(Hey all. Apologies for my inactivity on Felicifia of late. I've been working on other things, but I hope to return in a few months. However, I thought I would post this in the meantime while it's on my mind.)

tl;dr
Depending on your income, expected rate of return, and purchase transactions fees, it makes sense to buy a new stock about every half month or every month, assuming you manually buy stocks as your vehicle for continuous investment.

Problem
Say you earn $800 post-tax per week, and you want to invest those earnings by buying some random stock. Suppose you think the stock will earn 7% per year in expectation, and for convenience, let's use simple interest. When you buy a stock, there's a fixed transactions fee of $8 per purchase. How many weeks should you wait before you buy a stock in order to optimally trade off the cost of missed investment income against the transaction cost of buying a new stock?

Notation
To generalize, let's say your wage is w = $800, and that you earn n = 52 wages in a year. You expect an annual return of r = 0.07 with simple interest. When buying stocks, the transaction fee is c = $8.

Cost formula
Every wage period when you don't use w dollars to buy stock, you lose out on expected investment income of wr/n, since r/n is the expected interest rate for one wage period.

If you wait two wage periods to buy a stock, w dollars lie idle for one week, and w dollars are used immediately. The only lost income is from the first wage period in the amount of wr/n.

If you wait three wage periods, w dollars lie idle for two weeks, w dollars lie idle for one week, and w dollars are used immediately. The lost income is then (wr/n)(2) + (wr/n)(1).

If you wait k wage periods, the lost income is (wr/n)(k-1) + (wr/n)(k-2) + ... (wr/n)(1) = (wr/n)[k(k-1)/2] using the formula for triangular numbers.

Say you follow this pattern of waiting k wage periods for a long time T, where T is sufficiently big that we can assume it's a multiple of k and ignore the irregularities at the end of the time period. You would then buy stocks T/k times.

The purchase cost of the stocks would be c(T/k). The lost income would be (wr/n)[k(k-1)/2](T/k). The total cost is

c(T/k) + (wr/n)[k(k-1)/2](T/k),

which is what we want to minimize.

Minimizing the cost
T factors out of the cost, so it suffices to minimize

F(k) := c/k + (wr/n)[k(k-1)/2](1/k) = c/k + (wr)(k-1)/(2n).

While k can only take on discrete values, we can pretend for the moment that this function is continuous so that we can differentiate and set equal to 0:

dF/dk = -c/k^2 + wr/(2n) = 0
wr/(2n) = c/k^2
k^2 = 2cn/(wr)
k = sqrt(2cn/(wr)).

Note that the second derivative with respect to k is d^2F/dk^2 = 2c/k^3 > 0, so this is indeed a minimum point.

Plugging in values
Using the figures in the original problem, the continuous solution is k = sqrt(2 * 8 * 52 / (800 * 0.07)) = 3.85. Using Excel, I found the actual values of the cost at each k from 1 to 5:

F(1) = 8
F(2) = 4.5
F(3) = 3.7
F(4) = 3.6
F(5) = 3.8,

so k=4 is indeed the minimum. In other words, using these figures, you should buy a new stock about once a month.

Alternate values
Suppose instead you're paid twice a month (n = 24), with w = $2000 per pay period and an assumed rate of return of r = 0.10. Then the optimal continuous value of k = 1.4, which means it's better to buy a stock every pay period than to hold on for a second pay period.

Related problems
These situations of accumulating benefits for an action that has fixed cost crop up a lot. For example: How often should you cut your hair? I like to have short hair because it reduces shower time and saves on hot-water bills, but getting a haircut also costs money and time in itself. Every day that you have hair longer than it needs to be, you lose a small amount of money, say L(d), when it's been d days since your last hair cut. L(0) = 0, and then L(d) grows as d grows. The accumulated cost after waiting k days will be sum_i=1^k L(i). The minimization will be the same as before, with the [k(k-1)/2] term replaced by this summation.

Thanks for this information, but have you ever thought about the possibility that being productive to earn this money in the first place could do more harm than good, even including the good your donations could ever make?

If the spectre of space colonization or hyperefficient consciousness is realistic, then its cause will be human technological progress, and by adding expertise and economic productivity to this process, you are speeding it up. And if the suffering surplus (bad is stronger than good, minorities torture helpless majorities) remains a reality in this future, slowing this process down deliberately could actually be the expected-utility-maximizing choice.

Your assumption that you can donate so much money for meme spreading and people-convincing that it outweighs this harm requires that people can actually be convinced to become more ethical people, to such a degree. I'm not sure how realistic that is, given that real people only rarely actually change their minds based on moralizing or appeals to empathy, and certainly not in the long run. Remember, there will be a point when those in power can simply switch off and on their empathy as it suits their strategic goals. Already, civilization contains crude mechanisms to counter the effects of empathy in specific functional ways (propaganda, shielding from authentic emotional stimulus input, creating fake input etc.)

Maybe the most ethical choice would be to defect from the process of productivity and earn less, especially in areas of sought-after tech expertise, in order to slow down this species of torturous apes in their pursuit to create more consciousness in the universe.

Thanks, HedonicTreader! I must say, I really don't share your intuition in this case.

First, there's the fact of replaceability in careers, which means the impact I make is (roughly speaking) only the amount by which I'm more productive than the next person would have been (plus some other side-effect considerations).

Second, even if human survival is net bad, it's not clear to me whether accelerating technology matters. The only thing that matters beyond rounding error is whether humans survive in the long run, not how quickly they develop technology. It's not obvious whether accelerating technological development increases or reduces extinction risk. Maybe a faster pace of innovation means less time for safety constraints?

Third, even if the non-replaced impact of my work were net bad in this way, I find it hard to imagine that this wouldn't be outweighed by the effect of donations. Intellectual arguments do make a difference (after all, what convinced you to think the way you do?), and even if you think they don't, there are lots of other things one could fund that would make a difference by your criteria. For example, how about changing demographics such that more people with negative-leaning-utilitarian inclinations are born? Or more directly opposing economic growth in various ways? (The latter isn't something I necessarily support; I'm just speaking hypothetically.) There would seem to be lots of options where funding could make a targeted impact compared with the blunt side-effects of having made that money. And of course, funding research on these questions could be extremely valuable too.

Hi Brian, thanks for the reply, and feel free to delay further replies if you have more important priorities. I won't take it personally.

Brian Tomasik wrote:First, there's the fact of replaceability in careers, which means the impact I make is (roughly speaking) only the amount by which I'm more productive than the next person would have been (plus some other side-effect considerations).

Yes, but I actually think the total number of people qualified to do a certain high-level task is limited - and the number of people who could become qualified through education cheaply is also limited. The more qualified you are and the more sophisticated your job is, the less likely it is that dropping out would be compensated by the replacability effect. If a qualified software engineer decides to drop out of the labor force completely, or even commit victimless crimes and bounce in and out of prison, spend time on welfare etc., this would remove significant amounts of money from the economy. I don't think you can create high-quality people out of thin air, and the replaceability effect is more relevant in deciding whether a particular job is ethical, rather than whether removing or adding a high-quality employee to/from the economy is ethical (no?). Of course, there's an element of self-sacrifice involved in this decision, as it affects the personal life significantly.

Second, even if human survival is net bad, it's not clear to me whether accelerating technology matters. The only thing that matters beyond rounding error is whether humans survive in the long run, not how quickly they develop technology.

I do think astronomical waste does matter, unless you think the probabilites of space colonization are very small to begin with. If the probabilities are non-trivial, even one day of astronomical waste does matter a lot.

It's not obvious whether accelerating technological development increases or reduces extinction risk. Maybe a faster pace of innovation means less time for safety constraints?

Exactly - it's not obvious. Maybe a slower pace of innovation increases existential risk because crucial resources can run out before we innovate into substituting it efficiently. Maybe an extinction event hits in the additional time it takes to innovate. Or maybe you're right and slower innovation means more safety constraints. But the point is this: If we have no reason to assume one way or another, we should be agnostic about the effect of innovation speed/economic growth on existential risk. In which case it seems the astronomical waste would dominate expected utility, unless the probabilities are vanishingly small to begin with.

Third, even if the non-replaced impact of my work were net bad in this way, I find it hard to imagine that this wouldn't be outweighed by the effect of donations. Intellectual arguments do make a difference (after all, what convinced you to think the way you do?), and even if you think they don't, there are lots of other things one could fund that would make a difference by your criteria.

It would be arrogant of me to assume I'm part of an elite who's convinced by rational argument. But the truth is, a lot of my emotional intuitions regarding morality have been shaped by interactions with other people. Some positively reinforcing, like in the utilitarian community. But a lot negatively reinforcing, by people who wanted to convince me of the opposite of the positions I ended up with. And I think by trying to get people to care more about suffering, I may have pushed a lot of them to rationalize acceptance of suffering instead. My old psychology professor once told us he doesn't discuss controversial topics with people who don't hold views very similar to his in the first place, because the most likely outcome would be to push them further away. I'm sure I picked up some memes by people paid for to spread them, but I don't actually think it's robust and predictable enough.

For example, how about changing demographics such that more people with negative-leaning-utilitarian inclinations are born?

I'm very curious how exactly you would spend money to accomplish this.

Or more directly opposing economic growth in various ways? (The latter isn't something I necessarily support; I'm just speaking hypothetically.) There would seem to be lots of options where funding could make a targeted impact compared with the blunt side-effects of having made that money.

Yes, like sabotage. The problem is, of course, that this takes criminal energy, strategic positioning, and a degree of self-(and potentially other)-sacrifice most of us won't be willing to make.

And of course, funding research on these questions could be extremely valuable too.

Sure, if we can actually translate more money into better knowledge. Again, the question is, where to put the money, and how to make sure it actually improves knowledge at the margin, rather than shifting pre-existing funding so we end up funding literature studies indirectly. There's also the question how much this knowledge is really worth to us. Giving such knowledge to people who don't share our values is worthless or potentially harmful. And if we have to pay so much money for the knowledge that it absorbs the realistic economic (altruistic) power of our small special-interest community, the knowledge will be useless to us as well.

I wonder what people think about LendingClub http://www.lendingclub.com/
They seem to have 9% annual return average for the majority of people using their investment service. The way it works is you lend money to people who are trying to consolidate their debt or just want a loan. Because some are paying credit-card debt at 20%+ it makes sense for them to take a loan from you at 18%. Few people default, but on average you get a decent annual return.

Over the past 3 years I've averaged about 13% and it seems to be one of the the-lowest-risk-&-highest-return around. I might be blinded by some (confirmation?) bias here - but it seems like this investment vehicle is amazingly awesome.

yboris wrote:I wonder what people think about LendingClub http://www.lendingclub.com/
They seem to have 9% annual return average for the majority of people using their investment service. The way it works is you lend money to people who are trying to consolidate their debt or just want a loan. Because some are paying credit-card debt at 20%+ it makes sense for them to take a loan from you at 18%. Few people default, but on average you get a decent annual return.

I'm suspicious. "Average return" statistics can be juiced with survivorship bias, ignoring transaction fees (and taxes) and other shenanigans.

http://wealthpilgrim.com/lending-club-reviews/

It looks like the claimed 9% return comes from a sample that is disproportionately young loans, i.e. people who have just borrowed the loans, and paid back less in interest than they borrowed in the first place. Default rates will probably go higher with time. I also wonder about defenses against Ponzi schemes: what if borrowers take out multiple personal loans like this, then pay the interest using the loans, and after leveraging up just walk away?

OTOH, in principle this sort of disintermediation could produce net value if the borrowers feel worse about not repaying the peer-to-peer loans than not paying a credit card company, or if credit card rates are higher than necessary because of clever company exploitation of behavioral economics (teaser rates and the like), etc.

Also, selection bias: would you have recommended/mentioned it if defaults had wiped out much of your investment already?

Plus they are taxable as regular income (unless in a tax-advantaged account), much higher than other interest income: http://www.mint.com/blog/investing/shou ... ding-club/

This isn't to say that it's necessarily a bad investment, just that there's lots of room for critical scrutiny (and claimed free-lunch investment returns deserve scrutiny).