Transcript How Much Money Do You (or Your Parents) Need for Retirement?

How Much Money Do You
(or Your Parents) Need for
Retirement?
Natalia A. Humphreys
Based on the article by James W. Daniel, UT Austin
How Much Money Do You (or Your
Parents) Need for Retirement?
 $100,000 per year?
 A lump sum of a million dollars?
 As much as possible?
Let us refine our question…
Different Individuals-Different
Goals
Suppose:
 An individual wishes to receive an income of $I after
taxes at the start of each year of life, starting at the
moment of retirement (this type of annuity is called
an annuity due).
 How much would be required in investments at
retirement in order to provide that stream of
payments?
Consider $1 for Simplicity
To receive yearly payments of $I, the
retiree would need I times as much as
needed for a stream of $1 yearly
payments.
The Amount Required in
Investments depends on:
 Yearly rate of return on the investments
after taxes
 Whether the retiree wants the $1
payments to increase over time to
account for inflation.
The Real Growth Factor
 If r – after-tax yearly rate of return, then
 $1 would grow to $(1+r) at the end of the year
But…
 If g – yearly inflation rate, then
 At the end of the year $1 would only buy 1/(1+g) times
as much as at the start
So…
The Real Growth Factor (cont.)
(1+r)/(1+g)=1+i, where
i=(r-g)/(1+g)
i- the real yearly rate of return (after
expected taxes and inflation)
The Original Question:
Assuming a real yearly rate of return i,
how much does a retiree need to
have invested in order to provide $1
at the start of each year for life,
starting at the moment of retirement?
Our Analysis of This
Retirement Problem
Will illustrate the kinds of modeling
that actuaries perform.
Actuaries
Business people who use
mathematical and statistical
techniques from actuarial science to
analyze how to provide financially
now for future costs of various risks.
The Answer to Our Problem
It depends.
Future Lifetimes: K
 The answer to our question depends on how long the
retiree lives
 Let K be the whole number of years our retiree lives
after retirement
 The retiree will always receive K+1 payments
 But… K is unknown
Future Lifetimes:
Conservative Approach
 Have enough invested to provide $1 per year
regardless of the size of K, i.e. forever (this type of
annuity is called perpetuity):
 This requires a fund of $(1+i)/i
 At the start of the first year: $(1+i)/i-1 =$1/i
 This would grow to $(1+i)/i
 If i=4%, then this approach would require a $26 initial
fund.
Future Lifetimes: Realistic
Approach
 We don’t live forever, so we only require
enough invested for (K+1) payments
 But… K is unknown
 So let us examine the data to help us
understand the various values of K that
typically occur in real life.
Returning to Our Investment
Problem
 How much is needed on average to provide the
payments?
Beware!
 This is not the same as asking: How large a
fund would be needed for an average person
who survives an average time into the future?
The Average-Future-Lifetime
Mistake
Observe the number of whole future years lived by each
of 50,000 typical 65-year-old female retirees
Then average these future lifetimes over 50,000 retirees
The average will fall between 20.83 and 20.99 future
years
Let’s use 20.9 as the value for the average female (the
corresponding average for males is 17.9 years).
The Average-Future-Lifetime
Mistake (cont.)
Consider a simpler retirement plan that provides a single
$1,000,000 payment to any retiree who survives to age
87.
Since K=20.9, an average retiree would die between ages
85 and 86, so would not live to qualify for the payment
at age 87.
Thus, $0 is needed on an average retiree with the
average future lifetime!
The Average-Future-Lifetime
Mistake: Nonsense!
Clearly, investing just enough
(namely, zero) to pay the benefits
for a retiree who survives the
average number of years cannot
be the right approach… Why
not?
The Average-Future-Lifetime
Mistake (cont.)
If you start with 50,000 retirees, some of them will outlive
the average and collect their $1,000,000 at age 87.
Had a fund started with $P for such a person, it would
have grown with interest to
$P(1+i)^22 by that time
P(1+i)^22 = 1,000,000
P=1,000,000 v^22
The Average-Future-Lifetime
Mistake (cont.)
With i=4%, P=$421,955
About 25,866 of the 50,000 are expected to survive to age
87. Thus, on average, we need to invest
25,866 (1,000,000) v^22/50,000
With i=4%, this is about $218,285 per original retiree
This is different from $0!
Average Amounts Needed
 The technique used to analyze the single lump-sum
payment pension can be applied to the original pension
of $1 yearly for life.
 Start with L_{65}=50,000 65-year-old retirees
 Use actuarial data to estimate L_{65+k} of those alive k
years later to receive a $1 payment for k=0, 1, 2, …
Average Amounts Needed To
Fund All Retirees
 The amt needed in an investment at age 65 that would
grow to enough to pay $1 to each of the
survivors k years later is
L_{65+k}
$L_{65+k} v^k, the Present Value (PV) of money
needed at age 65+k.
 The initial amt needed to fund all the pmts for the
lifetime of all the original retirees is then
L_{65+0} v^0+L_{65+1} v^1+L_{65+2} v^2+…
Average Amounts Needed
Actuarial Present Value
 Divide by L_{65} to get the average number of dollars
needed per original retiree – Actuarial Present Value of
the pmts for the lifetime of one 65-year-old:
APV=L_{65+0}/L_{65} v^0+L_{65+1} /L_{65}
v^1+L_{65+2} /L_{65} v^2+…
Average Amounts Needed:
Example
 If L_{65} =50,000, then the first few values of L_{65+k}:
L_{66} = 49,543, L_{67} = 49,360, L_{68} = 48,483
 Had at most four payments been promised to survivors,
the average needed would be
$(L_{65}+L_{66} v^1+L_{67} v^2+L_{68} v^3)/L_{65}=
=$3.72
 With lifelong payments and i=4%, the average
investment is $14.25 (compare with $26)
Does This Settle the
Problem?
 Not entirely…
 The analysis has been from the viewpoint of an
insurance company that guarantees pmt for life to a
group of retirees.
 An individual who had invested only $14.25 at
retirement would exhaust her acct if she lived much
beyond the average age for her cohort.
Same Problem: Another Point
of View
Let us look at the problem from the
individual retiree’s
point of view
A Probability Theory
Perspective
Lurking in the background of the preceding analysis are
both
Probability
And
Statistics,
Two fundamental tools for actuaries
A Probability Theory
Perspective (cont.)
 Let X be the future lifetime of each of a large number
L_{0} newborns
 Assume X has the same probability distribution for
each newborn.
 This does not mean that each newborn’s future lifetime
is the same
 This means they all have the same chance behavior:
the probability that a newborn dies in some particular
age range is the same for all of the newborns
A Probability Theory Perspective: Cumulative
Distribution and Survival Functions
 Mathematicians describe the random behavior by the
cumulative distribution function
F(x)=Pr[X<=x] – the probability that the newborn dies
by age x
 Actuaries look at the bright side and describe the
random behavior by the survival function
S(x)=1-F(x)=Pr[X>x] – the probability that the newborn
survives beyond age x
A Probability Theory
Perspective: Survivors
 The expected number L_{x} of survivors to age x from
among the L_{0} newborns is:
L_{x}=s(x) L_{0}
Both L_{x} and F(x) describe the distribution:
F(x)=1-s(x)=1-L_{x}/L_{0}
Models of Survival Functions
Actuaries regularly collect statistics on large
number of human lives in various categories in
order to build models of survival functions s(x):
age, sex, smoker, non-smoker, geographical
region, special occupations, retired or preretired, widows and widowers, people with or
without certain diseases or disabilities, urban
vs. rural populations.
Surviving another k years
 Probability that a 65-year-old survives at least another
k years (p_k) is :
p_k=s(65+k)/s(65)=L_{65+k}/L_{65}
 Recall APV=L_{65+0}/L_{65} v^0+L_{65+1} /L_{65}
v^1+L_{65+2} /L_{65} v^2+…
 Re-writing
APV=p_0 v^0+p_1 v^1+p_2 v^2+…
 The expected value of the present value of pmts made
so long as the retiree survives
True vs. Expected PV
 True present value of the K+1 pmts
TPV=1+v+v^2+…+v^K=(1-v^{K+1})/(1-v)
 With i=4% TPV>$14.25 if K+1>20
 p_{20}=L_{85}/L_{65}=0.6, i.e. about 60% of the
retirees who start out with APV – the ave amt needed
for a lifetime of pmts – will run out of money before
running out of life.
Confidence in Having Enough
Money
 Greater confidence in having enough
money requires a greater initial fund.
 For 99% confidence (for only 1% of
retirees to run out of money), the initial
fund needed for an individual should be
about $20.58.
How to Protect at Lower
Cost?
Retirees could pool
their risks
Risk Pooling
 The fund needed to provide lifelong
payments to an individual can vary
depending on the individual’s future
lifetime
 In large groups these variations tend to
average out
Risk Pooling (cont.)
 Retirees who live a long time and require a
large initial fund are offset by those living a
short time
 Large corporate pension plans and insurance
companies provide the opportunity for
individuals to pool their risk and thereby benefit
from the more regular behavior of large groups.
Normal Distribution
 For a large group of N 65-year-old retirees, the
adequacy of the total initial fund is governed by the
sum over all the retirees of the PV of pmts to each
retiree
 The sum of a large number of independent random
variables is well approximated by a normal random
variable (“bell-shaped curve”)
 The larger N, the thinner and taller the bell (the values
are heavily concentrated near the average)
Fund Amount for 99%
Confidence
 N retiring 65-year-old females
 Each depositing $P_N into the fund earning i=4%
 How large need P_N be in order that we can be 99%
confident that the total fund will be able to provide
lifelong pmts to all N retirees?
P_N=14.25+[10.34/N^{0.5}],
where 14.25 is the APV – the average amt needed.
Fund Amount for 99%
Confidence (cont.)
P_N=14.25+[10.34/N^{0.5}]
N=100, then P_{100}=15.29
N=1000, then P_{1000}=14.35
 These numbers compare favorably with the $20.58
needed for a single individual not in such a group to be
99% confident
Generalizations
Ideas of actuarial science:
 How investments grow
 Effects of inflation
 Present value
 Probability
 Statistics
Can be used to analyze a wide range of similar
problems
Examples
 How much companies should contribute regularly to
special funds to meet their future pension and healthcare obligations to retirees
 How high premiums need be for life or health or auto or
homeowners insurance
 How the costs of leasing equipment compare to those
of buying
 How a new disease or treatment will impact health-care
costs
How Much You (or Your
Parents) need for retirement?
It depends, but now
we know more about
what it depends on
and how
References
 James W. Daniel, How Much Money Do You (or Your
Parents) Need for Retirement?, The College
Mathematics Journal, volume 29, number 4, 1998, pp.
278–283
 Etti Baranoff, Patrick Lee Brockett, Yehoda Kahane,
Risk Management for Enterprises and Individuals, 2009