Multi-Period Portfolio Selection: A Practical Simulation ...

Multi-Period Portfolio Selection:A Practical Simulation-BasedFrameworkJuly 2021

Nicholas Savoulides, PhD., CFAHead of Investment SolutionsResearch & Analytics

For Q-Group use only. Not for further distribution.

1 Literature review

2 Requisites for practical solutions

3 Problem setup

4 Analytical framework & guiding principles

5 Simulation-based framework & observations

6 Conclusion

7 Next steps

Appendix

Agenda

For Q-Group use only. Not for further distribution. 2

Literature review



Literature ReviewThe long road to multi-period solutions

• Markowitz (1959): Discusses how multi-period portfolio selection solution using dynamic programming could be approached but indicated that practical solutions were “far beyond the available and foreseeable computers.”

• Mossin (1968), Merton (1969), and Samuelson (1969): Provide the foundational constructs for multi-period portfolio selection using dynamic programming in both continuous and discrete time (under simplifying assumptions).

• Markowitz (1991): Proposes “Game of Life” simulation for modeling complex investment planning problems.

• Langetieg et al. (1990) and Liebowitz et al. (2014) : Provide insights regarding immunization of duration risk across multi-period investment horizons.

• Blay and Markowitz (2016): Present a simulation-based Net Present Value portfolio analysis approach that accounts for tax implications across a multi-period investment horizon.

• Das, et al. (2019): Present a practical dynamic programming approach for goals-based wealth management but limit investment selection to a finite set of MV efficient portfolios.

Note: Campbell and Viceira (2002) provide an excellent overview of advances in multi-period portfolio selection as well as discussion regarding the manychallenges faced by long-horizon investors.

Requisites for practical solutions



Three Key Requisites……Simulation-based approach allowed for required flexibility and practicality

A hold to maturity bond is the risk-free asset at its maturity

Price paths of a 10-year Treasury Strip

3Allocation & DurationProfile Alignment

1 2

Equity markets mean-revert, interest rates at all time low, etc.

1-std cumulative equity returns

Taxes are based on basis cost which are path dependent

Various investment taxes:

Capital Gains Tax= TCG* ( Price – Cost Basis)

Income Tax= TINC* ( Dividends)

Real-WorldAsset Dynamics

Investment Frictionsand Illiquidities

2 3

For Illustrative purposes only.

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0 1 2 3 4 5 6 7 8 9 10

Pric

e ($

)

Years to Maturity

-75%

-50%

-25%

0%

25%

50%

75%

0 5 10 15 20

Cum

ulat

ive

retu

rn

Years

Problem setup



Problem SetupNotations and portfolio wealth equation

• Investor has investment horizon of T periods indexed by t = 0, 1,…,T

• Investor has access to N assets

• N x 1 random vectors rt for t = 0, 1,…,T-1 represents asset returns observed between time t and t+1

• Investor has initial capital C0 at time t = 0 with series of future net cash flows C1, C2,…,CT at time t = 1, 2,…,T (where inflows are positive and outflows are negative)

• Investment strategy given by sequence of N x 1 portfolio weight vectors wt for t = 0, 1,…,T-1 represented as W = (w0,w1,…,wT-1)

• Portfolio wealth through time is Pt for t = 0,1,…, T with P0 = C0

𝑃 𝑾 𝑃 𝐶 1 𝑤 𝑟 … 1 𝑤 𝑟 𝐶


Problem SetupExamine two types of investor objectives

Growth Investor’s Objective Income Investor’s Objective1 2

Maximize terminal wealth while reducing uncertainty around terminal wealth

Where 𝑄 𝑾 and 𝑄 𝑾 are the 𝛼-th and 𝛽-th quantiles of the probability distribution of terminal wealth

Parameter 𝜆 0 balances reward (i.e., median) against risk (i.e., median minus 5th percentile)

𝐽 𝑾 𝑄 𝑾 𝜆 𝑄 𝑾 𝑄 𝑾

Maximize series of additional consumption cash flows, A starting at t = TA, for a given probability of remaining solvent

Where 𝜏 𝑾 is the timing of the insolvency (i.e.point in time when wealth falls below zero for first time)

Resulting cash flows:𝐶 ,𝐶 , … ,𝐶 𝐴, … ,𝐶 𝐴

𝐽 𝑾 𝑃𝑟𝑜𝑏 𝜏 𝑾 𝑇 1

Analytical framework& guiding principles



Analytical framework simplifying approximationsAssume small portfolio returns and mean-variance objective

Assume small portfolio returns, 1 𝑟 𝑟 ⋯ 𝑟 = 1 𝑟 1 𝑟 … 1 𝑟

𝑃 𝑾 𝐶 1 𝒘𝒕 𝒓𝒕 ⋯ 𝒘𝑻 𝟏𝒓𝑻 𝟏 𝐶 𝑋 𝒘𝒕 𝒓𝒕 𝑋

where 𝑋 𝐶 ⋯ 𝐶 are the partial sums of the cash flows for 𝑡 0, 1, … ,𝑇.

Proxy the percentile-based objective as a mean-variance objective:

𝐽 𝑾 𝐸 𝑃 𝑾 𝜆𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑃 𝑾

1

2


Analytical framework without duration assetsClosed form solution when assuming i.i.d. normal distributions

Assumed Asset Dynamics Optimal Solution

𝒘𝒕𝑰𝟏 𝑨

12𝜆𝑋 𝒎 𝒃 𝒆𝑵

where:𝑰 is the 𝑁 1 𝑁 1 identity matrix,𝟏 is the 𝑁 1 1 vector of ones, 𝒆𝑵 is the 𝑁 1 index vector with one at the 𝑁-th position and zero everywhere else𝒎 is the vector of excess returns of the first 𝑁 1assets over the 𝑁-th asset,𝑨 𝑰 𝟏 𝚺 𝑰 𝟏 is the relative covariance matrix of the first 𝑁 1 assets with respect to the 𝑁-th asset, and finally𝒃 𝑰 𝟏 𝚺𝒆𝑵 captures diversification benefit of the first 𝑁 1 assets with the 𝑁-th asset.

N assets assumed to be independently and

identically distributed (i.i.d.) normal random

variables with mean mean 𝝁 and covariance

matrix 𝜮.


Analytical framework with duration assetsAssumed normally distributed yield process with sloped yield curve

Duration Asset Dynamics

Assume following discrete yield process:

𝑦 𝑦 Δ𝑦 Δ𝑦 ⋯ Δ𝑦

Where ∆𝑦 are independently and normally distributed as 𝑁 𝜇 , ,𝜎 for 𝑡 0, 1, … , T and the time dependent parameter 𝜇 , represents the drift of the instantaneous yield process

The parameter 𝜎 captures the volatility of the instantaneous yield process

Allow for Sloped Yield Curve

𝑦 𝑦𝑇𝑠𝑇

Note: Growth asset assumed i.i.d. 𝑁 𝜇 ,𝜎 , and at any time 𝑡 the growth asset return 𝑟 and the change in instantaneous yield Δ𝑦 has a correlation 𝜌 , , i.e., 𝑐𝑜𝑣 𝑟 ,Δ𝑦 𝜌 , 𝜎 𝜎 . The correlation between the duration asset and the growth asset is approximately 𝜌 , 𝜌 , .

0

Yiel

d

TmaxT

yt

𝑦𝑇𝑠𝑇

𝑦 𝑠


Analytical framework with duration assetsClosed form solutions for optimal asset weights and durations

When the optimal durations 𝐷 ,𝐷 , … ,𝐷 are known, then the optimal growth asset weights through time 𝒘 𝑤 ,𝑤 , … ,𝑤 are:

𝒘 𝑼1

2𝜆𝒎 𝒗

when the optimal growth asset weights 𝑤 ,𝑤 , … ,𝑤 are known, the optimal duration is:

𝐷𝑋 1 𝑤

𝑋 1 𝑤

𝜌 , 𝜎 𝑤

𝜎 1 𝑤

𝜇 ,

2𝜆𝑋 1 𝑤 𝜎

𝑠

𝜆𝑋 𝑇 1 𝑤 𝜎

Note: 𝑼 𝜎 𝑿𝒅 𝜎 𝑿𝒖𝑿𝒖 2𝜌 , 𝜎 𝜎 𝑿𝒅𝑿𝒖; 𝟏;𝒎 𝜇 𝑿𝒅𝟏 𝑿𝒅𝝁𝑫; 𝒗 𝜌 , 𝜎 𝜎 𝑿𝒅𝑿𝒖 𝜎 𝑿𝒖𝑻𝑿𝒖 ; see Appendix 1 for 𝑿𝒅 and 𝑿𝒖


Analytical ExamplesExamined two cases each with two assets

Growth+

Growth

Duration+

Growth

• return 𝜇 6%• volatility 𝜎 15%


High GrowthAsset

Low GrowthAsset

• yield 𝑦 2%; no drift• volatility 𝜎 1%


DurationAsset

GrowthAsset


Analytical examplesAnalyze two problems, without and with interim cash flows

Maximize terminal wealthwithout interim cash flows

Maximize terminal wealthwith interim cash flows1 2

• 10-year investment horizon

• Initial investment of $1,000

• No interim cash flows



• Annual inflows of $500 over the first 5 years and annual outflows of $500 over the last 5 years

Note: Selected points on frontiers with an average 50%/50% allocation between assets

?-1,500

-500

500

1,500

0 1 2 3 4 5 6 7 8 9 10

Cas

h Fl

ows

Time (years)

?

-1,500

-500

500

1,500

0 1 2 3 4 5 6 7 8 9 10

Cas

h Fl

ows

Time (years)


Analytical example 1: Without interim cash flowsD

urat

ion

+ G

row

th

0255075

100g,d = - 0.25

Allo

catio

n %

Slop

e =

2%

Slop

e =

0%

Slop

e =

-2%

0255075

100

0255075

100

0 2 4 6 8

0

10

20

30

0

10

20

30

0255075

100

Gro

wth

+ G

row

th

Time (years)

0

10

20

30

0 2 4 6 8

Dur

atio

n (y

ears

)

% Low Growth Asset

% High Growth Asset

% Duration Asset

% Growth Asset

Asset DurationPortfolio Duration

= -1 = 0 = 1

0

10

20

30

0 2 4 6 8

g,d = 0 g,d = 0.25


Analytical example 2: With interim cash flows

0255075

100g,d = - 0.25

Allo

catio

n %

Slop

e =

2%

Slop

e =

0%

Slop

e =

-2%

0255075

100

0255075

100

0 2 4 6 8

0

10

20

30

0

10

20

30

0255075

100

Time (years)

0

10

20

30

0 2 4 6 8

Dur

atio

n (y

ears

)

% Low Growth Asset

% High Growth Asset

% Duration Asset

% Growth Asset

Asset DurationPortfolio Duration

= -1 = 0 = 1

0

10

20

30

0 2 4 6 8

g,d = 0 g,d = 0.25

Dur

atio

n+

Gro

wth

Gro

wth

+ G

row

th


Analytical framework guiding principles I/IIAllocations should aim for roughly constant dollar risk through time

Solutions should shift to safer (riskier) allocations as inflows (outflows) occur

Simplifying for two assets:

𝑤 ,1

2𝜆𝑋𝜇 𝜇

𝜎 2𝜌𝜎 𝜎 𝜎𝜎 𝜌𝜎 𝜎

𝜎 2𝜌𝜎 𝜎 𝜎

𝑤 ,1

2𝜆𝑋𝜇 𝜇

𝜎 2𝜌𝜎 𝜎 𝜎𝜎 𝜌𝜎 𝜎

𝜎 2𝜌𝜎 𝜎 𝜎

1

𝑋 𝑤 , 𝑋 𝑤 𝑐 𝜆

𝑋 𝑤 , 𝑋 𝑤 𝑐 𝜆

Note: 𝑤 and 𝑤 are lowest risk allocations which are constant through time obtained by setting 𝜆 ∞; 𝑐 𝜆 is a constant proportional to 𝜆.

The allocation to dollar risk through time should remain roughly constant,resulting in the typical U-shaped glidepath as inflows precede later outflows


Analytical framework guiding principles II/IIDuration should be dictated by future inflows/outflows and yield curve expectations

Asset duration should match the investment horizon when there are no cash flows

𝐷 1 𝑇 𝑡

2

Asset duration should be extended (shortened) with expected inflows (outflows)

𝐷 𝑇 𝑡1

1 𝑤 𝐶 ⋯ 𝐶 1 𝑤 𝐶 𝑇 𝑘

3

Duration should be adjusted based on the slope of the yield curve, expected correlation of duration and growth assets, and expected yield curve changes

4

Simulation-based framework& observations



Simulation-based framework methodologyOptimization algorithm

Setup and Central Ideas ADAM Steps

The optimization process considers the totality of the allocation decisions a.k.a. weight vectors. In other words, it updates all the weight vectors (w0, w1,…, wT-1)=W simultaneously.

Adaptive Moment (ADAM) based gradient search algorithm

Compute gradient g(n) at the current step-nmaximize𝑾 𝒘 ,𝒘 ,…,𝒘𝑻

𝐽 𝑾Arbitrary Objective

𝒈𝜕𝐽 𝑾𝜕𝑾

𝜕𝐽 𝒘𝟎 ,𝒘𝟏 , … ,𝒘𝑻 𝟏

𝜕𝒘𝟎 𝜕𝒘𝟏 …𝜕𝒘𝑻 𝟏

1

2

3

Use exponentially weighted moving average of gradient and squared gradient to update weights

𝒎 𝛽 𝒎 1 𝛽 𝒈

𝒗 𝛽 𝒗 1 𝛽 𝒈

𝒘 ,𝒘 , … ,𝒘𝑻 𝒘 ,𝒘 , … ,𝒘𝑻 ℎ𝒎

𝒗 𝜀

Update weights1 at step-n

(1) Bias corrected values of m and v are used instead of the raw m and v. h is the step size. See appendix 1 for details.


Simulation-based asset modelingSimulated five asset groups using Moody’s Analytics ESG

Simulation Assets

Central to the solutions produced by SBPS are the underlying stochastic processes used to simulate the evolution of asset prices across the investment horizon.

We assume practitioners can employ a simulation model that reflects real-world price dynamics.

For this analysis, we used the Moody’s Analytics Economic Scenario Generator (ESG) engine to simulate the time series.

• US equities

• Developed ex US equities

• Emerging market equities

• US Treasury zero coupon bonds (1-yr, 2-yr, ..., 30-yr constant maturity TR indices)

• TIPS (1-yr, 2-yr, ..., 30-yr constant maturity TR indices)


Maximize terminal wealthwithout interim cash flows

Maximize terminal wealthwith interim cash flows1 2



• No interim cash flows



• Annual inflows of $500 over the first 5 years and annual outflows of $500 over the last 5 years

Maximize real consumption



• Annual nominal inflows of $50,000 for first 20 years

• Annual real outflows years 20–40

-100-50

050

100

40 45 50 55 60 65 70 75 80

AccumulationConsumption

Cas

h Fl

ows

Age (years)

?-1,500

-500

500

1,500

0 1 2 3 4 5 6 7 8 9 10

Cas

h Fl

ows

Time (years)

?

-1,500

-500

500

1,500

0 1 2 3 4 5 6 7 8 9 10

Cas

h Fl

ows

Time (years)

3

Simulation-based examplesAnalyze two terminal wealth and one real consumption problem


Simulation-based example: Maximize wealth without cash flowsAl

loca

tion

% % Fixed Income (zero coupon)

% Equity

Fixed Income Duration

0

25

50

75

100

0 2 4 6 80

25

50

75

100

0 2 4 6 8

Buy and hold

0

25

50

75

100

0 2 4 6 8

Developed

US

EM10

9

21

10+6-10 5 4 3 2 1

05

1015

0 2 4 6 8

Dur

atio

n (y

rs)

Time (years)

05

1015

0 2 4 6 805

1015

0 2 4 6 8

Low Risk Medium Risk High Risk

Low Risk Medium Risk

High Risk

1,200

1,400

1,600

1,800

0 200 400 600 800 1,000 1,200Median - 5th Percentile of Terminal Wealth ($)

Med

ian

Term

inal

Wea

lth ($

)


Simulation-based example: Maximize wealth with cash flowsAl

loca

tion

% % Fixed Income (zero coupon)

% Equity


109

21

10+6-10

5 4 3 2 1

Dur

atio

n (y

rs)

Time (years)

0102030

0 2 4 6 8

Median - 5th Percentile of Terminal Wealth ($)

Med

ian

Term

inal

Wea

lth ($

)

Low Risk Medium Risk

High Risk

1,500

2,000

2,500

0 500 1,000 1,500 2,000 2,500 3,000

0

25

50

75

100

0 2 4 6 8

10+6-10 5 4 3

0

25

50

75

100

0 2 4 6 8

21

10+6-10 5 4 3 2 1

0

25

50

75

100

0 2 4 6 8

Developed

US

EM

0102030

0 2 4 6 80

102030

0 2 4 6 8



Simulation-based wealth examples observationsConsistent with analytical framework, including real-asset dynamics and illiquidities

SBPS generates allocations and duration profiles aligned with analytical expectations

Allocations shift to safer (riskier) allocations as inflows (outflows occur); duration profiles align with investment horizons when there are no cash flows; duration is extended (shortened) in the presence of future inflows (outflows) (1)

SBPS allows for long-term real-world asset dynamics

Fixed income instruments are priced off of simulated interest rate curves capturing unique characteristics of current low yield environment. In the no-interim cash flows, medium risk allocation solution we notice a shortening of duration early on which is aligned with a simulated initial rise in rates

SBPS allows for the consideration of investment frictions and illiquidities

The presented analyses included 50bps of transaction costs to better reflect trading costs. One of the impacts of this can be seen in the no-interim cash flows, low risk allocation solution where we notice that the holdings reflect a buy a hold strategy of a 10-year Treasury STRIP.

1

2

3

(1) For more details, please see a direct comparison of SBPS and the analytical framework in the appendix of accompanying paper.


Simulation-based example: Maximize consumptionAl

loca

tion

% % Fixed Income (TIPS)

% Equity


109

21

10+6-10

5 4 3 2 1

Dur

atio

n (y

rs)

Time (years)

1- Probability of Success (%)

Dis

tribu

tions

(Rea

l $00

0’s)

21

10+6-10 5 4 3 2 1

0

25

50

75

100

0 10 20 30

11-2020+

0

25

50

75

100

0 10 20 30

20+11-20 6-10

0

25

50

75

100

0 10 20 30

Developed

20+

0102030

0 10 20 30


0102030

0 10 20 300

10

20

30

0 10 20 30

EM

US

Low RiskMedium Risk High Risk

0

100

200

300

0 10 20 30 40 50 60 70 80 90 100


Simulation-based consumption example observationsDuration early-on for low-risk and later-on for high-risk solutions

Lowest risk solutions seek duration early on and diversify with growth assets in later years

Observing the lowest risk allocation, the solution focuses on duration assets with extended duration early on seeking to eliminate real rate risk associated with future purchases (and dispositions) of bonds. As we move more towards outflows, duration shortens, and some growth assets are introduced for diversification.

Highest risk solutions focus on growth assets early with duration being added in later years

Observing the high-risk allocation, we notice that the solution does not maintain 100% allocation to growth assets through time, but instead moves to safer assets as outflows begin. The reason for this is that once outflows begin, they are assumed to be constant and there is no utility for any remaining balance. For a given outflow, the more successful paths will only be exposed to downside. This downside risk is mitigated by moving to more duration assets once the outflows begin.

1

2

Conclusion



Conclusion Key innovations of this research

1. Development of an analytical framework for the multi-period mean-variance problem that provides a theoretical foundation for multi-period portfolio selection and provides intuition for how portfolio allocations and duration should evolve through time given a specific investor objective.

2. The development of a flexible simulation-based multi-period portfolio selection framework that addresses the practical realities of implementing and managing multi-period solutions and allows for the incorporation of individual hold-to-maturity fixed income investments alongside traditional investments used for strategic asset allocation.

3. Methods for leveraging of advances in computing power and machine learning algorithms to solve multi-period portfolio selection problems using a simulation-based approach.

Next steps



Next stepsInclude state dependence in deriving optimal solution

t=2 t=T-1…

W1(1)

w0

W1(k)

…

WT-1(1)

WT-1(k)

W2(1)

W2(k)

…

t=Tt=1

… …

t=0

N s

imul

ated

pat

hs

of p

ortfo

lio v

alue

kst

ates

Setup The state-space grid

Current approach allows for allocations to vary through time but not across states i.e., all paths at a given future time will have the same allocation

If we consider k possible states at each future time, we will solve for 1+k(T-1) weight vectors.

maximize𝑾 , ,…, ,…, ,…,

𝐽 𝑾

State dependence can be incorporated in existing framework

Appendix

34For Q-Group use only. Not for further distribution.


Appendix 1Analytical framework with duration assets: and

𝑿𝒅

𝑋 0 ⋯ 00 𝑋 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 … 𝑋

𝑎𝑛𝑑 𝑿𝒖

𝑋 1 𝐷 𝑋 ⋯ 𝑋

0 𝑋 1 𝐷 ⋯ 𝑋⋮ ⋮ ⋱ ⋮0 0 … 𝑋 1 𝐷

Bias corrected exponentially weighted moving average (EWMA) of gradient at step-n is


Appendix 2ADAM bias correction

𝒎𝒎

1 𝛽

Bias corrected EWMA of squared gradient at step-n is

𝒗𝒗

1 𝛽

The raw exponentially weighted moving average (EWMA) of gradient and squared gradient is biased towards the starting values 𝒎 and 𝒗 which are 𝟎. If not bias corrected, the algorithm will require a large number of iterations to mitigate the bias when 𝛽 and 𝛽 are close to 1


Appendix 3Sample simulation data summary

Sample US Yield Curve

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 10 20 30

Yiel

d

Maturity (years)

Arithmetic mean return (µ) and standard deviation (σ) of simulated returns

Asset Statistic t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 7 t = 10 t = 20 t = 30 t = 40

US Equities µ 6.8 6.6 6.5 6.7 6.8 7.1 7.3 7.6 8.1 8.4 8.5σ 20.2 20.7 20.7 20.8 20.8 20.9 20.9 21.1 21.2 21.3 21.3

Developed Equities

µ 6.0 5.8 5.9 6.0 6.1 6.4 6.6 6.8 7.4 7.6 7.8σ 18.4 18.5 18.4 18.6 18.6 18.5 18.7 18.8 18.9 18.9 18.9

EM Equities µ 7.7 7.5 7.6 7.7 7.8 8.1 8.4 8.6 9.0 9.4 9.6σ 26.7 27.5 27.7 27.7 27.9 27.8 28.0 28.0 28.2 28.3 28.4

STRIPS 1-Year µ 1.9 1.7 1.8 1.9 2.1 2.2 2.5 2.8 3.3 3.5 3.7σ 0.0 1.1 1.4 1.6 1.8 2.0 2.2 2.5 3.0 3.2 3.3

STRIPS 10-Year µ 1.9 1.5 1.6 1.7 1.9 2.1 2.4 2.9 3.9 4.5 4.8σ 5.4 5.7 5.9 6.1 6.3 6.5 6.8 7.2 8.2 8.6 8.8

STRIPS 30-Year µ 2.4 1.8 1.6 1.5 1.5 1.7 2.1 2.6 3.3 4.5 5.1σ 9.9 10.0 10.3 10.5 10.7 10.9 11.3 11.7 12.7 13.6 13.9

TIPS 1-Year µ 1.9 1.7 1.8 1.9 2.1 2.2 2.5 2.8 3.3 3.5 3.7σ 0.9 1.4 1.7 1.9 2.0 2.2 2.4 2.6 3.2 3.4 3.5

TIPS 10-Year µ 2.0 1.6 1.6 1.8 1.9 2.1 2.5 3.0 4.1 4.6 4.8σ 5.7 5.8 5.9 6.0 6.0 6.1 6.2 6.3 6.6 6.7 6.7

TIPS 30-Year µ 2.2 1.7 1.7 1.9 2.0 2.2 2.6 3.1 4.4 5.1 5.4σ 11.0 10.9 11.0 11.0 11.0 11.1 11.1 11.2 11.4 11.5 11.5

Inflation µ 1.4 1.3 1.5 1.7 1.8 1.8 1.9 2.0 2.2 2.3 2.5σ 0.9 1.4 1.7 1.9 2.0 2.2 2.4 2.6 3.2 3.3 3.4

Source: https://home.treasury.gov/

Source: Moody’s Analytics, Invesco

Blay, K. and Markowitz, H. (2016). “Tax-Cognizant Portfolio Analysis: A Methodology for Maximizing after-Tax Wealth,” Journal of InvestmentManagement 14(1), 26–64.

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References

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For Q Group use only. This presentation is for informational and educational purposes only. All material presented is compiled from sources believed to be reliable and current, but accuracycannot be guaranteed. This is not to be construed as an offer to buy or sell any financial instruments and should not be relied upon as the sole factor in an investment making decision. Aswith all investments there are associated inherent risks. This should not be considered a recommendation to purchase any investment product. This does not constitute a recommendation ofany investment strategy for a particular investor. Investors should consult a financial professional before making any investment decisions if they are uncertain whether an investment issuitable for them. Please obtain and review all financial material carefully before investing. The opinions expressed are those of the author, are based on current market conditions and aresubject to change without notice. These opinions may differ from those of other Invesco investment professionals. Invesco Advisers, Inc. NA6219 – 07/21.

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