Note: This piece is overly simplistic. For a more nuanced discussion, see "When Should Altruists Be Financially Risk-Averse?".
Summary
In many cases, the good accomplished by money is approximately proportional to the amount donated, so that traditional arguments for being risk averse with respect to wealth don't apply. In such circumstances, utilitarians should theoretically take advantage of economic risk premia, such as those that accrue to riskier stocks. In the context of the traditional Capital Asset Pricing Model, "riskier" means "higher beta," i.e., higher scaled covariance with market returns. However, empirically the opposite trend may be the case: Higher betas may give lower average returns. See, e.g., Figures 1 and 2 of "Re-Thinking Risk: What the Beta Puzzle Tells Us about Investing" by David Cowan and Sam Wilderman. Likewise, Eric G. Falkenstein notes that "Empirically, standard, intuitive measures of risk like volatility and beta do not generate a positive correlation with average returns in most asset classes."
Introduction
One of the reasons for which people often focus on direct charity instead of longer-term advocacy is that the results of charity are certain. A person can immediately see pets being taken care of when she volunteers at an animal shelter; it's less clear that she'll actually prevent animal suffering by handing out a few leaflets on factory farming.
Yet, I think the latter option is the better of the two. Indeed, in this essay, I'll argue that high-risk, high-payoff projects can often be ethically optimal and that small probabilities deserve greater consideration than they are usually afforded.
Definitions
To begin, it will be helpful to introduce a few basic concepts about probability. A random variable, intuitively, is a quantity that takes one of several possible values according to chance. For instance, one might roll a die twice and define the random variable X to represent the number of times the die came up six. X might take one of three different values: 0 if a six was never rolled, 1 if a six was rolled once, and 2 if a six was rolled both times.
The expected value of a random variable is the average of the values that the variable might take, weighted in proportion to their likelihood. It's calculated by multiplying each value of the variable by its corresponding probability and taking the sum.
Example. Alice is taking the SAT test. She comes upon a question for which she has no clue about the answer, so she decides to guess randomly. By doing so, will her score increase, decrease, or stay the same on average? If Alice guesses correctly, she'll gain one point. With five choices, there's a 1/5 probability that this will happen, so her expected gain is (1)(1/5) = 1/5. However, if Alice guesses a wrong answer, she'll lose 1/4 of a point. There's a 4/5 chance of that outcome, so her expected loss is (-1/4)(4/5) = -1/5. Alice's overall expected value is her expected gain plus her expected loss: 1/5 - 1/5 = 0.
Economists typically assume that, as people earn more and more income, the additional utility (satisfaction) that each dollar gives them decreases. For instance, if you make $100 a year and get a raise to $101 a year, your increase in utility is, according to the assumption, bigger than if you earned $1,000,000 a year and got a raise to $1,000,001 a year. This is called the law of diminishing marginal utility of income.
Risk Aversion
Economists also typically assume that people act so as to maximize the expected value of their utility, or "expected utility." (See Why Maximize Expected Value?) Combined with diminishing marginal utility of income, this implies that people are financially risk averse—that is, when they're presented with two options that have the same expected value of income, they choose the one with lower risk.
Example. Bob currently doesn't have health insurance. If he doesn't get sick this year, he'll earn $30,000. But if he does get sick, he'll have to pay for hospital expenses of $10,000. There's a 1/2 chance that he'll get sick. Bob's expected income is thus (1/2)($30,000)+(1/2)($20,000) = $25,000. Bob stumbles upon a health-insurance plan that costs $5,000. If insured, Bob won't have to pay any expenses for getting sick, so his income will be $25,000 regardless of whether he gets sick. Thus, if Bob buys the insurance, his expected income will be $25,000, and if he doesn't buy the insurance, his expected income will also be $25,000. Will he buy the insurance?
Imagine that Bob has $25,000. If Bob has diminishing marginal utility of income, a $5,000 drop in income is more of a loss than a $5,000 increase in income is a gain. Since the loss and gain are equally likely, Bob's expected utility is lowered by risk. Thus, if Bob acts to maximize expected utility, he will buy the insurance.
Risk Non-aversion
The law of diminishing marginal utility usually doesn't apply in the case of charity. For instance, suppose donating $1 to Vegan Outreach prevents an expected year of suffering in factory farms. Well, presumably $100 donated will prevent an expected 100 years of suffering. In this case, marginal societal utility doesn't diminish with respect to dollars donated. This fact leads to interesting conclusions.
Example. Charles is investing in the stock market to accumulate wealth that he will donate to Vegan Outreach. Should he choose a low-risk or high-risk stock?
Most investors in the stock market are risk averse. Therefore, riskier stocks have risk premiums—i.e., they have higher expected payoffs to compensate for greater variability. But if the marginal societal utility of donating to Vegan Outreach doesn't decrease, then expected suffering prevented is directly proportional to the stock's expected payoff. The high-risk stock is thus the option of greater expected societal utility.
Question. But what happens when the securities we invest in don't do well? What happens when a stock that we've invested in heavily tanks? Then we will have gambled away good that we could have done with our money.
Response. I think there is diminishing marginal utility on the part of individuals who want to effect change with respect to the magnitude of change effected. That is, bringing about ten times as much change probably doesn't feel ten times as good to those who accomplish it. So there is diminishing marginal individual utility for donating income. But utilitarianism is not about maximizing your own utility; it's about maximizing societal utility. And the option of higher expected societal utility is the one that's truly most compassionate.
I should note that one's individual utility, while negligible by comparison to the changes in societal utility that one can effect, may be important for instrumental reasons. For example, a person may be more productive if she feels good about her accomplishments; she may give up entirely if she tries to work toward a project with a tiny probability of making a difference. This effect needs to be incorporated into actual consideration of the expected values of risky and nonrisky endeavors.
Why Expected Value?
The reader may be wondering, "Okay, risky investments offer a higher expected value for reduction of suffering, but why should I care about expected value anyway?" This question is addressed in "Why Maximize Expected Value?"