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Unformatted text preview: Outlines of Lecture 3 Objectives
1. Introduce matrix calculation in portfolio
expected return and variance
2. Discuss the best feasible investment
opportunity set
3. Construct an efficient frontier portfolio and the
optimal risky portfolio by using Excel Solver References
Chapter 7 & 8, BKM; Chapter 13, EGBG; Handouts
Hsiulang Chen Portfolio Analysis 1 What is a portfolio?
ω1 1st security
ω2 2nd security
ω3 3rd security ·
·
· ωN Nth security Hsiulang Chen A portfolio is a group of
assets.
A portfolio is described
by the weights ωi
allocated to securities. Portfolio Analysis 2 Portfolio Expected Returns E ( ~ ) = ∑ϖ E ( ~)
r
r
N i =1 P i i Portfolio Variance (Portfolio Risk) Var ( ~ ) = Cov ( ~ , ~ )
rP
rP rP
N N i =1 j =1 N N = Cov (∑ϖ i ~ , ∑ϖ j ~j ) = ∑ ∑ϖ iϖ j Cov ( ~ , ~j )
ri
r
ri r
N i =1 j =1 N N = ∑ϖ i2Var ( ~ ) + ∑ ∑ϖ iϖ j Cov ( ~ , ~j ) ∀ i ≠ j
ri
ri r
i =1
N i =1 j =1 N N = ∑ϖ i2σ i2 + ∑ ∑ϖ iϖ jσ i j ∀ i ≠ j
i =1 Hsiulang Chen i =1 j =1 Portfolio Analysis 3 In Terms of Matrix Expression
E (~ ) = [ϖ 1 ϖ 2
rP Var (~ ) = [ϖ 1 ϖ 2
rP r1 E (~) E (~ ) r 2 • • ϖ N ] • • E (~ ) rN σ 11 σ 12
σ
σ 22 21
• • ϖ N ] •
• • •
σ N 1 σ N 2 • σ 1 N ϖ 1 • σ 2 N ϖ 2 • • • • • • • • σ NN ϖ N •
•
•
• How to do matrix multiplication in Excel?
Hsiulang Chen Portfolio Analysis 4 Portfolio Theory
(The Optimal Asset Allocation) What is the best feasible investment
opportunity set given risky assets?
−A minimum variance frontier (MVF)− ~)
Var ( r
ϖ
s.t.
E (~ ) = r
r Min P P N ∑ ϖ =1 i =1
Hsiulang Chen i Portfolio Analysis 5 Construction of MVF What are all feasible investment combinations
given two risky assets? E(r1)=25%,E(r2)=10%,σ1=75%,σ2=25%, ρ12 In a plane of {σP, E(rp)}, a portfolio of these two
assets can be described by
E(rp)=wE(r1)+(1w)E(r2)
σP = [w2σ12 +2w(1w) ρ12σ1 σ2+ (1w)2σ22]0.5 Hsiulang Chen Portfolio Analysis 6 The minimum variance frontier (MVF) can
be traced by varying w.
Case 1: ρ= 1
E(rp)=wE(r1)+(1w)E(r2)
σp=wσ1+(1w) σ2
Case 2: ρ= 1
E(rp)=wE(r1)+(1w)E(r2)
σp=wσ1(1w) σ2
Hsiulang Chen Portfolio Analysis 7 MVF of two risky assets
E(r1)=25%,E(r2)=10%,σ1=75%,σ2=25%,ρ12 1.0 What does MVF tell us? How to construct it?
Hsiulang Chen Portfolio Analysis 8 In Excel
1. Randomly choosing w
2. Trace all feasible investments in a plane of
E(rP) and σP for the four cases: (1) ρ12 =1,
(2)ρ12 = 0.2, (3) ρ12 =  0.2, (4) ρ12 = 1
3. What do you see the MVF?
4. Are all feasible investment combinations
on the MVF?
Hsiulang Chen Portfolio Analysis 9 Extend to a portfolio of three assets
Expected
Returns Return Covariance Matrix
Asset A Asset B Asset C Asset A 0.05 Asset A 0.04 0.02 0 Asset B 0.1 Asset B 0.02 0.1 0.06 Asset C 0.15 Asset C 0 0.06 0.16 1. Trace all feasible investments in a plane of {σP,
E(rp)} in Excel for two cases: (1) Shortselling is
allowed, (2) Short selling is prohibited.
2. What do you see the MVF ?
Hsiulang Chen Portfolio Analysis 10 Three risky assets
E(r) Add 0.1 to the σ coordinates
Hsiulang Chen Portfolio Analysis 11 MVF of Three Risky Assets
0.4 E(r) How to construct a portfolio on
the efficient frontier in Excel? 0.35
0.3
0.25 No Shortselling Var (~ )
r
s.t.
E (~ ) = r
r 0.2 Min
ϖ 0.15 P P 0.1 N ∑ 0.05 ϖ =1 i =1 i σ 0
0.1 Hsiulang Chen 0.1 0.3 0.5 0.7 0.9 Portfolio Analysis 1.1 1.3 1.5 12 MVF of Risky Assets
E(r)
Efficient The best feasible investment
opportunity set given risky assets
frontier
Individual
assets Global
minimum
variance
portfolio Hsiulang Chen Minimum
variance
frontier
Portfolio Analysis St. Dev. 13 Concepts What are the required input estimates
to generate the efficient frontier? What would be the matrix expression of
E(r̃p) & σp2 ? N expected returns; N Variances;
N(N1)/2 Correlations How to estimate them in the real world?
Hsiulang Chen Portfolio Analysis 14 Without a Prior Belief Construct return sample distribution Sample Mean
1 T
r =
∑ rt
T t =1 Sample Variance 2 = 1 T ( r − r )2
σ
T − 1 t∑1 t
= Sample Covariance 1 T (r − r )(r − r )
σ XY =
T − 1 t∑1 xt x yt y
=
Hsiulang Chen Portfolio Analysis 15 With a Prior Belief Construct return distribution based on
some models The market index model Fama and French 3factor model plus
the Momentum factor The macroeconomic factor model
Hsiulang Chen Portfolio Analysis 16 Another Example of Construction of an
Efficient Frontier in Excel Using the stock market data from seven
countries in the file “ch7a_BKM7.xls” to
construct the global efficient frontier. Examining the impact of shortselling prohibition
on the efficient frontier. Constructing the optimal risky portfolio given the
risk free rate of 5% when (1) the shortselling is
prohibited; (2) the shortselling is allowed. (To
be discussed next!)
Hsiulang Chen Portfolio Analysis 17 What is the best feasible
investment opportunity set
given risky assets and a
riskfree asset?
Case 1: Lending rates and borrowing rates are identical.
Case 2: Lending rates and borrowing rates are different. Hsiulang Chen Portfolio Analysis 18 Rf The CAL, not the efficient frontier, describes the best
feasible investment opportunity now! The optimal
risky portfolio is the tangency portfolio.
Hsiulang Chen Portfolio Analysis 19 Construction of the Optimal
Risky Portfolio Max
ϖ SP = E ( ~ ) − rf
rP σP i N s.t. ∑ϖ
i =1 i =1 Equation (7.13) at BKM (8th & 9th Edition) on the next
slide gives the formula to construct the optimal risky
portfolio in a case of two risky assets.
Hsiulang Chen Portfolio Analysis 20 The optimal risky portfolio in the
case of two risky assets B & C Solutions for the general case?
Hsiulang Chen Portfolio Analysis 21 Construction of the Optimal Risky
Portfolio (ShortSelling Allowed) The solution involves solving Zi for the
following system of N simultaneous
equations.
E ( ~) − rf = Z1σ i1 + Z 2σ i 2 + Z 3σ i 3 + ⋅ ⋅ ⋅ + Z N σ iN , i = 1,2,3,..., N
ri
Where ϖ k = Zk
N ∑Z
i =1 Hsiulang Chen i Portfolio Analysis 22 In Terms of Matrix Expression
r1 E (~ ) − rf E (~ ) − r
r2
f • • E (~ ) − rf rN σ 11 σ 21
= • • σ N 1 Z1 σ 11 Z σ 2 21
⇒ • = • • • Z N σ N 1 Hsiulang Chen σ 12
σ 22 σ N2 •
•
•
•
• •
•
•
•
• •
•
•
•
• •
• σ 12
σ 22
•
• σ N2 Portfolio Analysis • σ 1 N Z1 • σ 2N Z2 •
• • •
• • • σ NN Z N −1
r1
σ 1N E (~ ) − rf σ 2 N E (~ ) − rf r2 •
• •
• σ NN E (~ ) − rf rN 23 The mutual fund theorem
(the separation property) A portfolio choice problem can be separated into
two independent tasks.
#1: Determining the optimal risky portfolio, which
is purely technical. Given the manager’s input
list, the best risky portfolio is the same for all
clients, regardless of risk aversion.
#2: Allocating of the complete portfolio to T bills
versus the optimal risky portfolio depending on
personal preference.
Hsiulang Chen Portfolio Analysis 24 In practice, however, different managers
will estimate different input lists, thus
deriving different efficient frontiers, and
offer different “optimal” portfolios to their
clients. The source of the disparity mainly
lies in the security analysis. Hsiulang Chen Portfolio Analysis 25 The Derivation of CAPM
• If there are homogeneous expectations, then
all investors must select the same optimal
risky portfolio, the market portfolio. [EGBG, p.
291][BKM, p. 293]
E ( ~) − r = λCov ( ~, ~ ), i = 1,2,3,..., N
r
r r
E ( ~ ) − r = λCov ( ~ , ~ ) = λσ
r
r r
i f i M 2 M f M ⇒ E ( ~) − r =
r
i M M E (~ ) − r
r
M σ f f Cov ( ~, ~ )
r r
i 2
M ⇒ E ( ~) = r + β (E ( ~ ) − r
r
r
i Hsiulang Chen f i M Portfolio Analysis f M )
26 Construction of the Optimal Risky
Portfolio (ShortSelling Prohibited) Max
ϖ θ= E (~P ) − r f
r σP N s.t. ∑ϖ i =1 i =1 ϖ i ≥ 0, ∀i This is a quadratic programming
problem! You can accomplish this in
the Excel or other statistical software.
Hsiulang Chen Portfolio Analysis 27 On Diversification
N
2σ 2 + N N ϖ ϖ σ ∀ i ≠ j
Var (~ ) == ∑ ϖ
r
∑ ∑
P
i i
i j ij
i =1
i =1 j =1
For an equallyweighted portfolio (ωi =1/N), Var ( ~ ) =
rP 1
N σ + 2 N −1
N C O V as N → C O V →∞ N Where σ 2 ≡ σ i2
∑
i =1 N COV ≡ N 1
N ( N −1) N ∑ ∑σ
i =1 j =1 ij ∀i ≠ j Thus only the average covariance matter for large portfolios. If the
average covariance is zero, then the portfolio variance is close to zero
for large portfolios!
Hsiulang Chen Portfolio Analysis 28 On Diversification The component of risk that can be diversified away
we call the diversifiable or nonsystematic risk. Empirical facts:
– The average (annual) return standard deviation is
49%.
– The average (annual) covariance between stocks is
0.037, and the average correlation is about 39%. Since the average covariance is positive, even a
very large portfolio of stocks will be risky. We call
the risk that cannot be diversified away the
systematic risk.
Hsiulang Chen Portfolio Analysis 29 Multiple Risky Securities Basic Message: Your risk/return tradeoff is
improved by holding many assets with less than
perfect correlation. Far from everybody agrees: Hsiulang Chen BERKSHIRE HATHAWAY INC.
Portfolio Analysis BRK/A: $104k; BRK/B: $69.93 09/07/201130 All eyes were on Warren Buffett at Berkshire Hathaway's 2008 annual
meeting at the Qwest Center in Omaha, Neb. Hsiulang Chen Portfolio Analysis 31 Less Hedging, Bigger Fees SingleStock Funds Defy Diversification Doctrine,
Make Ackman a Winner June 7, 2007; WSJ; Page C12 Hsiulang Chen Portfolio Analysis 32 Illustration of Ackman Hedge Fund
Hedge funds typically collect 20% of profits they generate. Suppose an investor
want to put the entire $1 billion into two different fund investments, a typical
hedge fund or Ackman hedge fund. A Typical Hedge Fund Ackman Hedge Fund Invest $100m in each of 10 stocks
(5: +50%; 2: +0%; 3: 50%) Invest $100m in each of 10 funds
held by Ackman, where each fund
holds one stock only
(5: +50%; 2: +0%; 3: 50%) Overall Profit:
$100m*2*50%=$100m
20% Profits: 0.2*$100m=$20m Profits on 5 Funds:
$100m*5*50%=$250m
20% Profits: 0.2*$250m=$50m Hsiulang Chen Portfolio Analysis 33 Focused Funds Find Less Can Be More
Ups, and Downs, of Concentrated Investing
By JONATHAN BURTON; WSJ; AUGUST 18, 2009; C9 Focused funds  portfolios with only a
couple of dozen holdings  are getting
attention in a market where stock selection
is more important than ever. Diversification's downside is that it limits a
fund's chance to meaningfully outperform
its index. Moreover, an overly diversified
portfolio can mimic an index fund, but at a
much higher cost.
Hsiulang Chen Portfolio Analysis 34 Appendix  STATS FACTS Hsiulang Chen Portfolio Analysis 35 ...
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 Fall '11
 HengChen

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