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Unformatted text preview: Outlines of Lecture 4 Objectives
1. Determine the optimal asset allocation given a
client’s risk preference
2. Estimate a stock’ market beta empirically
3. Apply the market index model in practice References
Chapter 68, EGBG; Chapter 79, BKM;
Handouts
Hsiulang Chen Portfolio Analysis 1 2nd Task in 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 Tbills versus the optimal risky portfolio
depending on personal preference.
Hsiulang Chen Portfolio Analysis 2 Portfolio Selection & Risk Aversion
U’’’ U’’ U’ E(r) S
P
Q Less
riskaverse
investor More
riskaverse
investor
Hsiulang Chen Efficient
frontier of
risky assets Portfolio Analysis St. Dev 3 Efficient Frontier with Lending &
Borrowing
CAL E(r)
B
Q
P 1. What is the expression of CAL?
A
rf 2. How to construct the tangency
portfolio A? Equation 7.14 of BKM F Hsiulang Chen Portfolio Analysis 4 Optimal Asset Allocation Given Risk
Tolerance
Construction of the Tangency Portfolio (Optimal Complete
Portfolio) between the indifference curve and the capital
allocation line—a combination of a risky asset (P) and a
riskfree asset (rf) MaxU (r C ) = E (rC ) − 0.5 Aσ 2
C y = rf + y[ E (rP ) − rf ] − 0.5 Ay σ
2 ⇒ y* = E (rP ) − rf
Aσ 2
P 2
P I ntuition? (y*: the proportion allocated to P)
Hsiulang Chen Portfolio Analysis 5 Example
• Suppose that S&P 500 index has an
expected return of 13% and the standard
deviation of 25%. The riskfree rate (Rf) is
5% but the borrowing rate that your client
faces is 9%. What is the range of risk
aversion for which a client will (1) always
borrow, (2) always lend, and (3) neither
borrow nor lend (i.e. for which y=1)?
Hsiulang Chen Portfolio Analysis 6 CAL with Higher Borrowing Rate
E(r) S&P 500
) S = .16 13% 9%
5% ) S = .32 σ
σp = 25%
Hsiulang Chen Portfolio Analysis 7 Markowitz Portfolio Selection vs. CAPM So far, we have looked at the problem of how to
select a portfolio, given that we know what the
expected returns and return covariances are. Expected returns are very hard to measure, so it
would be useful to have a model of what returns
should be (absent any special information). Such a model would provide baseline estimates
to be modified based on specific information
about particular assets. The CAPM is an equilibrium model of the
relation between the expected rate of return and
the return covariances for all assets.
Hsiulang Chen Portfolio Analysis 8 CAPM Basics  Assumptions No transaction costs.
Assets are all tradable and are all infinitely divisible.
No taxes.
No individual can affect security prices (perfect
competition).
Decisions are made solely in terms of expected
returns and variances.
Unlimited short sales.
Unlimited borrowing and lending at the riskfree
rate.
Homogeneous expectations.
Hsiulang Chen Portfolio Analysis 9 CAPM Basics  The market portfolio What do we know from the passive portfolio
problem?
Everyone holds two portfolios: the riskfree
security and the tangency portfolio. If everyone sees the same CAL, then everyone
has the same tangency portfolio. So, what must the tangency portfolio be? It must be the market portfolio. What is it?
– A portfolio of all risky securities held in
proportion to market cap. ( Important: “all”
include stocks, bonds, realestate, human
capital, etc.)
Hsiulang Chen Portfolio Analysis 10 CAPM Basics  The Capital Market Line
2
E (~ ) − rf ≈ A σ M
rM • Note that this says that all investors should only hold combinations of
the market and the riskfree asset.
• How does this relate to the increased popularity of index funds?
Hsiulang Chen Portfolio Analysis 11 CAPM Basics  The Security Market Line Describe the relationship between expected returns
and systematic risk of individual assets. Investors will only want to hold a security in their
portfolio if it provides a reasonable amount of extra
reward. For all securities, what a security adds to the risk of a
portfolio will be just offset by what it adds in terms of
expected return. The ratio of marginal return contribution to marginal
variance contribution must be the same for all assets.
– What it adds in expected return is E(ri)rf.
– What it adds in terms of risk is proportional to its
covariance with the market portfolio (i.e. its β). ~) − r = β (E ( ~ ) − r )
E ( ri
rM
f
i
f Hsiulang Chen Portfolio Analysis 12 CAPM Basics  CML vs. SML Hsiulang Chen Difference?
Portfolio Analysis 13 Disequilibrium in CAPM
E(r) Undervalued
15% SML +α{ Rm=11% α Overvalued 9% rf=3%
β
1.0
Hsiulang Chen 1.25 Portfolio Analysis 14 Hsiulang Chen Portfolio Analysis 15 β  Estimation Beta is usually estimated using a Security
Characteristic Line (SCL) regression: ri,t –rf,t = αi + βi[rm,t –rf,t ] + εi,t The part of rirf that is “explained” by the market
return is βi[rmrf], a component related to the
systematic or market risk of the asset. The part of rirf not explained by the market return
is εi, a component related the unsystematic or
idiosyncratic risk.
What’s the difference between SML and SCL?
What are the expected return and variance/covariance of stocks
under this specification?
Hsiulang Chen Portfolio Analysis 16 Example Hsiulang Chen Portfolio Analysis 17 ßEstimation Can we read off αˆi, βˆi, and εˆi,t from this scatter plot?
Hsiulang Chen Portfolio Analysis 18 Regression Results
rGM  rf = α + ß(rm  rf) α Estimated coefficient
2.590
Std error of estimate
(1.547)
Variance of residuals = 12.601
Std dev of residuals = 3.550
RSQR = 0.575
Hsiulang Chen Portfolio Analysis ß
1.1357
(0.309) 19 ß  Estimation The systematic variance is βi2σm2. The unsystematic/idiosyncratic variance is σε2. The CAPM says: only the systematic component is
priced. Note that the total return variance is βi2σm2+σε2.
Why is the correlation between the two parts equal to zero? Note that the return covariance is βiβjσm2.
Why is the correlation between εi and εj equal to zero? Why don't investors care about unsystematic risk? What must αi be, according to the CAPM? What is the R2 of the regression?
Hsiulang Chen Portfolio Analysis 20 Hsiulang Chen Portfolio Analysis 21 ßEstimation To estimate ß, typically we could use monthly data. It is standard use 5 years (60 months) of monthly
data or sometimes weekly data. Why not use more data? (10 or 20 years).
• Parameter stationarity. Can we use daily or intraday data?
• Nonsynchronous prices (Scholes & Williams, JFE 1977).
• Bidask bounce.
• Weekend/holidays. Sumbetas: to capture the potential delayed reaction
of a stock to the market. Analysis
Hsiulang Chen
Portfolio
22 Regression Output in Excel
Regression rPNC = α + β rS&P500 +ε
estimated from Jan 1998 to Dec 2010. Hsiulang Chen Portfolio Analysis 23 The Single Index Model for Asset Allocation
R =α +β R +e
i
i
i M
i The expected return on a portfolio:
N
N
E(R ) = ∑ϖ α + ∑ϖ β E(R )
M
P
i =1 i i i =1 i i The variance on the portfolio:
N N
N 2 2
2
2
σ
= ∑ ∑ ϖ ϖ β β σ
+ ∑ϖ σ
P
i =1 j =1 i j i j M
i =1 i ei How many input estimates are required to
generate the efficient frontier now? 3N+2!
Hsiulang Chen Portfolio Analysis 24 How does an index model reduce # of
input estimates for the efficient frontier?
An efficient frontier of 500 stocks
• Data collection:
• Input estimates in expected returns
• Input estimates in return covariance matrix
Hsiulang Chen Portfolio Analysis 25 ßEstimation There are a number of Institutions that
supply β's:
– ValueLine uses the past five years (with
weekly data) with the ValueWeighted
NYSE as the market.
Merrill Lynch uses 5 years of monthly
data with the S&P 500 as the market.
Hsiulang Chen Portfolio Analysis 26 ßEstimation Note: Merrill uses adjusted β's, which are equal to:
βiAdj ≈ 1/ 3 + (2/3)βˆi Why do we need an adjustment? What is the Intuition?
– Statistical Bias.
– diversification of operations. Why 1/3 and 2/3 ?
– What should these numbers be based on?
– Other more advanced ``shrinkage'' techniques.
– Using characteristics (e.g. industry) to estimate/forecast’s.
• BARRA and Ibbotson forecast β based on firm size,
growth, leverage etc.
– How would you estimate the β of a new company?
Hsiulang Chen Portfolio Analysis 27 Time Varying Beta
• Typical time series of beta estimate
• Here shown for AT&T using a rolling 60 month
window Hsiulang Chen Portfolio Analysis 28 Time Varying Beta Suppose CAPM holds year by year Consider only 2 stocks in the economy: A and B Betas and the expected risk premium on the market change form
year to year: If we only look at average betas, we’ll mistakenly
conclude that the risk premium is independent of beta.
Hsiulang Chen Portfolio Analysis 29 Accuracy of Historical Betas Hsiulang Chen Portfolio Analysis 30 Why might observed betas in one period differ
from betas in an adjacent period? [EGBG, p. 139] The risk (beta) of a security or portfolio
might change Fundamental Betas.
β i = a0 + a1 X 1 + a2 X 2 + a3 X 3 + ⋅ ⋅ ⋅
Where Xi could be market variability, earning variability,
firm characteristics, industry dummy variables, and so
on. The beta in each period is measured with
a random error Portfolio grouping
technique, and Vasicek’s technique.
Hsiulang Chen Portfolio Analysis 31 Vasicek's Technique
2
2
σ β1
σ βi1
βi2 = 2
β1 + 2
β i1
2
2
σ β 1 + σ βi1
σ β 1 + σ βi1 Where βit is the ith security’s beta at time t.
β¯1 is the crosssectional mean beta. This weighting procedure adjusts observations
with large standard errors further toward the
mean than it adjusts observations with small
standard errors. It is a Bayesian estimation
technique.
Hsiulang Chen Portfolio Analysis 32 A "Bayesian" • Greenspan's Legacy Rests On Results, Not
Theories
•
• Fed Chief's Biggest Idea Was to Avoid Having One; Embracing the Ambiguity
By GREG IP Staff Reporter of THE WALL STREET JOURNAL
January 31, 2006; Page A1 Thomas Bayes was an 18thcentury British Presbyterian
minister who had early insights into making decisions
when key determinants of the outcome are unknown. A
Bayesian makes a decision based not on the most
probable outcome but on a range of possible outcomes.
Hsiulang Chen Portfolio Analysis 33 'He Has Set a Standard'
By MILTON FRIEDMAN
January 31, 2006; Page A14 Hsiulang Chen Portfolio Analysis 34 “I was praised for
things I didn’t do.” “I am now being
blamed for things
that I didn’t do.”
His Legacy Tarnished,
Greenspan Goes on Defensive
Future of U.S. Financial Reform
Is at Stake; 'I Am Right'
By GREG IP; W SJ, April 8, 2008; P age A1 Hsiulang Chen Portfolio Analysis 35 ...
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