Practical issues in applying modern portfolio theory and mean-variance optimization (mainly Week 3) Flashcards
3 approaches to asset allocation
–Static strategic asset allocation (SAA)
–Dynamic asset allocation (DSAA / strategic tilting)
–Tactical asset allocation (TAA)
–Static strategic asset allocation (SAA)
Assume what we have now is going to last very long term. think of long term investment horizon in average conditions
- unconditional
- don’t assume that australian economy is going to outperform the rest of the world in next 3-5 years. Australia equity market is going to have the same expected return as world equity. Don’t consider dynamic nature of financial market or individual circumstances.
- focus on information i have right now. do not consider investment opportunities The average expectation right now. Given the set of info and what you understand of financial market and economy, we construct a portfolio this way to help client to achieve objectives
Dynamic asset allocation
–Tactical asset allocation (TAA)
features
last few days or months, dynamic and static are long term
conditional; portfolio manager makes positive excess return in short period of time taking adv of temporary market inefficiencies
usually not used alone. it is used together with others to temporarily boost return
Dynamic asset allocation
features
they have target date funds or dynamic funds
- Conditional - e.g. asset risk/return, investor horizon, preferences
- Investors and investment opportunities can evolve through time
Modern Portfolio Theory
Perfect capital market assumptions:
frictionless and efficient
single period. investment horizon is not defined. don’t care about next period
rational and risk averse investors - only care about risk and return trade off. they are price takers. have no influence on market price. objective to maximise utility.
with homogeneous expectations of assets expected return and variance, covariance metrics
all assets are liquid, tradable and accessible by all investors
same access to market price and information.
•Separation pertains. separate risky and risk free assets
Theoretical issues with traditional MPT
- Portfolio optimization across all available assets
In practice, don’t throw thousands of investable assets
in mean variance optimisation model to let it calculate the market portfolio. we use proxies
fixed income, property, asset classes as a close division of assets available.
if we do run mean variance optimisation problem using MDT, it only returns a very close asset allocation e.g. equity, fixed income and other assets. no instruction on how much you are going to hold in BHP, ANZ etc
Theoretical issues with traditional MPT
- Portfolio optimization across all available assets
Not necessarily feasible (curse of dimensionality); multi-factor models may be more effective
if you consider stock index in today’s financial market, all of the listed australian run mathetical problem of having this optical trade off of return and risk. not a trivial problem.
have to include government bonds, derivatives, properties, gold and other investable assets all into the calculation. will impose great pressure on calculation ability
Theoretical issues with traditional MPT
3.
Assumes risk aversion is only investor difference that matters
(‘separation’ => investors hold combination of M and Rf)
Other investor differences matter, e.g. objectives, liabilities, investment horizon, opportunities, costs, taxes
(=> separation unlikely to hold in practice, i.e. different portfolios may be optimal)
Theoretical issues with traditional MPT
- Single Period model
Real world is multi-period, with stochastically changing investment opportunities. does not address dynamic nature of financial market and investors circumstances
Theoretical issues with traditional MPT
- Investment horizon is undefined
yet in real world it matters. how to mitigate this problem
extending data frequency. instead of using weekly and monthly. use quarterly and yearly data. correlation between quaterly and yearly is very low.
some asset classes is zero; some is high
Theoretical issues with traditional MPT
- Investment horizon is undefined
Yet it matters in the real world, especially if returns are not iid (identically and independently distributed)
identically distributed = following same probability distribution. from same distribution
independent = each return of this period is not correlated with return of previous and future periods
Long term investment horizon. long term view of asset return and std.
How portfolios are built in practice
Analysis tool
computer program does not understand
regret, worry about medical circumstances, worries about retirement, increasing utility.
mathetmical process can provide us a tool. final decision making is made by portfolio manager given his/her understanding of the investor’s circumstances.
How portfolios are built in practice
•Real-world influences matter: liquidity, fees & costs, business risk, resources, decision-making structures
telstra has own super fund management team
call special meeting of board to get things done over few weeks
others have external consultants and limited resources in fund to exploit all investment opportunities.when consultant propose and calls for meeting with board, everywhere in australia. call for special meeting, some cannot come. without vote of directors cannot go ahead with new investment opp. what is directors have questions about risks, portfolio manager speaks to consultant. this process can go on for weeks then investment opportunity is gone
How portfolios are built in practice
•Widespread pursuit of active returns (‘alpha’)
popular way to construct portfolio and to understand performance especially outperformance.
active managers construct a baseline portfolio. e.g. active australian equity has asset allocation in difference sectors banking 25% resource 25%. generate alpha through active return, deviate from industry average. have baseline portfolio similar to asset allocation of peers.
setlight portfolio. small, controlled, underweight positions in market you believe may generate excess return. from small cap start up high tech companies, start up pharamaceutical
How portfolios are built in practice
•Tiered approach
Tiered to portfolio construction (division into buckets: portfolio => asset classes => asset managers => assets)
large superannuation funds. e.g. uni super, retail super funds. just because of size under management, no way to plug in all expected return, standard variation and covariance metrics in the mean variance optmisation model.
asset allocation into equity alone. will have equity managers of australia equity, world quity. geographic division of equity portfolio, e.g. japanese equity, . emerging market equity
How portfolios are built in practice
industry norms
for similar funds australian equity funds hold large proportion of assets in australian banks because within australian market top 100 companies are a few finance companies.
asset allocation will be very similar to your peers. want to beat peers, have to come up with ideas to generate excess return and certain level of tracking error deviating from industry peers.
How portfolios are built in practice
legacy portfolio
super funds have a mandate determined by board. they all agree date. have certain asset allocation in this way. do not invest in coal mining or tobacco companies. social responsibilities in investment
have context of existing porfolio and mandate in place.
Mathetmatical function does not consider
qualitative risk e.g. liquidity risk and investor circustances
One way to make mean-variance useful
add constraints e.g. if illiquidity is risk, put illquid assets as a constraint
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
modified Black litterman approach second type
imposing probability function on market equilibrium and investor’s views are hard to apply so practitioners come up with two methods
First: extracts spirits from market equilibrium model. assume that what we observe the average asset allocation of all my peers is the efficient portfolio and that is the most efficient asset allocation of risky portfolio. from there i work backwards of mean variation optimisation process and let model decide itself the expected return, covaraince, STD of all assets within the portfolio so that the portfolio i observe in the market is the most efficient one. model decides what set of ER, STD so this asset allocation is optimal.
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
modified Black litterman approach one type
Use the spirit CAPM model of the single market factor.
Impose on investors own view in the E(R) of each asset class e.g. i believe Australian equity should have a spread over Australian cash of 5% and i have expected return and all other classes this is based on investor’s views. Not probability function.
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
modified Black litterman approach gets over
hypersensitivity of mean-variance optimisation
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
Constrain risk relative to benchmark
many managers are given a capitalization-weighted index as performance benchmark, it is relevant to optimize the portfolio construction by constraining the tracking error, or standard deviation of the portfolio excess returns.
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
Black litterman assume
asset returns do deviate from its equilibrium level and imbalances in market will push them back.
Two distinct sources of information about future excess returns – investor views and market equilibrium; both sources of information are uncertain and are best expressed as probability distributions.
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
Black litterman approach
asset allocation should not be solved in CAPM is single market equilibrium.
expected asset returns reflects combination of market equilibrium model like CAPM and investor’s expectations. both are represented by probability function.
Implied views’ in text: Black-Litterman without investor expectations
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
Sensitivity analysis is another tool to make mean-variance model useful
What if i change some key inputs into mean-variance optimisation problem. e.g. small change and observe dramatic change in portfolio performance, be careful in allocating assets into these assets because they wil have significant impact on the performance
A) M-V optimization - in practice
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
- Impose weighting constraints
- Constrain risk relative to benchmark, i.e. TE (tracking error)
- Use approach of Black & Litterman (1989)
- Use re-sampling to generate the distribution of optimal portfolios
If M-V optimization is going to be used, a method will be needed to ensure sensible portfolio weightings:
- Use re-sampling to generate the distribution of optimal portfolios
run mean-variance optimisation given constraints, tracking error so i can produce reasonable asset allocation thousands of times.
Generate distribution of asset weights. each asset is going to have a distribution. better ideas how that asset allocation in Australian equity can change in this distribution
M-V as portfolio analysis workhorse
5.Be aware of parameter uncertainty
be aware of parameter uncertainty
M-V as portfolio analysis workhorse
4.Investigate impact of changes, e.g. alter the asset allocation, add new assets, different inputs, etc
Consider other implementation issues
requires client to sell all their properties including their home. do they agree?
all of this practical implementation issues should be considered before you make final decision of asset allocation.
M-V as portfolio analysis workhorse
4.Investigate impact of changes, e.g. alter the asset allocation, add new assets, different inputs, etc
having assset allocation and portfolio outcome is not enough.
run sensitive analysis. key inputs into model. E(r), STD or covariance assumptions in key assets. if beating market index or benchmark portfolio, introduce an overweight, a strategic tilting into your own portfolio to generate excess return.
M-V as portfolio analysis workhorse
- Estimate portfolio outcomes: E(Rp), σp, Sharpe ratio, shortfall
Data-based
run mean-variance optimisation and given you have weighting constrained and tracking error in place to generate set of asset weights, portfolio outcome, STD and other measures of risk.
- Examine hypothetical historical performance
- Simulate by ‘bootstrapping’ from the data
M-V as portfolio analysis workhorse
- Represent the distribution of asset returns
forcast E(R), standard deviation and covariance matrix of all the possible asset classes in the portfolio
a)Data-based (‘non-parametric’)
- do not impose distribution assumptions. normal distribution and constant return
- use historical available data
b)Model-based (‘parametric’; E(R)s and covariance matrix)
- to forecast E(r) and covariance matrix
- requires strong assumptions of asset returns. for some assets these assumptions do not hold
- Australian cash are going to be highly correlated
M-V as portfolio analysis workhorse
1.Nominate a baseline portfolio
asset allocation of average holding of peers of benchmark.
M-V as portfolio analysis workhorse
- Nominate a baseline portfolio
- Represent the distribution of asset returns
a) Data-based (‘non-parametric’)
b) Model-based (‘parametric’; E(R)s and covariance matrix) - Estimate portfolio outcomes: E(Rp), σp, Sharpe ratio, shortfall
a) Data-based:
- Examine hypothetical historical performance
- Simulate by ‘bootstrapping’ from the data
b)Model-based:
- Directly estimate portfolio statistics (single period only)
- Simulate by drawing from the model (facilitates multiple periods)
- Investigate impact of changes, e.g. alter the asset allocation, add new assets, different inputs, etc
- Be aware of parameter uncertainty
Applying M-V analysis: Considerations
Missing variables e.g. liquidity, intra-period cash flows
quantifiable measure of risk e.g. probability of lost, STD, we have way to evaluate to finetune.
other risks that are important for investors to achieve their objectives e.g. business risk, own career, property. these should be considered before you make your final decision of asset allocation.
M-V framework can provide the basis for a portfolio analysis workhorse:
–Generate basic measures of portfolio risk and return
–Conduct sensitivity analysis, e.g. asset weight changes
–Basis for simulations
mean-variance model mechanically allocates
allocates funds to achieve a marginally higher expected return at a given level portfolio variance.
It doesn’t understand diversification across asset classes, let alone fundamental risk factors.
It doesn’t consider investors’ specific circumstances, for instance accessibility to financial products/market sectors.
mean-variance model
hypersensitivity
If your estimation of expected return of one asset class is biased,
If your estimation of expected return of one asset class is biased, that one “trivial” error, optimization model can amplify that one parameter error and change asset allocation completely.
Mean-variance optimization tend to overweight securities of large estimated returns, negative correlations and small standard deviations.
How much of portfolio return variations can be explained by asset allocation decisions ON AVERAGE?
90%
CFS Freedom Fund (hyperthetical) has an asset allocation target of 80-20 mix of growth assets and fixed income. The manager is allowed a 5% band either way around the target asset allocation. Occasionally the actual asset allocation may deviate from the defined band of asset allocation. If appproved by the board, the manager may take on long or short positions in some assets over a short period of time to generate excess returns. Which one of the following best describe the asset allocation apporach of the fund?
static strategic asset allocation overlayed with tactical asset allocation
An optimizer is used to generate mean-variance optimal portfolios. Some of the portfolios contain weightings of over 100% in international fixed income, the majority of which is offset by large short positions in local fixed income. What is the most likely source of this result?
international fixed income has been given a higher expected return, while both asset classes make a similar contribution to total portfolio risk.
How portfolios are built in practice? Choose all correct statements.
Tiered approach to portfolio construction along the dimensions of: asset classes, sub-classes, asset managers, asset selection
Industry “norms”, eg. average asset allocation target of all similar funds
Legacy portfolio as starting point with marginal changes to improve performance or other particular objectives
Considerations of liquidity need, tax, cost, asset availablity, fund governance may all play important role in the asset allocation decision.
Possible measures to mitigate the hyper-sensitivity problem of Optimizer may include
weighting constraint, constraint risk relative to a benchmark (so that the portfolio weights don’t wander away too much from the peers), or approach that incorporates investor’s expectations.
Problems with MPT
Single Period Model
Does not incorporate investor’s objectives. does not incorporate dynamics of investment opportunities
CFS Freedom Fund (hyperthetical) has an asset allocation target of 80-20 mix of growth assets and fixed income. The manager is allowed a 5% band either way around the target asset allocation. Occasionally the actual asset allocation may deviate from the defined band of asset allocation. If appproved by the board, the manager may take on long or short positions in some assets over a short period of time to generate excess returns. Which one of the following best describe the asset allocation apporach of the fund
band: deviate from this band to take advantage of temporary market inefficiences
static strategic asset allocation overlayed with tactical asset allocation.
static strategi asset allocation is for long term investment horizon. tactical is for short period.
An optimizer is used to generate mean-variance optimal portfolios. Some of the portfolios contain weightings of over 100% in international fixed income, the majority of which is offset by large short positions in local fixed income. What is the most likely source of this result?
International fixed income has been given a higher expected return, while both asset classes make a similar contribution to total portfolio risk.
optimisier is a mathematical process chases highest level of portfolio return given level of risk.
How portfolios are built in practice?
B
Tiered approach to portfolio construction along the dimensions of: asset classes, sub-classes, asset managers, asset selection. “Fill the bucket”
C
Industry “norms”, eg. average asset allocation target of all similar funds. esp for managers given a benchmark or performance evaluated against a performance
D
Legacy portfolio as starting point with marginal changes to improve performance or other particular objectives. take over portfolio, mandate already there. certain assets you can invest in, band.
E
Considerations of liquidity need, tax, cost, asset availablity, fund governance may all play important role in the asset allocation decision.
How to use mean-variance optimisation?
input historical returns of all asset classes and run “solver” in Excel so that the optimal portfoio weights are generated to produce the highest portfolio return or minimum portfolio variance.
Given historial returns….
run mean-variance optimisatoin to generate aset weights
How does shifting the time horizon to 5 years under the bootstrap influence the risk measures?
Use of bootstrap methods can be inappropriate when returns are NOT independent through time.
Serial correlation of 0.3 for the investor’s portfolio suggests this is indeed the case. Implication is that the bootstrap in particular may under-estimate the risk of the investor’s portfolio over multiple years, i.e. the 5 year portfolio risk may be greater than the bootstrap implie
Below are the historical means. Current AC yields are around 1%-2%, so how can 7% returns on these assets be expected going forward? Also, it is probably unreasonable to expect WE to earn 1- 2% less than AE and only slightly more than WFI going forward
Historical mean returns can be questionable as measures for expected returns looking forward b/c
Risk associated with events not captured in the data history
About other forms of risk: illiquidity, business risk, governance failures, etc
How the portfolio meets the needs of the end-investor, e.g. is it utility maximizing? what if the investor has other assets? where is the investor in their life cycle?
Should investors attempt to eliminate risk, at least as far as possible?
Investor circumstances may matter. Does the investor have some comparative advantage (disadvantage) that makes places them in a better (worse) position to accept certain types of risk with a view to increasing return?
For instance, long-term investors are better placed to buy illiquid assets, and capture ‘illiquidity premiums’
Scrutinize the unconstrained weightings. What is the optimizer doing?
Optimizer willingly goes short
. − Starts with relatively balanced minimum variance portfolio of mainly defensive asset classes.
− To secure higher returns, the optimizer increasingly goes long AE and WFI, funded by shorting all other assets classes (especially AFI).
− This captures differential returns for asset classes with similar risk profiles. For instance, it favors AE as the highest returning ‘risk’ asset; and goes long WFI/short AFI. Weightings are very impractical.
Scrutinize the long-only weightings. What is the optimizer doing?
Again starts with relatively balanced minimum variance portfolio of mainly defensive asset classes.
− Optimizer captures additional returns by moving towards WFI and then AE. These are the defensive and growth assets with the highest (historic) returns.
- Also, WFI is favored over AFI due to a lower correlation with AE plus lower standard deviation. − Other assets such as WE and LP barely get a look in. Portfolio is imbalanced.
− Solver can’t satisfy highest return requirements once return > highest asset class return (AE) as unable to go short. Stays at 100% equity investment from return = minimum variance return + 0.25% onwards.
The inputs to mean variance optimization are
the expected returns, variances and covariances of each asset class.
Optimization using adjusted returns suggested weightings now make sense for most part.
except
Direct property is the key exception. This is due to the fact that its contribution to portfolio risk is understated given smoothed and lagged (appraisal-based) returns. The 10% constraint is imposed to avoid DP emerging as a dominant asset, which is clearly counter-factual.
Without the 10% constraint, direct property is 32.7% of the minimum variance portfolio
Optimization using adjusted returns
suggested weightings now make sense for most part. This arises because the expected returns are in better alignment with the contribution of assets to portfolio risk
NB: WFI is still being favored over AFI, at least at lower levels of returns. Standard deviation gives the clue why: WFI 3.44%, AFI 4.30% pa
Usefulness of analysis
The aim is to use the ‘Solver’ add-in in Excel to estimate the mean-variance efficient frontier of portfolios under the three conditions listed below.
(a) Unconstrained portfolios i.e. (short positions are permitted), with inputs based on historic data without adjustment.
(b) Long-only portfolios, with inputs based on historic data without adjustment.
(c) Unconstrained portfolios, with expected returns imposed, i,e. the mean return for each asset class is to be adjusted towards certain target compound expected returns. The target returns and associated monthly means for ln(1+R) can be found in the worksheet named ‘E(R) adjust’.
Generated weightings are sensitive to assumptions – assumed returns and the overall set-up (e.g. constraints) – and hence need to be used with caution.
Sample means give highly unreliable results. The output based on adjusted returns makes more sense, and might be (cautiously) used as a reference point when building portfolios.
Problems with using traditional mean-variance optimization to set asset allocation in practice
- Investment is typically a multi-period, dynamic process, not a one-shot event
A single period approach is simplistic. Investment typically continues beyond any assumed holding period. Further, setting a defined holding period ignores any opportunities to respond to changes in the interim.
Problems with using traditional mean-variance optimization to set asset allocation in practice
- Hyper-sensitivity to uncertain inputs
- Objectives
- Key risks may be missing from the analysis
- Investment is typically a multi-period, dynamic process, not a one-shot event
things that don’t invalidate use of optimizers, but lead to difficulties in doing so
asset forecasting
Generating forecasts is a non-trivial exercise. Only limited guidance or distorted data may be available for some assets. Costs like taxes and transaction costs matter, but are tricky to incorporate. The output relies heavily on the inputs (see point 1), but good inputs are hard to generate.
things that don’t invalidate use of optimizers, but lead to difficulties in doing so
Deciding which portfolio you want on the efficient frontier
The trade-off risk and return is often difficult to characterize, i.e. determining risk tolerance, and optimizing ‘utility’. The MPT notion that all investors will hold the ‘tangent portfolio’ with maximum Sharpe ratio assume the existence of a risk-free asset and the ability to borrow (i.e. leverage the portfolio). This assumption is tenuous in practice.
things that don’t invalidate use of optimizers, but lead to difficulties in doing so
investment horizon
Need to settle on a specific investment horizon. As an essentially single-period model, a choice of specific investment horizon needs to be made to do the analysis, e.g. quarterly, yearly, multiple years, etc. This choice is not straightforward, and can often be arbitrary.
Further, the choice matters to the results if returns are not iid1
Measurement of risk and diversification potential using historically-based statistics like standard deviation and correlation has a whole range of shortcomings.
Correlation and standard deviations can be distorted downwards
Correlation and standard deviations can be distorted downwards when prices adjust with a lag due to aspects like thin trading, appraisal valuations.
In all these cases, diversification benefits can be over-stated and risk potentially under-stated.
Measurement of risk and diversification potential using historically-based statistics like standard deviation and correlation has a whole range of shortcomings.
measurement period
Historically-based statistics represent what happened on average over a particular measurement period. Both standard deviation and correlation can be unstable over time
Measurement of risk and diversification potential using historically-based statistics like standard deviation and correlation has a whole range of shortcomings.
normal distribution
Asset returns are not well described by a single normal distribution. It is well known that returns have fat tails, and are sometimes skewed.
portfolio optimisation based on historical measures is not reliable because T
They don’t reveal the full story.
Supplementing with an alternative approach can be very useful, if not essential.
strategic asset allocation
fixed
fixed strategic asset allocation would be consistent with the constant investment opportunities and risk rolerances
strategic asset allocation
based on long-term goals. e.g. need asset allocation today for the next 30 years.
tactical asset allocation respnds to
short-term changes in investment opportunites
how do some active allocation managers use tactical asset allocation?
they may define a band around strategic weights within which weights my be set in the short term but never deviate by so much so that tactical bets overwhelm the strategic allocation
e.g. long-term average target weight of 50% stocks with a plus minus 10 percent band yielding a 40-60% range would incorporate both strategic and tactical allocations
when will allocations be set tactically
investors who frequently adjust their exposure to stocks, bonds and cash i.e. market timer
they seek to profit from short-term market movements in the market, expect to change their asset weights in the near future and may not worry much about the long-term implications of their average weights
dynamic asset allocation is driven by
changes in risk tolerance typically inducated by cumulative performance relative to investment goals or an approaching investment horizon
Why are the results from the optimiser so sensitive
Small changes in ER , optimiser will switch from one combination to another
weights on assets in the portfolio may be unstable or take extreme values
go long in one and short in the other
optimiser does not see this as a problem because it does not attach any significance to wehther a particular asset is held long, sold short or omitted.
real problem with optimiser
optimiser’s sensitivity to the inputs
we do not know the true values of the necessary model inputs: ER, variances and covariances
if we knew, then we would be confident that optimiser is giving accurate advice
classic MV optimisatoin assumes that
investor prefers a portfolio of securities that offers maximum ER for a given level of risk
Potential benefits of MV optimisers
- Convenient framework to integrate simple but important client constraints and objectives within portfolio
- Efficient use of investment info: designed to use info optimically while ad hoc weighting can be counterproductive
- Portfolio optimisers can process large amounts of info quickly, good for large institutional that needs to determine impact of new info on its portfolios quickly
MV optimisers overweight?
optimisers significantly overweight (underweight) securities that have large (small) ER, negative (positive) correlation) and small (high) variances
MV optimisers are
because
estimation error maximisions because risk and return are subject to estimation error
MV optimisation ignores fundamental factors including
liquidity i.e. percentage of company’s market capitalisation represented by portfolio holding
Asset Allocation With Respect to a Benchmark
How can this enhance MV optimisation
benchmark can significantly alter the characteristics of MV-optimal asset allocations. In general, benchmark asset allocation is strongly dependent on the economic characteristics of the liabilities.
It is consequently far less time-period dependent or unstable and appears to have substantially more practical investment value.
