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

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### 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 L_{65+k} survivors k years later is

$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 pre-retired, 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 health-care 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
- EttiBaranoff, Patrick Lee Brockett, YehodaKahane, Risk Management for Enterprises and Individuals, 2009

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