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 riskreturn tradeoff 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
weightedaverage 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
riskfree 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’n1)
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
worsethanexpected returns from one asset by the
betterthanexpected 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  Economywide sources of risk that
affect a large number of assets (e.g., the overall stock
market). Also called “nondiversifiable 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 extraordinarily 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 detread,
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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 Spring '08
 Stephens
 Variance, Modern portfolio theory, systematic risk

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