individual 401k portfolio asset allocation optimization

Texas State University QMST 5332

Individual 401K Portfolio Asset Allocation Optimization

Rachel Sampalli

Individual 401K Portfolio Asset Allocation Optimization | Rachel Sampalli

ContentsProblem.......................................................................................................................................................3

Significance..................................................................................................................................................3

Data.............................................................................................................................................................3

Type of Model.............................................................................................................................................3

Literature.....................................................................................................................................................3

Determinants of Portfolio Performance..................................................................................................3

Asset Allocation Hoax..............................................................................................................................3

Rethinking Portfolio Rebalancing.............................................................................................................4

Robust Allocation....................................................................................................................................4

Tactical Asset...........................................................................................................................................4

Decision Variables.......................................................................................................................................4

Constraints..................................................................................................................................................5

Allocation Must Equal 100%....................................................................................................................5

Asset Allocation.......................................................................................................................................5

No More than 15% per Fund...............................................................................................................5

60% in Domestic Stocks.......................................................................................................................5

30% in Foreign Stocks.........................................................................................................................5

Non-negativity.........................................................................................................................................5

Objective Function.......................................................................................................................................6

Formulation.................................................................................................................................................6

Optimal Objective Function.........................................................................................................................6

Decision Variable Values.............................................................................................................................6

Sensitivity Analysis......................................................................................................................................6

Objective Function Sensitivity Analysis....................................................................................................6

Reduced Cost...........................................................................................................................................6

Shadow Price...........................................................................................................................................6

100% Allocation Shadow Price...........................................................................................................7

No More than 15% per Fund...............................................................................................................7

60% in Domestic Stocks.......................................................................................................................7

30% in Foreign Stocks.........................................................................................................................7

Non-negativity.....................................................................................................................................7

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Limitations...................................................................................................................................................8

Future Work................................................................................................................................................8

Learning.......................................................................................................................................................8

Appendix.....................................................................................................................................................9

Table 1 Decision Variables......................................................................................................................9

Equation 1: Allocation Must Equal 100%................................................................................................9

Table 2: Asset Allocation within Each Fund..........................................................................................10

Equation 2: No More than 15% per Fund.............................................................................................11

Equation 3: 60% in Domestic Stocks.....................................................................................................11

Equation 4: 30% in Foreign Stocks........................................................................................................11

Equation 5: Non-negativity...................................................................................................................11

Equation 6: Objective Function.............................................................................................................12

Table 3: Optimal Solution.....................................................................................................................12

Table 4: Objective Function Sensitivity................................................................................................13

Table 5: 100% Allocation Shadow Price................................................................................................13

Table 6: 15% per Fund Shadow Price....................................................................................................14

Table 7: 60% in Domestic Stocks Shadow Price....................................................................................14

Table 8: 30% in Foreign Stocks Shadow Price.......................................................................................15

Table 9: Non-negativity Shadow Price..................................................................................................15

R Code...................................................................................................................................................15

R Code Solution.....................................................................................................................................17

Bibliography...........................................................................................................................................21

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ProblemThis report provides information on the author’s optimal portfolio mix model for an individual 401k plan.

SignificanceWhen individuals employees determine which funds to invest in for their 401k plan they often have little knowledge in investments and are under a deadline to select a portfolio mix. As an inexperienced investor, he or she may arbitrarily select a fund based on the name alone. Allowing inexperienced employees to create their retirement fund in this manner is concerning, as this will be income when the investor is unable to work. This model is significant because provides inexperienced investors with a methodical approach to maximize their retirement income.

DataThe author’s personal 401K fund options are the source for this data set. Each fund segments into various asset classes (Paychex, Inc, 2015). The fiver year average return, retrieved from Yahoo! Finance, indicates the potential value (Morningstar, Inc., 2015). The asset allocation of each fund and average return is continuous numerical data measured in percentages.

Type of ModelThis is a portfolio and product mix model. Each fund derives from a percentage of various asset classes. This model provides a recommended percent of investment to each fund, given the asset breakdown.

LiteratureThis optimization portrays the traditional, passive model as discussed in the first literature source. The other sources attempt to build upon this original model.

Determinants of Portfolio PerformanceBy Gary P. Brinson, L. Randolph Hood and Gilbert L. Beebower

This is the industry standard for the traditional asset allocation theory most commonly used (Jahnke, 2004). Although, it is worth noting, the authors updated this report in 1991 and 2006 (Gary P. Brinson B. D., 1991) (Mark Kritzman, 2006). This article reports a study analyzing the effects of investment policy, marketing timing, and security selections on total plan return. The study defines investment policy as the long-term selection of weighted asset classes. The conclusion indicates investment policy explains 93.6% of total plan return. The study encourages a passive strategy focused on weighted asset allocation and less active management from investment managers (Gary P. Brinson L. R., 1986). The conclusion mirrors the author’s personal recommendations from investment advisors.

Asset Allocation HoaxBy William Jahnke

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The purpose of this article is to demonstrate the fallacies within the study described in the article Determinants of Portfolio Performance (Jahnke, 2004). The study did not analyze the decision making process used to determine the asset allocation policy. Furthermore, although the study claims to explain returns, it actually attempts to explain the portfolio variance. The asset allocation policy explains only 14.6% of policy returns, a large difference from the 93.6% cited in the study. In terms of volatility, standard deviation is a more appropriate measure compared to variance as standard deviation uses the same units of measurements as returns. Finally, the transaction cost variable is missing. Fees, commissions and other trading costs differ between asset classes and impact portfolio return. This article recommends using expectation based asset allocation and changing allocation as the market changes with new opportunities, instead of the fixed plan described in study (Jahnke, 2004).

Rethinking Portfolio RebalancingBy Alexander Köhler and Hagen Wittig

In portfolio management, weights of portfolio’s various asset classes balance the risk and return to the investor’s preference. Since asset classes grow at different rates, the portfolio will drift from the investor’s original risk and return plan over time. Advisors rebalance by setting upper and lower bandwidths weights around the asset‘s weights. If the asset weight goes over or declines to the respective bandwidths, the weights automatically readjust to the preferred amounts of risk and return. Traditionally, weight parameters are value-based by dividing the dollar value of an asset by the dollar value of the portfolio. This works for expected returns but does not handle risk. This article discusses rebalancing focusing on risk instead of return (Wittig,2014).

Robust AllocationBy Nalan Gulpınar, Kabir Katata and Dessislava A. Pachamanova

Uncertainty is unavoidable, as models need investors’ personal preferences and historical data to forecast. This article reports on a model that mathematically incorporates the uncertainty of the expected return into their model (Nalan Gulpınar, 2011).

Tactical AssetBy Michael E. Kitces, Mebane Faber, Jerry Miccolis and Ken Solow

In strategic investing, investors change allocation between asset classes to take advantage of opportunities in the market of each asset class. The process involves creating market assumptions, asset class assumptions, and resulting optimization models every year to 18 months. Portfolio management should be forward looking. Unfortunately, an investor only has past data to work with. This article recommends segmenting historical data into market segments, such as when the market had high or low inflation, or when it had high or low growth, to provide a more accurate estimated return (Michael E. Kitces, 2013).

Decision VariablesEach variable indicates the investment percentage allocated to each fund. Please see x on page 9 for a list of decision variables used in this model.

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ConstraintsAllocation Must Equal 100%Since each variable indicates a percentage of investment, the total investment must be 100 percent allocated throughout the funds. Equation 1: Allocation Must Equal 100% on page 9 states this numerically.

Asset AllocationThe constraints concerning asset allocation came from investments tips from the founding editor of Money under 30 website, David Weliver. Weliver recommends a diversified portfolio with a mix of domestic stocks, foreign stocks, bonds, and alternative investments (Weliver). This supports the conclusions in the study described in Determinants of Portfolio Performance summarized on page 3. Additionally, he recommended 60% domestic stock, 40% foreign stock, and 0% in bonds and alternative investments for an individual between 20 and 30 years old (Weliver). However, the author of this model wanted a less risky portfolio and opted for no more than 60% in domestic stock, no more than 30% in foreign stock and the rest in bonds and other investments. The funds available were a blend of domestic stock, foreign stock, and various types of bonds and other investments. One cannot only invest in one asset within a fund. Therefore, this model described each fund as the percentages of their various asset classes. Every asset that was not domestic or foreign stock was categorized as bonds and other. Please see Table2: Asset Allocation within Each Fund on page 10 for the asset allocation within each fund.

No More than 15% per FundIn order to diversify, each fund cannot contain more than 15% of the investment. Equation 2: No More than 15% per Fund on page 11 describes this numerically.

60% in Domestic StocksAs recommended by Weliver, 60 % of the total portfolio should be in domestic stocks. As each fund is a blend of many assets, the domestic stock percentage is the coefficient for each variable in this constraint. This reads as 86.40% of the percentage of investment allocated to the Allianzgi Nfj Dividend Value A fund, plus 60.26% of the percentage of investment allocated to the American Funds Growth Fund Of America R3 fund, and so on until the last mutual fund, must be less than or equal to 60%. Equation 3: 60% in Domestic Stocks states this numerically on page 11.

30% in Foreign StocksAlthough Weliver recommended 40% in foreign stock, the author decided to decrease the risk in the portfolio by reducing the asset allocation to 30% in foreign stocks. As each fund is a blend of many assets, the foreign stock percentage is the coefficient for each variable in this constraint. This can be read as 12.26% of the percentage of investment allocated to the Allianzgi Nfj Dividend Value A fund, plus 13.59% of the percentage of investment allocated to the American Funds Growth Fund Of America R3 fund, and so on until the last mutual fund, must be less than or equal to 30%. Please refer to Equation 4: 30% in Foreign Stocks on page 11.

Non-negativity With this passive portfolio management model, one cannot have a negative percentage allocated to (or short sell) a fund. Therefore, the author added a non-negativity constraint. Equation 5: Non-negativity on page 12 states this numerically.

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Objective FunctionThe goal of this model is to maximize portfolio return using 5-year average return as the determinant of value. The objective function coefficient is the 5-year average return. Please refer to Equation 6: Objective Function on page 12.

FormulationThis model transforms the objective coefficient and constraints listed above into R. Please see on page 14 for the full R script.

Optimal Objective FunctionThe optimal objective function is 9.78%. This means the estimated portfolio return for one year of investing in the optimal solution is 9.78%. Please refer to Error: Reference source not found on page Error: Reference source not found for the full R script.

Decision Variable ValuesThe optimal solution is 15% of an individual’s investment in American Funds Growth Fund of America R3, American Funds New Economy R3, Federated Kaufmann Small Cap A, Fidelity Advisor Balanced T, John Hancock Lifestyle Aggressive Portfolio A and John Hancock Lifestyle Growth Portfolio A. The 10% remaining allocates as 2.84% in Allianzgi Nfj Dividend Value A and 7.16% in Loomis Sayles Bond Admin. Table 3: Optimal Solution on page 13 displays the optimal solution.

Sensitivity AnalysisObjective Function Reduced CostThe Objective Function reduced costs shows ranges in which the optimal solution will not change, ceteris paribus. Please refer to Table 4: Objective Function Sensitivity on page 13 for a full list of ranges for each fund. The solution will not change as long as the fund is within its lower and higher limit. For example, as long as Allianzgi Nfj Dividend Value A’s 5 year annual return is between 9.57% and 10.17% this model will recommend the investor allocate 2.84% of their investment into the fund, all else held constant. It also informs the investor how much a mutual fund’s average return would have to increase or decrease before it would be included in the optimal solution. For example, Eaton Vance Government Obligations A’s 5 year average return would have to increase by 3.98% before it would be included and change the optimal solution. Table 4

Shadow PriceShadow Prices describe how much one more unit increase or decrease in the constraint is worth to the objective function. The lower and higher limits give the range to which the shadow price is accurate. From the base to the lower limit, a 1-unit decrease in the constraint would decrease the portfolio return by the shadow price. From the base to the higher limit, a 1-unit increase in the constraint would increase the portfolio return by the shadow price. A shadow price of 0% indicates that constraint is not binding and does not affect the portfolio return.

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100% AllocationIf one could invest more than 100%, for every percent increase, the portfolio return would increase by 3.98%, until 107.34%. If one allocated less than 100%, the portfolio return would decrease by 3.98% per percentage unallocated, until the investor decreased the amount allocated to 93.30%. After 93.30%, the portfolio return decreased per unit would change. Please refer to on page 14.

No More than 15% per FundPlease refer Table 6: 15% per Fund Shadow Price on page 14. From 15% to the lower limit, a 1% decrease allocated to the fund would decrease portfolio return by the shadow price. From 15% to the higher limit, a 1% increase allocated to the fund would increase portfolio return by the shadow price. For example, if the investor could invest more than 15% in American Funds Growth Fund of America R3, each additional percent allocated to the fund would increase portfolio return by 3.07%, until one allocated up to 18.14%. Similarly, if the investor mandated a maximum less than 15% for American Funds Growth Fund of America R3, each percentage decrease would decrease portfolio return by 3.07%, until the investor decreased the percentage allowed to 1.57%.

60% in Domestic StocksThis is a binding constraint and defined the optimal solution. If the investor allowed more domestic stock, each percent increase would increase portfolio return by 6.60%, until the investor allocated 65.79%. On the other hand, if the investor required less than 60% in domestic stock, a 1% decrease allocated to domestic stocks would decrease portfolio return by 6.60%, until the investor decreased it 57.70%. Please refer to Table 7: 60% in Domestic Stocks Shadow Price on page 15.

30% in Foreign Stocks This constraint is not binding. Any increase or decrease will not change the portfolio return. Please refer to Table 8: 30% in Foreign Stocks Shadow Price on page 15.

Non-negativityIf the investor could have less than 0% allocated to a mutual fund, or short sell the fund, the portfolio return would increase, on certain funds. Please refer to Table 9: Non-negativity ShadowPrice on page 16. If the investor mandated to have more than 0% in each individual fund, the portfolio return would increase by the shadow price. Since the shadow price is negative, this indicates mandating more than 0% per fund would decrease the portfolio return by the absolute value of the shadow price, until the higher limit. If the investor decreased the percent invested in past 0%, or short sold the fund, for every percent the investor short sold, the portfolio return would increase by the absolute value of the shadow price, until the investor short sold to the lower limit. For example, if the investor was mandated to have more than 0% in Eaton Vance Government Obligations A, for every percent allocated to the fund, the portfolio return would decrease by 2.62%, until the mandated allocation increased to 6.7%. If the investor short sold Eaton Vance Government Obligations A, for every percent short sold, the portfolio return would increase by 2.62%, until the investor short sold 7.34%.

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LimitationsThere are numerous limitations to this model. The use of 5-year average return is not a sure indicator of future success in the fund. As discussed in Tactical Asset on page 4, it would be more appropriate to segment the entire life of the fund into market regimes, re-calculate the average price, and choose the average price from the market regime that most similarly resembles the current market condition. Furthermore, personal preference on risk dictates the asset allocation and therefore this could model is difficult to standardize.

Future Work The author of this report intends to use this model immediately as the asset allocation of her personal 401K plan. She also plans to share the model amongst her colleagues to who have access to the same funds and update the constraints to their personal preferences and needs. This model will be kept on file and used to update asset allocation yearly for the author and her colleagues if they so wish.

LearningThe author learned a real world lesson in asset allocation that will secure a stronger retirement fund. Asset allocation is a valuable skill that is relevant in other investments as well. Most importantly, the author learned a methodical formulation needed for safer and more reliable decision-making.

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Appendix

X1 = Percentage of investment allocated to Allianzgi Nfj Dividend Value A X2 = Percentage of investment allocated to American Funds Growth Fund Of America R3 X3 = Percentage of investment allocated to American Funds New Economy R3 X4 = Percentage of investment allocated to Eaton Vance Government Obligations A X5 = Percentage of investment allocated to Federated Kaufmann Small Cap A X6 = Percentage of investment allocated to Fidelity Advisor Balanced T X7 = Percentage of investment allocated to Fidelity Advisor Freedom 2040 T X8 = Percentage of investment allocated to Fidelity Advisor Freedom 2045 T X9 = Percentage of investment allocated to Fidelity Advisor Freedom Income T X10 = Percentage of investment allocated to Janus Overseas S X11 = Percentage of investment allocated to John Hancock Lifestyle Aggressive Portfolio A X12 = Percentage of investment allocated to John Hancock Lifestyle Balanced Portfolio A X13 = Percentage of investment allocated to John Hancock Lifestyle Conservative Portfolio A X14 = Percentage of investment allocated to John Hancock Lifestyle Growth Portfolio A X15 = Percentage of investment allocated to John Hancock Lifestyle Moderate Portfolio A X16 = Percentage of investment allocated to Loomis Sayles Bond Admin X17 = Percentage of investment allocated to Oppenheimer Global Strategic Income A X18 = Percentage of investment allocated to Oppenheimer Limited Term Government A X19 = Percentage of investment allocated to Pioneer Disciplined Value A X20 = Percentage of investment allocated to T. Rowe Price Intl Growth & Income R X21 = Percentage of investment allocated to Victory Fund For Income A

Equation 1: Allocation Must Equal 100%X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12+X13+X14+X15+X16+X17+X18+X19+X20+X21=100

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Table 1 Decision Variables


Table 2: Asset Allocation within Each FundVariable

NameFund Name

Domestic Stock

Foreign Stock

Bonds & Other

X1 Allianzgi Nfj Dividend Value A 86.40% 12.26% 1.34%X2 American Funds Growth Fund Of America R3 78.72% 13.59% 7.69%X3 American Funds New Economy R3 60.26% 27.89% 11.85%X4 Eaton Vance Government Obligations A 0.00% 0.00% 100.00%X5 Federated Kaufmann Small Cap A 75.02% 11.12% 13.86%X6 Fidelity Advisor Balanced T 61.02% 4.40% 34.58%X7 Fidelity Advisor Freedom 2040 T 62.38% 28.60% 9.02%X8 Fidelity Advisor Freedom 2045 T 62.39% 28.60% 9.01%X9 Fidelity Advisor Freedom Income T 17.51% 7.74% 74.75%X10 Janus Overseas S 10.99% 86.24% 2.77%X11 John Hancock Lifestyle Aggressive Portfolio A 56.87% 33.49% 9.64%X12 John Hancock Lifestyle Balanced Portfolio A 37.05% 20.62% 42.33%X13 John Hancock Lifestyle Conservative Portfolio A 13.58% 8.48% 77.94%X14 John Hancock Lifestyle Growth Portfolio A 49.10% 26.88% 24.02%X15 John Hancock Lifestyle Moderate Portfolio A 25.40% 13.64% 60.96%X16 Loomis Sayles Bond Admin 5.51% 0.63% 93.86%X17 Oppenheimer Global Strategic Income A 0.07% 0.17% 99.76%X18 Oppenheimer Limited Term Government A 0.00% 0.00% 100.00%X19 Pioneer Disciplined Value A 96.81% 2.61% 0.58%X20 T. Rowe Price Intl Growth & Income R 1.21% 92.36% 6.43%X21 Victory Fund For Income A 0.00% 0.00% 100.00%

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Equation 2: No More than 15% per Fund

Equation 3: 60% in Domestic Stocks0.864X1+0.7872X2+0.6026X3+0X4+0.7502X5+0.6102X6+0.6238X7+0.6239X8+0.1751X9+0.1099X10+0.5687X11+0.3705X12+0.1358X13+0.491X14+0.254X15+0.0551X16+0.0007X17+0X18+0.9681X19+0.0121X20+0X21<=60

Equation 4: 30% in Foreign Stocks0.1226X1+0.1359X2+0.2789X3+0X4+0.1112X5+0.044X6+0.286X7+0.286X8+0.0774X9+0.8624X10+0.3349X11+0.2062X12+0.0848X13+0.2688X14+0.1364X15+0.0063X16+0.0017X17+0X18+0.0261X19+0.9236X20+0X21 <=30

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X1 <=15X2 <=15X3 <=15X4 <=15X5 <=15X6 <=15X7 <=15X8 <=15X9 <=15X10 <=15X11 <=15X12 <=15X13 <=15X14 <=15X15 <=15X16 <=15X17 <=15X18 <=15X19 <=15X20 <=15X21 <=15


Equation 5: Non-negativityX1 <=0X2 <=0X3 <=0X4 <=0X5 <=0X6 <=0X7 <=0X8 <=0X9 <=0X10 <=0X11 <=0X12 <=0X13 <=0X14 <=0X15 <=0X16 <=0X17 <=0X18 <=0X19 <=0X20 <=0X21 <=0

Equation 6: Objective FunctionMAX= 0.0968x1+ 0.1224x2+ 0.1284x3+ 0.0136x4+ 0.1184x5

+ 0.0881x6+ 0.0725x7+ 0.0740x8+ 0.0299x9+ -0.0795x10+ 0.0808x11+ 0.0638x12+ 0.0410x13+ 0.0748x14+ 0.0538x15+ 0.0434x16+ 0.0310x17+ 0.0121x18+ 0.0942x19+ 0.0345x20+ 0.0189x21

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Table 3: Optimal SolutionInvestment Allocated

Variable Name

Fund Name Domestic Stock

Foreign Stock

Bonds & Other

2.84% X1 Allianzgi Nfj Dividend Value A 86.40% 12.26% 1.34%15% X2 American Funds Growth Fund of America R3 78.72% 13.59% 7.69%15% X3 American Funds New Economy R3 60.26% 27.89% 11.85%15% X5 Federated Kaufmann Small Cap A 75.02% 11.12% 13.86%15% X6 Fidelity Advisor Balanced T 61.02% 4.40% 34.58%15% X11 John Hancock Lifestyle Aggressive Portfolio A 56.87% 33.49% 9.64%15% X14 John Hancock Lifestyle Growth Portfolio A 49.10% 26.88% 24.02%

7.16% X16 Loomis Sayles Bond Admin 5.51% 0.63% 93.86%

Portfolio Total: 100.00% 60% 18% 22%

Table 4: Objective Function SensitivityVariable

NameFund Name

5 Year Average Return

Monthly Investment

Average Return Lower Limit

Average Return Higher Limit

X1 Allianzgi Nfj Dividend Value A 9.68% 2.84% 9.57% 10.17%X2 American Funds Growth Fund Of America R3 12.24% 15.00% 9.17% Positive InfinityX3 American Funds New Economy R3 12.84% 15.00% 7.95% Positive InfinityX4 Eaton Vance Government Obligations A 1.36% 0.00% Negative Infinity 3.98%X5 Federated Kaufmann Small Cap A 11.84% 15.00% 0.09% Positive InfinityX6 Fidelity Advisor Balanced T 8.81% 15.00% 8.00% Positive InfinityX7 Fidelity Advisor Freedom 2040 T 7.25% 0.00% Negative Infinity 8.94%X8 Fidelity Advisor Freedom 2045 T 7.40% 0.00% Negative Infinity 8.09%X9 Fidelity Advisor Freedom Income T 2.99% 0.00% Negative Infinity 5.13%X10 Janus Overseas S -7.95% 0.00% Negative Infinity 4.70%X11 John Hancock Lifestyle Aggressive Portfolio A 8.08% 15.00% 7.73% Positive InfinityX12 John Hancock Lifestyle Balanced Portfolio A 6.38% 0.00% Negative Infinity 6.42%X13 John Hancock Lifestyle Conservative Portfolio A 4.10% 0.00% Negative Infinity 4.87%X14 John Hancock Lifestyle Growth Portfolio A 7.48% 15.00% 7.22% Positive InfinityX15 John Hancock Lifestyle Moderate Portfolio A 5.38% 0.00% Negative Infinity 5.65%X16 Loomis Sayles Bond Admin 4.34% 7.16% 4.27% 4.91%X17 Oppenheimer Global Strategic Income A 3.10% 0.00% Negative Infinity 3.98%X18 Oppenheimer Limited Term Government A 1.21% 0.00% Negative Infinity 3.98%X19 Pioneer Disciplined Value A 9.42% 0.00% Negative Infinity 1.04%X20 T. Rowe Price Intl Growth & Income R 3.45% 0.00% Negative Infinity 4.06%X21 Victory Fund For Income A 1.89% 0.00% Negative Infinity 3.98%

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Table 5: 100% Allocation Shadow Price

Constraint

Shadow Price [Change in Portfolio Return Given 1

Unit Change in Constraint]

Lower Limit [Decrease by Shadow Price]

Higher Limit [Increase by

Shadow Price]

100% Allocation 3.98% 93.30% 107.34%

Table 6: 15% per Fund Shadow PriceShadow Price [Change in Portfolio Return Given 1




Shadow Price]

15% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinitity15% in American Funds Growth Fund Of America R3 3.07% 1.57% 18.14%15% in American Funds New Economy R3 4.89% 0.00% 19.20%15% in Eaton Vance Government Obligations A 0.00% Negative Infinity Positive Infinitity15% in Federated Kaufmann Small Cap A 2.91% 0.85% 18.31%15% in Fidelity Advisor Balanced T 0.81% 0.00% 19.14%15% in Fidelity Advisor Freedom 2040 T 0.00% Negative Infinity Positive Infinitity15% in Fidelity Advisor Freedom 2045 T 0.00% Negative Infinity Positive Infinitity15% in Fidelity Advisor Freedom Income T 0.00% Negative Infinity Positive Infinitity15% in Janus Overseas S 0.00% Negative Infinity Positive Infinitity15% in John Hancock Lifestyle Aggressive Portfolio A 0.35% 0.00% 19.48%15% in John Hancock Lifestyle Balanced Portfolio A 0.00% Negative Infinity Positive Infinitity15% in John Hancock Lifestyle Conservative Portfolio A 0.00% Negative Infinity Positive Infinitity15% in John Hancock Lifestyle Growth Portfolio A 2.62% 0.00% 20.28%15% in John Hancock Lifestyle Moderate Portfolio A 0.00% Negative Infinity Positive Infinitity15% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinitity15% in Oppenheimer Global Strategic Income A 0.00% Negative Infinity Positive Infinitity15% in Oppenheimer Limited Term Government A 0.00% Negative Infinity Positive Infinitity15% in Pioneer Disciplined Value A 0.00% Negative Infinity Positive Infinitity15% in T. Rowe Price Intl Growth & Income R 0.00% Negative Infinity Positive Infinitity15% in Victory Fund For Income A 0.00% Negative Infinity Positive Infinitity

No More than 15% per Fund

Constraint

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Table 7: 60% in Domestic Stocks Shadow PriceShadow Price [Change in Portfolio Return Given 1




Shadow Price]

60% in Domestic Stocks 6.60% 57.70% 65.79%

Constraint

Table 8: 30% in Foreign Stocks Shadow PriceShadow Price [Change in Portfolio Return Given 1




Shadow Price]

30% in Foreign Stocks 0.00% Negative Infinity Positive Infinity

Constraint

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Table 9: Non-negativity Shadow PriceShadow Price [Change in Portfolio Return Given 1




Shadow Price]

0% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinity0% in American Funds Growth Fund Of America R3 0.00% Negative Infinity Positive Infinity0% in American Funds New Economy R3 0.00% Negative Infinity Positive Infinity0% in Eaton Vance Government Obligations A -2.62% -7.34% 6.70%0% in Federated Kaufmann Small Cap A 0.00% Negative Infinity Positive Infinity0% in Fidelity Advisor Balanced T 0.00% Negative Infinity Positive Infinity0% in Fidelity Advisor Freedom 2040 T -0.84% -17.29% 4.05%0% in Fidelity Advisor Freedom 2045 T -0.69% -17.29% 4.04%0% in Fidelity Advisor Freedom Income T -2.14% -9.21% 8.40%0% in Janus Overseas S -12.65% -8.41% 7.68%0% in John Hancock Lifestyle Aggressive Portfolio A 0.00% Negative Infinity Positive Infinity0% in John Hancock Lifestyle Balanced Portfolio A -0.04% 12.86% 7.29%0% in John Hancock Lifestyle Conservative Portfolio A -0.77% -8.71% 7.95%0% in John Hancock Lifestyle Growth Portfolio A 0.00% Negative Infinity Positive Infinity0% in John Hancock Lifestyle Moderate Portfolio A -0.27% -10.40% 9.49%0% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinity0% in Oppenheimer Global Strategic Income A -0.88% -7.35% 0.07%0% in Oppenheimer Limited Term Government A -2.77% -7.34% 6.70%0% in Pioneer Disciplined Value A -0.95% -10.77% 2.52%0% in T. Rowe Price Intl Growth & Income R -0.61% -7.45% 6.79%0% in Victory Fund For Income A -2.09% -7.34% 6.70%

Constraint

Non-negativity

R Code

library(lpSolve)

library(lpSolveAPI)

mylp6=make.lp(0,21)

lp.control(mylp6,sense="maximize") #the default is min

set.objfn(mylp6,c(0.0968, 0.1224, 0.1284, 0.0136, 0.1184, 0.0881, 0.0725, 0.0740, 0.0299, -0.0795, 0.0808, 0.0638, 0.0410, 0.0748, 0.0538, 0.0434, 0.0310, 0.0121, 0.0942, 0.0345, 0.0189))

#Obj Function maximize profit using average 5 year return

# X1+ X2+ X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13, X14 X15 X16 X17 X18 X19 X20 X21

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#Constraints

#Total Porfolio = 100%

add.constraint(mylp6, c(1, 1,1,1,1,1, 1,1,1,1,1, 1,1,1,1,1, 1,1,1,1,1),"=",100)

#No More than 15 % in 1 fund

add.constraint(mylp6, c(1, 0,0,0,0,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,0,0,0),"<=",15)





















#60% in Domestic Stocks

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add.constraint(mylp6, c(0.864, 0.7872, 0.6026, 0, 0.7502, 0.6102, 0.6238, 0.6239, 0.1751, 0.1099,0.5687, 0.3705, 0.1358, 0.491, 0.254, 0.0551, 0.0007, 0, 0.9681, 0.0121, 0),"<=",60)

#30% in Foreign Stocks

add.constraint(mylp6, c(0.1226, 0.1359, 0.2789, 0, 0.1112, 0.044, 0.286, 0.286, 0.0774, 0.8624,0.3349, 0.2062, 0.0848, 0.2688, 0.1364, 0.0063, 0.0017, 0, 0.0261, 0.9236, 0),"<=",30)

#Nonnegativity

set.bounds(mylp6, lower=c(rep(0,21)))

#Solution

solve(mylp6)

get.objective(mylp6)

get.variables(mylp6)

#Sensitivity Analysis

get.sensitivity.obj(mylp6)

get.sensitivity.objex(mylp6) #reduced costs

get.sensitivity.rhs(mylp6) #shadow prices / duals

R Code Solution

> #Solution

> solve(mylp6)

[1] 0

> get.objective(mylp6)

[1] 9.779369

> get.variables(mylp6)

[1] 2.843986 15.000000 15.000000 0.000000 15.000000 15.000000 0.000000

[8] 0.000000 0.000000 0.000000 15.000000 0.000000 0.000000 15.000000

[15] 0.000000 7.156014 0.000000 0.000000 0.000000 0.000000 0.000000

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>

> #Sensitivity Analysis

> get.sensitivity.obj(mylp6)

$objfrom

[1] 9.571947e-02 9.173000e-02 7.954353e-02 -1.000000e+30 8.928743e-02

[6] 8.004525e-02 -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30

[11] 7.730560e-02 -1.000000e+30 -1.000000e+30 7.217619e-02 -1.000000e+30

[16] 4.270942e-02 -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30

[21] -1.000000e+30

$objtill

[1] 1.016690e-01 1.000000e+30 1.000000e+30 3.976254e-02 1.000000e+30

[6] 1.000000e+30 8.094306e-02 8.094966e-02 5.132187e-02 4.701765e-02

[11] 1.000000e+30 6.422131e-02 4.872746e-02 1.000000e+30 5.653050e-02

[16] 4.909008e-02 3.980875e-02 3.976254e-02 1.036722e-01 4.056133e-02

[21] 3.976254e-02

> get.sensitivity.objex(mylp6) #reduced costs

$objfrom

[1] 9.571947e-02 9.173000e-02 7.954353e-02 -1.000000e+30 8.928743e-02

[6] 8.004525e-02 -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30

[11] 7.730560e-02 -1.000000e+30 -1.000000e+30 7.217619e-02 -1.000000e+30

[16] 4.270942e-02 -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30

[21] -1.000000e+30

$objtill

[1] 1.016690e-01 1.000000e+30 1.000000e+30 3.976254e-02 1.000000e+30

[6] 1.000000e+30 8.094306e-02 8.094966e-02 5.132187e-02 4.701765e-02

[11] 1.000000e+30 6.422131e-02 4.872746e-02 1.000000e+30 5.653050e-02

[16] 4.909008e-02 3.980875e-02 3.976254e-02 1.036722e-01 4.056133e-02

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[21] 3.976254e-02

$objfromvalue

[1] -1.000000e+30 -1.000000e+30 -1.000000e+30 6.699653e+00 -1.000000e+30

[6] -1.000000e+30 4.045191e+00 4.044480e+00 8.402526e+00 7.676038e+00

[11] -1.000000e+30 7.293912e+00 7.949052e+00 -1.000000e+30 9.489344e+00

[16] -1.000000e+30 6.705085e+00 6.699653e+00 2.519715e+00 6.794812e+00

[21] 6.699653e+00

$objtillvalue

[1] 3.133042e-294 6.382566e-314 1.358077e-309 2.052769e-289 7.566989e-307

[6] 2.803072e-309 1.251613e-308 1.310429e-306 2.803072e-309 1.251613e-308

[11] 2.508662e-310 1.379807e-309 5.432309e-312 2.172924e-311 2.172924e-310

[16] 5.432309e-312 3.259386e-311 3.212612e-319 4.889111e-311 4.880603e+252

[21] 2.781342e-309

> get.sensitivity.rhs(mylp6) #shadow prices / duals

$duals

[1] 0.0397625417 0.0000000000 0.0306699963 0.0488564718 0.0000000000

[6] 0.0291125726 0.0080547534 0.0000000000 0.0000000000 0.0000000000

[11] 0.0000000000 0.0034943998 0.0000000000 0.0000000000 0.0026238101

[16] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000

[21] 0.0000000000 0.0000000000 0.0660155767 0.0000000000 0.0000000000

[26] 0.0000000000 0.0000000000 -0.0261625417 0.0000000000 0.0000000000

[31] -0.0084430585 -0.0069496600 -0.0214218692 -0.1265176536 0.0000000000

[36] -0.0004213129 -0.0077274570 0.0000000000 -0.0027304982 0.0000000000

[41] -0.0088087526 -0.0276625417 -0.0094722215 -0.0060613302 -0.0208625417

$dualsfrom

[1] 9.330035e+01 -1.000000e+30 1.568775e+00 -1.776357e-15 -1.000000e+30

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[6] 8.538340e-01 -1.776357e-15 -1.000000e+30 -1.000000e+30 -1.000000e+30

[11] -1.000000e+30 -1.776357e-15 -1.000000e+30 -1.000000e+30 1.776357e-15

[16] -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30 -1.000000e+30

[21] -1.000000e+30 -1.000000e+30 5.769950e+01 -1.000000e+30 -1.000000e+30

[26] -1.000000e+30 -1.000000e+30 -7.343750e+00 -1.000000e+30 -1.000000e+30

[31] -1.729031e+01 -1.728727e+01 -9.210335e+00 -8.414003e+00 -1.000000e+30

[36] -1.285714e+01 -8.713266e+00 -1.000000e+30 -1.040164e+01 -1.000000e+30

[41] -7.349705e+00 -7.343750e+00 -1.076999e+01 -7.448057e+00 -7.343750e+00

$dualstill

[1] 1.073437e+02 1.000000e+30 1.814233e+01 1.920183e+01 1.000000e+30

[6] 1.830960e+01 1.914430e+01 1.000000e+30 1.000000e+30 1.000000e+30

[11] 1.000000e+30 1.947917e+01 1.000000e+30 1.000000e+30 2.027759e+01

[16] 1.000000e+30 1.000000e+30 1.000000e+30 1.000000e+30 1.000000e+30

[21] 1.000000e+30 1.000000e+30 6.578850e+01 1.000000e+30 1.000000e+30

[26] 1.000000e+30 1.000000e+30 6.699653e+00 1.000000e+30 1.000000e+30

[31] 4.045191e+00 4.044480e+00 8.402526e+00 7.676038e+00 1.000000e+30

[36] 7.293912e+00 7.949052e+00 1.000000e+30 9.489344e+00 1.000000e+30

[41] 6.705085e+00 6.699653e+00 2.519715e+00 6.794812e+00 6.699653e+00

BibliographyGary P. Brinson, B. D. (1991). Determinants of Portfolio Performance II: An Update. Financial Analysts

Journal, 40.

Gary P. Brinson, L. R. (1986). Determinants of Portfolio Performance. Financial Analysts Journal, 39-44.

Jahnke, W. W. (2004). The Asset Allocation Hoax. Journal of Financial Planning, 64-71.

Mark Kritzman, L. R. (2006). Determinants of Portfolio Performance: 20 Years Later. Financial Analysts Journal, Vol. 62, No. 1, 10-13.

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Michael E. Kitces, M. F. (2013). Tactical Asset Allocation. Journal Of Financial Planning, 30-36.

Morningstar, Inc. (2015, October 11). Performance Overview. Retrieved from Yahoo! Finance: finance.yahoo.com

Nalan Gulpınar, K. K. (2011). Robust portfolio allocation under discrete asset choice constraints. Journal Of Asset Management, 67-83.

Paychex, Inc. (2015, October 8). Research funds. Retrieved from Paychex Flex: https://myapps.paychex.com

Weliver, D. (n.d.). 401k Asset Allocation: Two Methods for Headache-Free Diversification. Retrieved from www.moneyunder30.com: http://www.moneyunder30.com/401k-asset-allocation

Wittig, A. K. (2014). Rethinking Portfolio Rebalancing: Introducing Risk Contribution Rebalancing as an Alternative Approach to Traditional Value-Based Rebalancing Strategies. Journal Of Portfolio Management, 34-46.

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individual 401k portfolio asset allocation optimization

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