Ch 2 Measurement Issues and Implications

Question 1

Utility Theory - measuring risk aversion
Absolute Risk Aversion (ARA) =
Relative Risk Aversion (RRA) =

Answer 1

Absolute Risk Aversion (ARA) = −(𝑈′′(𝑊))/(𝑈′(𝑊)) (Coefficient of ARA in notes)
Relative Risk Aversion (RRA) = W×𝐴𝑅𝐴 (Coefficient of RRA in notes)
Note the division by the first derivative to make them level invariant and negation is used for interpretation convenience

Question 2

Interpretation:

1. ARA increasing ->
2. ARA decreasing -> more ‘risky’ dollars are held as wealth increases
3. ARA constant ->
4. RRA increasing ->
5. RRA decreasing ->
6. RRA constant ->

Answer 2

Interpretation:

1. ARA increasing -> less ‘risky’ dollars are held as wealth increases
2. ARA decreasing -> more ‘risky’ dollars are held as wealth increases
3. ARA constant -> risk dollars remains unchanged, even as wealth increases … no wealth effect
4. RRA increasing -> less ‘risky’ percentage …
5. RRA decreasing -> more ‘risky’ percentage …
6. RRA constant -> ‘risky’ percentage remains unchanged … no change in bet percentages regardless of wealth

Question 3

Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs

Quadratic

• Equation
• ARA
• RRA

Answer 3

Important Wealth Utility Functions

Quadratic

• Equation 𝑎𝑊−𝑏/2 𝑊^2
• ARA 𝑏/(𝑎−𝑏𝑊)≈−1/𝑊
• RRA −1+𝑎/(𝑎−𝑏𝑊)≈−1

Question 4

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

Answer 4

Important Wealth Utility Functions

Exponential

• Equation −𝑒^(−𝑎𝑊)
• ARA −𝑎
• RRA −𝑎𝑊

Question 5

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

Answer 5

Important Wealth Utility Functions

Power

• Equation (𝑊^(1−𝑘)−1)/(1−𝑘)
• ARA 𝑘⁄𝑊
• RRA 𝑘

Question 6

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

Answer 6

Important Wealth Utility Functions

Log
- Equation Important Wealth Utility Functions

Power

• Equation 〖𝑙𝑜𝑔〗_𝑒 (𝑊)
• ARA 1⁄𝑊
• RRA 1

Question 7

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

Answer 7

Important Wealth Utility Functions (do not need to know equations for exam; do need to know how to make tradeoffs

Prospect W> benchmark

• Equation ((𝑊−𝑏𝑚𝑘)^𝑎)/𝑎
• ARA (1−𝑎)/(𝑊−𝑏𝑚𝑘)
• RRA 𝑊×(1−𝑎)/(𝑊−𝑏𝑚𝑘)

Question 8

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

Answer 8

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 −𝑘×(𝑏𝑚𝑘−𝑊)^𝑏/𝑏
• ARA (1−𝑏)/(𝑊−𝑏𝑚𝑘)
• RRA 𝑊×(1−𝑏)/(𝑊−𝑏𝑚𝑘)

Question 9

How Utility Theory relates to Portfolio COnstruction

Answer 9

1. Markowitz mean-variance problem
   max 𝑈= 𝛼′ 𝑤−𝜆/2 𝑤′Σw
   is described as a ‘quadratic problem’ in optimisation speak
2. It’s ‘consistent’ with quadratic utility in utility theory speak; and consistent with other utility functions.
3. The reason why there isn’t 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 … they are related but not identical
4. 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.

Question 10

Why isn’t htere a direct link between the mean variance problem and utility

Answer 10

1. Even though MV problem is described as ‘quadratic’, and there is ‘quadratic’ 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
2. They are therefore consistent in only a few situations

Question 11

In what situations is Markowitz MV consistent with Utility Theory

Answer 11

1. Quadratic Utility in all return distributions, but as wealth rises the utility roles over and less wealth perversely becomes more attractive
2. Exponential Utility but only when returns are normally distributed, but relative risk aversion is increasing, implying wealth is never sated
3. Power Utility but only when returns are lognormally distributed

Question 12

If MV is not completely consistent with Utility Theory, why is it so ingrained

Answer 12

1. Only two things are needed: the expected returns and the expected variance
2. A ‘Quadratic Program’ is computationally simple to solve
3. The 80/20 rule – the 80% solution is easy, the next 20% consumes considerably more time & effort

Question 13

Optimisation

• Grid Search
• Gradient Ascent
• Interior Point (Matlab)
• GRG Non linear (Excel)

Answer 13

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)

Question 14

Challenges with Optimisation

Answer 14

1. Multiple local optima - optimisers get stuck, especially for complex utility functions. (MV has a single local max). eg optimising for tax, utility, ESG etc.
2. Constraints restrict the solutions space (sometimes greatly - eg long only, non negative weights

Question 15

Issues in Optimisation

Answer 15

1. Multiple Solutions – optimisers stop when the ‘improvement’ is within some small tolerance, this different starting points can results in slightly different solutions
2. 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

Question 16

Multi Collinearity

Answer 16

• 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

Question 17

Measurement Issues and Statistical Nuances

Types of Errors (3)

Answer 17

1. Standard Error ≠ Standard Deviation
2. Estimation Error
3. Specification Error = Model Risk

Question 18

Measurement Issues and Statistical Nuances

High level, 3

Answer 18

1. Types of Errors
2. Variation due to fidelity, frequency, horizon (Measurement of risk using daily data vs monthly data often ends up with quite different outcomes)
3. Realised vs Forecast measurements (ie exp post vs ex ante)

Question 19

Turner and Hensel paper - testing variances and correlations

Answer 19

1. Sample means for stocks and bonds are statistically indistinguishable from each other
2. Sample variances are so large that detection of a differences in means would be extremely difficult
3. There is some evidence of variance differences, but it likely due to economic & industry factors
4. Correlations are indistinguishable from each other

Question 20

Shrinkage Estimators

Shrinking a naive estimate towards a base value yields a better estimate.
1. Equation:

Answer 20

Shrinking a naive estimate towards a base value yields a better estimate.
1. Equation: 𝑥̂ ̂=𝛿𝑥̂+(1−𝛿)𝐵 for some 0≤𝛿≤1 (usually)
𝑥 ̂ = central return of market
𝑥 ̂ = central return of another
B = base value (usually over a global mean)

Question 21

Shrinkate Estimators

Shrinking a naive estimate towards a base value yields a better estimate.
1. Examples:

Answer 21

2. Examples:
   - Shrinking correlations towards 0
   - Shrinking forecast returns towards their grand mean (James Stein)
   - Using n-1 instead of n to compute sample variance: (𝑥−𝑥 ̅ )^2/(𝑛−1)

Question 22

Taxation

why is tax difficult

Answer 22

1. Income (e.g. dividends, coupons) is payable regularly (quarterly)
2. Capital gains are payable annual after netting off losses
3. The ‘problem’ is path dependent – the timing of gains greatly affects after tax fund value
4. Tax rates vary by client – consideration of tax in comingled products is especially difficult

Question 23

Taxation

After-tax benchmarks are either (2)

Tax is important:

Answer 23

1. Cheap with simplistic assumptions, e.g. assumed rates of turnover – 20% in FTSE ASFA Series
2. Expensive & highly tailored – 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

Question 24

Factor/Risk Models: Ex Ante Forecasts

Why is there noise in returns

Answer 24

1. factor models decompose stock returns into a small number of common factors
2. 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

Question 25

Factor/Risk Models: Ex Ante Forecasts

Three types of risk models

Answer 25

1. Fundamental models include country-industry-currency & styles
2. Statistical/Stochastic models identify the combination of stocks that explains the variability of the data … no linkage to real world characteristics, a purely data driven decomposition
3. Macro-economic model include macro factors: e.g. economic growth, inflation, commodity prices, credit spreads, fx

Question 26

Define

1. Heteroskedasticity
2. Autocorrelation

Answer 26

1. Heteroskedasticity: ● Variances and covariances may be changing over time. Should not blindly assume historical estimates are best to forecast future.
2. 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

Question 27

3 types of risk models

Answer 27

● Fundamental models include country-industry-currency & styles
● Statistical/Stochastic models identify the combination of stocks that explains the variability of the data … 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

Question 28

What impacts does autocorrelation have on portfolio analysis

Answer 28

● 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.
● 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
• Ratio should hold if independent
• If ratio is higher, positive autocorrelation. If lower, negative autocorrelation
● 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.
● 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
● Could be a momentum factor

Question 29

Summary of Reading (Forecasting Covariances/Chan, Karceski, Lakonishok)

1. Comment on Factors re min var
2. Comment on optimisation
3. Comment on TE (another slide)

Answer 29

Factors (min variance)
● A few factors capture general covariance structure – generally market, size and book to market. Expanding to a number of factors does not necessarily improve ability to predict covariance
● Min Var helps with risk control. Biggest benefits arise from reducing exposure to market (ie lower beta). Therefore utility stocks favoured.

Optimisation
● Portfolio optimisation helps for risk control; A 3 factor model is adequate for selecting the MV portfolio

Question 30

Forecasting Covariances/Chan, Karceski, Lakonishok)

1. Comment on Factors re min var (other slide)
2. Comment on optimisation (other slide)
3. Comment on TE

Answer 30

Tracking Error
● Practical application: TE. TE focus for global min variance portfolios or global min TE.
● For TE; impact of market factor is diminished
● TE criterion: larger differences across models. More factors are necessary when the objective is to minimise TE vol
● Sizeable reduction in TE volatility through optimisation highlights the importance of optimisation.
● Better to match a portfolio to attributes like size and book to market ratio.