using lifecycle funds in a suitable and optimal …6 life cycle funds: the theory ugerd infanger and...

Using Lifecycle Funds in a Suitable and Optimal Portfolio

Sandy Warrick, CFANorthfield Information Services

Newport SeminarJune 2006

2

Life Cycle Fundsu In recent years, “life cycle” funds have become an important

financial vehicle for retail investors – as of May 2006 we estimate that there is at least $100 Billion invested in Lifecycle funds.

u Lifecycle funds are designed to address the needs of individuals who are planning on retiring at a particular future date

u During the course of the life cycle, the investment policies of the fund vary from:– Emphasizing capital accumulation during the early years– Emphasizing preservation of capital, income and liquidity

approaching and in retirement.u These funds save the individual investor from having to

periodically reformulate their asset allocation policies as theyapproach retirement.

3

Fund Managers and Target Dates

72141061061161134Grand Total

5XXXXXWF Advisors

7XXXXXXXVantagepoint

5XXXXXVanguard

9XXXXXXXXXT. Rowe Price

4XXXXState Farm

2XXSchwab

4XXXXRussell

8XXXXXXXXPutnam

5XXXXXPrincipal Investors

5XXXXXMass Mutual

9XXXXXXXXXFidelity

5XXXXXBarclays Global

4XXXXAmerican Century

Total20502045204020352030202520202015201020052000Company

72141061061161134Grand Total

5XXXXXWF Advisors

7XXXXXXXVantagepoint

5XXXXXVanguard

9XXXXXXXXXT. Rowe Price

4XXXXState Farm

2XXSchwab

4XXXXRussell

8XXXXXXXXPutnam

5XXXXXPrincipal Investors

5XXXXXMass Mutual

9XXXXXXXXXFidelity

5XXXXXBarclays Global

4XXXXAmerican Century

Total20502045204020352030202520202015201020052000Company

4

Fund Families and Assets Under Management – Target Date 2020

5

Fidelity: Assets Under Management vs. Target Date

6

Life Cycle Funds: The Theory

u Gerd Infanger and George Dantzig developed many of the underlying ideas behind a dynamic approach to life-cycle investing: “Multi-stage Stochastic Linear Programs for Portfolio Optimization,” Annals of Operations Research, December 1993

u Jarrod Wilcox introduced the idea of how an investors’ balance sheet (assets, liabilities and discretionary wealth) affects their optimum portfolio: – “Harry Markowitz and the Discretionary Wealth Hypothesis,”

Journal of Portfolio Management, Spring 2003– diBartolomeo, Horvitz and Wilcox, “Life-Cycle Investing,”

Chapter 3: Investment Management for Taxable Investors, CFA Institute, 2006

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Discretionary Wealth = Assets - Liabilities

8

Life-Cycle Investing

uUtility is the median expected discretionary wealth.– Leverage = Assets / Liabilities– Maximize discretionary wealth.– If you include human capital as part of

financial assets, discretionary wealth usually decreases as an investor approaches retirement.

9

Financial Leverage and the Life Cycle

u Utility = LE[r] – L2σ2/2 + SL3σ3/3 – KL4σ4/4– L = Leverage = Assets / Liabilities– s 2 = Risk (variance)– S = skew– K = kurtosis. Normal K = 3. Fat tails K > 3.

v For a “typical” diversified portfolio, 4<K<5 over the last 15 years

v 4<K<5 is similar to a t-distribution with 7-10 degrees of freedom.

10

Financial Leverage and the Life Cycle

u Utility = LE[r] – L2σ2/2 + SL3σ3/3 – KL4σ4/4– Ignoring higher moments and dividing both sides by L:

Utility = E[r] – Lσ2/2Utility = E[r] - λσ2

λ = L / 2u If leverage is less than 1.5, we have a high net worth investor.

– Life cycle funds are not appropriate because the high net worth investor has a multi-generation or charitable horizon.

u If leverage (< 3) and volatility (< 9%/year) are reasonably lowfor volatility), then ignoring higher moments is probably OK.

u Higher moments are a important if volatility is too high for theleverage.– In other words, risky portfolios with negative skew or excess

kurtosis are even riskier (with regard to asset/liability shortfall) than they appear in a world where only mean and variance count.

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What We Are Going to Do1. We are going to develop 40 different model and investment portfolios using five

fund companies, five target dates and three different levels of investor aggression.

2. Use the Analytic Hierarchy Process to propose a suitable (model) portfolio for a “typical” 50 year old investor who plans to retire in 2020.

3. Select the fund management company’s 2020 target fund to form the core holding for the portfolio.

4. Select two more complementary funds to accommodate varying investor risk tolerance and reduce the tracking error between the model portfolio and the proposed portfolio.

5. We will use mean-variance optimization to minimize the tracking error between the model portfolio and the proposed portfolio.

6. We will repeat steps 3 & 4 for two other investors who will retire in 2020. One is more conservative and one is more aggressive than our initial study.

7. We will Repeat steps 3 and 4 for investors who are will (or have) retired in 2000, 2010, 2030 and 2040 at 65 to 70 years of age.

8. Repeat steps 1-6 for five different fund families.

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The Analytical Hierarchy Process

u A wide body of literature indicates the AHP is useful when making complex decisions involving multiple criteria.

u Thomas Saaty, a professor at the University of Pittsburgh, developed the AHP as a way to improve complex decision making and to identify and weight selection criteria.

u AHP can capture both subjective and objective evaluation measures.

u AHP provides an effective means to deal with, analyzing the data collected for the criteria and expediting the decision-making process.

u AHP has been proven in practice to be a useful mechanism for checking the consistency of the evaluation measures and reducing bias in decision making.

13

The Analytic Hierarchy Process: Suitability and Optimality

uPaul Bolster, Janjigian, and Trahan, “Determining Investor Suitability Using the Analytic Hierarchy Process,” Financial Analyst’s Journal, July/August 1995

uBolster and Warrick, “Suitability and Optimality in the Asset Allocation Process,” www.northinfo.com/papersearch.cfm - This is the version of AHP we used in this study

14

Questions to Determine Objectives

15

Questions to Determine Income and Saving Ability

16

Questions about Client’s Current Investments

17

Questions to Assess Risk Profile

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Suitable Portfolio 2020 Retirement Date“Typical Investor”

Cash: 1%High Quality Bonds: 21%High Yield Bonds: 13%International Bonds: 8%Income Stock: 27%Growth Stock: 15%International Stock: 9%Small Cap Stock: 6%

19

Five Families’ 2020 Target Funds 2002-2005

1.121.161.339.4916.19Gambino0.981.021.498.2914.38Bonnano0.940.962.037.8212.84Colombo0.930.831.277.7311.53Lucchese0.810.831.906.9313.02Genovese

β2β1YieldRiskReturnFamily

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Market Proxy for Calculating β1

5Citi Emerging Bond Index

5Citi High Yield Corporate Bond Index

10Citi Investment Grade Corporate Bond Index

15Citi Currency Hedged World Government Bond Index

65Citi/S&P Broad Market Global Index

WeightIndex

21

Market Proxy for Calculating β2

25FTSE World w/o USA40Russell 1000 Index5Lehman Emerging Markets5Lehman Global High Yield

25Lehman Global AggregateWeightIndex

22

Comparison of Style Weights2020 Target Funds, 2001-2005

4.20.20.06.82.56.3Emerging / Total, %

10068.080.954.469.769.1Equity / Total,%

51.382.983.673.274.772.8Domestic / Equity, %

6.31.88.927.80.012.4Small Cap / Equity, %

52.652.345.445.864.878.2Growth / Equity, %

0.00.20.00.01.63.5Emerging Market Bonds

0.01.78.840.43.24.0Non US Bonds, Hedged

0.00.00.04.30.70.0Non US Bonds

0.00.010.30.00.03.9Domestic High Yield Bonds

0.030.00.040.924.819.6Domestic Quality Bonds

3.80.00.06.80.82.8Emerging Market Stock

20.93.76.87.80.03.8International Value

23.77.96.40.016.812.2International Growth

3.81.23.215.20.05.3Small Cap Value

2.404.00.00.03.3Small Cap Growth

20.627.534.22.924.25.4Large Cap Value

24.127.726.221.827.836.4Large Cap Growth

BMI GlobalLuccheseGenoveseGambinoColomboBonnanoStyle Index

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Comparison of Style WeightsBonnano Funds, Varying Target Dates

4.76.25.84.40.00.0Emerging / Total

89.378.567.446.027.020.8Equity / Total

70.076.377.578.686.586.4Domestic / Equity

11.317.517.017.322.421.5Small Cap / Equity

81.874.975.379.280.568.1Growth / Equity

1.16.25.84.40.030.0Emerging Market Bonds

4.58.56.112.032.044.3Non US Bonds, Hedged

0.00.00.00.00.00.0Non US Bonds

5.10.00.00.00.00.0Domestic High Yield Bonds

0.06.720.837.740.934.9Domestic Quality Bonds

3.50.00.00.00.00.0Emerging Market Stock

3.22.93.41.90.61.0International Value

20.115.811.87.93.11.8International Growth

5.61.70.20.00.02.8Small Cap Value

4.512.011.37.96.11.7Small Cap Growth

6.915.113.17.64.72.8Large Cap Value

45.531.027.720.512.610.7Large Cap Growth

204020302020201020001995Fund Name

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Style Analysis vs. Target DateBonnano Funds

25

Beta vs. Target DateBonnano Funds

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Beta vs. Target Date: Five Fund FamiliesBeta Decreases 2% Per Year

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Complementary Fundsu Investors of the same age and retirement target differ in their ability and

willingness to assume risk in pursuit of higher returnu Complementary funds:

– Allow the financial planner to vary the level of risk while keeping the retirement target constant.

– Reduce the tracking error between the suitable portfolio and the lifecycle fund’s asset allocation.

– Provide additional diversification or opportunity for active return enhancement.– If investor’s risk tolerance varies with wealth (and other factors that determine

leverage), then the investor can keep the lifecycle allocation unchanged while rebalancing the complementary funds between stocks and bonds. This will normally happen as returns vary from their long-term expectation.

u Trial and error (lots of error) indicates that good candidate complementary funds invest in:

– Global Stock– Global Bonds– Equity Income – to balance AHP’s desired value tilt against a growth tilt that we see in

some (but not most) life-cycle funds, see slides 18, 20 and 21.

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Style Analysis: Global Equity

2.07.0Emerging Markets

5.223.6Small Cap Europe

5.720.0Large Cap Europe

3.70.2Small Cap Asia Pacific

4.410.4Large Cap Asia Pacific

4.49.9US Small Cap Growth

2.54.0US Small Cap Value

3.615.4US Large Cap Growth

4.19.6US Large Cap Value

ErrorWeightFund Name

Tracking Error: 3.9Alpha: -1.2

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Style Analysis: Global Bonds

1.1Alpha:1.0Tracking Error

1.814.7Emerging Market Bonds5.55.5International Bonds, Hedged1.913.6International Bonds5.734.5High Yield Bonds1.931.7High Quality Bonds

ErrorWeightFund Name

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Optimal Fund Allocation2020 Retirement Horizon

Life Cycle

Global Income

Global Stock

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A More Aggressive Investor with 2020 Retirement Horizon

u Return to slides 8-10 and increase the income, savings rate, wealth and loss tolerance by one level to model a more aggressive investor.

u Calculate the model portfolio using the AHP.u Return to the optimizationu Recalculate statistics vs. the new model portfolio.u Calculate optimum portfolio for more aggressive

investor.

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A More Conservative Investor with 2020 Retirement Horizon

u Return to slides 8-10 and decrease the income, savings rate, wealth and loss tolerance by one level to model a more conservative investor.

u Calculate the model portfolio using the AHP.u Return to the optimizationu Recalculate statistics vs. the new model portfolio.u Calculate optimum portfolio for more conservative

investor.

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Vary the Retirement Horizon from 2000 to 2040

Assume:1. All investors intend to retire between 65 and 70.2. Increasing life spans and pressure on social security may

extend this even further for investors currently 40 years of ageand younger.

3. The 30 year old investor has a near term spending goal: purchasing a first house.

4. Older investors who are still working have incomes and savings that increase by one level per decade of age, except that real income does not increase from age 50 to 60.

5. Younger investors are more optimistic about future income growth relative to inflation.

6. Younger investors will prefer a more aggressive allocation consistent with their higher level of optimism.

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AHP Model Portfolios vs. Target Date

0

25

50

75

100

2000 2010 2020 2030 2040

Small Cap Stock

International Stock

Growth Stock

Income Stock

International Bonds

High Yield Bonds

High Quality Bonds

Cash

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Model PortfolioExpected Return vs. Risk

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Comparisons of the Fund Families

u The next three pages of slides in your handouts compare the five families with regard to:– The relative allocation between the 2020 lifecycle fund, the

bond fund and the stock fund vs. the investor’s ability and willingness to assume investment risk.

– The relative allocation between the target date lifecycle fund, the bond fund and the stock fund vs. the fund’s target date.

– The performance of the 2020 target fund for the “typical” investor vs. the AHP portfolio, which is constant for all fund families. Although this is in-sample optimize performance, these graphs give us the feel for the effects of tracking error and alpha on cumulative performance.

37

Bonnano Fund Family Weightsvs. Investor Aggressiveness

0

20

40

60

80

100

Conservative Typical Aggressive

Global Stock

Global Bonds

Life Cycle

38

Colombo Fund Family Weightsvs. Investor Aggressiveness

0

20

40

60

80

100

Conservative Typical Aggressive

Global Stock

Global Bonds

Life Cycle

39

Gambino Fund Family Weightsvs. Investor Aggressiveness

0

20

40

60

80

100

Conservative Typical Aggressive

Global Stock

Global Bonds

Life Cycle

40

Genovese Fund Family Weightsvs. Investor Aggressiveness

0

20

40

60

80

100

Conservative Typical Aggressive

Global Stock

Global Bonds

Life Cycle

41

Lucchese Fund Family Weightsvs. Investor Aggressiveness

0

20

40

60

80

100

Conservative Typical Aggressive

Global Stock

Global Bonds

Life Cycle

42

Bonnano Fund Family Weightsvs. Retirement Date

0

2 0

4 0

6 0

8 0

100

2000 2010 2020 2030 2040

Global Stock

Global Bonds

Life Cycle

43

Colombo Fund Family Weightsvs. Retirement Date

0

2 0

4 0

6 0

8 0

100

2000 2010 2020 2030 2040

Globa l S tock

Globa l Income

Life Cycle

44

Gambino Fund Family Weightsvs. Retirement Date

0

20

40

60

80

100

2000 2010 2020 2030 2040

Global Stock

Global Income

Life Cycle

45

Genovese Fund Family Weightsvs. Retirement Date

0

2 0

4 0

6 0

8 0

1 0 0

2 0 0 0 2010 2 0 2 0 2030 2 0 4 0

G l o b a l S t o c k

G l o b a l B o n d s

L i fe Cyc le

46

Lucchese Fund Family Weightsvs. Retirement Date

0

20

40

60

80

100

2000 2010 2020 2030 2040

Global Stock

Global Bonds

Life Cycle

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Bonnano 2020+ vs. AHPP e r f o m a n c e C o m p a r i s o n G r a p h

A n n u a l R e b a l a n c i n g N o R e b a l a n c i n g * A H P * A H P F u n d

Date2005/102 0 0 5 / 0 42 0 0 4 / 1 02004/042003/102003/042002/102002 /042001 /102001/042000 /10

Inde

x1 3 0

1 2 8

1 2 6

1 2 4

1 2 2

1 2 0

1 1 8

1 1 6

1 1 4

1 1 2

1 1 0

1 0 8

1 0 6

1 0 4

1 0 2

1 0 0

98

96

94

92

90

88

86

84

48

Colombo 2020+ vs. AHPP e r f o m a n c e C o m p a r i s o n G r a p h

A n n u a l R e b a l a n c i n g N o R e b a l a n c i n g * A H P * A H P F u n d

Date2006/032005/122 0 0 5 / 0 92005/062 0 0 5 / 0 32004/12

Inde

x

1 1 0

1 0 8

1 0 6

1 0 4

1 0 2

1 0 0

49

Gambino 2020+ vs. AHPPerfomance Comparison Graph

Annual Rebalancing No Rebalancing *AHP* AHP Fund

Date2006/032005/122005/092005/062005/032004/122004/092004/062004/03

Inde

x

118

116

114

112

110

108

106

104

102

100

98

50

Genovese 2020+ Fund vs. AHPP e r f o m a n c e C o m p a r i s o n G r a p h

A n n u a l R e b a l a n c i n g N o R e b a l a n c i n g * A H P * A H P F u n d

Date2 0 0 6 / 0 22005/112005/082 0 0 5 / 0 52005/022004/112004/082004/052004/022003/11

Inde

x1 2 6

1 2 4

1 2 2

1 2 0

1 1 8

1 1 6

1 1 4

1 1 2

1 1 0

1 0 8

1 0 6

1 0 4

1 0 2

1 0 0

51

Lucchese 2020+ Fund vs. AHPP e r f o m a n c e C o m p a r i s o n G r a p h

A n n u a l R e b a l a n c i n g N o R e b a l a n c i n g * A H P * A H P F u n d

Date2 0 0 5 / 1 02005/042004/102004/042003/102 0 0 3 / 0 42002/102002/042001/102 0 0 1 / 0 42 0 0 0 / 1 02000/041999/101 9 9 9 / 0 41 9 9 8 / 1 01998/041997/10

Inde

x1 6 6

1 6 4

1 6 2

1 6 0

1 5 8

1 5 6

1 5 4

1 5 2

1 5 0

1 4 8

1 4 6

1 4 4

1 4 2

1 4 0

1 3 8

1 3 6

1 3 4

1 3 2

1 3 0

1 2 8

1 2 6

1 2 4

1 2 2

1 2 0

1 1 8

1 1 6

1 1 4

1 1 2

1 1 0

1 0 8

1 0 6

1 0 4

1 0 2

1 0 0

52

Conclusionsu The Analytic Hierarchy Process can be used to quickly create a model

portfolio for an investor whose main goal is a retirement planning.u Portfolio optimization using the model portfolio as a benchmark can create a

portfolio that combines a lifecycle fund and two or more complementary funds to best fit the model portfolio. Complementary funds:– Vary the risk for investors that are more or less aggressive than the typical

investor with the same retirement horizon.– Allow the investor to leave the allocation to the lifecycle fund unchanged while

varying the allocation between the complementary funds as wealth changes due to investment returns or other factors unique to the investor.

– Improve active performance vs. the model portfolio.u Life cycle funds will comprise between 40% and 60% of the weight of

portfolio for most investors between 30 and 70 years old, depending on:– fund company– target date– the investor’s specific ability and willingness to assume investment risk.