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