rj)]2 = 60%(50%-22%)2 + 40%(-20%-22%)2 = 11.76% Standard Deviation: σj = √σ2j = √11.76% = 34.293%

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

Portfolio Theory – a bit of history Modern portfolio theory (MPT)—or portfolio theory—was 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) Link to his Nobel Prize lecture if you are interested: http://nobelprize.org/economics/laureates/1990/markowitz-lecture.pdf

Illustration: 2 risky assets Assume you have 2 risky assets (x & y) to choose from, both are normally distributed. rx ~ N(E(rx), σ2x) & ry ~ N(E(ry), σ2y) 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), σ2x) & ry ~ N(E(ry), σ2y) Portfolio Variance: σ2p = 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 - E[rx])(ry - E[ry])] = a2 σ2x + b2 σ2y + 2abCov(rx, ry) σ2p = a2 σ2x + b2 σ2y + 2abσxσyρxy σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)

Illustration: 2 risky assets σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy) σp increases as ρxy increase. Implication: given a (and thus b), if ρxy 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.

Diversification Proposition: portfolio of less than perfectly correlated assets always offer better risk-return opportunities than the individual component assets on their own. Proof: If ρxy = 1 (perfectly positively correlated) then, σp = a σx + b σy If < 1 (less than perfectly correlated) then, σp < a σx + b σy

Varying the portion on X & Y Suppose: rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2) E(rp) = E[arx + bry]=aE(rx)+ bE(ry) E(rp) 13% %8 a 0% 100%

Varying the portion on X & Y Suppose: rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2) σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy) σp 20% ρxy=1 ρxy=-1 ρxy=0.3 12% a 0% 100%

Min-Variance opportunity set with the 2 risky assets E(rp) 13%  = -1  = .3  = -1  = 1 %8 σp 12% 20%

Min-Variance opportunity set with the Many risky assets E(rp) Efficient frontier Individual risky assets Min-variance opp. set σp

Min-Variance opportunity set 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 E(rp) σp

Efficient set 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. E(rp) σp

Individual’s decision making with 2 risky assets, no risk-free asset E(rp) U’’ U’ U’’’ Efficient set S P Q Less risk-averse investor More risk-averse investor σp

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 set E(rp) M E(Rm) 5%=Rf σp σm

Individual’s decision making with 2 risky assets, with risk-free asset E(rp) CML B Q M A rf σp

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.