Concepts Description
Introduction to asset allocation
Economic balance sheet 1. Economic balance sheet = financial assets and liabilities + nonfinancial assets and liabilities
2. Nonfinancial assets and liabilities = extended assets and liabilities
Examples :
‐ human capital : PV of expected earnings
‐ PV of pension income
‐ PV of expected intellectual properties royalties
‐ PV of expected underground resources
3. Life cycle balance fund : consider change in human capital and financial capital for individual investor
‐ Entering workforce → human capital >> financial capital → 85% equity ; 15% bonds (human capital has bond‐like characteris cs)
‐ Retirement → total wealth = financial capital → 50% quity, 50% bonds
Approaches to asset allocation 1. Asset‐only approaches : asset allocation decision 100% based on investor's assets
‐ Objectives : max the expected return per unit of risk
‐ Should consider investor constraints + risk tolerance
‐ Risk concept : asset class risk + create effective asset class combination
‐ Risk measure : standard deviation of portfolio, relative risk to a benchmark (tracking error), downside risk (VAR, semivariance, maximum drawdown)
2. Liability‐relative approaches : asset allocation decision based on funding liabilities
‐ Objectives : pay liabilities when it due
‐ Risk concept : (1) not having enough assets to pay liabilities when due; (2) volatility of contributions
‐ Risk measure : standard deviation of surplus
3. Goals‐based approaches : asset allocation for subportfolios, that help investor to achieve the lifestyle/aspirational financial objectives
‐ Need to specify : (1) type of CF needed, (2) time horizons, (3) evel of risk tolerance (probability of achieving certain goal)
‐ Risk concept : unable to achieve financial goals
‐ Portfolio risk = weighted sum of risk of each goal
Allocation by asset classes Asset classes : group of assets with similar characteristics
3 main super asset classes : (1) capital assets (bond, equity) ; (2) consumable / transformable assets (commodities) ; (3) store‐of‐value assets (currency, art)
Criteria to specify asset classes :
‐ Same asset class → similar a ributes (both descrip ve and sta s cal)
‐ Assets cannot be classified in more than 1 asset class
‐ Asset classes should cover all investable assets
‐ Asset classes should contain mostly liquid assets
Use of risk factors in asset Investor selects asset classes based on desired exposure to common risk factors (volatility, liquidity, inflation, interest rate, duration, FX, default)
allocation Factor‐based asset allocation : Risk factor may be overla across multiple asset classes → should focus on risk factor by iden fying risk factors + desired exposure to each
Limitation : cannot directly invest in all risk factors
Strategic asset allocation Strategic asset allocation : combine market expectations with investor's risk, return and constraints
Steps in select strategic asset allocation :
‐ Step 1 ‐ Determine objectives : how investor use these assets? What investor want to achieve? What is the liabilities/goals? How to measure the objectives?
‐ Step 2 ‐ Determine risk tolerance : What is the sensitivities to risk? How to measure risk?
‐ Step 3 ‐ Determine time horizon
‐ Step 4 ‐ Determine constraints : tax situation? Social, environmental or governance considerations? Legal, regulatory, political considerations?
‐ Step 5 ‐ Select aset allocation approach : asset‐only / liability‐relative / goals‐based
‐ Step 6 ‐ Specify asset classes
‐ Step 7 ‐ Develop potential asset allocation
‐ Step 8 ‐ Test the results of potential asset allocation : to see if results align with objectives and risk tolerance for chosen horizon
‐ Step 9 ‐ Repeat step 7 + 8 untill discover optimal asset allocation
Global market portfolio Global market portfolio :
‐ all available risky assets (global equity, global fixed income, global real estates, etc.)
‐ Most diversified portfolio → minimum diversifiable risk
‐ weight could be adjusted to meet specific investor objectives, constraints and desires
‐ some asset classes are not practical for individual investors (e.g.: residential real estate, private equity) → invest via proxy such as ETFs
,Passive / Active choice For weighting asset classes :
1. Tactical asset allocation : active management strategy, seek for additional risk and return (alpha) to take advantes of ST opportunities
‐ Deviation should be restricted by risk budgets / rebalancing
‐ Reasons for deviation : forecasted asset class valuation, business cycle state, stock price momentum
‐ Trade off between potential outperformance vs tracking error
‐ Limitation : additional trading + monitoring cost; additional capital gain taxes
2. Dynamic asset allocation : multiperiod view of investment horizon, where asset performance in one period affects required rate of return and acceptable level of risk in subsequent
periods
For selection within asset classes :
1. Passive management : investor insight / expectations does not impact portfolio composition
2. Active management : change investor insight / expectations → change porfolio composi on → to earn risk‐adjusted return > benchmark
Factors for selecting between active and passive return :
‐ availability of appropriate investments
‐ active management scalability (value added from each active decision)
‐ investor constraints when using passive approach
‐ investor belief in efficient markets → discourage use of ac ve management
‐ cost‐benefit tradeoff between additional costs vs. excess return
‐ tax status of investor
Risk budgeting Risk budgeting : specify how risk should be distributed among portfolio assets without regad to asset expected returns
Active risk budgeting : how much additional risk investor willing to take compared to benchmark
‐ for asset classes allocation : active risk is deterimined by strategic asset allocation vs. benchmark
‐ for selection within asset classes : active risk is determined by asset class selection vs. asset class benchmark
Rebalancing Benefits :
‐ maintain portfolio's risk and return characteristics as specified in IPS
‐ provide discipline
Drawbacks :
‐ related tax liability
‐ transaction costs
Rebalancing approaches Calendar rebalancing Percentage‐range rebalancing
Definition Rebalance the portfolio to its strategic allocation on predetermined, regular Manager set a tolerance band / corridors (±5%) that are considered optimal
basis (monthly, quarterly). Rebalancing frequency depends on portfolio for each asset class. If any asset class goes outside of that corridor, the
volatility portfolio is rebalanced
Benefits No need for constant monitoring Minimize the degree that asset classes can violate their allocation corridor
Increase time and expenses of constantly monitoring and trading
Costs Subject to hard movement between rebalancing dates
Considerating when setting 1. Transaction costs : more expensive to trade → should trade less frequently → wider corridor
corridors 2. Risk tolerance : Less risk‐averse investor → wider corridor
3. Correlations : more highly correlated assets in a portfolio → require less frequent rebalancing → ghter corridors
4. Momentum : believe that current trend will continue → wider corridor ; believe in mean reversion → ghter rebalancing
5. Liquidity : Illiquidity assets (private equity, real estate) → larger trading costs → wider corridor
6. Derivatives : If derivatives could be used in rebalancing → lower transac on costs, lower taxes, execute quicker and easier → ghter corridor
7. Taxes : Taxable portfolio → wider corridor ; tax‐exempt por olio → gher corridor
8. Volatility : higher volatility asset → ghter corridor
,Concepts Description
Principles of asset allocation
Mean‐variance optimisation Mean‐variance optimisation (MVO) : identify portfolio allocations that maximise return for every level of risk
Assumptions : assume investors are risk adverse → prefer more return for for same level of risk
Approach to find the most efficient portfolio :
𝑈𝑡𝑖𝑙𝑖𝑡𝑦 𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑎𝑡𝑖𝑜𝑛 : 𝑈 𝐸 𝑅 0.5 λ 𝑉𝑎𝑟
𝑈 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟′𝑠 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑓𝑟𝑜𝑚 𝑖𝑛𝑣𝑒𝑠𝑡𝑖𝑛𝑔 𝑖𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
𝐸 𝑅 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
λ 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟′𝑠 𝑟𝑖𝑠𝑘 𝑎𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑉𝑎𝑟 𝜎 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
λ : Investor's preference for risk ‐ return trade off
‐ Higher expected return for same level of risk : Higher λ
‐ Lower expected return for same level of risk : Lower λ
‐ Range of λ : From 1 to 10, with risk neutral investor's λ = 0.
‐ Average λ = 4
Common constraints :
‐ Budget constraints/Utility constraint : Sum of weights of all asset classes = 100%
‐ Non‐negativity constraint : Weight of each asset class is between 0% and 100%
Criticisms of MVO 1. Garbage‐in‐garbage‐out (GIGO) : quality of output is sensitive to quality of inputs
2. Concentrated asset allocation : MVO often identifies efficient portfolios that are highly concentrated in a subset of asset classes, and zero allocation to others → lowest standard
deviation does not mean diversification
3. Skewness / Kurtosis : asset returns are not normally distributed, but MVO ignore skewness and kurtosis
4. Risk diversification : MVO diversify asset allocation across asset classes, but may not diversify source of risk
5. Ignore liabilities of investors
6. Single‐period framework : ignore interim CF / serial correlatio of asset returns from each period → ignore poten al costs/benefits of rebalancing
Improve inputs quality ‐
Inputs
Reverse optimisation
‐ Assumed optimal asset allocations
Reverse MVO
‐ Variances Output
‐ Maximise utility
‐ Covariances (Correlation)
‐ Risk aversion factor
Output
Inputs ‐ Implied returns
MVO
‐ Revised optimal asset allocations ‐ Variances
‐ Maximise utility
‐ Portfolio expected return ‐ Covariances
(Correlations)
Advantages :
‐ Adress criticism #4 ‐ Risk diversification : Already reflect highly diversified portfolio
‐ Address criticism #1 ‐ GIGO : Improve return estimtes
Improve inputs quality ‐ Black Litterman model : extension of reverse optimisation, with implied returns are adjusted to reflect investor's view of future returns
Black Litterman model
Inputs
‐ Assumed optimal asset allocations
Reverse MVO
‐ Variances Output
‐ Maximise utility
‐ Covariances (Correlation)
‐ Risk aversion factor
Black Litterman
Output
Inputs ‐ Adjusted implied returns
MVO
‐ Revised optimal asset allocations ‐ Variances
‐ Maximise utility
‐ Portfolio expected return ‐ Covariances
(Correlations)
Improve inputs quality ‐ Adding more constraints : to include / exclude certain asset classes
Add more constraints Examples :
‐ Specifiy a fixed allocation to one or more assets (e.g.: human capital, nontradable assets)
‐ Set asset allocation range for asset class
‐ Set upper limit allocated for asset class to address liquidity issues
‐ Specify relative allocation between 2 or more classes
‐ Include constraints to require allocation to assets that hedge the liability in liability‐relative setting
Improve inputs quality ‐ Step 1 : Best estimates of expected returns, sigma, correlation → MVO → Efficient fron er of Op mal alloca ons
Resampled MVO Step 2 : Monte Carlo simulations to generate random variations for inputs around initial estimtes → Efficient fron er + asset alloca on for each point in the fron er
Step 3 : Average simulated efficient frontier + asset allocation
Improve inputs quality ‐ Directly incorporate skewness / kurtosis in utility function, using asymmetric definition of risk (Value‐at‐Risk) instead of variance
Non‐normal distributions
(Skewness and Kurtosis)
, Incorporate human capital and 1. Human Capital
residential real estate to MVO ‐ Stable job + consistent wage that increase with inflation = CF from inflation‐linked bond
framework ‐ Less certain and more volatile future wage = mix of inflation‐linked bond, equities and corporate bonds
‐ Add human capital → increase individual's capacity to take on addi onal risk
‐ Constraints : Human capital is not tradable → % allocated to human capital = current value of human capital / total por olio value
2. Residential real estate
‐ Can be add to individual's portfolio
‐ Risks and returns inputs are estimated using residential real estate property index for that specific region
‐ Add residential real estate → increase individual's capacity to take on addi onal risk
‐ Constraints : % allocated to residential real estate = current value of residential real estate / total portfolio value
Use of Monte Carlo simulation and 1. Address limitation of MVO as single‐period model :
scenario analysis in MVO ‐ Rebalancing → buy/sell investments → trigger taxable capital gains/losses
‐ Investors may add to /withdraw from its portfolio → interim CFs
2. Guide investors to identify their risk tolerance level by showing the range and likelihood of possible outcomes, with various assumptions
Liquidity considerations in asset Less‐liquid asset classes require liquidity return premium
allocation
Issue with less liquid asset classes in MVO :
‐ Few indexes available to track illiquid investments → harder to find data to es mate return, risk and correla on
‐ Not investable as passive alternative
‐ Risk‐return characteristics of each specific real estate, private equity is different from other assets in its asset class
Solutions :
‐ Exclude illiquid asset classes when MVO, but use them to meet separate target asset allocation
‐ Include illiquid asset classes in MVO ; generate inputs from the specific investments
‐ Include illiquid asset classes in MVO ; use highly diversified asset class inputs. Input estimates are usually from alternative investment indexes (which include characteristics of
other asset classes)
Risk budgeting Goal : maximise return per unit of risk (porfolio risk / active risk / residual risk)
Marginal contribution to total risk (MCTR) : Change in total portfolio risk for a small change in asset allocation of a specific asset class
Absolute contribution to total risk (ACTR) : Asset classes' contribution to total risk
𝑀𝐶𝑇𝑅 𝛽 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑐𝑙𝑎𝑠𝑠 𝑖 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑖𝑠𝑘 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑏𝑦 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝐴𝐶𝑇𝑅 𝑤 𝑀𝐶𝑇𝑅
𝐴𝐶𝑇𝑅
% 𝑟𝑖𝑠𝑘 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑏𝑦 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑖
𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑖𝑠𝑘
Incorporate risk preference in 1. Specify additional constraints (e.g.: limit allocations to risky asset classes ; ceiling on portfolio risk)
asset allocation 2. Specify risk aversion factor (lamda)
3. Use Monte Carlo to show various wealth outcomes from assuming allocations with different levels of risk
Use of investment factors in asset Investor could define the opportunity as set of factors (e.g.: market exposure, size, valuation, momentum, liquidity, duration, credit, volatility)
allocation
Factors are zero‐dollar investment portfolio, which long the better performing attribute + short underperforming attribute
Example :
‐ Zero‐dollar portfolio long small stocks + short large stocks
‐ Zero‐dollar portfolio long value + short growth
Factors are not highly correlated with each other and other portfolio → improve risk‐return tradeoff from op mal por olio + expands efficient fron er
Characteristics of liabilities that 1. Fixed vs contigent :
are relevant to asset allocation ‐ Fixed liabilities : CF and timing are specified in advance (e.g.: Fixed rate Cbond)
‐ Contingent liabilities : CF depends on uncertain future events (e.g.: pension liabilities of a defined pension plan)
2. Legal vs Quasi‐legal :
‐ Legal liabilities : obligations defined in a legal agreement
‐ Quasi‐legal liabilities : not legal obligated, but CF are expected to occur in future, which are essential to the mission of the institution
3. Duration and convexity : measure change in liability value for a given change in interest rates
4. Liability value vs size of sponsoring organisation :
‐ large liability compared with the size of sponsoring organisation → necessary to be accounted in asset alloca on decision
‐ small liability : could be ignored
5. Factors affecting future CF : inflation, interest rates, risk premiums, etc.
6. Timing considerations (longevity risk)
7. Regulations affecting the determination of liability value
Introduction to asset allocation
Economic balance sheet 1. Economic balance sheet = financial assets and liabilities + nonfinancial assets and liabilities
2. Nonfinancial assets and liabilities = extended assets and liabilities
Examples :
‐ human capital : PV of expected earnings
‐ PV of pension income
‐ PV of expected intellectual properties royalties
‐ PV of expected underground resources
3. Life cycle balance fund : consider change in human capital and financial capital for individual investor
‐ Entering workforce → human capital >> financial capital → 85% equity ; 15% bonds (human capital has bond‐like characteris cs)
‐ Retirement → total wealth = financial capital → 50% quity, 50% bonds
Approaches to asset allocation 1. Asset‐only approaches : asset allocation decision 100% based on investor's assets
‐ Objectives : max the expected return per unit of risk
‐ Should consider investor constraints + risk tolerance
‐ Risk concept : asset class risk + create effective asset class combination
‐ Risk measure : standard deviation of portfolio, relative risk to a benchmark (tracking error), downside risk (VAR, semivariance, maximum drawdown)
2. Liability‐relative approaches : asset allocation decision based on funding liabilities
‐ Objectives : pay liabilities when it due
‐ Risk concept : (1) not having enough assets to pay liabilities when due; (2) volatility of contributions
‐ Risk measure : standard deviation of surplus
3. Goals‐based approaches : asset allocation for subportfolios, that help investor to achieve the lifestyle/aspirational financial objectives
‐ Need to specify : (1) type of CF needed, (2) time horizons, (3) evel of risk tolerance (probability of achieving certain goal)
‐ Risk concept : unable to achieve financial goals
‐ Portfolio risk = weighted sum of risk of each goal
Allocation by asset classes Asset classes : group of assets with similar characteristics
3 main super asset classes : (1) capital assets (bond, equity) ; (2) consumable / transformable assets (commodities) ; (3) store‐of‐value assets (currency, art)
Criteria to specify asset classes :
‐ Same asset class → similar a ributes (both descrip ve and sta s cal)
‐ Assets cannot be classified in more than 1 asset class
‐ Asset classes should cover all investable assets
‐ Asset classes should contain mostly liquid assets
Use of risk factors in asset Investor selects asset classes based on desired exposure to common risk factors (volatility, liquidity, inflation, interest rate, duration, FX, default)
allocation Factor‐based asset allocation : Risk factor may be overla across multiple asset classes → should focus on risk factor by iden fying risk factors + desired exposure to each
Limitation : cannot directly invest in all risk factors
Strategic asset allocation Strategic asset allocation : combine market expectations with investor's risk, return and constraints
Steps in select strategic asset allocation :
‐ Step 1 ‐ Determine objectives : how investor use these assets? What investor want to achieve? What is the liabilities/goals? How to measure the objectives?
‐ Step 2 ‐ Determine risk tolerance : What is the sensitivities to risk? How to measure risk?
‐ Step 3 ‐ Determine time horizon
‐ Step 4 ‐ Determine constraints : tax situation? Social, environmental or governance considerations? Legal, regulatory, political considerations?
‐ Step 5 ‐ Select aset allocation approach : asset‐only / liability‐relative / goals‐based
‐ Step 6 ‐ Specify asset classes
‐ Step 7 ‐ Develop potential asset allocation
‐ Step 8 ‐ Test the results of potential asset allocation : to see if results align with objectives and risk tolerance for chosen horizon
‐ Step 9 ‐ Repeat step 7 + 8 untill discover optimal asset allocation
Global market portfolio Global market portfolio :
‐ all available risky assets (global equity, global fixed income, global real estates, etc.)
‐ Most diversified portfolio → minimum diversifiable risk
‐ weight could be adjusted to meet specific investor objectives, constraints and desires
‐ some asset classes are not practical for individual investors (e.g.: residential real estate, private equity) → invest via proxy such as ETFs
,Passive / Active choice For weighting asset classes :
1. Tactical asset allocation : active management strategy, seek for additional risk and return (alpha) to take advantes of ST opportunities
‐ Deviation should be restricted by risk budgets / rebalancing
‐ Reasons for deviation : forecasted asset class valuation, business cycle state, stock price momentum
‐ Trade off between potential outperformance vs tracking error
‐ Limitation : additional trading + monitoring cost; additional capital gain taxes
2. Dynamic asset allocation : multiperiod view of investment horizon, where asset performance in one period affects required rate of return and acceptable level of risk in subsequent
periods
For selection within asset classes :
1. Passive management : investor insight / expectations does not impact portfolio composition
2. Active management : change investor insight / expectations → change porfolio composi on → to earn risk‐adjusted return > benchmark
Factors for selecting between active and passive return :
‐ availability of appropriate investments
‐ active management scalability (value added from each active decision)
‐ investor constraints when using passive approach
‐ investor belief in efficient markets → discourage use of ac ve management
‐ cost‐benefit tradeoff between additional costs vs. excess return
‐ tax status of investor
Risk budgeting Risk budgeting : specify how risk should be distributed among portfolio assets without regad to asset expected returns
Active risk budgeting : how much additional risk investor willing to take compared to benchmark
‐ for asset classes allocation : active risk is deterimined by strategic asset allocation vs. benchmark
‐ for selection within asset classes : active risk is determined by asset class selection vs. asset class benchmark
Rebalancing Benefits :
‐ maintain portfolio's risk and return characteristics as specified in IPS
‐ provide discipline
Drawbacks :
‐ related tax liability
‐ transaction costs
Rebalancing approaches Calendar rebalancing Percentage‐range rebalancing
Definition Rebalance the portfolio to its strategic allocation on predetermined, regular Manager set a tolerance band / corridors (±5%) that are considered optimal
basis (monthly, quarterly). Rebalancing frequency depends on portfolio for each asset class. If any asset class goes outside of that corridor, the
volatility portfolio is rebalanced
Benefits No need for constant monitoring Minimize the degree that asset classes can violate their allocation corridor
Increase time and expenses of constantly monitoring and trading
Costs Subject to hard movement between rebalancing dates
Considerating when setting 1. Transaction costs : more expensive to trade → should trade less frequently → wider corridor
corridors 2. Risk tolerance : Less risk‐averse investor → wider corridor
3. Correlations : more highly correlated assets in a portfolio → require less frequent rebalancing → ghter corridors
4. Momentum : believe that current trend will continue → wider corridor ; believe in mean reversion → ghter rebalancing
5. Liquidity : Illiquidity assets (private equity, real estate) → larger trading costs → wider corridor
6. Derivatives : If derivatives could be used in rebalancing → lower transac on costs, lower taxes, execute quicker and easier → ghter corridor
7. Taxes : Taxable portfolio → wider corridor ; tax‐exempt por olio → gher corridor
8. Volatility : higher volatility asset → ghter corridor
,Concepts Description
Principles of asset allocation
Mean‐variance optimisation Mean‐variance optimisation (MVO) : identify portfolio allocations that maximise return for every level of risk
Assumptions : assume investors are risk adverse → prefer more return for for same level of risk
Approach to find the most efficient portfolio :
𝑈𝑡𝑖𝑙𝑖𝑡𝑦 𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑎𝑡𝑖𝑜𝑛 : 𝑈 𝐸 𝑅 0.5 λ 𝑉𝑎𝑟
𝑈 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟′𝑠 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑓𝑟𝑜𝑚 𝑖𝑛𝑣𝑒𝑠𝑡𝑖𝑛𝑔 𝑖𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
𝐸 𝑅 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
λ 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟′𝑠 𝑟𝑖𝑠𝑘 𝑎𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑉𝑎𝑟 𝜎 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑎𝑠𝑠𝑒𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑖
λ : Investor's preference for risk ‐ return trade off
‐ Higher expected return for same level of risk : Higher λ
‐ Lower expected return for same level of risk : Lower λ
‐ Range of λ : From 1 to 10, with risk neutral investor's λ = 0.
‐ Average λ = 4
Common constraints :
‐ Budget constraints/Utility constraint : Sum of weights of all asset classes = 100%
‐ Non‐negativity constraint : Weight of each asset class is between 0% and 100%
Criticisms of MVO 1. Garbage‐in‐garbage‐out (GIGO) : quality of output is sensitive to quality of inputs
2. Concentrated asset allocation : MVO often identifies efficient portfolios that are highly concentrated in a subset of asset classes, and zero allocation to others → lowest standard
deviation does not mean diversification
3. Skewness / Kurtosis : asset returns are not normally distributed, but MVO ignore skewness and kurtosis
4. Risk diversification : MVO diversify asset allocation across asset classes, but may not diversify source of risk
5. Ignore liabilities of investors
6. Single‐period framework : ignore interim CF / serial correlatio of asset returns from each period → ignore poten al costs/benefits of rebalancing
Improve inputs quality ‐
Inputs
Reverse optimisation
‐ Assumed optimal asset allocations
Reverse MVO
‐ Variances Output
‐ Maximise utility
‐ Covariances (Correlation)
‐ Risk aversion factor
Output
Inputs ‐ Implied returns
MVO
‐ Revised optimal asset allocations ‐ Variances
‐ Maximise utility
‐ Portfolio expected return ‐ Covariances
(Correlations)
Advantages :
‐ Adress criticism #4 ‐ Risk diversification : Already reflect highly diversified portfolio
‐ Address criticism #1 ‐ GIGO : Improve return estimtes
Improve inputs quality ‐ Black Litterman model : extension of reverse optimisation, with implied returns are adjusted to reflect investor's view of future returns
Black Litterman model
Inputs
‐ Assumed optimal asset allocations
Reverse MVO
‐ Variances Output
‐ Maximise utility
‐ Covariances (Correlation)
‐ Risk aversion factor
Black Litterman
Output
Inputs ‐ Adjusted implied returns
MVO
‐ Revised optimal asset allocations ‐ Variances
‐ Maximise utility
‐ Portfolio expected return ‐ Covariances
(Correlations)
Improve inputs quality ‐ Adding more constraints : to include / exclude certain asset classes
Add more constraints Examples :
‐ Specifiy a fixed allocation to one or more assets (e.g.: human capital, nontradable assets)
‐ Set asset allocation range for asset class
‐ Set upper limit allocated for asset class to address liquidity issues
‐ Specify relative allocation between 2 or more classes
‐ Include constraints to require allocation to assets that hedge the liability in liability‐relative setting
Improve inputs quality ‐ Step 1 : Best estimates of expected returns, sigma, correlation → MVO → Efficient fron er of Op mal alloca ons
Resampled MVO Step 2 : Monte Carlo simulations to generate random variations for inputs around initial estimtes → Efficient fron er + asset alloca on for each point in the fron er
Step 3 : Average simulated efficient frontier + asset allocation
Improve inputs quality ‐ Directly incorporate skewness / kurtosis in utility function, using asymmetric definition of risk (Value‐at‐Risk) instead of variance
Non‐normal distributions
(Skewness and Kurtosis)
, Incorporate human capital and 1. Human Capital
residential real estate to MVO ‐ Stable job + consistent wage that increase with inflation = CF from inflation‐linked bond
framework ‐ Less certain and more volatile future wage = mix of inflation‐linked bond, equities and corporate bonds
‐ Add human capital → increase individual's capacity to take on addi onal risk
‐ Constraints : Human capital is not tradable → % allocated to human capital = current value of human capital / total por olio value
2. Residential real estate
‐ Can be add to individual's portfolio
‐ Risks and returns inputs are estimated using residential real estate property index for that specific region
‐ Add residential real estate → increase individual's capacity to take on addi onal risk
‐ Constraints : % allocated to residential real estate = current value of residential real estate / total portfolio value
Use of Monte Carlo simulation and 1. Address limitation of MVO as single‐period model :
scenario analysis in MVO ‐ Rebalancing → buy/sell investments → trigger taxable capital gains/losses
‐ Investors may add to /withdraw from its portfolio → interim CFs
2. Guide investors to identify their risk tolerance level by showing the range and likelihood of possible outcomes, with various assumptions
Liquidity considerations in asset Less‐liquid asset classes require liquidity return premium
allocation
Issue with less liquid asset classes in MVO :
‐ Few indexes available to track illiquid investments → harder to find data to es mate return, risk and correla on
‐ Not investable as passive alternative
‐ Risk‐return characteristics of each specific real estate, private equity is different from other assets in its asset class
Solutions :
‐ Exclude illiquid asset classes when MVO, but use them to meet separate target asset allocation
‐ Include illiquid asset classes in MVO ; generate inputs from the specific investments
‐ Include illiquid asset classes in MVO ; use highly diversified asset class inputs. Input estimates are usually from alternative investment indexes (which include characteristics of
other asset classes)
Risk budgeting Goal : maximise return per unit of risk (porfolio risk / active risk / residual risk)
Marginal contribution to total risk (MCTR) : Change in total portfolio risk for a small change in asset allocation of a specific asset class
Absolute contribution to total risk (ACTR) : Asset classes' contribution to total risk
𝑀𝐶𝑇𝑅 𝛽 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑐𝑙𝑎𝑠𝑠 𝑖 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑖𝑠𝑘 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑏𝑦 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝐴𝐶𝑇𝑅 𝑤 𝑀𝐶𝑇𝑅
𝐴𝐶𝑇𝑅
% 𝑟𝑖𝑠𝑘 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑏𝑦 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑖
𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑖𝑠𝑘
Incorporate risk preference in 1. Specify additional constraints (e.g.: limit allocations to risky asset classes ; ceiling on portfolio risk)
asset allocation 2. Specify risk aversion factor (lamda)
3. Use Monte Carlo to show various wealth outcomes from assuming allocations with different levels of risk
Use of investment factors in asset Investor could define the opportunity as set of factors (e.g.: market exposure, size, valuation, momentum, liquidity, duration, credit, volatility)
allocation
Factors are zero‐dollar investment portfolio, which long the better performing attribute + short underperforming attribute
Example :
‐ Zero‐dollar portfolio long small stocks + short large stocks
‐ Zero‐dollar portfolio long value + short growth
Factors are not highly correlated with each other and other portfolio → improve risk‐return tradeoff from op mal por olio + expands efficient fron er
Characteristics of liabilities that 1. Fixed vs contigent :
are relevant to asset allocation ‐ Fixed liabilities : CF and timing are specified in advance (e.g.: Fixed rate Cbond)
‐ Contingent liabilities : CF depends on uncertain future events (e.g.: pension liabilities of a defined pension plan)
2. Legal vs Quasi‐legal :
‐ Legal liabilities : obligations defined in a legal agreement
‐ Quasi‐legal liabilities : not legal obligated, but CF are expected to occur in future, which are essential to the mission of the institution
3. Duration and convexity : measure change in liability value for a given change in interest rates
4. Liability value vs size of sponsoring organisation :
‐ large liability compared with the size of sponsoring organisation → necessary to be accounted in asset alloca on decision
‐ small liability : could be ignored
5. Factors affecting future CF : inflation, interest rates, risk premiums, etc.
6. Timing considerations (longevity risk)
7. Regulations affecting the determination of liability value
