4Equilibrium投资学课件

7
4 Equilibrium
μ-r γσ2
Measuring Components of Risk
i2 = i2 m2 + 2(ei)
where;
i2 = total variance i2 m2 = systematic s stematic variance ariance 2(ei) = unsystematic variance


riskless part: αj risky part: βj log F + εj return due to sensitivity βj to common factor log F additional return with an expected value of zero

The risky part can be further decomposed into:
i
ei
=the stock’s expected return if the market’s excess return is zero
=firm specific component, not due to market movements
Ri i i Rm ei
6
4 Equilibrium


16
4 Equilibrium
μ-r γσ2
The Optimal p Risky y Portfolio of the Singleg Index Model
N 1 i 1
P wi i P wi i
i 1 N 1
( eP ) wi2 2 ( ei )
1
4 Equilibrium
μ-r γσ2
4 Equilibrium
Fangyi Jin
Chinese Academy of Finance and Development Central University of Finance and Economics fangyi jin@cufe edu cn fangyi.jin@cufe.edu.cn
N 1
N 1
17
4 Equilibrium
μ-r γσ2
Solution to the Risky Portfolio

The optimal risky portfolio is composed of two component portfolio

An active portfolio The market index portfolio, the (n+1)th asset
19
4 Equilibrium
μ-r γσ2
Using g the Single-Index g Model with Active Management

The single-index model can be extended to optimize the portfolio with active management The portfolio consists of an active portfolio and a passive i or index i d portfolio tf li The weight of the active portfolio is determined by the information ratio
2
4 Equilibrium
μ-r γσ2
Single g Factor Models
Specification

For securities j = 1, 2, ..., m and time t = 1, 2, ..., T:
Log g (r ( jt/r) ) = j + j log g Ft + jt
13
4 Equilibrium
μ-r γσ2
Regression g Statistics for the SCL of HewlettPackard
14
4 Equilibrium
μ-r γσ2
Estimates Needed for the Single-Index g Model

3n+2 estimates needed; otherwise?
0 w A w* A 0 1 (1 A ) w A
A w 2 A
0 A
E ( RM ) / 2 M
18
4 Equilibrium
μ-r γσ2
Sharpe Ratio for the Combined Portfolio
2 2 A sP sM (e) A
5
4 Equilibrium
μ-r γσ2
Single Index Model
(ri rf ) i i (rm rf ) ei
Market Risk Prem or Index Risk Prem
Risk Prem
rm rf =0 i (rm rf ) =the component of return due to movements in the market index


3
4 Equilibrium
μ-r γσ2
Specification (continued)

E(εj) = 0 can be achieved by setting [ g( j/r)] )] – βj E(log ( g F) ). αj = E[log(r The return of a security can be decomposed into two p components:
2 i 1
N 1
E ( RP ) wi i E ( RM ) wi i
i 1 i 1
N 1
N 1
P [
2 P
2 M
( eP )]
2
1/ 2
[ ( wi i ) wi2 2 ( ei )]1/ 2
2 M 2 i 1 i 1


A (e) A
20
4 Equilibrium
μ-r γσ2
Optimal Weight for Each Security
wi* w* A

i / 2 (ei )

i 1
n
i
/ 2 ( ei )
The contribution of each security to the information ratio of the active portfolio depends on its own information ratio 2 2
11
4 Equilibrium
μ-r γσ2
Excess Returns on HP and S&P 500 April p 2001 – March 2006
12
4 Equilibrium
μ-r γσ2
Scatter Diagram g of HP, S&P 500, and Security y Characteristic Line (SCL) for HP
i2 m2
9
4 Equilibrium
μ-r γσ2
Index Model and Diversification
R P P P RM eP
P 1 N i i 1 P 1
eP 1 N i 1
N N
N
i
N i 1
N
ei
2 2 2 P P M 2 ( eP )
n i A ( e ) ( e ) i 1 A i
15
4 Equilibrium
μ-r γσ2
Preparing the Input List

Macroeconomic analysis is used to estimate the risk premium and risk of the market index Estimate beta coefficients of all securities and their residual variances, 2(ei) Compute the market market-driven driven expected return Derive security alphas forecasts
8
4 Equilibrium
μ-r γσ2
Examining Percentage of Variance
Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2
ßi2 m2
/ 2 = 2 / [i2 m2 + 2(ei)] = 2
( eP ) (
2 i 1
1 2 2 1 ) ( ei ) 2 ( e ) N N
10
4 Equilibrium
μ-r γσ2
The Variance of a Portfolio with Risk Coefficient Beta in the Single-Factor Economy

For example (dropping the time subscript), for three securities: log(r1/r) = α1 + β1 log F + ε1 log(r2/r) = α2 + β2 log F + ε2 log(r3/r) = α3 + β3 log F + ε3 So far, since the j could be anything, these can hold as tautologies. Now, impose some structure. Assume (j, log l F) = 0 Provided now that we choose the j properly, again the equations are still tautologies tautologies. j = Cov(log rj, log F)/Var(log F)
μ-r γσ2
Components of Risk

Market or systematic risk: risk related to the macro economic factor or market index index. Unsystematic or firm specific risk: risk not related to the macro factor or market index. T t l risk Total i k = Systematic S t ti + Unsystematic U t ti


Important assumption: for each pair of securities j and k: ρ(εj, εk) = 0 for j k
4Βιβλιοθήκη Baidu
4 Equilibrium
μ-r γσ2
Single Factor Security Model
ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F = some macro factor; f t in i this thi case F is i unanticipated ti i t d movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor. Advantages: reduces the number of inputs for di diversification; ifi ti easier i for f security it analysts l t to t specialize. i li