Suitability and Optimality in the Asset Allocation Process Conflict and Resolution Paul Bolster, Northeastern University Sandy Warrick, S&S Software.

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

Conclusions

Minor alterations in AHP rule base (or minor change in inferred RAP) can close gap

AHP shortfall is always greatest for highest risk levels

Suitability and Optimality are not distant cousins