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Contents
Problem.............................................................................................................................................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......................................................................................Error! Bookmark not defined.
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
Limitations.........................................................................................................................................8
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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...........................................................................................................12
Equation 6: Objective Function.....................................................................................................12
Table 3: Optimal Solution.............................................................................................................13
Table 4: Objective Function Sensitivity..........................................................................................13
Table 5: 100% Allocation Shadow Price.........................................................................................14
Table 6: 15% per Fund Shadow Price.............................................................................................14
Table 7: 60% in Domestic Stocks Shadow Price..............................................................................15
Table 8: 30% in Foreign Stocks Shadow Price.................................................................................15
Table 9: Non-negativity Shadow Price...........................................................................................16
R Code .........................................................................................................................................16
R Code Solution ............................................................................................................................18
Bibliography.................................................................................................................................21
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Problem
This report provides information on the author’s optimal portfolio mix model for an individual
401k plan.
Significance
When 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.
Data
The 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 Model
This 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.
Literature
This 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 Performance
By 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 Hoax
By William Jahnke
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
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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 Rebalancing
By 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).
RobustAllocation
By 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 Asset
By 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 Variables
Each variable indicates the investment percentage allocated to each fund. Please see x
Table 1 Decision Variables
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on page 9 for a list of decision variables used in this model.
Constraints
Allocation 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 Allocation
The 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
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
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asset that was not domestic or foreign stock was categorized as bonds and other. Please see Table
2: Asset Allocation within Each Fund on page 10 for the asset allocation within each fund.
No More than 15% per Fund
In 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 Stocks
As 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 Stocks
Although 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.
Objective Function
The 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.
Formulation
This model transforms the objective coefficient and constraints listed above into R. Please see
Table 5: 100% AllocationShadow Price
Constraint
ShadowPrice [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
ShadowPrice]
Higher Limit
[Increase by
ShadowPrice]
100% Allocation 3.98% 93.30% 107.34%
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Table 6: 15% per Fund Shadow Price
Table 7: 60% in Domestic Stocks Shadow Price
Shadow Price [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
Shadow Price]
Higher Limit
[Increase by
Shadow Price]
15% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% 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 Infinitity
15% in Fidelity Advisor Freedom 2045 T 0.00% Negative Infinity Positive Infinitity
15% in Fidelity Advisor Freedom Income T 0.00% Negative Infinity Positive Infinitity
15% in Janus Overseas S 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% in John Hancock Lifestyle Conservative Portfolio A 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinitity
15% in Oppenheimer Global Strategic Income A 0.00% Negative Infinity Positive Infinitity
15% in Oppenheimer Limited Term Government A 0.00% Negative Infinity Positive Infinitity
15% in Pioneer Disciplined Value A 0.00% Negative Infinity Positive Infinitity
15% in T. Rowe Price Intl Growth & Income R 0.00% Negative Infinity Positive Infinitity
15% in Victory Fund For Income A 0.00% Negative Infinity Positive Infinitity
No More than 15% per Fund
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]
60% in Domestic
Stocks 6.60% 57.70% 65.79%
Constraint
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Table 8: 30% in ForeignStocks Shadow Price
Table 9: Non-negativityShadow Price
R Code on page 14 for the full R script.
Optimal Objective Function
The 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! Bookmark not defined. for the full R script.
Shadow Price [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
Shadow Price]
Higher Limit
[Increase by
Shadow Price]
30% in Foreign
Stocks 0.00% Negative Infinity Positive Infinity
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]
0% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinity
0% in American Funds Growth Fund Of America R3 0.00% Negative Infinity Positive Infinity
0% in American Funds New Economy R3 0.00% Negative Infinity Positive Infinity
0% in Eaton Vance Government Obligations A -2.62% -7.34% 6.70%
0% in Federated Kaufmann Small Cap A 0.00% Negative Infinity Positive Infinity
0% in Fidelity Advisor Balanced T 0.00% Negative Infinity Positive Infinity
0% 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 Infinity
0% 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 Infinity
0% in John Hancock Lifestyle Moderate Portfolio A -0.27% -10.40% 9.49%
0% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinity
0% 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
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Decision Variable Values
The 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 Analysis
Objective Function Reduced Cost
The 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 Price
Shadow 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.
100% Allocation
If 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
Table 5: 100% AllocationShadow Price on page 14.
No More than 15% per Fund
Please 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
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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 Stocks
This 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 DomesticStocks 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 ForeignStocks Shadow Price on page 15.
Non-negativity
If 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-negativityShadow
Price 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%.
Limitations
There 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
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model will be kept on file and used to update asset allocation yearly for the author and her
colleagues if they so wish.
Learning
The 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
Equation 1: AllocationMust Equal 100%
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
X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12+X13+X14+X15+X16+X17+X18+X19+X20+X21=100
Table 1 DecisionVariables
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Table 2: AssetAllocationwithin EachFund
Variable
Name
Fund 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 Stocks
Equation 4: 30% in ForeignStocks
0.864X1+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
0.1226X1+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
X1 <=15
X2 <=15
X3 <=15
X4 <=15
X5 <=15
X6 <=15
X7 <=15
X8 <=15
X9 <=15
X10 <=15
X11 <=15
X12 <=15
X13 <=15
X14 <=15
X15 <=15
X16 <=15
X17 <=15
X18 <=15
X19 <=15
X20 <=15
X21 <=15
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Table 3: Optimal Solution
Table 4: Objective Function Sensitivity
Investment
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%
Variable
Name
Fund 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 Infinity
X3 American Funds New Economy R3 12.84% 15.00% 7.95% Positive Infinity
X4 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 Infinity
X6 Fidelity Advisor Balanced T 8.81% 15.00% 8.00% Positive Infinity
X7 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 Infinity
X12 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 Infinity
X15 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%
18. Individual 401KPortfolioAssetAllocationOptimization |Rachel Sampalli
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Table 5: 100%Allocation Shadow Price
Table 6: 15% per Fund Shadow Price
Constraint
ShadowPrice [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
ShadowPrice]
Higher Limit
[Increase by
ShadowPrice]
100% Allocation 3.98% 93.30% 107.34%
Shadow Price [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
Shadow Price]
Higher Limit
[Increase by
Shadow Price]
15% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% 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 Infinitity
15% in Fidelity Advisor Freedom 2045 T 0.00% Negative Infinity Positive Infinitity
15% in Fidelity Advisor Freedom Income T 0.00% Negative Infinity Positive Infinitity
15% in Janus Overseas S 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% in John Hancock Lifestyle Conservative Portfolio A 0.00% Negative Infinity Positive Infinitity
15% 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 Infinitity
15% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinitity
15% in Oppenheimer Global Strategic Income A 0.00% Negative Infinity Positive Infinitity
15% in Oppenheimer Limited Term Government A 0.00% Negative Infinity Positive Infinitity
15% in Pioneer Disciplined Value A 0.00% Negative Infinity Positive Infinitity
15% in T. Rowe Price Intl Growth & Income R 0.00% Negative Infinity Positive Infinitity
15% in Victory Fund For Income A 0.00% Negative Infinity Positive Infinitity
No More than 15% per Fund
Constraint
19. Individual 401KPortfolioAssetAllocationOptimization |Rachel Sampalli
18 | P a g e
Table 7: 60% in Domestic Stocks Shadow Price
Table 8: 30% in ForeignStocks Shadow Price
Shadow Price [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
Shadow Price]
Higher Limit
[Increase by
Shadow Price]
60% in Domestic
Stocks 6.60% 57.70% 65.79%
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]
30% in Foreign
Stocks 0.00% Negative Infinity Positive Infinity
Constraint
20. Individual 401KPortfolioAssetAllocationOptimization |Rachel Sampalli
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Table 9: Non-negativityShadow Price
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
Shadow Price [Change in
Portfolio Return Given 1
Unit Change in
Constraint]
Lower Limit
[Decrease by
Shadow Price]
Higher Limit
[Increase by
Shadow Price]
0% in Allianzgi Nfj Dividend Value A 0.00% Negative Infinity Positive Infinity
0% in American Funds Growth Fund Of America R3 0.00% Negative Infinity Positive Infinity
0% in American Funds New Economy R3 0.00% Negative Infinity Positive Infinity
0% in Eaton Vance Government Obligations A -2.62% -7.34% 6.70%
0% in Federated Kaufmann Small Cap A 0.00% Negative Infinity Positive Infinity
0% in Fidelity Advisor Balanced T 0.00% Negative Infinity Positive Infinity
0% 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 Infinity
0% 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 Infinity
0% in John Hancock Lifestyle Moderate Portfolio A -0.27% -10.40% 9.49%
0% in Loomis Sayles Bond Admin 0.00% Negative Infinity Positive Infinity
0% 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
