Ch 2 Measurement Issues and Implications Flashcards
Utility Theory - measuring risk aversion
Absolute Risk Aversion (ARA) =
Relative Risk Aversion (RRA) =
Absolute Risk Aversion (ARA) = β(πβ²β²(π))/(πβ²(π)) (Coefficient of ARA in notes)
Relative Risk Aversion (RRA) = WΓπ΄π
π΄ (Coefficient of RRA in notes)
Note the division by the first derivative to make them level invariant and negation is used for interpretation convenience
Interpretation:
- ARA increasing ->
- ARA decreasing -> more βriskyβ dollars are held as wealth increases
- ARA constant ->
- RRA increasing ->
- RRA decreasing ->
- RRA constant ->
Interpretation:
- ARA increasing -> less βriskyβ dollars are held as wealth increases
- ARA decreasing -> more βriskyβ dollars are held as wealth increases
- ARA constant -> risk dollars remains unchanged, even as wealth increases β¦ no wealth effect
- RRA increasing -> less βriskyβ percentage β¦
- RRA decreasing -> more βriskyβ percentage β¦
- RRA constant -> βriskyβ percentage remains unchanged β¦ no change in bet percentages regardless of wealth
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Quadratic
- Equation
- ARA
- RRA
Important Wealth Utility Functions
Quadratic
- Equation ππβπ/2 π^2
- ARA π/(πβππ)ββ1/π
- RRA β1+π/(πβππ)ββ1
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Exponential (Exp, Power & Log are special examples of same thing)
- Equation
- ARA
- RRA
Important Wealth Utility Functions
Exponential
- Equation βπ^(βππ)
- ARA βπ
- RRA βππ
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Power (Exp, Power & Log are special examples of same thing)
- Equation
- ARA
- RRA
Important Wealth Utility Functions
Power
- Equation (π^(1βπ)β1)/(1βπ)
- ARA πβπ
- RRA π
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Log (Exp, Power & Log are special examples of same thing)
- Equation
- ARA
- RRA
Important Wealth Utility Functions
Log
- Equation Important Wealth Utility Functions
Power
- Equation γπππγ_π (π)
- ARA 1βπ
- RRA 1
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Prospect W> benchmark
- Equation
- ARA
- RRA
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Prospect W> benchmark
- Equation ((πβπππ)^π)/π
- ARA (1βπ)/(πβπππ)
- RRA πΓ(1βπ)/(πβπππ)
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Prospect if W < benchmark
- Equation
- ARA
- RRA
Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs
Prospect if W < benchmark
- Equation βπΓ(πππβπ)^π/π
- ARA (1βπ)/(πβπππ)
- RRA πΓ(1βπ)/(πβπππ)
How Utility Theory relates to Portfolio COnstruction
- Markowitz mean-variance problem
max π= πΌβ² π€βπ/2 π€β²Ξ£w
is described as a βquadratic problemβ in optimisation speak - Itβs βconsistentβ with quadratic utility in utility theory speak; and consistent with other utility functions.
- The reason why there isnβt a direct link is due to the Markowitz mean-variance problem being posed in terms of risk, return and portfolio weights, whereas the Quadratic Utility is posed in terms of wealth β¦ they are related but not identical
- Importantly, for the same return forecasts, risk aversion and variance, a Markowitz mean-variance problem will always have the same solution. It is independent of wealth.
Why isnβt htere a direct link between the mean variance problem and utility
- Even though MV problem is described as βquadraticβ, and there is βquadraticβ utility, the MV problem is posed in terms of risk, return & portfolio weights
The Quadratic Utility is posed in terms of wealth.
Therefore they are related but not identical - They are therefore consistent in only a few situations
In what situations is Markowitz MV consistent with Utility Theory
- Quadratic Utility in all return distributions, but as wealth rises the utility roles over and less wealth perversely becomes more attractive
- Exponential Utility but only when returns are normally distributed, but relative risk aversion is increasing, implying wealth is never sated
- Power Utility but only when returns are lognormally distributed
If MV is not completely consistent with Utility Theory, why is it so ingrained
- Only two things are needed: the expected returns and the expected variance
- A βQuadratic Programβ is computationally simple to solve
- The 80/20 rule β the 80% solution is easy, the next 20% consumes considerably more time & effort
Optimisation
- Grid Search
- Gradient Ascent
- Interior Point (Matlab)
- GRG Non linear (Excel)
Optimisation
- Grid Search (1000s): evaluate objective at a range of possible solutions, repeat over small regions with finer grids, stop when rate of improvement is in tolerance. Very cery slow but guaranteed a solution
- Gradient Ascent (100): Given starting point (random or choose), choose another point in the direction of the steepest slope. Continue until rate of improvement is within tolerance and/or gradient is zero. Much faster but suitable only in certain types of objectives (eg MV)
- Interior Point (Matlab) (2)
- GRG Non linear (Excel) (Generalised Reduced Gradient). (3)
Challenges with Optimisation
- Multiple local optima - optimisers get stuck, especially for complex utility functions. (MV has a single local max). eg optimising for tax, utility, ESG etc.
- Constraints restrict the solutions space (sometimes greatly - eg long only, non negative weights
Issues in Optimisation
- Multiple Solutions β optimisers stop when the βimprovementβ is within some small tolerance, this different starting points can results in slightly different solutions
- Multi-collinearity - this has many names, ill-conditioning, non-positive definite, non-invertibility
the issue arises when one asset is quite similar to linear combination of other assets
Multi Collinearity
- If every asset has the same return and variance expectation, the mean-variance objective is flat
- Multicollinearity can be a major problem when there are lots of assets, such as individual equities
Measurement Issues and Statistical Nuances
Types of Errors (3)
- Standard Error β Standard Deviation
- Estimation Error
- Specification Error = Model Risk
Measurement Issues and Statistical Nuances
High level, 3
- Types of Errors
- Variation due to fidelity, frequency, horizon (Measurement of risk using daily data vs monthly data often ends up with quite different outcomes)
- Realised vs Forecast measurements (ie exp post vs ex ante)
Turner and Hensel paper - testing variances and correlations
- Sample means for stocks and bonds are statistically indistinguishable from each other
- Sample variances are so large that detection of a differences in means would be extremely difficult
- There is some evidence of variance differences, but it likely due to economic & industry factors
- Correlations are indistinguishable from each other
Shrinkage Estimators
Shrinking a naive estimate towards a base value yields a better estimate.
1. Equation:
Shrinking a naive estimate towards a base value yields a better estimate.
1. Equation: π₯Μ Μ=πΏπ₯Μ+(1βπΏ)π΅ for some 0β€πΏβ€1 (usually)
π₯ Μ = central return of market
π₯ Μ = central return of another
B = base value (usually over a global mean)
Shrinkate Estimators
Shrinking a naive estimate towards a base value yields a better estimate.
1. Examples:
- Examples:
- Shrinking correlations towards 0
- Shrinking forecast returns towards their grand mean (James Stein)
- Using n-1 instead of n to compute sample variance: (π₯βπ₯ Μ )^2/(πβ1)
Taxation
why is tax difficult
- Income (e.g. dividends, coupons) is payable regularly (quarterly)
- Capital gains are payable annual after netting off losses
- The βproblemβ is path dependent β the timing of gains greatly affects after tax fund value
- Tax rates vary by client β consideration of tax in comingled products is especially difficult
Taxation
After-tax benchmarks are either (2)
Tax is important:
- Cheap with simplistic assumptions, e.g. assumed rates of turnover β 20% in FTSE ASFA Series
- Expensive & highly tailored β some superfunds subscribe to tax benchmarks incorporating their precise starting tax lots
Tax is important: eg Australian franking credits add about 1% extra to returns
Factor/Risk Models: Ex Ante Forecasts
Why is there noise in returns
- factor models decompose stock returns into a small number of common factors
- Factors are expected to covary together. The components of stock return that is not explained by the factors is the idiosyncratic or stock specific component
Factor/Risk Models: Ex Ante Forecasts
Three types of risk models
- Fundamental models include country-industry-currency & styles
- Statistical/Stochastic models identify the combination of stocks that explains the variability of the data β¦ no linkage to real world characteristics, a purely data driven decomposition
- Macro-economic model include macro factors: e.g. economic growth, inflation, commodity prices, credit spreads, fx
Define
- Heteroskedasticity
- Autocorrelation
- Heteroskedasticity: β Variances and covariances may be changing over time. Should not blindly assume historical estimates are best to forecast future.
- Autocorrelation: assumption is that time periods are independent of each other. However returns from one month to the next can be correlated with each other
3 types of risk models
β Fundamental models include country-industry-currency & styles
β Statistical/Stochastic models identify the combination of stocks that explains the variability of the data β¦ no linkage to real world characteristics, a purely data driven decomposition
β Macro-economic model include macro factors: e.g. economic growth, inflation, commodity prices, credit spreads, fx
What impacts does autocorrelation have on portfolio analysis
β Autocorrelation: assumption is that time periods are independent of each other. However returns from one month to the next can be correlated with each other.
β Estimating volatility using different units of time, eg compare 1 month units to 6 month units. Ratio of variances will not always be the same as the basic time period; eg 1:6
β’ Ratio should hold if independent
β’ If ratio is higher, positive autocorrelation. If lower, negative autocorrelation
β If all monthly data is independent; then unconditional estimate of volatility is constant. If the value of the estimate changes as the measurement period changes; there is a possibility of autocorrelation.
β Same as assessing correlation between two different time series, except that the same time series is used twice; once in original form and once lagged by 1 or more time periods
β Could be a momentum factor
Summary of Reading (Forecasting Covariances/Chan, Karceski, Lakonishok)
- Comment on Factors re min var
- Comment on optimisation
- Comment on TE (another slide)
Factors (min variance)
β A few factors capture general covariance structure β generally market, size and book to market. Expanding to a number of factors does not necessarily improve ability to predict covariance
β Min Var helps with risk control. Biggest benefits arise from reducing exposure to market (ie lower beta). Therefore utility stocks favoured.
Optimisation
β Portfolio optimisation helps for risk control; A 3 factor model is adequate for selecting the MV portfolio
Forecasting Covariances/Chan, Karceski, Lakonishok)
- Comment on Factors re min var (other slide)
- Comment on optimisation (other slide)
- Comment on TE
Tracking Error
β Practical application: TE. TE focus for global min variance portfolios or global min TE.
β For TE; impact of market factor is diminished
β TE criterion: larger differences across models. More factors are necessary when the objective is to minimise TE vol
β Sizeable reduction in TE volatility through optimisation highlights the importance of optimisation.
β Better to match a portfolio to attributes like size and book to market ratio.