Optimal Retirement Financial Planning Model
Justin Xu, Zhiguo Wang
September 9, 2016
Background
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About 4 million people retire every
year
In 20 years, about 80 million people
are expected to retire (By
reference.com)
22.4% of seniors saved 300,000 and
more (By time.com 2016)
$1 million can roughly yield $40,000
per year until age 95 without proper
retirement investment.
Retirement Consumption
1,200,000
1,000,000
800,000
Account Value
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600,000
400,000
200,000
65
70
75
80
Age
85
90
95
Strategy Comparison
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What is the maximum likelihood
of having at least $1 at age 95?
Strategy Comparison
$66,000
$64,000
$62,000
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What is the maximum annual
spending with 95% likelihood of
having at least $1 at age 95?
$60,000
$58,000
$65,000
$56,000
$61,000
$54,000
$52,000
$55,000
$50,000
Current Strategy
Including Annuity
Including Annuity, Revised Allocation
Complexity
Modeling Variations
Real life Cash flows
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Over 500 inputs
5 Accounts:
– Non Qualified
– Qualified - Client & Spouse
– ROTH IRA - Client & Spouse
Up to 10 investment assets
Flexible assets’ allocations
Infusion of capital
Annuities (SPIA & DIA)
…
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Rules of withdrawing from
accounts
Spending curve: inflation and taper
Tax variation upon account type,
asset type, withdrawing source
– Ordinary income tax
– Capital gain tax
– Federal tax
Case Study
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Male (65), average health. Divorced, no kids
Has saved $2M (Non-qualified account: $1.2M, Qualified account: $ 800K)
$30K of social security (start from 70), no pension
Moderate allocation
CASH
U.S. Inv. Grade Corporate Bonds
U.S. High Yield Bonds
U.S. Large Cap
U.S. Mid Cap
U.S. Small Cap
Non-Qual
2.0%
35.0%
23.0%
24.0%
8.0%
8.0%
IRA – A
2.0%
30.0%
25.0%
27.0%
8.0%
8.0%
Retiring this year; wants to know the maximum safe income level
Targeting a 95% confidence of having at least a dollar of assets when he is 95
Modeling
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Discrete optimization problem
Maximize
Subject to
Retirement Living Needs
$180,000
$160,000
$140,000
Notations
x: the vector of annual retirement living needs
$120,000
$100,000
$80,000
$60,000
$40,000
$20,000
Actually, we are maximizing the initial retirement living
needs
𝑔(𝑥,r): the ending value of the individual’s account at time
n. It is a random variable and it depends on the annual
retirement living needs (𝑥) and the returns of portfolios (r).
$66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
Soc Sec
Pension
Annuities
After Tax Acct Draw
Before Tax Acct Draw
Roth Acct Draw
Need
RMD
96
98
100
Modeling
Objective Function
Equality Constraint
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To maximize the initial retirement
living needs (IRLN)
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The equality constraint is a
probability equation
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The sequence of annual retirement
living needs depends on IRLN,
inflation and taper
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The ending value of account
𝑔(𝑥, r) is a stochastic process of
cash flows, which is driven by the
random returns of portfolios
Use Monte Carlo method to
simulate the distribution of 𝑔(𝑥, r)
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Monte Carlo method
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Simulate yields curves
Case study
– Iteration number: 2000
– Period: 95 – 65 = 30
– The number of assets: 6
CASH
U.S. Inv. Grade Corporate Bonds
U.S. High Yield Bonds
⋯
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⋯
U.S. Large Cap
U.S. Mid Cap
U.S. Small Cap
⋯
After we get these matrices of expected returns, for a given IRLN (𝑜𝑟 𝑥1 ), we can get the distribution of 𝑔(𝑥, r)
by running the cash flow model
Methodology
Try and Error Method
Formula-based Method
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Since the cash flow model includes a lot
of “rules” (i.e. the rules of calculating
taxes and the rules of withdraw, in
programming they are if statements), it is
really time consuming to get the
distribution of 𝑔(𝑥, r)
3 – 4 minutes for one trial.
If we want to satisfy the constraint (a
specific ruin target) by using try and error
method, it may take hours or even the
whole day to get the optimal solution
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Do not try it, just solve it
Incorporate the rules into recursive
formulas
Derive the general form of 𝑔 𝑥, r for each
scenario from recursive formulas
Solve 𝑔 𝑥, r = 0 directly for each scenario
and then get 2000 IRLNs
Choose the quantile (1 – α%) of these
2000 IRLNs
Use matrix calculation instead of loop
statement
Formula-based Model
Process of formulating
Only one account
without pension, soc.
sec., IOC, annuities or
taxes
Two accounts
(qual./non-qual.)
without pension,
soc.sec., IOC, annuities
or taxes
Two accounts with
pension, soc.sec., IOC,
annuities and all taxes
Only one account with
pension, soc. sec., IOC
and annuities
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R model with only one account without pension, social security,
infusion of capital, annuities or taxes
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R model with only one account with pension, social security,
infusion of capital and annuities
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R model with two one accounts with pension, social security,
infusion of capital, annuities and taxes (not including taxes related
to yield)
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R model with two one accounts with pension, social security,
infusion of capital, annuities and all taxes
Two accounts with
pension, soc.sec., IOC,
annuities and taxes (not
including taxes related to
yield)
Formula-based Model
By creating this formula-based model
and using matrix calculation in R,
now we can solve the maximized
retirement living needs within 0.5
second!
Summary & Next Step
Summary
Next Step
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Optimization problem
– Maximize the retirement living
needs subject to a specific ruin
probability
Monte Carlo Application
– Formula method VS Try and
error method
– Matrix calculation VS Iterative
computations
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Solve the optimal allocation of
annuities
Incorporate mortality and morbidity
into the model
Use healthy life expectancy to
adjust the curve of retirement living
needs
Thank you！
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If people do not believe that mathematics is simple, it is only
because they do not realize how complicated life is.
——John von Neumann