suitability and optimality in the asset allocation process conflict and resolution paul bolster,...
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Suitability and Optimality in the Asset Allocation Process
Conflict and Resolution
Paul Bolster, Northeastern University
Sandy Warrick, S&S Software
Objectives
Develop a suitable asset allocation model using a robust methodology. Suitability: The appropriateness of particular
investments or portfolios of investments for specific investors.
Evaluate suitable asset allocation for mean-variance optimality.
Propose resolution if results conflict.
Distinguishing Suitability from Optimality
Suitability Practical, Legal concept Portfolio’s risk
exposure is paramount Correlations between
asset classes considered in a subjective manner
A suitable portfolio need not be optimal
Optimality Econonmic, Statistical
concept Portfolio’s risk
exposure is paramount Correlations between
asset classes considered explicitly.
An optimal portfolio need not be suitable
Modeling Suitability
NYSE Rule 405AMEX Rule 411AIMR materials
Risk Tolerance Time Horizon Liquidity Unique Factors (legal, tax, etc.)
The Analytic Hierarchy Process
Developed by Saaty (1980) Useful for evaluating relative value
(conflicting ) qualitative criteriaDecomposes a complex decision into
smaller components that are easier to associate with specific alternatives
Allows subjective judgements to be weighed consistently
The Analytic Hierarchy Process
1. Define the problem as a hierarchy.
2. Assess the relative importance of factors at each level of the hierarchy using pairwise comparisons
3. Evaluation of pairwise comparison matricies and determination of best alternative(s).
AHP Step 1: Forming the Hierarchy
S u b fac to r 1 1 S u b fac to r 1 2 S u b fac to r 1 3
F ac to r 1
S u b fac to r 2 1 S u b fac to r 2 2
F ac to r 2
S u b fac to r 3 1 S u b fac to r 3 2 S u b fac to r 3 3
F ac to r 3
G oa l
Alt 1 Alt 2 Alt 3
AHP Step 2: Pairwise Comparisons (Saaty)
The 9-point comparison scale: X to Y = 1 Equal importance X to Y = 3 X mod. favored X to Y = 5 X strongly favored X to Y = 7 X clearly dominant X to Y = 9 X super dominant
Note: X to Y = 3 implies Y to X = 1/3
AHP Step 3: Evaluation of Pairwise Comparisons
Extract standardized eigenvector for each group of factors or subfactors.
The eigenvector can be interpreted as the weight, or importance of a specific factor relative to all other factors.
These weights reflect the full information contained in the pairwise comparison matrix
AHP Applications
Corporate site selection Alternative uses for public
land Choice of environmental
plan Selection of R&D projects
Prediction of bond rating (Srinivasan & Bolster, 1990)
Mutual fund selection (Khasiri, et. al., 1989)
Asset allocation (Bolster, Janjigian, & Trahan, 1995)
The Suitability Hierarchy
In com e an d S avin g s In ves tm en t O b jec tives In ves tin g E xp erien ce
S U ITA B IL ITY
1. Income2. Source3. Savings4. Savings Rate5. Cash Holdings6. Fixed Income Holdings7. Equity Holdings
1. Age2. Dependents3. Time Horizon4. Investments Consumed5. Loss Tolerance6. Liquidity7. Risk Attitude
1. Money Market2. Fixed Income3. Equity
18 Asset Classes
The Suitability Hierarchy: AssetsPrecious MetalsMoney Mkt., Govt.Money Mkt., TaxableMoney Mkt., Tax-FreeMortgage BackedGovernment BondsBonds- HiGrade Corp.Bonds- High YieldBonds- Global
Convertible BondsUtility Stocks Income Stocks International EquityGrowth and IncomeGrowthSmall Cap.Aggressive GrowthSpecialty
Pairwise Comparisons
Table 1: Pairwise comparisons for Suitability Factors
IS IO IE Weight
IS 1 1 3 0.4286IO 1/3 1 3 0.4286IE 1/3 1/3 1 0.1429
Pairwise Comparisons
Table 2: Pairwise comparisons for Income & Savings Subfactors
INCOME SOURCE SAVINGS SVG.RATE
CASHHOLD
FIHOLD
EQUITYHOLD
WEIGHTS
INCOME 1 5 1/2 3 6 4 2 0.2399SOURCE 1/5 1 1/6 1/3 2 1/2 1/4 0.0448SAVINGS 2 6 1 4 7 5 3 0.3543SVG.RATE
1/3 3 1/4 1 4 2 1/2 0.1036CASHHOLD
1/6 1/2 1/7 1/4 1 1/3 1/5 0.0312FIHOLD
1/4 2 1/5 1/2 3 1 1/3 0.0676EQUITYHOLD
1/2 4 1/3 2 5 3 1 0.1587
Data Requirements
Each matrix requires n(n-1)/2 comparisonsThe “hardwired” portion of hierarchy
requires evaluation of matricies of rank 7, 7, and 3. This represents 48 pairwise comparisons.
But each of the 17 subfactors spawns an 18x18 matrix => 18(18-1)/2 = 153 comparisons.
5 levels per subfactor x 17 x 153 = 13,005!
Data: Asset Class Proxies
Identify MF with 10 years of history (120 months) Choose 75th percentile fund using Sharpe ratio Use CAPM return estimate for forecast MF style should be consistent with fund
classification
Data: Asset Class ProxiesTable 3 Funds Chosen for Each Asset Class
Predominant AssetClass1
Fund Type Default Fund Return Std. Dev. Sharpe* Beta
Aggressive Equity Precious Metals Handy&Harman, Gold -4.40 11.40 -0.87 -0.06Cash Money Market, Gov Dreyfus 100% US Treas 4.70 0.32 -2.50 0.00Cash Money Market, taxable ONE Fund Money Mkt 4.60 0.31 -2.90 0.00Cash Money Market, tax free Prudential Tax Free 3.90 0.16 -10.00 0.00Aggressive Bond Mortgage Backed Fidelity Mortgage Sec 8.60 3.22 0.96 0.17Conservative Bond Government Bonds Strong Government Sec 9.31 3.88 0.98 0.18Conservative Bond Bond-High Quality Fidelity U.S. Bond Index 8.86 4.11 0.82 0.23Aggressive Bond Bond-High Yield Fidelity Advisor High Yield 12.66 6.86 1.04 0.31Aggressive Bond Global Bonds Paine Webber Global Income 7.92 4.92 0.49 0.32Aggressive Bond Convertible Bonds Value Line Convertible 11.92 9.40 0.68 0.56Conservative Equity Utility Prudential Utility/A 13.86 9.19 0.91 0.67Conservative Equity Income Riverfront Income Equity 15.95 11.94 0.88 0.88Conservative Equity International Equity Templeton Foreign/1 14.05 9.65 0.89 0.94Conservative Equity Growth and Income Vanguard Growth and Income 17.48 13.40 0.89 1.01Aggressive Equity Growth MFS Large Cap Growth 18.71 15.23 0.87 1.07Aggressive Equity Small Cap Janus: Smal Cap 22.36 17.40 0.97 1.19Aggressive Equity Aggressive Growth Spectra Fund 20.74 14.71 1.04 1.31Aggressive Equity Specialty T Rowe Price Sci&Tech 24.33 19.01 0.99 1.37
1 The predominant asset class is not necessarily the only asset class represented by a fund. For
example, convertible bonds and balanced funds are a mix of bond and equity asset classes.Growth and income funds (such as an S&P index fund) represent a mix of both conservativeand aggressive equities.
Data: Investor Questionnaire
17 questions (1 per subfactor)2-5 categorical responsesThere are 76 distinct responses
76 “suitability vectors” with 18 elements each Total of 1368 pairwise comparisons Remaining pairwise comparisons are inferred
Data: Pairwise ComparisonsEvaluation of a moderately aggressive investor with above average savings (above $500,000)
Table 4: Example of Pairwise comparisons and Resulting Weights of Asset ClassesResulting from a Single Questionnaire Response
PreciousMetals
MoneyMarket,
Gov
MoneyMarket,taxable
MoneyMarket,tax free
Govern-ment
Bonds
Mort-gage
Backed
Bond-High
Quality
Bond-HighYield
GlobalBonds
Conver-tible
Bonds
Utility Income Intern-ationalEquity
Growthand
Income
Growth SmallCap
Aggres-sive
Growth
Specialty
8 8 8 8 1 3 2 5 5 5 4 3 2 1 1 2 3 40.0100 0.0100 0.0100 0.0100 0.1287 0.0559 0.0854 0.0253 0.0253 0.0253 0.0372 0.0559 0.0854 0.1287 0.1287 0.0854 0.0559 0.03720.0036 0.0036 0.0036 0.0036 0.0456 0.0198 0.0303 0.0089 0.0089 0.0089 0.0132 0.0198 0.0303 0.0456 0.0456 0.0303 0.0198 0.01320.0015 0.0015 0.0015 0.0015 0.0195 0.0085 0.0130 0.0038 0.0038 0.0038 0.0056 0.0085 0.0130 0.0195 0.0195 0.0130 0.0085 0.0056
Results from Suitability Model
0%
25%
50%
75%
100%
Billy Broke Larry Lunchpail Norma Smith Wanda Wannabe Lance Rich
Money Mkt Cons Bond Agg Bond Cons Equity Agg. Equity
Mean-Variance Optimization
We estimate returns using a CAPM (single factor) model
The “market” is 30% US Equities (70/15/15 large, mid, small) 20% US Bonds 30% Non-US Equities (EAFE) 20% Non-US Bonds
Asset class betas derived from 10-yr. hist.
Mean-Variance Optimization
MVO produces a smooth efficient frontierDefine Risk Acceptance Parameter
RAP = Var / E(Rp) Higher RAP => Higher risk tolerance Need to map questionnaire responses to RAP
and identify the MVO portfolio with same RAP.
Mean-Variance Optimization
0%
25%
50%
75%
100%
Billy Broke Larry Lunchpail Norma Smith Wanda Wannabe Lance Rich
Money Mkt Cons Bond Agg Bond Cons Equity Agg. Equity
Reconciling Suitability and Optimality
6
7
8
9
10
11
2 3 4 5 6 7 8 9 10 11 12
Mean Variance Optimal AHP
Reconciling Suitability and Optimality
AHP underperforms marginally with an increase in shortfall as risk tolerance increases
How to reconcile? alter pairwise comparisons? alter RAPs? alter CAPM parameters? live with it?
Reconciling Suitability and Optimality: Implied Returns
4
6
8
10
12
14
4 6 8 10 12 14Expected Return, CAPM
Implie
d R
etu
rn, A
HP W
eig
hts RAP=7.5
RAP = 12
RAP = 25
RAP = 40
RAP=49