TOPIC 5 - DIVERSIFICATION AND PORTFOLIO RISK Flashcards
Market risk (systematic risk) features
1) attributable to market-wide risk sources
2) remains even after diversification
firm-specific risk
risk that can be eliminated by diversification
diminishing marginal reduction in risk occurs from
increasing portfolio diversification
expected return of a portfolio made of equity and debt
rp= wdE(r)d + weE(r)e
w = weight in the asset
just a weighted average
covariance is used in portfolio theory to determine what assets to include to reduce the overall risk for a portfolio. It’s a
statistical measure of the directional relationship between 2 asset prices.
+ve covariance =
assets generally move in same direction
-ve covariance =
assets generally move in opposite directions
modern portfolio theory uses covariance how
optimises returns by including assets in their portfolio that have a -ve covariance
covariance helps investors create a portfolio that includes a mix of distinct asset types, thus employing
a diversification strategy to reduce risk
what does modern portfolio theory aim to do?
attempts to determine an efficient frontier for a mix of assets in a portfolio.
the efficient frontier aims to optimise the maximum return vs the degree of risk for the overall combined assets in the portfolio.
In MPT the goal is is to choose assets that have a
lower SD for the combined portfolio than the SD of the individual assets to reduce portfolio’s volatility.
MPT seeks to create an optimal mix of
higher volatility assets with lower volatility assets.
By diversifying assets in a portfolio, investors can reduce risk and still allow for a positive return.
what sort of relationship must assets have with eachother to be selected for a portfolio?
a negative covariance.
Analysts use historical price data to determine the measure of covariance between different stocks. This assumes that the same statistical relationship between the asset prices will continue into the future, which is not always the case.
drawbacks of using covariance to reduce portfolio risk
1) can only measure the directional relationship btw 2 assets.
- -> can’t show the strength of relationship btw assets. (correlation coefficient a better measure of that strength)
another drawback of covariance in terms of its calculation
2) calculation of covariance is sensitive to higher volatility returns –> more volatile assets incl returns that are farther from the mean (outlying returns can have undue influence on resulting covariance calc)
MPT is a theory on:
a) how risk-averse investors can construct portfolios to maximize E(r) based on a given level of market risk.
b) can also be used to construct a portfolio that minimises risk for a given level of E(r)
what does MPT recommend in terms of correlation coefficients? 1
1) that an investor measure the correlation coefficients between the returns of various assets in order to strategically select those that are less likely to lose value at the same time.
By measuring the orrelation coefficients between the returns of various assets in order to strategically select those that are less likely to lose value at the same time WHAT DOES THIS MEAN?
determining to what extent the prices of the assets tend to move in the same direction in response to macroeconomic trends.
what do you use correlation coefficients to do in MPT?
seek a 0 or near 0 correlation in price movements of the various assets in a portfolio.
This means seeking assets that respond to macroeconomic trends in distinctly different patterns.
covariance vs correlation
Covariance tells you that 2 variables change the same way
Correlation: reveals how a change in one variable affects a change in the other.
Diversification works best when assets are
uncorrelated or negatively correlated with one another, so that as some parts of the portfolio fall, others rise.
When it comes to diversified portfolios, correlation represents the
degree of relationship between the price movements of different assets included in the portfolio.
correlation of 1 =
perfectly +vely correlated –> asset prices move in tandem
correlation of -1 =
prices move in opposite directions
perfectly -vely correlated
correlation of 0 =
uncorrelated –> price movement of one asset has no effect on the price movement of the other asset.
different macroeconomic factors affect
different assets differently!
Not all assets created equal
Statisticians use price data to find out how the prices of two assets have moved in the past in relation to each other. Each pair of assets is assigned a
number that represents the degree of correlation in their price movements. This number can be used for constructing what is called a “correlation matrix” for different assets
How does a correlation matrix make task of choosing dif assets easier?
presenting their correlation with each other in a tabular form. Once you have the matrix, you can use it for choosing a wide variety of assets having different correlations with each other.
once you’ve got a correlation matrix, what do you do
start with broad categories (asset classes - stocks, bonds, gov securities etc, then narrow down to subclasses - consumer goods, energy tech etc).
What is the risk in terms of correlation and using levergae?
using leverage to make investments cuts both ways.
The strategy of taking high exposure by using borrowed money is good at increasing upside potential, but increases exposure to downside risk. You have to pay back the money that you owe from some other source.
When price of an asset is collapsing, the level of leverage may
force a trader to liquidate even his good assets. When a trader is selling his good assets to cover his losses, he hardly has time to distinguish between correlated and uncorrelated assets. He sells whatever is there in his hands.
our optimal risky portfolio is found from
the Sharpe Ratio (portfolio’s risk premium (rp - rf)/SDp
trying to find weights of D and E that give the highest slop of the CAL
4 Steps to determine portfolio construction
1) id risk-return combos available from set of risky assets
2) id optimal portfolio of risky assets by finding portfolio weights that result in the steepest CAL
3) choose an appropriate complete portfolio by mixing the risk-free assets by finding the portfolio weights that result in the steepest CAL
4) choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio
security characteristic line (SCL)
Excess return of security i =
expected excess return when market excess return is 0 + Market B*Excess return of market + error term
information ratio measures what
the extra returns we can obtain from security analysis
Measurement of portfolio returns beyond the returns of a benchmark, usually an index, compared to the volatility of those returns.
Sharpe ratio of an optimally constructed risky portfolio will
exceed that of the index portfolio (passive strategy)
Extent to which SD of a combined portfolio of dif risky assets can be reduced through diversification really depends on
correlation btw the returns on those two assets and how high/low that correlation happens to be
The minimum-variance portfolio has a standard deviation smaller than that of
either of the individual component assets
risk reduction for min variance portfolio depends on
correlation
If ρ=+1.0, no risk reduction is possible
If ρ = 0, σp may be less than the standard deviation of either component asset
If ρ=- 1.0, a riskless hedge is possible
in terms of asset allocation with stocks, bonds and bills, we want to combine the major asset classes that provide
the HIGHEST POSSIBLE SHARPE RATIO
once you’ve got the optimal risky portfolio based on the asset classes, you can use the optimal risk portfolio P’s expected return and volatility along with
the individual investor’s degree of risk aversion A to calculate the optimal proportion of the complete portfolio to invest in this risky component.
single index model vs Markowitz
SIM - easier to approach and consider portfolio and estimating returns
Markowitz = data intensive, computationally difficult
SIM makes a critical assumption that
there is a single common source of risk and once that is controlled there is no residual risk
what’s good and about about the SIM’s assumption of there being a single common source of risk?
simplifies calculations, it’s unrealistic to assume there’s no residual risk
–> means that risk isn’t necessarily minimised and utility isn’t maximised when the SIN is used
The Markowitz model makes no assumptions about
the returns generating process and is generally more accurate in assuming correct inputs
expected return of a portfolio =
weighted average of the component security expected returns with the investment proportions as weights
variance of a portfolio =
weighted sum of elements of the covariance matrix with the product of the investment proportions as weights.
The variance of each assset is weighted by
the square of its investment proportion.
Since the covariance of each pair of assets appears twice in the covariance matrix, the portfolio variance includes
twice each each covariance weighted by the product of the investment proportions in each of the 2 assets
Even if the covariances are positive, the portfolio standard deviation is
less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated
Portfolio diversification is of value as long as
assets are less than perfectly correlated.
The greater an asset’s covariance with the other assets in the portfolio, the more
it contributes to portfolio variance.
An asset that is perfectly negatively correlated with a portfolio can serve as
a perfect hedge. That perfect hedge asset can reduce the portfolio variance to zero.
The efficient frontier is the graphical representation of
a set of portfolios that maximize expected return for each level of portfolio risk.
Rational investors will choose a portfolio on the efficient frontier.
A portfolio manager identifies the efficient frontier by
first establishing estimates for asset expected returns and the covariance matrix.
This input list is then fed into an optimization program that produces as outputs the investment proportions, expected returns, and standard deviations of the portfolios on the efficient frontier.
In general, portfolio managers will arrive at different efficient portfolios because of
differences in methods and quality of security analysis.
Managers compete on the quality of their security analysis relative to their management fees.
If a risk-free asset is available and input lists are identical, all investors will choose
the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL.
All investors with identical input lists will hold an
identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction.
Diversification is based on the allocation of a portfolio of fixed size
across several assets, limiting the exposure to any one source of risk.
Does adding additional risky assets to a portfolio, thereby increasing the total amount invested reduce dollar risk?
No. Even if it makes the rate of return more predictable. This is because that uncertainty is applied to a larger investment base. Nor does investing over longer horizons reduce risk.
Increasing the investment horizon is analogous to
investing in more assets. It increases total risk.
A single-factor model of the economy classifies sources of uncertainty as
systematic (macroeconomic) factors or firm-specific (microeconomic) factors.
The index model assumes that
the macro factor can be represented by a broad index of stock returns
Single-index model drastically
reduces the necessary inputs in the Markowitz portfolio selection procedure. It also aids in specialisation of labour in security analysis
According to the index model specification, the systematic risk of a portfolio or asset equals
beta of the portfolio x beta of market x variance of market
The index model is estimated by applying regression analysis to excess rates of return. The slope of the regression curve is the
beta of an asset, whereas the intercept is the asset’s alpha during the sample period.
the regression line is also called
security characteristic line
Practioners routinely estimate the index model using total rather than excess rates of return. This makes their estimate of alpha
= (alpha + rf(1-Beta))
Betas show a tendency to evolve toward
1 over time. Beta forecasting rules attempt to predict this drift.
Optimal active portfolios include
analysed securities in direct proportion to their alpha and in inverse proportion to their firm-specific variance.
The full risky portfolio is a mix of the
active portfolio and the passive market-index portfolio.
The index portfolio is used to enhance
the diversification of the overall risky position.
If you require that your portfolio yield an expected return of 14%, then you
can find the corresponding standard deviation from
the optimal CAL. The equation
for this CAL is: rf + Sharpe ratiox SDportfolio
To find the proportion invested in the T-bill fund, remember that Let y be the proportion invested in the portfolio
P.
the mean of
the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal
combination of stocks and bonds (P).
beta is the slope of the
security characteristic line,
with beta being the measure of systematic risk
the R^2 (squared correlation coefficient) of the SCL is the
ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the unexplained variance (stock’s residual variance)
in terms of the security characteristic line, alpha is
the intercept of the SCL with the expected return axis.
in a two index model regression, firm specific risk is measured by
residual SD
in a two index model regression, market risk is measured by
beta, the slope coefficient of the regression
in a two index model regression, what does R^2 measure?
the fracion of total variance of return explained by the market return.
the variance of a portfolio
is a weighted sum of covariances and each weight is the product of the product of the portfolio proportions of the pair of assets
If we know correlation coefficient of 2 individual assets and their individual standard deviations then we can easily calculate
covariance
