X-Message-Number: 1491
Date: Tue, 22 Dec 92 20:01:11 -0800
From:
Subject: CRYONICS
THE TRANS TIMES
Life Extension through Cryonic Suspension
Volume 1 Number 3 December 1992
STAYING COLD
Providing Sufficient Maintenance Funding
by Art Quaife, Ph.D.
Persons to be placed in cryonic suspension provide a trust fund to
pay for their ongoing maintenance. It is important that the total
return on the trust fund usually exceeds the yearly cost of
storage, so that the trust fund grows rather than diminishes.
Clearly the larger the fund, the less likely it is ever to run out.
But how much is enough? How large an initial fund is needed to
insure that the fund is nearly certain to never become exhausted?
In this article, I will attempt to determine the probability that
trust funds of various sizes and expected returns will eventually
go broke. In probability theory, this problem is known as the
*gambler's ruin*, or more generally as the problem of *first
passage times*.
The three main determinants of how large a fund is required are:
1. The total return on investment *roi* (dividends and interest
plus capital gains) that the trust can obtain.
2. The yearly charge *LTS* for maintaining the patient in long
term storage. A large component of this charge will be the cost of
liquid nitrogen. I will assume that the cost of maintaining the
patient increases yearly by *inf*, the rate of inflation as
measured by the Consumer Price Index.
3. Whether or not the fund is taxable. TRANS TIME can help you
set up a trust fund that is tax-free, avoiding the large initial
and yearly bites that the IRS will otherwise take. Thus I will
ignore taxation in this article.
*Funding when the returns are certain*
If we know the value of our total return on investment *for
certain* (i.e., if *roi* has standard deviation 0) and similarly
the rate of inflation is certain to be *inf*, then the return
*after inflation* is given by their geometric difference:
1 + roi roi - inf
i = ------- - 1 = ---------
1 + inf 1 + inf
approx. = roi - inf
It is easy to see that if *i* > 0, then a fund of size *F* =
*LTS*/*i* will produce exactly enough return each year to pay the
long term storage bill, and still grow with the rate of inflation.
However, we cannot be certain that *inf* will remain the same for
any time into the future. If we invest the fund in the stock
market, our rate of return *roi* will also be uncertain even in the
short term. Even investing in long-term government bonds and
holding them to maturity gives an uncertain return over a long
enough time scale, since we do not know what the renewal interest
rate will be.
*Funding when the returns are uncertain*
A very conservative management policy is to invest the trust
principal in U.S. Treasury bills. According to the extensive
studies presented in [1], over the 65 year period 1926-1990,
Treasury bills returned 3.7% +- 3.4% (mean +- standard deviation)
per year. However over the same period, the rate of inflation was
3.2% +- 4.7%. After adjusting out the inflationary increase each
year, the T-bills returned only 0.6% +- 4.4%!
Suppose that *LTS* = $5,000/year, close to TRANS TIME's current
charge. Then using T-bills for funding we have *LTS*/.006 =
$833,333! And even a fund this large is not guaranteed to last
indefinitely; because of the variation in the returns, it could
still go broke.
Let us suppose, instead, that we invest the fund in the stock
market. The return during 1926-1990 on the Standard & Poor 500 was
12.1% +- 20.8% per year, and after adjusting out inflation was 8.8%
+- 21.0% per year [1]. While this return was substantially higher
than that of T-bills, so was the standard deviation. This accords
with the general rule: to get a higher expected return, one usually
has to accept greater volatility.
One can buy mutual index funds that invest in exactly the S&P 500
stocks, and obtain their return, less about .2% per year for fund
overhead in the best of them. Buying an index fund is a very
reasonable investment strategy; only a small fraction of mutual
funds manages to beat the S&P 500 over any extended period of time.
With this strategy, the minimum suggested fund size *LTS*/*i* is
$56,818. This is more like it. But note the very large standard
deviation. How likely is a fund of this size to go broke?
*Distribution of stock prices*
Let *SP(t)* be the value of the S&P 500 index, with dividends
reinvested, at time *t*. A weak form of the random walk hypothesis
asserts that for times *t* < *u* <= *v* < *w*, the differences
*SP(u)* - *SP(t)* and *SP(w)* - *SP(v)* are independent random
variables. To first approximation, for a fixed interval of time
*u* - *t*, these changes are normally distributed with a constant
mean and standard deviation. But this approximation ignores
scaling: it is about as likely for a stock to increase from $10.00
to $11.00 as it is for it to increase from $100.00 to $110.00, not
to $101.00. This leads to our second approximation: for fixed
difference *u* - *t*, the *ratios* *SP(u)*/*SP(t)* are normally
distributed with constant mean and standard deviation.
But this better approximation still cannot be quite correct. It
gives a non-zero probability of changes to *negative* values of
*SP(t)*, which are impossible. This and other theoretical
arguments suggest our best approximation: the ratios of changes
have the *lognormal* distribution, which means that the changes in
the *logarithms* of the index have the *normal* distribution. Both
the mean and the variance of this lognormal distribution are
proportional to the time difference *u* - *t*. This model bears
out well under testing [1]. The equations for the evolution of the
logarithm of stock prices are the same as for the diffusion process
consisting of Brownian motion with a drift.
*The lognormal distribution*
Let the random variable *X* have the lognormal distribution with
mean *m* and standard deviation *s*. Then *ln(X)* has the normal
distribution with (say) mean *mu* and standard deviation *sigma*.
These statistics are related as follows:
mu = ln(m / (1 + (s/m)^2)^.5)
sigma = (ln(1 + (s/m)^2))^.5
m = exp(mu + .5 sigma^2)
s = (exp(sigma^2) - 1)^.5 exp(mu + .5 sigma^2)
In the case of the S&P 500, where *X* = *SP(t[sub 0] + 1 yr.)* /
*SP(t[sub 0])*, [1] gives the values *m[sub 1]* = 1.088, *s[sub 1]*
= .21. Thus we calculate *mu[sub 1]* = .0661, *sigma[sub 1]* =
.1913.
When the time interval is *t*, we have:
m[sub t] = m[sub 1]^t
s[sub t] = s[sub 1] t^.5 m[sub 1]^(t-1)
In the graph below, the illustrated parameter values emphasize the
skewness of the lognormal distribution. But the lognormal
distribution with the S&P 500 parameter values calculated above is
much less skew, and harder to distinguish visually from the normal
distribution.
[GRAPH OF LOGNORMAL DISTRIBUTION OMITTED]
*The probability of fund death using the S&P 500 for funding*
I have written a program in C++ that provides a Monte Carlo
simulation of the problem. For each of various initial levels of
funding, I ran 30,000 trials. In each trial the portfolio begins
with the specified level of funding. At the end of each year, the
program uses a random number generator and the lognormal
distribution to post the portfolio's return for the year. The
program then deducts the yearly storage charge. If the deduction
reduces the portfolio to <= 0, the fund is declared dead, and the
patient is now without funding. Each trial was run for 400 years.
The results are presented below.
S&P 500 Funding
After 200 years After 400 years
Initial Fraction of Mean Fraction of Mean
Funding Portfolios Alive Portfolios Alive
_________________________________________________________________
$ $ $
56,818 .26 2.6 y 10^11 .26 7.9 y 10^18
80,000 .50 8.3 y 10^11 .50 1.8 y 10^19
100,000 .65 1.0 y 10^12 .65 2.5 y 10^19
120,000 .74 2.2 y 10^12 .74 3.8 y 10^19
150,000 .84 2.9 y 10^12 .84 4.2 y 10^19
200,000 .92 3.5 y 10^12 .93 4.9 y 10^19
250,000 .96 4.3 y 10^12 .96 8.9 y 10^19
300,000 .97 6.9 y 10^12 .97 1.2 y 10^20
400,000 .99 8.4 y 10^12 .99 1.6 y 10^20
500,000 .99 1.1 y 10^13 .99 2.6 y 10^20
The Mean columns average all portfolios, with those that died
valued at 0.
Almost all of the portfolios that died did so within the first 100
years. Those that were still alive at the end of 200 years were
now so large that there was very little chance of them dying even
in the next 1,000,000,000 years.
It is discouraging to see how poorly our minimum recommended
funding fares. With only $56,818 of funding, one stands a 74%
chance of running out of money. One needs almost $250,000 of
funding to reduce this chance of failure to 5%.
*The probability of fund death using T-bills for funding*
If we invest the fund in T-bills, the data in [1] yields *mu* =
.0050, *sigma* = .0437. Simulation as above yields the results
below.
Here *many* funds were still marginal after 200 years, and died out
in the next 200 years. Many are still marginal even after 400
years.
T-Bill Funding
After 200 Years After 400 Years
Initial Fraction of Mean Fraction of Mean
Funding Portfolios Alive Portfolios Alive
__________________________________________________________________
$ $ $
500,000 .20 104,000 .05 142,000
833,333 .79 881,000 .43 1,754,000
1,000,000 .91 1,404,000 .61 3,098,000
1,500,000 .99 3,045,000 .90 8,140,000
*Trust Management Fees*
Banks typically charge 1% to 3% per year to manage a trust,
depending upon its size. I have not deducted any such charges in
computing the two tables above, so the actual results will be
worse. Indeed with T-bill funding, the yearly return would become
*negative*!
If the portfolio is kept simple enoughDsuch as investment in an S&P
500 index fundDa cryonics organization might manage the trust for
a lower fee, or perhaps as part of their long term funding charge.
I have also ignored the much smaller .2% management fee on least-
expensive S&P 500 index funds.
*Funding reanimation*
Not only must the patient have sufficient funding to insure
indefinite long term storage, he must also be able to fund
reanimation. Speculations as to the cost of reanimation range from
very cheap to very expensive. I will not explore that question
further, except to note that the funds that grew like the S&P 500
and survived for 200 years were then generally *very large* [2].
Thus if reanimation is available at any price, these funds should
be able to pay for it.
Once the yearly storage charge becomes negligible with respect to
the size of the fund, during the course of a century every $1.00 of
S&P 500 funding is expected to grow to $4,601 in current dollars!
On the other hand, each dollar of T-bill funding only grows to
$1.82.
If your S&P 500 funding grows like the S&P 500 has grown
historically, and if reanimation proves possible at *some* price,
then it can be at most a few score years more until your fund will
be able to pay for it. But the situation is strikingly worse with
T-bill funding.
*Establishing an estate: Should one *ever* purchase T-bills?*
Most cryonicists take out life insurance policies to fund their
eventual suspension. Term life insurance can inexpensively create
an "instant estate" for a young person in good health. But as you
become older, eventually the insurance premiums will become
prohibitive. By that time you must have accumulated an actual
estate sufficient to fund suspension.
To build such an estate, it is hard to imagine circumstances in
which it is advisable to invest in T-bills. Surely if you are
attempting to maximize your expected return, you will invest in the
S&P 500 (or similar stock market investment) and expect to receive
8.8% +- 21.0% inflation rather than accept the minuscule 0.6% +-
4.4% per year return on T-bills (all percentages after inflation).
The "1 - Cumulative Distribution" graph shows the probability of
obtaining a yearly return of at *least* the X-axis value minus 1.
Of course the investor wants this probability to be as high as
possible. We see that the probability using S&P 500 investment
exceeds that from T-bill investment, unless X < .97.
[GRAPH OF 1 - CUMULATIVE DISTRIBUTION OMITTED.]
For a person to prefer T-bill investment to S&P 500 investment, he
would have to have *very* strong aversion to risk. This means that
his personal utility as a function of money is sharply concave
(curves downward).
It is difficult to contrive a situation in which it is advisable to
invest solely in T-bills, but I will try. If an orphan widow
currently has $100,000, and must have exactly $90,000 available one
year from now to meet the balloon payment on the mortgage or else
the heartless banker will foreclose and throw her in the gutter
where she will have to beg for crumbs, one might argue that she
should invest in T-bills to minimize that possibility. In this
contrived example, the widow's personal utility is not a linear
function of money. In particular, $90,000 one year from now is
worth *much* more to her than $89,900, which won't pay off the
mortgage.
But few of us face such a circumstance. We are in the situation
where we know that the more we expect to make over the long term,
the better off we will be. We know that we cannot build a
significant estate investing in T-bills. So for us the answer is
clearly *no*, do not invest in T-bills, invest in the S&P 500.
If our investment horizon is ten or more years rather than one
year, the conclusion is even clearer. For if we replot the "1 -
Cumulative Distribution" graph over that time scale, the region
where the T-bill probability significantly exceeds the S&P
probability nearly vanishes.
*The Kelly criterion*
We don't have to invest all of our funds in a single investment;
we can diversify. Suppose that the only two available investments
are T-bills and the S&P 500. Even though the T-bills are an
inferior investment, perhaps we should diversify and put *some* of
our bankroll into them. But what fraction?
It is intuitively plausible that the subjective value of an
additional dollar received is inversely proportional to how many
dollars one currently has. In this case, the utility of money is
given by the *logarithm* of one's total wealth. Daniel Bernoulli
first proposed this way back in 1730; to this day, the logarithm
remains the best prototype for everyman's utility function. Since
the logarithm function is concave, persons with logarithmic utility
will be averse to risk. In maximizing the logarithm of their
capital they may buy insurance, which a person maximizing his pure
monetary return will not do because he expects to pay more in
premiums than he will ever get back in claims (insurance companies
make money).
To select a portfolio (a mixture of investments) for a period of
time (such as a year), the Kelly criterion is to maximize the
expected value of the logarithm of wealth one will have at the end
of the period. One then reevaluates the situation, and repeatedly
invests the same way. This strategy has two very desirable
properties:
(1) Maximizing the expected logarithm of wealth asymptotically
maximizes the rate of asset growth; and
(2) The expected time to reach a fixed preassigned wealth W is,
asymptotically as W increases, least with this strategy.
Note that both conclusions concern the accumulation of *wealth*,
not the logarithm of wealth. At the risk of oversimplifying, the
Kelly criterion it is the best long-term investment strategy for
getting rich. See Thorp [2] for further discussion and references.
The Kelly criterion determines the proper tradeoff between risk and
return. It allows us to compute exactly what percent of our funds
we should put in T-bills, and hence what remaining percent in the
S&P 500. I have done the computation, and the answer is: 0% in T-
bills, 100% in the S&P 500. The decreased risk of the T-bills
still does not justify any investment at such a miserable return.
It isn't even close. We would have to increase the standard
deviation of the S&P 500 from .21 to .31 before the Kelly criterion
would begin investing a tiny fraction of the bankroll in T-bills.
Alternatively, we would have to reduce the expected return from the
S&P 500 from 8.8% to 4.4% (after inflation) before T-bills could be
considered for a small part of the portfolio.
The same conclusion applies to just about all other short-term
financial instruments, such as savings accounts, money-market
funds, or certificates of deposit. Such instruments may offer a
slightly higher return than T-bills, accompanied by slightly higher
risk. They are not close to belonging in a portfolio with the S&P
500.
Reference [1] also gives mean and standard deviation figures for
intermediate government bonds, long-term government bonds, and
long-term corporate bonds. None of these investments qualify as
part of a portfolio with the S&P 500.
*Conclusions*
We assume that the goal of the cryonics fund manager is to keep the
portfolio alive until it is possible to reanimate the patient. It
is hard to find any circumstances under which it would be advisable
to invest any of a patient's portfolio in T-bills, or even in
corporate bonds. At every level of funding and at every time until
attempted reanimation, the portfolio has a greater chance of
surviving if invested in the S&P 500 than if invested in T-bills
[3]. Nor should T-bills be part of an investment strategy for
building a cryonics estate.
If we calculate the needed funding using only the expected return
on investment and ignore the *variance* of the investment, we will
seriously underestimate the needed funding. The amount of funding
required for the patient's fund to survive at the .05 confidence
level is about $250,000. This required funding level may turn out
to be too high if the cost of long term storage does not increase
as rapidly as inflation. This seems likely to be the case if we
benefit from large economies of scale. But the game you are
playing is *You Bet Your Life.* How much risk do you wish to take
on losing?
*Notes*
1. Edward Thorp advises me that deviations of the actual
distribution from this model distribution may make the situation
slightly worse than my predictions.
2. If we extrapolate exponential growth into the distant future,
the predictions are likely to be too high. Eventually limits are
approached that are not part of the model. Thus the precise Mean
figures appearing in the S&P 500 Funding table should not be taken
too seriously.
3. To be completely precise, I should add "except possibly for
funding levels where both probabilities of failure are vanishingly
small." For example, if the portfolio size is $100,000,000, the
probability of failure with S&P 500 funding might be 10^-50, while
the T-bill probability of failure might be 10^-75. In both cases
these probabilities are so small that they can be safely rounded to
0. Such extremely small probabilities of failure are only within
the limits of our model assumptions; other factors such as possible
government intervention make the real probability of failure
higher.
*References*
[1] *Stocks, Bonds, Bills, and Inflation 1991 Yearbook*.
Chicago: Ibbotson Associates (1991).
[2] Thorp, E. Portfolio Choice and the Kelly Criterion.
Reprinted in *Stochastic Optimization Models in Finance*, W. Ziemba
and R. Vickson eds., New York: Academic Press (1975).
Automated Development of
Fundamental Mathematical Theories
This work, by TRANS TIME President Art Quaife, was just released by
Kluwer Academic Publishers. It is a revised version of his Ph.D.
thesis, and the second volume in their Automated Reasoning Series.
Of particular interest to cryonicists is the Preface, which is a
strong expression of cryonicist/immortalist goals and philosophy.
Art had to fight hard to get that into the book. Both the managing
editor and the senior editor strongly urged him to remove it,
fearing it would detract from the serious nature of the work. But
Art successfully insisted that it stay in.
Art provides an introduction to automated reasoning, and in
particular to resolution theorem proving using the prover OTTER.
He presents a new clausal version of von Neumann-Bernays-Gdel set
theory, and lists over 400 theorems proved semiautomatically in
elementary set theory. He presents a semiautomated proof that the
composition of homomorphisms is a homomorphism, thus solving a
challenge problem.
Art next develops Peano's Arithmetic, and gives more than 1200
definitions and theorems in elementary number theory. He gives
part of the proof of the fundamental theorem of arithmetic (unique
factorization), and gives an OTTER-generated proof of Euler's
generalization of Fermat's theorem.
Next he develops Tarski's geometry within OTTER. He obtains proofs
of most of the challenge problems appearing in the literature, and
offers further challenges. He then formalizes the modal logic
calculus K4, in order to obtain very high level automated proofs of
Lb's theorem, and of Gdel's two incompleteness theorems. Finally
he offers thirty-one unsolved problems in elementary number theory
as challenge problems.
The publishers have priced the book at a hefty $123.00. This may
briefly slow its rise up the New York Times bestseller list.
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