Lecture 4 Flashcards
The Input list of Markowitz Model.
The success of portfolio rule depends on the quality of the input list. Includes the estimates of expected security returns and the covariance matrix.
In the long run, efficient portfolios will beat portfolios with less reliable input lists and consequently inferior reward-to-risk trade-offs.
(n^2 - n)/2
- Quadratic growth of the number of inputs
Estimations
- expected return on each asset
- Variance of each asset
- Correlations between each asset pair
Difficulty in applying the Markowitz model to portfolio optimisation
- The errors in the assessment or estimation of correlation coefficients can lead to nonsensical results. The overwhelming number of parameter estimates required to implement it.
- Portfolio variances are always positive. If we know true covariances/correlations between returns, we will
always obtain positive portfolio variances. However, in practice, we only have estimated covariances/correlations
that may result in negative portfolio variances. We conclude that the inputs in the estimated correlation matrix must be mutually inconsistent. This is due to estimation error from using historical data to estimate
correlations.
Single Input Model
Allows a simpler model as it uses smaller, consistent set of estimates of risk parameters and risk premiums.
Reduces the number of parameters that must be estimated.
Correlations between securities might be due to a common response to market changes. Thus, we might be able to obtain a measure of this correlation by relating the return on a stock to the return on the stock market index.
A single-factor model of the economy
- classifies sources of uncertainty as systematic (macroeconomic) factors or firm-specific (microeconomic) factors.
- assumes that the macro factor can be represented by a broad index of stock returns.
- drastically reduces the necessary inputs in the Markowitz portfolio selection
procedure. It also aids in specialisation of labor in security analysis.
ei (residual)
is the unexpected return, that measures the uncertainty about the security return of the firm in particular.
It is a zero-mean, firm-specific surprise in the security return in time T.
mean = 0 sd = oi
Rm
The sources of uncertainty about the economy.
It is the re rate of excess return on market index.
ei & m relationship
They are uncorrelated.
ei = firm specific, it is independent of shocks to the common factor that affect the entire economy.
m = has no subscript because the same common factor affects all securities.
Variance & Covariance
variance arises from two uncorrelated sources, systematic and firm specific.
Both variances and covariances are determined by the security betas and the properties of the market index.
Beta : sensitivity coefficient
Demoted because some securities will be more sensitive than others to macro-economic shocks.
- Cyclical firms have a greater sensitivity to the market and therefore higher systematic risk.
- is the slope coefficient of the regression equation.
- Beta is the sensitivity analysis of the index, the amount by which the security tends to increase or decrease for every 1% change in the return on the index.
- Market risk remains the same regardless of the number of firms combined into the portfolio
Single Index Model
It uses the market index to proxy for the common factor.
- assumes that covariances between asset pairs are driven by a single factor.
- separates an assets total risk into two parts: 1. systematic risk, 2. non-systematic risk.
The single-index model simplifies mean-variance analysis substantially. While it is simple, it is not necessarily inferior to the full-covariance model.
Alpha
It is the component of excess return that is independent of the market’s performance. It is the security’s expected excess return when the market excess return is 0.
Is the non-market premium.
Therefore, a may be large if you think a security is underpriced and therefore offers an attractive expected return.
As the number of stocks included in the portfolio increases, the part of the portfolio risk attributable to non-market factors becomes even smaller. This part of the risk is diversified away.
Ai is the component of the security of security i’s return that is independent of the market’s performance.
ai = expected component
ei = random component
The Expected Return–Beta Relationship
As E(ei) = 0, we take the expected value of E(R), that we obtain from the expected return-beta relationship of the single-index model.
The security’s risk premium is due to the risk premium of the index.
The market risk premium is multiplied by the relative sensitivity or beta of the individual security. Also known as systematic risk premium
Alpha: is the non-market premium.
Estimates in the Single-Index Model
Imply that the set of parameter estimates needed for the single index
model consists of only a , b , and o(e) for the individual securities, plus the risk premium and variance of the market index.
- n estimates of the extra-market expected excess returns, a i
- n estimates of the sensitivity coefficients, b i
- n estimates of the firm-specific variances, o^2(ei)
- 1 estimate for the market risk premium, E(RM)
- 1 estimate for the variance of the (common) macroeconomic factor, o^2(M)
For (3n+2) estimates
Total risk
systematic risk + firm-specific risk
As N increases, the effect of individual risk diminishes whereas the effect of market risk remains. Individual assets contribute to portfolio risk through their beta.
The markets common exposure regardless of the amount of assets is represented by portfolio systemic risk.
Covariance
product of betas x market-index risk
Correlation
Product of correlations with the market index
Estimates of Single Index Model
Thus for a 50-security portfolio we will need 152 estimates rather than 1,325;
For the entire New York Stock Exchange about 3,000 securities, we will need 9,002 estimates rather than approximately 4.5 million
The Index Model was first suggested by
Sharpe
The key characteristics of the single model index.
- ei and Rm are uncorrelated how well this model performs does not depend on what
the market return happens to be. - ei and ej are uncorrelated
stock returns are only correlated because of the market.
Estimating Beta and Alpha
use a simple regression analysis to estimate both beta and alpha.
Diversification
As diversification increases, the total variance of a portfolio approaches
the systematic variance, defined as the variance of the market factor multiplied by the square of the portfolio sensitivity coefficient, Bp^2 .
As more and more securities are combined, the portfolio variance decreases because of the diversification of firm-specific risk.
The market risk is always there. Systemic risk is non- diversifiable.
The R squared explanation, highlights the practical importance of diversification.
In a diversified portfolio, only the assets systematic risk matters, which is measured by its beta.
High residual standard deviations show how important diversification is.
- These are usually securities with tremendous firm-specific risk.
- These securities would have the unnecessary high volatility and inferior Sharpe ratio.
Alpha estimations
Alphas are not constant over time
Alpha estimates are ex post (after the fact), it does not mean that alphas can be forecasted ex ante (before the fact).
Ex Post: alpha is able to earn extra return, evident through historical mispricing. This is a indication of how much we could have earned, the return on the stock
Ex Ante: Future, based on current mispricing.
Security analysis: to forecast alpha values ahead of time.
Beta estimations
Compare to 1. T value is comparing it to 0. 0.98 is different from 1, but statistically it isn’t very different.
Beta estimates are data sensitive as they are a look at the historical values of the beta.
Firm Cycle Explanation:
As its one stock’s beta, it is idiosyncratic risk A firm usually starts by producing a specific product or service and may be more unconventional than established firms. Over time, the firm diversifies and become more conventional. It starts to resemble the rest of the economy. B gets closer to 1.
Statistical Explanation:
The average beta over all security is 1, this is our best guess of a firm’s beta even before we do any estimation.
A beta estimated from a sample contains sampling error, the greater the difference between the estimated beta and 1, the greater is the chance we incurred a large estimation error.
Raw Returns over Excess Returns
We assume that risk free rate is constant over time we will obtain a different intercept.
In reality, the risk free rate is not constant over time but the variation is low. In this case, the beta estimates will be different but they will be very similar.
Root-mean-square (Root MSE)
Firm Specific Risk - idiosyncratic risk
This is the measure of the firm specific risk variance. (the error)
- This is your idiosyncratic risk.
- It represents the amount of variance is due to idiosyncratic risk
- This value is basically the standard deviation of the residual from your market model regression.
- the variance of the unexplained portion of stock i return, that is the portion of the return that is independent of the market index
Security Characteristics line
The equation describes the linear dependence of stock i excess return on changes in the state of economy as represented by the excess returns of the market index portfolio.
SCL formed using regression analysis, summarises a particular security’s systematic risk, s&p and rate of return.
As the index model is linear we can estimate the beta coefficient of a security on the index using a single-variable linear regression.
Security Characteristics line elements
To estimate the regression, we collect a historical sample of paired observations. The scatter diagram includes the regression line.
X - axis: is the excess return of the market in general.
Y -axis: is the excess return on a security over the risk free return.
Intercept : alpha
Slope coefficient : Beta
Vertical distance: from the fitted line is the residual Ei
Correlation between regression forecasts and realisations of out-of-sample data is almost always considerably lower than in sample correlation.
Sum of Squares
The sum of squares (SS) of the regression is the portion of the variance of the dependent variable (stock’s return) that is explained by the independent variable (S&P 500 return)
Residual
Beta is the security’s sensitivity to the index. It is the amount by which the security return tends to increase or decrease for every 1% increase of decrease in the return of the index. The zero-mean, firm-specific surprise in the security returns in time t is called the residual.
Where the regression line is drawn through the scatter. The vertical distance of each point from the regression line is the value of HP’s residual, e HP ( t ),
- the regression of that particular month
The standard deviation of the error of the regression is the standard deviation of the residual.
Root MSE
Shows the variance of the unexplained portion of the stock’s return. The portion of return that is independent of the market index.
Root-mean-squared (RMSE^2)
The square root of RM is the standard error (SE) of the regression.
It is a standard way to measure the error of the model. It is the standard error of the estimation.
R square
R^2 = is the portion of the market risk from the total risk (market risk+ firm risk).
The ratio of the explained to total variance. This equals the explained regression.
Tells us that variation in the S&P 500 excess returns explains of the variation in the HP series.
T-statistic
Shows if the value of the coefficient or the slope are statistically significant.
This is the ratio of the regression parameter to its standard error.
This statistic equals the number of standard errors by which our estimate exceeds zero, and therefore can be used to assess the likelihood that the trey but unobserved value might actually equal to zero.
Large t-statistics imply low probabilities that the true value is zero.
Alpha: We are interested in the average value of the stock’s return net of the impact of the market movements.
Beta: the t-statistic would measure how many standard errors separate the estimated beta from a hypothesised value of 1.
Is the index model inferior to the full covariance model?
Single Index Model Disadvantage:
Applies when replace with a full- covariance model or even a multi-index model of security returns. To add another index,
we need both a forecast of the risk premium of the additional index portfolio and estimates of security betas with respect to that additional factor
Single Index Model Advantage:
The single-index model is simpler: covariances between assets pairs are driven by a single factor. There are only 2 assumptions.
Markowitz model Disadvantage:
- Add more variables, we introduce errors
- The advantage can be illusory if we cant estimate covariance terms with any degree
any degree of confidence. Using the full-covariance matrix invokes estimation risk of thousands of terms.
Markowitz model Advantage:
Is more flexible in our modelling of asset covariance structure compared to single-index model: each covariance term can be estimated separately. Estimation error is an issue.
Even if the full Markowitz model would be better in principle, it is very possible that the cumulative effect of so many estimation errors will result in a portfolio that is actually inferior to that derived from the single-index model.
Index Models and Tracking Portfolio
Assuming a portfolio manager has identified an underpriced portfolio with estimates.
The manager is confident in her security analysis but is wary about the performance of the broad market in the near term. If she buys the portfolio, and the market as a whole turns down, she still could lose money on her investment.
She would like a like a position that takes advantage of her analysis but is
independent of the performance of the overall market. To this end, she constructs a tracking portfolio to match the systematic
component of P (i.e. the same beta) but has as little nonsystematic risk as possible, beta capture. She can do with a combination between a risk-free asset and the index.
If we long P and short T, can construct N:
This is hedged portfolio produces the market neutral portfolio. This is a market neutral portfolio, its performance does not depend on the performance of the market.
Tracking Portfolio
tracking portfolio for portfolio
P is a portfolio designed to match the systematic component of P ’s return.
The idea is for the portfolio to “track” the market-sensitive component of P ’s return. This means the tracking portfolio must have the same beta on the index portfolio as P and as little nonsystematic risk as possible. This procedure is also called beta capture
Tracking funds are the vehicle used to hedge the market risk to which the hedge fund managers do not want exposure.
Underpriced Tracking Portfolio
Assuming a portfolio manager has identified an underpriced portfolio with estimates.
- Buying portfolio P but at the same time offsetting systematic risk by
- assuming a short position in the tracking portfolio.
The short position in T cancels out the systematic exposure of the long position in P: the overall combined position is thus market neutral. Therefore, even if the market does poorly, the combined position should not be affected. But the alpha on portfolio P will remain intact.
While this portfolio is still risky (due to the residual risk, e P ), the systematic risk has been eliminated, and if P is reasonably well-diversified, the remaining nonsystematic risk will be small
Alpha Transport
The manager can take advantage of the 4% alpha without inadvertently taking on market exposure.
The process of separating the search for alpha from the choice of market exposure is called alpha transport. Such as If we long P and short T, can construct N or vice versa.
Long Short Strategy
Is a characteristic of activity of many hedge funds.
If a hedge fund manager detects an underpriced security, she can try to attain a “pure play” on the perceived underpricing. That is, she hedges out all extraneous risk, focusing the bet only on the perceived “alpha”.
Tracking funds are the vehicle used to hedge the market risk to which the hedge fund managers do not want exposure.
