Copyright © 2010 by Moshe A. Milevsky
Optimal
p
“Retirement Income” Planning:
g
Can we Reconcile Financial Economics
with Financial Planning?
Moshe A. Milevsky, Ph.D.
Fi
Finance
Professor
P f
/ York
Y k University
U i
it
(Joint work with Prof. Huaxiong Huang)
1
Two differing views on “retirement
income” planning…
2
1
Copyright © 2010 by Moshe A. Milevsky
The Financial Planner Says Things Like:
• Target 65% to 85% of your working
income once yyou retire.
• Spend and/or withdraw between 4% and
5% of your retirement nest egg and adjust
that for inflation each year.
• You might live to 95 so make sure not to
spend too much money at 65.
65
• Buy a variable annuity (VA) with a GLWB
or you might starve to death on cat food…
3
The Financial Economist Says:
Smooth consumption, taking into account
your “patience” and survival probabilities….
4
2
Copyright © 2010 by Moshe A. Milevsky
The optimal (total) consumption rate over
time will satisfy the following equation:
5
. You can derive a closed-form analytic
expression for the initial consumption
rate at retirement….
6
3
Copyright © 2010 by Moshe A. Milevsky
Ok. Very elegant.
Can we do anything useful with this math?
I think yes and offer seven lessons.
7
Assume we live in a world in which the
only source of uncertainty is longevity…
How does Mr. Spock behave if his entire
portfolio is invested in real-return bonds?
8
4
Copyright © 2010 by Moshe A. Milevsky
Longevity Risk Aversion vs.
Financial Risk Aversion
Coefficient of Relative Risk Aversion (CRRA) (
)
Allocation to “Stocks” in Asset Allocation model
γ=1
150%
γ=2
80%
γ=4
40%
γ=8
20%
9
Once again, there are no “risky assets”
allowed on Planet Vulcan…
10
5
Copyright © 2010 by Moshe A. Milevsky
11
Economic Question #1:
• What is the optimal initial portfolio
withdrawal
ithd
l and
d consumption
ti
rate
t during
d i
retirement?
• How do pre-existing pensions and/or risk
aversion impact these spending rates?
12
6
Copyright © 2010 by Moshe A. Milevsky
Net-Withdrawal Rates from $100 at age 65
Realistic Investment Assumption: v = 2.5%
Increasing Degree of Risk Aversion….
Pre‐Existing
Pension Annuity
γ=1
γ=2
γ=4
γ=8
π = $0
6.33%
5.30%
4.60%
4.12%
π = $1
6.80%
5.65%
4.87%
4.32%
π = $2
7.16%
5.92%
5.08%
4.48%
π = $5
8.02%
6.55%
5.55%
4.83%
Note: Assumes 5% Survival to Age 100, 25% Survival to Age 93 and 50% to Age 87. Subjective Discount Rate (ρ) assumed equivalent to real investment rate. 13
Ok, what if I am more optimistic about
the rate of return my portfolio can earn?
What if the REAL investment assumption
is increased by 100 basis points?
Answer: Increase spending rates by
about 40 to 80 basis points.
14
7
Copyright © 2010 by Moshe A. Milevsky
Takeaway #1
• Optimal consumption rates (total) and
withdrawal
ithd
l rates
t
(f
(from
portfolio)
tf li ) depend
d
d
on risk aversion and pre-existing pension
income in addition to other “background”
variables like mortality assumptions.
• Rational rates can not be determined
based only on the age at retirement and/or
size of the nest egg.
15
Economic Question #2:
• How does the “optimal” consumption rate
evolve over time?
• Does Mr. Spock withdraw the same
amount each year (adjusted for inflation)
forever?
16
8
Copyright © 2010 by Moshe A. Milevsky
17
Intuitively…
95%
Age 70
5%
Age 100
18
9
Copyright © 2010 by Moshe A. Milevsky
Irving Fisher (1930)
The Theory of Interest
…The shortness of life thus tends powerfully
to increase the degree of impatience or rate
of time preference beyond what it otherwise
might be…
…Everyone at some point in his life doubtless
changes his degree of impatience for
income…”
…He expects to die and he thinks: Instead of
pilling up for the remote future, why
shouldn’t I enjoy myself during the few
years that remain…
19
Takeaway #2
• The optimal consumption rate declines as
you age and
d progress thru
th retirement.
ti
t
• The greater your level of risk aversion, the
slower the rate of decline because you are
afraid you might live to an advanced age.
• New way to think about longevity risk
aversion: People who are worried they are
younger than they really are.
20
10
Copyright © 2010 by Moshe A. Milevsky
Economic Question #3
• How exactly does risk aversion interact
with
ith optimal
ti l retirement
ti
t spending
di rates?
t ?
21
Mathematically Speaking:
(t px )1/ γ = exp((x + t − m* ) / b) / b
m* = m + b ln[γ ]
22
11
Copyright © 2010 by Moshe A. Milevsky
Risk-Adjusted Survival Probability: p^(1/γ)
It makes you feel younger
Survive to Survive to Survive to Age 80
Age 80
Age 90
Age 90
Age 100
Age 100
Survive to Age 115
Age 115
True Probability
74%
37%
5%
4x10^(‐7)
γ=4
92%
78%
47%
2%
γ=8
96%
88%
69%
15%
23
Takeaway #3
• The probability of survival is intertwined
with
ith risk
i k aversion.
i
• You shouldn’t tell people (ok, Vulcan’s) to
worry about living to age 100 (i.e. they
should behave as if they are very risk
averse) and then advise them to invest in
the stock market (which is only suitable for
people with low risk aversion.)
24
12
Copyright © 2010 by Moshe A. Milevsky
Economic Question #4
• What is the optimal trajectory of wealth
over the
th entire
ti
retirement
ti
t horizon,
h i
if I
maximize my utility of consumption?
25
26
13
Copyright © 2010 by Moshe A. Milevsky
Takeaway #4
• There are bag-ladies on Planet Vulcan.
• It is optimal to spend down your wealth and
eventually
ll
only
l
consume your pension
i
income (if you have any.)
• The greater your risk aversion the more
wealth you have at any point during
retirement.
risk-averse
averse parents,
parents because they
• Kids love risk
leave more leftovers.
• In some cases it might be optimal to borrow
against pension income if you can.
27
Low gamma behavior…
National Post, August 20, 2005
14
Copyright © 2010 by Moshe A. Milevsky
Economic Question #5
• What is the optimal reaction to shocks and
unforeseen
f
market
k t events?
t ? By
B how
h
much
h
do you adjust your consumption rate (total)
and withdrawal rate (from the portfolio)?
29
Five years later (age 70) unexpected things happen…
By how much do you adjust portfolio spending?
‐30%
Shock
‐15% Shock
+15% Shock
+30% Shock
No Pension
Income
$2 Pension
‐30.0%
‐15.0%
+15.0%
+30.0%
‐27.1%
‐13.5%
+13.5%
+27.1%
$5 Pension
‐23.5%
‐11.7%
+11.7%
+23.5%
Notes: Assumes a high degree of risk‐aversion (γ=8). The investment rate is equal to the subjective discount rate (SDR) which is assumed to be 2.5% inflation‐adjusted.
30
15
Copyright © 2010 by Moshe A. Milevsky
Takeaway #5
• When unexpected things happen (to your
portfolio)
tf li ) you should
h ld adjust
dj t your spending
di
plans by less than the change in wealth
assuming you have pension income.
• With no pension income it is one-for-one.
• Vulcan’s
Vulcan s don
don’tt panic and then reduce
spending by 30% because their portfolio
declined by 30%.
31
Economic Question #6
• What is the utility/value of pension
(i
(income)
) annuities
iti
exactly
tl and
d why
h do
d
financial economist like them so much?
• Is it worthwhile to spend some of my $100
nest egg to purchase a pension annuity?
Why not take my chances?
32
16
Copyright © 2010 by Moshe A. Milevsky
How Does “Pensionization” Impact
Retirement Consumption at age 65?
Percent of $100
Pensionized
0%
20%
40%
60%
100%
Medium Risk Aversion High Risk Aversion
(CRRA = 4)
(CRRA 4)
(CRRA = 8)
(CRRA 8)
$4.605
$5.263
$5.795
$6.227
$6.330
$4.121
$4.801
$5.385
$5.937
$6.330
Note: Cost of $1 lifetime income annuity is $15.791 at age 65, assuming a real pricing rate of 2.5% per annum.
33
Yeah, but the money is gone!
• Obviously,
y, if you
y “pensionize”
p
ap
portion of
your nest egg, you have less wealth to
bequeath and your legacy is reduced.
• Remember. If you die the beneficiaries
only get the financial capital.
• This is why you must first decide where
you sit on the retirement frontier.
34
17
Copyright © 2010 by Moshe A. Milevsky
Finan
ncial Legacy
Economic Tradeoffs at Retirement:
…or the kids
It’s either you…
Retirement Sustainability
35
Takeaway #6
• Pension annuities are valuable because
they don
don’tt “waste”
waste investment returns.
returns
• They increase consumption at all future
ages (and also increase discounted utility.)
• Obviously, “pensionizing” (a portion of)
your nest egg will reduce the financial
capital and hence your legacy/bequest.
• But remember, the objective was to
maximize utility of lifetime consumption.
36
18
Copyright © 2010 by Moshe A. Milevsky
Economic Question #7
• It seems like these models suggest that at
g
and in some cases the
advanced ages
financial capital trajectory is negative.
• Does this mean that it might be optimal to
borrow -- possibly with life insurance to
protect the lender – at advanced ages?
• ANSWER: Yes.
Yes If you have a very secure
and relatively high pension, go ahead and
use you credit cards (on Planet Vulcan.)
37
Concluding Remarks
• Longevity risk aversion leads to reduced
spending and withdrawal rates,
rates just like
financial risk aversion leads to reduced
equity in the investment asset allocation.
• Pensions are valuable because they allow
you to withdraw from portfolio at a greater
rate all else being equal.
rate,
equal
• Don’t let the ruin, shortfall or success
probability distort retirement planning.
38
19
Copyright © 2010 by Moshe A. Milevsky
Letter from Professor Paul Samuelson
…Take my cold water with many grains of salt. Max
Planck said: “Science p
progresses
g
funeral by
y funeral”
…You could avert in advance criticism by wording
your paper as a reductio ad absurdum…
…I regard the shortfall probability criterion as a
determiner of stochastic portfolio decision making
to be arbitrary and unappealing…
39
Download copy of paper from
www ifid ca
www.ifid.ca
• Thank you!
40
20