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# Sess14 new_1 - Lecture 14 Portfolio Theory Reading RWJ...

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Unformatted text preview: Lecture 14: Portfolio Theory Reading: RWJ Chapter 13 Outline: Portfolio risk and returns Diversification Systematic and unique Risk Historical risk and return Fin3715 – Fall 07 – Kayhan 1 Portfolios A portfolio is a group of assets. A security’s portfolio weight is the percentage of the portfolio’s total value invested in that particular asset. Portfolio weights can be positive (a “long” position) or negative (a “short” position). Portfolio weights must sum to 1. The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets Fin3715 – Fall 07 – Kayhan 2 Portfolio Returns The return of a portfolio is simply the weightedaverage of the returns of assets in the portfolio. n ~ = rp ∑wi ~ ri i= 1 where n is the number of assets in the portfolio, wi is the portfolio weight of asset i, and ri is the return of asset i. Similarly, the expected return of a portfolio is the weighted-average of the expected returns of assets. n rp = E[ ~ ] = ∑wi ri rp i= 1 Fin3715 – Fall 07 – Kayhan 3 Example 1: Portfolio Returns Assume the expected returns on IBM and Merck stocks are 18% and 12% respectively. Suppose we have a portfolio with total value \$150. What is the expected return of the portfolio, if we invest \$50 in IBM and \$100 in Merck? if we invest \$50 each in IBM , Merck, and a risk-free bonds (rf = 3%)? if we invest \$200 in IBM and -\$50 in Merck? Fin3715 – Fall 07 – Kayhan 4 Example 1: Portfolio Returns (Sol’n) Use index 1 and 2 to denote IBM and Merck respectively. 50 1 100 2 (i) w1 = = , w2 = = 150 150 3 1 2 rP = w1 × r1 + w2 × r2 = (.18) + (.12) = .14 3 3 (ii) (iii) 3 50 1 50 1 50 1 = , w2 = = , w3 = = , 150 3 150 3 150 3 1 1 1 rP = (.18) + (.12) + (.03) = .11 3 3 3 w1 = 200 4 − 50 1 = , w2 = =− , 150 3 150 3 4 1 rP = (.18) + − (.12) = .20 3 3 w1 = Fin3715 – Fall 07 – Kayhan 5 Portfolio Variance: Two Assets σ p =Var[ ~p ] =Var [ w1~ + w1~ ] r r1 r2 2 σ p = w1 σ1 + w2 σ 2 + 2 w1w2σ12 2 2 2 2 2 where σ12 is the covariance between stocks 1 and 2: σ12 = ρ12σ1σ2 , - 1 ≤ ρ12 ≤1 Fin3715 – Fall 07 – Kayhan 6 Warning: Portfolio Variance The Variance of the return on a portfolio Does Not Equal the Weighted Average of the variances of the returns on individual stocks! Fin3715 – Fall 07 – Kayhan 7 Example 2: Portfolio Variance Assume volatilities (i.e., standard deviation) of IBM and Merck stocks are 30% and 20% respectively, and the correlation coefficient between the two stocks is .4. What is the standard deviation of a portfolio with 1/3 IBM and 2/3 Merck stock? Fin3715 – Fall 07 – Kayhan 8 Example 2: Portfolio Variance (Sol’n-1) Use index 1 and 2 to denote IBM and Merck respectively 1 2 w1 = , w2 = , σ 1 = .3, σ 2 = .2, ρ12 = .4 3 3 σ 12 = ρ12σ 1σ 2 = (.4)(.3)(.2) = .024 Method 1: apply the variance formula 2 σP 2 2 1 2 1 2 2 2 = (.3) + (.2) + 2 (.024) = .0384 3 3 3 3 σ P = .0384 = 19.6% Fin3715 – Fall 07 – Kayhan 9 Portfolio Risk and Return To achieve higher expected return, investors generally need to take higher risk. Forming portfolios can reduce risk without sacrificing too much expected return. As illustrated in example 1 and 2, IBM has expected return of 18% and volatility of 30% Merck has expected return of 12% with 20% volatility A portfolio of 1/3 IBM and 2/3 Merck has expected return of 14% and volatility of only 19.6%. The above portfolio has BOTH higher expected return and lower risk than Merck stock! Fin3715 – Fall 07 – Kayhan 10 Portfolio and Diversification Diversification - reduce risk without an equivalent reduction in expected returns by spreading the portfolio across many asset classes. The reduction in risk is achieved by offsetting the worse-than-expected returns from one asset by the better-than-expected returns from another. Diversification is easier to achieve with less correlated assets (negative correlation is the best). Diversification is easier to achieve with a larger number of assets. Fin3715 – Fall 07 – Kayhan 11 Example: Diversification and Correlation Assume asset 1 and 2 has expected return of 18% and 12%, and volatility of 30% and 20%, respectively. What are the expected return and volatility of a portfolio that invests 50% in each asset, if the correlation between the two assets is (i) 0 (ii) –1 (iii) 1? Fin3715 – Fall 07 – Kayhan 12 Example: Diversification and Correlation (Sol’n) (i) w1 = w2 = .5, ρ12 = 0 ⇒ σ 12 = ρ12σ 1σ 2 = 0 rp = (.5)(.18) + (.5)(.12) = .15 σ p = (.5) 2 (.3) 2 + (.5) 2 (.2) 2 + 0 = .0325 2 ⇒ σ p = 18% (ii) expected return is still .15 and ρ12 = −1 ⇒ σ 12 = ρ12σ 1σ 2 = (−1)(.3)(.2) = −.06 σ p = (.5) 2 (.3) 2 + (.5) 2 (.2) 2 + 2(.5)(.5)(−.06) = .0025 ⇒ σ p = 5% 2 (iii) expected return is still .15 and ρ12 = 1 ⇒ σ 12 = ρ12σ 1σ 2 = (1)(.3)(.2) = .06 σ p = (.5) 2 (.3) 2 + (.5) 2 (.2) 2 + 2(.5)(.5)(.06) = .0625 ⇒ σ p = 25% 2 Fin3715 – Fall 07 – Kayhan 13 Portfolio with N Assets Portfolio Mean n rp = E[ ~ ] = ∑wi ri rp i= 1 Portfolio Variance N N σ = ∑∑ wi w jσ ij 2 p i =1 j =1 Fin3715 – Fall 07 – Kayhan 14 Each Asset’s Contribution to Portfolio Variance An asset’s contribution to the risk of a welldiversified portfolio is determined by its average covariance with other assets, not by its own variance. An asset’s average covariance with other assets is called the systematic (or market) risk. Fin3715 – Fall 07 – Kayhan 15 Risk of Individual Assets Individual assets have two kinds of risk: n Market Risk - Economy-wide sources of risk that affect a large number of assets (e.g., the overall stock market). Also called “non-diversifiable risk” and/or “systematic risk.” n Unique Risk - Risk that affects at most a small number of assets. Also called “diversifiable risk,” “unsystematic risk,” and/or “idiosyncratic risk.” Fin3715 – Fall 07 – Kayhan 16 Examples: Market or Systematic Risk 1. 2. 3. An extra-ordinarily hot summer creates a spike in energy prices nationwide. Companies are discovered to be systematically abusing GAAP rules. A global recession occurs. => Systematic risk cannot be diversified away. Fin3715 – Fall 07 – Kayhan 17 Examples: Unique or Unsystematic Risk 1. 2. 3. A rogue currency trader racks up \$750M in losses before being discovered. A company is sued because the tires on a particular make of SUV tend to de-tread, resulting in fatalities. A company’s main manufacturing facility burns to the ground. => Idiosyncratic risk can be diversified away. Fin3715 – Fall 07 – Kayhan 18 Diversification and Risk: Historic Evidence (1) Fin3715 – Fall 07 – Kayhan 19 Diversification and Risk: Historic Evidence (2) Fin3715 – Fall 07 – Kayhan 20 Implication for Asset Returns Total risk = Systematic risk + Unsystematic risk Unsystematic risk can be diversified away, while systematic risk has to be held by investors. The market rewards investors for bearing risk, but only for bearing the necessary risk, i.e., the systematic risk. Bearing unsystematic risk is unnecessary since it can be eliminated via diversification, thus, is not rewarded. As a consequence of diversification, an asset’s expected return depends only on its systematic risk. Fin3715 – Fall 07 – Kayhan 21 ...
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