SocialSecurityBeamer Final

Social Security: Now or Later?
Kadie Clancy, Katie Linthicum, and Amanda Mummert
Washington & Jeﬀerson College
Washington, PA 15301
August 5, 2015
MathFest 2015, Washington D.C.

Table of Contents
1 Overview
2 Background of Social Security
History of S.S.
Regulations
3 A Continuous Model for S.S.
Notations and Constants
A Diﬀerential Equation Model
Solution to the IVP
Examples
4 Analysis on Catch up Time
Between P62 and P66
Between P66 and P70
5 Summary
6 Acknowledgement
7 Reference

Overview
Overview
Background on Social Security
A Continuous Model using diﬀerential equation
Solution to the IVP
Apply the Model at diﬀerent retirement ages: 62, 66, and 70
Some analysis on the catch up point

Background of Social Security
History of S.S.
History of Social Security
August 14, 1935 signed by Franklin D. Roosevelt
Started when welfare started
Originally to be retirement beneﬁts for workers
Help with disability, survivors and elderly

History of S.S.
Social Security
Foundation of economic security for millions of Americans
Retirees, disabled persons, and families of retired, disabled or
deceased workers
158 million Americans pay Social Security taxes
57 million collect monthly beneﬁts in 2013.
About one household in four receives income from Social
Security.

History of S.S.
Factors to Take Into Consideration
Taxes and living expenses (food, medical, house and utilities)
How long do you expect to live?
Other incomes: pensions, investments, and savings
Inflation
Adjustment of annual benefit from S.S. due to inflation

Regulations
Social Security Regulations
Currently full retirement age is at 66 but will continue to rise
over the years.

Regulations
over the years.
Can start beneﬁt as early as at age 62 or as late as at age 70.

Regulations
over the years.
Full (100%) retirement beneﬁt if retire at age 66.

Regulations
over the years.
75% of full retirement beneﬁt if retire at age 62.

Regulations
over the years.

Regulations
over the years.
Only 85% of beneﬁt are taxable.

Regulations
When calculating consider what is going into your bank
account and what is going out

Regulations
Going out: taxes and living expenses

Regulations
Going out: taxes and living expenses
Going in: annual beneﬁt from social security and interest
accrued through investments

A Continuous Model for S.S.
Assumption for Model
Assumptions:
The unit time in this continuous model is a year.

Assumptions:
Assume a ﬁxed annual S.S. beneﬁt for an individual.

Assumptions:
Assume a ﬁxed annual investment return rate, compounded
continuously.

Assumptions:
Assume a ﬁxed annual investment return rate, compounded
continuously.
Assume a ﬁxed individual tax rate.

t = time (in year)

t = time (in year)
P(t) = account balance at year t after retirement.
P62(t) if start retirement beneﬁt at age 62.

t = time (in year)
Assume no money is used from the account. This can be
modiﬁed by adding another parameter Withdrawal.

t = time (in year)
A = ﬁxed annual retirement beneﬁt. Assume A = $28, 800 at
full retirement age.

t = time (in year)
r = ﬁxed annual tax rate. For instance, r = 25%.

t = time (in year)
r = ﬁxed annual tax rate. For instance, r = 25%.
s = ﬁxed annual investment return rate. For instance,
s = 6%.

Modeling P(t) by a Diﬀerential Equation
Annual change of account value

= incoming money - outgoing money

= (annual beneﬁt + investment income) - taxes

The diﬀerential equation:
dP
dt
= A + (P + A)s − ((P + A)s + 0.85A)r
beneﬁt investment taxes

The differential equation:
dP
dt
= A + (P + A)s − ((P + A)s + 0.85A)r
benefit investment taxes
This is a first order linear ODE.
Initial condition: P(0) = 0.

Solution to the IVP
Solution to dP
dt = A + (P + A)s − ((P + A)s + 0.85A)r
Recall: A = annual beneﬁt, r = tax rate, s = investment rate

Solution to the IVP
Solution to dP
dt = A + (P + A)s − ((P + A)s + 0.85A)r
Using integrating factor, the general solution to the ODE can
be found to be
P(t) = Ce−st(r−1)
+ A
s − 0.85r − rs + 1
rs − s
, t ≥ 0.

Solution to the IVP
Solution to dP
dt = A + (P + A)s − ((P + A)s + 0.85A)r
be found to be
+ A
s − 0.85r − rs + 1
rs − s
, t ≥ 0.
With initial condition P(0) = 0,
C = −A
s − 0.85r − rs + 1
rs − s
.

Solution to the IVP
Solution to dP
dt = A + (P + A)s − ((P + A)s + 0.85A)r
be found to be
+ A
s − 0.85r − rs + 1
rs − s
, t ≥ 0.
With initial condition P(0) = 0,
C = −A
s − 0.85r − rs + 1
rs − s
.
The solution to the IVP is
P(t) = A
s − 0.85r − rs + 1
rs − s
(1 − est(1−r)
), t ≥ 0.

Examples
Example: tax rate 25%, investment rate 6%
Assume tax rate of r = 25% and investment rate of s = 6%

Examples
Retire at age 62, A = 0.75 × $28, 800 = $21, 600
P62(t) = 399600(e0.045t
− 1).

Examples
Retire at age 62, A = 0.75 × $28, 800 = $21, 600
P62(t) = 399600(e0.045t
− 1).
Retire at Age 66 (Full Retirement Age), A = $28, 800
P66(t) = 532800(e0.045t
− 1).

Examples
Retire at age 62, A = 0.75 × $28, 800 = $21, 600
P62(t) = 399600(e0.045t
− 1).
Retire at Age 66 (Full Retirement Age), A = $28, 800
P66(t) = 532800(e0.045t
− 1).
Retire at Age 70, A = 1.32 × $28, 800 = $39016
P70(t) = 703296(e0.045t
− 1).

Examples
Graphs of P62 and P66 with r = 25%, s = 6%

Examples

Analysis on Catch up Time
Between P62 and P66
Analysis of Age 62 and Age 66
For P66 to catch up with P62, we need
P62(t + 4) = P66(t).

Between P62 and P66
P62(t + 4) = P66(t).
P62(t + 4) = .75A
s − 0.85r − rs + 1
rs − s
(1 − es(t+4)(1−r)
), t ≥ 0.
P66(t) = A
s − 0.85r − rs + 1
rs − s
(1 − est(1−r)
), t ≥ 0.

Between P62 and P66
P62(t + 4) = P66(t).
P62(t + 4) = .75A
s − 0.85r − rs + 1
rs − s
(1 − es(t+4)(1−r)
), t ≥ 0.
P66(t) = A
s − 0.85r − rs + 1
rs − s
(1 − est(1−r)
), t ≥ 0.
Solving for t, we get
t =
ln(4 − 3e4s(1−r))
s(r − 1)
.

Between P62 and P66
Criteria for P66 to catch up with P62
Catch up time:
t =
ln(4 − 3e4s(1−r))
s(r − 1)
.
For t to have a solution (P66 can catch up with P62), we need
4 − 3e4s(1−r)
> 0.
Solving for s, we get
s <
ln 4
3
4(1 − r)
.

Between P62 and P66
Graph of s =
ln 4
3
4(1−r)

Between P62 and P66
Examples with ﬁxed tax rate
Catch up time:
t =
ln(4 − 3e4s(1−r))
s(r − 1)
.
r = 25%, s = 4% : t = 16.1 years
r = 25%, s = 5% : t = 17.7 years
r = 25%, s = 6% : t = 19.9 years
r = 25%, s = 10% : Never catch up
At a ﬁxed tax rate r, the larger the investment rate s, the
longer it takes for P66 to catch up with P62.
When s >
ln 4
3
4(1−r) , P66 never catches up with P62.

Between P62 and P66
Examples with fixed investment rate
Catch up time:
t =
ln(4 − 3e4s(1−r))
s(r − 1)
.
r = 15%, s = 6% : t = 22.3 years
r = 25%, s = 6% : t = 19.9 years
r = 35%, s = 6% : t = 18.1 years
r = 39.6%, s = 6% : t = 17.4 years.
At a fixed investment rate s, the higher the tax rate r, the
shorter time for P66 to catch up with P62. So if you are in a
low tax bracket, retire at 62 is more beneficial.

Between P66 and P70
P66(t + 4) = P70(t).

Between P66 and P70
P66(t + 4) = P70(t).
P66(t + 4) = A ×
s − 0.85r − rs + 1
rs − s
(1 − es(t+4)(1−r)
), t ≥ 0.

Between P66 and P70
P66(t + 4) = P70(t).
P66(t + 4) = A ×
s − 0.85r − rs + 1
rs − s
(1 − es(t+4)(1−r)
), t ≥ 0.
P70(t) = 1.32A ×
s − 0.85r − rs + 1
rs − s
(1 − est(1−r)
), t ≥ 0.

Between P66 and P70
P66(t + 4) = P70(t).
P66(t + 4) = A ×
s − 0.85r − rs + 1
rs − s
(1 − es(t+4)(1−r)
), t ≥ 0.
P70(t) = 1.32A ×
s − 0.85r − rs + 1
rs − s
(1 − est(1−r)
), t ≥ 0.
Solving for t, we get
t =
1
s(1 − r)
ln
8
25(1.32 − e4s(1−r))
.

Between P66 and P70
Criteria for P70 to catch up with P66
Catch up time:
t =
1
s(1 − r)
ln
8
25(1.32 − e4s(1−r))
.
For t to have a solution (P70 can catch up with P66), we need
1.32 − e4s(1−r)
> 0.
Solving for s, we get
s <
ln(1.32)
4(1 − r)
.

Between P66 and P70
Graph of s = ln(1.32)
4(1−r)

Between P66 and P70
Examples
Catch up time:
t =
1
s(1 − r)
ln
8
25(1.32 − e4s(1−r))
.
At r = 25%, s = 9.26% on the graph.
So if the expected investment rate is s < 9.26%, one should
retire at age 70 instead of at age 66.
At s = 6%, this will take 21 years at the age of 91 for P70 to
catch up with P66.
So the decision to wait till age 70 to retire also depends on
one’s life expectancy.

Summary
Summary
Used a diﬀerential equation model to calculate the best time
to start collecting social security

Summary
Summary
Compare P62, P66, and P70 at various tax brackets and
investment rates.

Summary
Summary
investment rates.
Analyze the catchup time relative to the tax rate and
investment rate.

Summary
Summary
investment rates.
investment rate.
Because of the cummulation from the extra four years,
at a ﬁxed tax bracket, the higher the investment rate, the
earlier one should retire.

Summary
Summary
investment rates.
investment rate.
Because of the cummulation from the extra four years,
at a ﬁxed tax bracket, the higher the investment rate, the
at a ﬁxed investment rate, the lower the tax bracket, the

Acknowledgement
Acknowledgement
Adviser: Dr. Wong
Washington & Jeﬀerson College Math Department for their
support

Reference
Reference
A First Course in Diﬀerential Equations with Modeling
Applications Dennis G. Zill
Tax Brackets
http://www.bankrate.com/ﬁnance/taxes/tax-brackets.aspx
Government social security web site: ssa.gov
http://www.merrilledge.com/Publish/Content/application
/pdf/GWMOL/ME TimeisMoney Topic

SocialSecurityBeamer Final

Recomendados

Recomendados

Mais conteúdo relacionado

Destaque

Destaque (15)

Semelhante a SocialSecurityBeamer Final

Semelhante a SocialSecurityBeamer Final (20)

SocialSecurityBeamer Final