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Date: Thu, 2 Sep 93 12:47:00 -0700
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Subject: CRYONICS Trans Time Newsletter
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THE TRANS TIMES
Life Extension through Cryonic Suspension
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Volume 2 Number 4 August 1993
Rational Portfolio Determination
by Art Quaife, Ph.D.
I was recently elected a Director of the International Cryonics
Foundation (ICF). The current plan is that I will be chairman of
a Suspension Funds Investment Committee. As such, I will have
primary responsibility for developing an investment policy for
Donor funds. That responsibility has prompted the considerations
in this article.
Kelly criterion
When your favorite financial guru tells you that you should
invest 50% in stocks, 30% in bonds, 10% in gold, and 10% in cash,
do you ever wonder if there is a *rational* way to determine what
those percentages should be? Yes there is, although your guru
almost surely did not use it. The method is to use the *Kelly
criterion*. I described this criterion briefly in my article [2],
and gave a reference to an article by Edward Thorp [4] with much
more detail.
Roughly stated, the Kelly criterion shows that to maximize the
rate of growth of your wealth in the *long* run, you should
maximize the *logarithm* of your wealth in the *short* run. The
short run is the length of time that you stick with your
investment decisions, before going through a re-evaluation and
making new decisions. For the purposes of this article, I will
assume this period is one year, although it could be done much
more frequently. The long run is, if all of our dreams come true,
forever.
For the mathematically inclined, let me show how to solve the
Kelly criterion equations in one case of interest. According to
the Ibbotson Associates program EnCORR, as quoted in The Hulbert
Financial Digest of June 1993, we have the following:
YEARS 1972-1991
Investment Mean - 1 Standard Deviation
1. Morgan Stanley EAFE Index .145 .241
2. S&P 500 .119 .175
3. Gold .110 .389
4. World Bond Portfolio .107 .101
5. U.S. Intermediate Term Bonds .094 .069
6. U.S. Treasury Bills .077 .026
CORRELATIONS
World Inter
EAFE S&P 500 Gold Bonds Bonds T-bills
1. EAFE 1.00 .57 -.17 .53 .06 -.33
2. S&P 500 .57 1.00 -.34 .19 .39 -.09
3. Gold -.17 -.34 1.00 -.17 -.35 -.08
4. World Bonds .53 .19 -.17 1.00 .36 -.45
5. Inter Bonds .06 .39 -.35 .36 1.00 .20
6. T-bills -.33 -.09 -.08 -.45 .20 1.00
Here the S&P 500 is an index of the stock values of 500 of the
largest U.S. companies, which is widely used to measure "the
market". The Morgan Stanley EAFE (Europe-Australia-Far East)
Index is a less well known index of about 1000 stocks used to
measure the international market. The domestic return on a
foreign investment depends not only on how well the company does,
but on how the foreign currency varies with respect to the
dollar. This additional source of variation may explain why the
standard deviation of the EAFE Index is higher than that of the
S&P 500. The random variable in each case is the ratio Value(year
+ 1) / Value(year).
Distribution
To use the Kelly criterion, we must know the joint probability
distribution of these six investments for the upcoming year. All
we have available from the quoted article are the averages for
the past 20 years. Has the nature of these markets remained the
same over the past 20 years? In technical terms, are these time
series *stationary*? It seems plausible that to predict the
market behavior next year, data from the current year is more
relevant than data from 20 years ago. If we had periodic (e.g.,
monthly) data available over the 20 year period, we could test an
exponential smoothing model. Lacking such periodic data, we will
simply use the 20 year historical averages as our assumed
parameter values for next year.
Let us determine an optimal portfolio P that includes fraction
x[i] between 0 and 1 of investment i, where the x[i]s add up to
1. We have:
6
Mean(P/P0) = SUM x[i] mean[i]
i=1
2 6 6
Std (P/P0) = SUM SUM x[i] x[j] std[i] std[j] corr[i,j]
i=1 j=1
Our goal is to find the values of the x[i] that maximize
(integral ln(P/P0) dPr), where Pr is the probability measure of
P. Above we have given the mean and standard deviation of P, but
those two moments do not fully specify its distribution.
It is reasonable to assume that the distribution of each of these
investments is *lognormal*, meaning that, e.g., ln(S&P 500) has
the *normal* distribution. The lognormal distribution is a widely
accepted model of the distribution of stock prices [Footnote 1].
Let me offer a plausible argument that we can approximate the
distribution Pr by the lognormal distribution. For if the
distribution of the components is not too skew (i.e., if
Std(ln(component)) is not too large), then each component could
be approximated almost as well by the normal distribution. Next
we can approximate the joint distribution of the six investments
by the multivariate normal distribution. Then the distribution of
P is obtained by a quintuple integral, which evaluates to another
normal distribution. This normal distribution can in turn be
approximated by the lognormal distribution with the same mean and
standard deviation.
It must be noted that several of the component standard
deviations are not that small, bringing into question how good
this approximation is. Thorp [3] argues that if we "rebalance"
(see below) the portfolio more frequently, in the limit of
continuous rebalancing this approximation becomes exact.
The beauty of this approximating assumption, that P has the
lognormal distribution, is that it allows us to easily calculate
the integral we want. Namely, if the random variable X has the
lognormal distribution with mean *mean* and standard deviation
std, then ln(X) has mean mu and standard deviation sigma given by
2 .5
mu = ln(mean / (1 + (std / mean) ) ),
2 .5
sigma = (ln(1 + (std / mean) ) ,
and this mean mu is exactly the value of the integral we are
trying to maximize! [Footnote 2]
Optimization
As a first step toward optimization, we can calculate the growth
rate mu for each of the components as stand-alone investments. We
get:
Investment Growth Rate
EAFE .114
S&P 500 .100
Gold .046
World Bonds .098
US Intermediate Bonds .088
US T-bills .074
Note that even though the mean return on gold was the third
highest at 11.0%, the Kelly criterion growth rate places it dead
last because its standard deviation was such a large 38.9%. In
other words, gold investment is just too risky. However, because
gold is negatively correlated with the other investments, it can
be useful in a *mixed* portfolio to reduce variance.
The growth rates determined above may be considered "risk-
adjusted returns". Clearly mu increases with increasing mean, and
decreases with increasing std. In these respects it is similar to
the Sharpe ratio (mean - mean(T-bills)) / std, which is sometimes
used to rank investments on a risk-adjusted basis. But mu is a
better measure of investment desirability than the Sharpe ratio.
In particular, the Sharpe ratio rates gold above T-bills as a
stand-alone investment on the above data, whereas the Kelly
criterion reverses that order.
All lognormally distributed investments with the same mu are
equivalent with respect to the Kelly criterion. Thus for fixed
mu, we can explicitly determine the tradeoff between the risk std
and the return mean as:
2
std = mean sqrt((mean / exp(mu)) - 1)
If we want to choose a pure portfolio of just one of these
investments, the Kelly criterion tells us to invest purely in the
EAFE. We could easily do this much of the computation on a hand
calculator.
Thorp [3] shows another special case that can be easily solved on
a hand calculator. Note that if we determine the optimal
coefficients x[i] and purchase the corresponding portfolio, even
one day later this will no longer be the optimal portfolio. This
is because the values of the component securities are changing
daily, so that a portfolio with 90% EAFE today may have 91% EAFE
tomorrow. To maintain the optimal mix, one then needs to
rebalance by selling some of the EAFE. Let us assume that we
rebalance the portfolio very frequently ("continuously"), and
ignore any transaction costs.
Note that the standard deviation of T-bills is a small .026.
With frequent rebalancing, we may assume that this standard
deviation is zero. For if we buy a 3 month T-bill, we can be
(nearly) certain that the government will redeem it at the quoted
interest rate. Of course at redemption time, the renewal
interest rate may be different.
The assumptions of Thorp's special case are thus that (a) we do
continuous rebalancing and (b) the standard deviation of T-
bills is zero. Then we can determine the optimal portfolio
consisting of fraction x of any of the of the other securities,
and fraction (1 - x) of T-bills. Use of the Kelly criterion in
this case reduces to maximizing a quadratic function (an upside
down parabola). We find:
2
x = ln(mean / mean(T-bills)) / sigma ,
mumax = .5 x + ln(mean(T-bills)).
Using the data in the above tables, we get:
Investment x mumax
EAFE 1.412 .117
S&P 500 1.583 .104
Gold .260 .078
World Bonds 3.314 .120
US Inter Bonds 3.945 .105
Note that in four cases we have x > 1. These results can only be
achieved if we can go on margin and borrow money at the T-bill
interest rate. We are unlikely to be able to borrow money at that
good a rate, and we are unlikely to be able to borrow anywhere
near the amounts required for the largest x coefficients in the
table. If we prohibit borrowing, we should set x = 1.0 in these
four cases, with the corresponding reduction in mumax.
General Case
To find the optimal weights for a mixed portfolio of all six
component investments, the computations are much more
complicated, and we need to use a digital computer. I programmed
the optimization of the growth rate in C++, using the downhill
simplex method of Nelder and Mead [1]. The values I determined
are:
Investment Fraction
EAFE .868
S&P 500 .000
Gold .132
World Bonds .000
US Intermediate Bonds .000
US T-bills .000
It is disappointing to determine that the growth rate of this
portfolio is .115, which is only .001 better than the pure EAFE
portfolio. To compare with the EAFE values in our first table
above, the optimal portfolio reduces (mean - 1) from .145 to
.140, but also reduces std from .241 to .206. Here, use of the
Kelly criterion has not improved our return significantly over
the simple-minded choice of investing in the component that has
the highest historical rate of return. But at least it has
confirmed that choice, on a sounder basis.
Thorp [3] also shows how to solve the general case of mixing all
six investments, with continuous rebalancing and unrestricted
lending and borrowing of securities. Of course this case can also
be solved using the downhill simplex method that I used above,
but Thorp shows that in this case the equations are linear, and
thus can be solved by simpler techniques such as Gauss-Jordan
elimination.
Other policy considerations
We have now solved the mathematical problem of optimizing the
return within the listed set of investment options. But ICF faces
other problems in determining the best investment policy, which
are practical, political, and legal. Namely,
1. The policy should be easily understood by the Suspension
Members,
2. The policy should be easily defensible against charges that
the incompetents running the investment committee are making dumb
or reckless investments that a prudent man would not make.
Over the past 20 years, I have invested in mutual funds which
have outperformed the S&P 500 by a few percent per year. I expect
to be able to continue choosing such funds for my personal
investments. But I will probably not recommend them to the
investment committee, as they are less defensible against irate
party complaints than is a pure S&P 500 investment if we
encounter a major bear market, and of course we will sometime
encounter another major bear market.
On the other hand, if the ICF Investment Committee adopts the
optimal strategy determined in this article, some people might
even charge us with being un-American for investing almost all of
the Donor Funds in foreign equities.
I have also solved the equations with unrestricted borrowing and
lending of securities permitted, using the data in the tables
above. The optimal growth rate increases by about .04 over the
.115 we determined above. This is a significant increase in
return, that one would try to implement in personal investing.
But in the practice of the ICF Investment Committee, purchasing
securities on margin is probably unacceptable. Many people
believe that going on margin (borrowing money to buy securities)
is too risky, and will not be persuaded otherwise by Kelly-
criterion computations that they do not understand.
For these reason, I will likely recommend that the investment
committee place 50% of Donor Funds in an S&P 500 index fund, and
50% in an EAFE index fund. This policy is only slightly inferior
to our optimal policy: it reduces the growth rate from .115 to
.111. Extrapolating from past performance, this policy should
handily beat the published investment policies of other cryonics
organizations that manage funds.
Footnotes
1. If we consider the ratio of price changes over a period of
one day, an even more accurate model is given by Student's t-
distribution with about 5 degrees of freedom. But as we increase
the interval between measurements, by the time this interval
becomes one year the lognormal model becomes quite accurate.
2. Thorp [4] carries out a Taylor series expansion of ln(P) and
obtains the approximation (in our notation):
2
mu = [approx] ln(mean) - .5 (std / mean) .
It is easy to see that his approximation follows directly from
our approximation, when std is small with respect to mean.
References
[1] Press, H. et al. Numerical Recipes in C. New York: Cambridge
University Press (1992).
[2] Quaife, A. Staying Cold: Providing Sufficient Maintenance
Funding. THE TRANS TIMES 1:3, 1-8 (1992).
[3] Thorp, E. Personal communication dated August 11, 1993.
[4] 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).
Research Update
by Hal Sternberg, Ph.D.
We have accelerated our efforts to improve solutions and
protocols for tissue, organ, and whole animal cryogenic storage.
Improvements were made on models to develop better freeze-
protecting technology. We now have in place tests to determine
cell viability, anatomical integrity, and function.
Whole animals (rats and hamsters) are perfused with newly
developed cryoprotective solutions and frozen to liquid nitrogen
temperatures. After thawing, we transplant full thickness skin
onto recipients. We then monitor hair growth and general
appearance. Additionally, reports from a certified pathologist
indicate the integrity of the tissue. We assess muscle cell
viability by observing and quantifying activity (movement/
contraction) using high power stereo-microscopy and monitors.
Also, we assess reperfusion injury by inspection of blood vessels
and the extent to which they refill, particularly in the brain.
Moreover, the leakage of fluid (i.e., blood or stain) from the
vessels is a further indication of damage.
We have successfully used new solutions that dramatically reduce
reperfusion injury (i.e., most of the brain can be reperfused
without noticeable leakage from blood vessels), and that maintain
a high degree of muscle cell viability. Most importantly, the new
solutions do not induce excessive dehydration, which we have
found is both functionally and anatomically disruptive.
Risk Probability
by H. Jackson Zinn
The Defendant was on trial in a Chicago Court for possession of
stolen property, a portable T.V. set.
"How did you acquire the T.V.?" asked the defense attorney of the
Defendant.
"I walked around the corner into the alley and I found it," the
Defendant replied.
The Judge looked upon the parties before him quite sternly.
"For 53 years I have walked around the streets and alleys of
Chicago, and I have never, ever found a T.V. set!"
Later the Judge got the defense attorney back in chambers:
"Why did you put that crazy story on in Court?"
"Well, that's what he told me," replied the defense attorney.
"And you believed it? You're goofier than he is!" opined the
Judge.
The Judge's disbelief sprang from two apparent factors:
1) The lack of observance of the occurrence, although not
scientifically quantifiable as to probability, and
2) The implausibility that anyone would abandon valuable
personal property in a public way.
Today cryonicists look at risk factors affecting their long-term
preservation. I have seen maps of the United States and of
California showing where the greatest risks are from various
natural disasters such as hurricanes, tornadoes, earthquakes,
etc. To date, I am aware of no studies differentiating between
the disasters (e.g., is an earthquake more or less likely to kill
someone than a tornado?). However, personal observation and
common sense lead me to believe that other risks are far greater.
I know many individuals and businesses that have become insolvent
or bankrupt, but I know of no individual killed in a hurricane,
tornado, or earthquake, or business ended by same.
The best way a cryonics enterprise can minimize long-term risk is
to practice long-term frugality, in my view. For example,
cryonics patients do not require an ocean view or other high-rent
property. The dramatic savings achieved through the economy of
low-cost real estate can be used to keep prices competitive and
to improve corporate profits and financial stability. It is
important as well to avoid legal risks in all of our actions. In
modern litigation, courts in the United States follow what is
called the American Rule. This rule states that each side pays
their own attorney's fees, regardless of the result of the case.
The exceptions to this rule are rare. I recall an item of
litigation a few years ago wherein TRANS TIME was sued for ten
million dollars, but the adversary was forced to settle for zero
dollars. However, each side had to pay its own attorney's fees. .
In the aforesaid case, the adversary was forced to face the fact
that TRANS TIME had ample low-cost legal representation, and the
fact that even if they should win, there was no ten million
dollar fund at the end of the rainbow. Further, much of TRANS
TIME's property at the time was of far less value to outsiders
(e.g., dewars), and would cost a lot of money just to transport
or store.
Nature's example of the porcupine should be emulated. The
porcupine is virtually impervious to predators because it is
protected by a large number of needles (substitute "attorneys").
Assuming an animal gets through the needles, porcupine meat is
still stringy and not very tasty. Thus, assuming TRANS TIME
remains armed with attorneys, the latter part of implementation
would then be to try to insulate TRANS TIME assets from outside
attachment, or from the effort to attach. Cryonics is still an
embryonic enterprise. One vigorous lawsuit, even if unfounded and
unsuccessful, could sink any cryonics organization at present.
Northern California cryonics organizations have always practiced
preventive law. The American Cryonics Society and the
International Cryonics Foundation have never been sued by anyone
for any reason. TRANS TIME, although it has been sued, has never
been adjudged liable for any willful or negligent conduct. In
today's society, anyone can be sued, and a lot of zeros can be
tacked onto the claim. Our mutual task is to make sure that all
of those zeros are preceded by zeros!
[Cartoon omitted]
Subscription information
THE TRANS TIMES is published bimonthly by Trans Time, Inc., 10208
Pearmain Street, Oakland, CA 94603, phone 510-639-1955.
Subscription price is $12.00 per year (6 issues).
[I may not be posting this newsletter to the CryoNet any more, as
it is quite tedious to convert typeset mathematics and other
characters to ASCII. AQ]
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