صفحه 1:
صفحه 2:
۴ 9
The Correlation
Structure of
Security Returns-
the Single-Index
Model
صفحه 3:
به نام خدا
۲۳6 60۳۲۵۱۵۱۵9 5۲۵6۵۲6 : عنوان
of Security Returns-the Single-
Index Model
صفحه 4:
۱۱36 میتی کت e Single-Index Model
یت یات ری
Title: Inputs to Portfolio Analysis
Key Points (Bullet Points):
Expected Returns: Forecasted average returns for each asset in the
portfolio. Crucial for assessing potential gains.
Variances: Measures of risk for individual assets, indicating the dispersion
of possible returns around the expected return. (e.g., standard deviation
Correlations: Measures of how the returns of different assets move in
Positive correlation: Assets tend to move in the same direction.
Negative correlation: Assets tend to move in opposite directions.
Zero correlation: No discernible relationship.
صفحه 5:
Inputs to Portfolio Analysis :: ادامه مطلب
_ The Challenge of Correlations:
For a portfolio of 'N' assets, the number of unique correlation coefficients
required isN * (N - 1)/2.
Example: For 50 assets,50 * 49 / 2 = 1225 correlation estimates are
needed. This becomes computationally intensive and prone to estimation
err
0
Title: Single-Index Model: A Simplification
_ Key Points (Bullet Points):
Core Idea: Assumes that the return of any security is primarily driven by
the return of a single common factor (e.g., the market) and a unique,
independent factor.
صفحه 6:
Inputs to Portfolio Analysis :: ادامه مطلب
Sea eal
Title: Single-Index Model: A Simplification
| Key Points (Bullet Points):
Core Idea: Assumes that the return of any security is primarily driven by
the return of a single common factor (e.g., the market) and a unique,
independent factor.
صفحه 7:
Single-Index Model : : ادامه مطلب
| Components:
R_i: Expected return of security ۰
@ i (Alpha): The security's expected return when the market return is zero.
Represents the security's idiosyncratic return independent of the market.
Often interpreted as a measure of abnormal return.
6_1 (Beta): The sensitivity of the security's return to changes in the market
return. A measure of systematic risk.
B gt 1: More volatile than the market.
Bit 1: Less volatile than the market.
8 = 1: Moves with the market.
R_m: Return of the market index (e.g., SampP 500, MSCI World).
3
صفحه 8:
Single-Index Model : : ادامه مطلب
e_i (Error Term/Residual): The unsystematic (firm-specific) portion of the
security's return that is not explained by the market. Assumed to have an
expected value of zero and be uncorrelated with the market return and
Equation:‘Ri=ai+Bi*Rmt+ei
* The single-index model simplifies portfolio analysis by relating security
returns to a single common factor: the market return.
* Equation: Ri = ai + BiRm + ef
* Ri: Return on security
* ai: The stock's expected return that is independent of the market
* Bi: Sensitivity of security / to market movements (Beta).
* Rm: Return on the market index
صفحه 9:
Single-Index Model : : ادامه مطلب
و *ei: The unsystematic return of the stock i, uncorrelated with the market
* *Key Assumption: The only reason stocks vary together systematically is
due to their common movement with the market (el is independent of ej).
اعم
ع5 6ه كأمعمهم ممع
Title: Deconstructing Security Return: Systematic vs. Unsystematic Risk *
Key Points (Bullet Points): |
Systematic Risk (Market Risk): _
Risk that affects all assets in the market. +
Cannot be eliminated through diversification. *
* Measured by Beta (fi).
صفحه 10:
Components of Security Return : lle el
Arises from macroeconomic factors, interest rate changes, political events,
5 in Model: i * يه
Unsystematic Risk (Idiosyncratic Risk / Firm-Specific Risk / Diversifiable Risk):
Risk unique to a particular company or industry.
Can be significantly reduced or eliminated by holding a well-diversified
Measured by the error term (ei).
Arises from company-specific events like management changes, product
recalls, strikes, or technological breakthroughs.
صفحه 11:
Components of Security Return ::sllae cba
+ Component in Model:e i
* Fundamental Betas
* Beta is a risk measure that arises from the
relationship between the return on a stock
and the return on the market. However, we
know that the risk of a firm should be
determinedby some combination of the firm's
fundamentals and the market characteristics
of the firm's stock. If these relationships could
be determined, they would help us to better
under- stand and forecast betas.
Figure 7.1 Plo ونام باسح اه vers mae rtm
صفحه 12:
Components of Security Return : : ادامه مطلب
One of the earliest attempts to relate the beta of a stock to fundamental
firm variables was performed by Beaver, Kettler, and Scholes (1970). They
examined the relationship betweenseven firm variables and the beta ona
company’s stock. The seven variables they used were: 1. dividend payout
2. asset growth (annual change in total assets)
3. leverage (senior securities divided by total assets)
4. liquidity (current assets divided by current liabilities) 5. asset size (total
6. earning variability (standard deviation of the earnings price ratio)
7. accounting beta (the beta that arises from a time series regression of
the earnings of the firm against average earnings for the economy, often
called the earnings beta)
صفحه 13:
ات ۱
Title: Calculating Security Variance with the Single-Index Model
Key Points (Bullet Points):
The total risk (variance) of a security, as per the Single-Index Model, is the
sum of its systematic risk and its unsystematic risk.
* Formula:o_i-2 = B12 *o_m*2 + 0_@i7*2
@_i*2: Total variance of security‘.
8.172 * o_m~2: Systematic Variance. This component represents the
portion of the security's total variance that is due to its exposure to the
market's variance.o_m~2 is the variance of the market index.
@_e 172: Unsystematic Variance (also known as Residual Variance). This
component represents the portion of the security's total variance that is
unique to the security and not explained by market movements. It is the
variance of the error terme_i.
صفحه 14:
Variance of a Security : : ادامه مطلب
Ione نك Ely
* Title: The Power of Diversification: Reducing Residual Risk
| Key Points (Bullet Points):
* Principle: By combining multiple assets in a portfolio, the unsystematic
(firm-specific) risks of individual assets tend to cancel each other out.
صفحه 15:
Portfolio Diversification : : ادامه مطلب
Re Ri ad Ree Se + Impact on Portfolio Risk: As the number of 12 ما
ا ۳ Toga a ag ape =
هت risk of the (EET وال i
Title: Conclusion: Benefits of the Single-Index * 3
“key Points (Bullet Points): سس اس سس
vidual ecu is Bo, + 92, Because the effet of مار مه
ok dos ot i
i Bris the me
insted by holding &
of esis sk
risk canbe made to
‘diversi
+ Simplified Inputs: Drastically reduces the
number of inputs required for portfolio
optimization (fromN * (N - 1) /2 correlations to
‘N' betas, ‘N' alphas, ‘N' residual variances,
and 1 market variance). This makes it practical
+ Computational Efficiency: Requires far fewer calculations, making portfolio
analysis faster and more manageable.
cnstat with spect al
مان مدمه لح 9
مهب ها نی حون sed a hem
صفحه 16:
Portfolio Diversification : : ادامه مطلب
Clear Risk Decomposition: Explicitly separates systematic and
unsystematic risk, providing a clearer understanding of risk sources.
Intuitive Understanding: Betas provide a straightforward measure of
market sensitivity for each asset.
Facilitates Diversification Strategy: Helps investors understand how
adding assets affects portfolio risk by distinguishing between diversifiable
Foundation for Performance Evaluation: Provides a basis for evaluating
manager performance (e.g., Jensen's Alpha).
Narration/Explanation: Reiterate that despite its simplifying assumptions,
the Single-Index Model offers a powerful and practical framework for
portfolio analysis, especially for institutional investors managing large
portfolios.
صفحه 17:
Betas as Forecasters of Correlation Coefficients
Elton, Gruber, and Urich (1978) have compared the ability of the following models to fore-
cast the correlation structure between securities:
1. the historical correlation matrix itself
2. forecasts of the correla
ical period
|. forecasts of the correlation matrix prepated by estimating betas from the prior two
periods and updating via the Blume technique
jon matrix prepared by estimating betas from the prior histor~
4, forecasts prepared as in the third model but where the updating is done via the Vasicek
Bayesian technique
One of the most striking results of the study was that the historical correlation matrix
itself was the poorest of all techniques. In most cases it was outperformed by all of the
beta forecasting techniques at a statistically significant level. This indicates that a large
part of the observed correlation structure between securities, not captured by the single~
index model, represents random noise with respect to forecasting. The point to note is that
the single-index model, developed to simplify the inputs to portfolio analysis and thought
to lose information because of the simplification involved, actually does a better job of
forecasting than the full set of historical data.
‘The comparison of the three beta techniques is more ambiguous. In each of two five-year
samples tested, the Blume adjustment technique outperformed both the unadjusted betas
and the betas adjusted via the Bayesian technique. The difference in the techniques was
statistically significant. However, the Bayesian adjustment technique performed better than
the unadjusted beta in one period and worse in a second. In both cases, the results were
statistically significant, This calls for some further analysis, The performance of any fore=
casting technique is, in part, a function of its forecast of the average correlation between all
stocks and, in part, a function of its forecast of previous differences from the mean,
صفحه 18:
