Summary

TLDR: According to my preliminary model, the altruistic investing portfolio should ultimately allocate 5–20% on a risk-adjusted basis to mission-correlated investing. But for the current EA portfolio, it’s better on the margin to increase its risk-adjusted return than to introduce mission-correlated investments.

Last updated 2022-04-06.

The purpose of mission-correlated investing is to earn more money in worlds where your money matters more. For instance, if you’re working to prevent climate change, you could buy stock in oil companies. In worlds where oil companies are more successful and climate change gets worse, you make more money.

Previous work by Roth Tran (2019)¹ proved that, under certain weak assumptions, philanthropists should invest more in so-called “evil” companies than they would from a pure profit-making standpoint. This result follows from the assumption that a philanthropist’s actions become more cost-effective when the world gets worse along some dimension.

That’s an interesting result. But all it says is altruists should invest more than zero in mission hedging. How much more? Am I supposed to allocate 1% of my wealth to mission-correlated assets? 5%? 100%?

To answer this question, I extended the standard portfolio choice problem to allow for mission-correlated investing. This model makes the same assumptions as the standard problem—asset prices follow lognormal distributions, people experience constant relative risk aversion, etc.—plus the assumption that utility of money increases linearly with the quantity of the mission target, e.g., because the more CO2 there is in the atmosphere, the cheaper it is to extract.

I used this model to find some preliminary results. Future work should further explore the model setup and the relevant empirical questions, which I discuss further in the future work section.

Here are the answers the model gives, with my all-things-considered confidence in each:

• Given no constraints, philanthropists should allocate somewhere between 2% and 40% to mission hedging on a risk-adjusted basis,² depending on what assumptions we make. Confidence: Somewhat likely. [More]
• Given no constraints, and using my best-guess input parameters:
  • Under this model, a philanthropist who wants to hedge a predictable outcome, such as CO2 emissions, should allocate ~5% (risk-adjusted) to mission hedging.
  • Under this model, a philanthropist who wants to hedge a more volatile outcome, for example AI progress, should allocate ~20% to mission hedging on a risk-adjusted basis.
• If you can’t use leverage, then you shouldn’t mission hedge unless mission hedging looks especially compelling. Confidence: Likely. [More]
• If you currently invest most of your money in a legacy investment that you’d like to reduce your exposure to, then it’s more important on the margin to seek high expected return than to mission hedge. Confidence: Likely. [More]
• The optimal allocation to mission hedging is proportional to: (Confidence: Likely)
  1. the correlation between the hedge and the mission target being hedged;
  2. the standard deviation of the mission target;
  3. your degree of risk tolerance;
  4. the inverse of the standard deviation of the hedge.

Cross-posted to the Effective Altruism Forum.

Contents

• Summary
• Contents
• Setup
  • The goal
  • The utility function
  • Some more assumptions
  • Table of variables
• Results
  • Unconstrained results
    • General solution
    • Specific numeric results
  • Results with a leverage constraint
  • Results with a legacy investment
• How accurate is this model?
• Future work
• Acknowledgments
• Appendices
  • Appendix A: Mission leveraging
  • Appendix B: Why use this approach?
  • Appendix C: Proof of analytic solution
  • Appendix D: Tables of optimization results
  • Appendix E: Numeric approximation of the optimal solution
• Notes

Setup

The goal

We have some utility function that says how much we value the state of the world. Our utility function cares about two things: (1) our wealth and (2) some mission target. The mission target is a measure of an outcome we care about. For example, if we want to hedge climate change, the mission target is the level of CO2 in the atmosphere. If we’re hedging factory farming, the mission target could be the number of animals on factory farms.

Suppose there are two things we can invest in: an MVO (mean-variance optimal) asset and a hedge asset. (Note: usually MVO stands for “mean-variance optimization”, but I’m using it to mean “mean-variance optimal”.) Let the MVO asset be the investment that maximizes expected utility of wealth without regard to the mission target³. Let the hedge asset be an investment that’s correlated with the mission target and uncorrelated with the MVO asset—for example, a climate change hedge could consist of buying oil stocks + shorting the S&P 500. We choose some proportions of our wealth to invest in each of the MVO asset and the hedge asset. This total could add up to more than 100% (by using leverage) or it could be negative (by short selling).

Then we want to find the asset proportions that maximize expected utility.

As an example, say we can invest in two securities: XOM (Exxon Mobil) and SPY (S&P 500 index fund). The MVO asset might be {90% SPY, 10% XOM}, and the hedge asset might be {100% XOM, –100% SPY}. That means, by assumption, {90% SPY, 10% XOM} maximizes expected utility when holding the mission target constant.

If we invest 90% in the MVO asset and 10% in the hedge asset, then our allocation becomes

        0.9 (90% SPY + 10% XOM) + 0.1 (100% XOM – 100% SPY)
        = 71% SPY + 19% XOM

That is, 71% long SPY and 19% long XOM, with the remaining 10% in cash.

The hedge asset is a combination of positions that (by definition) has no correlation to the MVO asset. But we might never invest directly in the hedge asset. In the example above, even though the hedge includes both long and short positions, our final allocation doesn’t require short-selling. And in practice, you probably wouldn’t think about how to allocate between (90% SPY, 10% XOM) and (100% XOM, -100% SPY). You’d just think about how to allocate between SPY and XOM. But for the sake of analyzing the problem, it’s easier to think in terms of a mean-variance optimal asset vs. an uncorrelated hedge asset.

Importantly, mission hedging climate change doesn’t just mean we allocate more than 0% to oil stocks. It means we allocate more to oil stocks than a typical investor would. In this example, a normal investor allocates 10% to XOM, while a mission hedger allocates 19%.

Note A: Mission hedging is a special case of mission-correlated investing. It’s the most intuitive case, so I will focus on it in this essay, but the model I present could be adapted to apply to other types of mission-correlated investing. See Appendix A for how this could be done.

Note B: This model of mission hedging uses an extension of mean-variance optimization, but it’s not obvious that this is a good approach. See Appendix B for more on why I used this method.

The utility function

Following Roth Tran (2019), we want our utility function to meet the following criteria. Plain descriptions on the left, mathematical definitions on the right, with U = utility, W = wealth, b = mission target (which Roth Tran called a “bad activity”, although in my model, it’s not necessarily bad):

More wealth is better	\(\displaystyle\frac{dU}{dW} > 0\)
More of the mission target is worse	\(\displaystyle\frac{dU}{db} < 0\)
More of the mission target makes wealth more valuable	\(\displaystyle\frac{d^2U}{dW db} > 0\)

This describes a broad class of utility functions—so broad that it can’t give much practical guidance. Let’s pin down some more requirements:

Wealth has diminishing marginal utility	\(\displaystyle\frac{d^2 U}{dW^2} < 0\)
Sufficient wealth can eliminate almost all of the mission target, but never quite all of it	\(\displaystyle\lim\limits_{W \rightarrow \infty} \displaystyle\frac{dU}{db} = 0\)
We have constant relative risk aversion with respect to wealth	\(-W \displaystyle\frac{d^2 U}{dW^2} / \frac{dU}{dW} = \gamma\)

A constant relative risk aversion (CRRA) utility function looks like

\begin{align} U(W) = \displaystyle\frac{W^{1 - \gamma} - 1}{1 - \gamma} \end{align}

where \(\gamma\) (that’s the Greek letter gamma) is a constant that’s greater than 1. For our utility function to account for the mission target, we need to ensure (a) utility decreases with the mission target and (b) wealth mitigates the mission target, up to a maximum of 100% mitigation.

So we can define

\begin{align} U(W, b) = f(b) \displaystyle\frac{W^{1 - \gamma} - 1}{1 - \gamma} - \frac{f(b)}{\gamma - 1} \end{align}

given some function \(f(b)\) that defines how utility scales with the mission target.⁴

The \(\frac{W^{1 - \gamma} - 1}{1 - \gamma}\) term says that the utility of wealth scales with \(f(b)\). The \(\frac{f(b)}{\gamma - 1}\) term says that total utility decreases with \(f(b)\).

What is the shape of the badness function? As an example, consider climate change. If the atmosphere contains twice as much CO2, you can remove it approximately twice as quickly.⁵ In general, interventions to prevent a bad activity are plausibly twice as effective when there’s twice as much of the bad thing. So we can say \(f(b) = b\).

Our utility function simplifies to

\begin{align} U(W, b) = b \displaystyle\frac{W^{1 - \gamma}}{1 - \gamma} \end{align}

For example, \(\gamma = 1.5\) gives

\begin{align} U(W, b) = \displaystyle\frac{-2 b}{\sqrt{W}} \end{align}

If we wanted to, we could generalize this to \(U(W, b) = \displaystyle\frac{b^\lambda W^{1 - \gamma}}{1 - \gamma}\) for some constant \(\lambda\). That would allow utility to change non-linearly with \(b\), which more accurately describes many real-world situations. For this essay, to keep things simple, I will stick with \(\lambda = 1\).

These assumptions imply that we should mission hedge rather than mission leverage. But there are related assumptions that suggest we should mission leverage. For example, if we change the third condition to \(\displaystyle\frac{d^2U}{dW db} < 0\), that means wealth becomes more valuable as the mission target increases, so we should mission leverage.

Why this utility function?

Most of the listed criteria make perfect sense—e.g., of course more wealth is better and more of the mission target is worse. It’s not always true that when the mission target gets worse, our spending becomes more cost-effective. But it’s true in some cases, and those are the cases where we’d want to consider mission hedging.

The weirdest thing about this utility function: it has an upper bound. Why is that? Couldn’t we produce arbitrarily high value for the world if we had sufficient wealth?

A bounded utility function makes sense if we think of ourselves as committed to minimizing the harm of a particular mission target. For instance, there’s an upper bound to how much climate change we can prevent, namely, all of it. If we mitigate 100% of climate change, that provides an upper bound on how much good we can do.

Note: This utility function implies that impact investing doesn’t work—the way you invest doesn’t directly change the world. That might be false, but I’ll treat it as true for now.

Some more assumptions

The philanthropist doesn’t care about anyone else’s investment portfolio.

In reality, philanthropists should value the investments of other like-minded people. Instead of maximizing the expected utility of their personal portfolio, they should consider the aggregate portfolio across all value-aligned investors. They should invest in whatever way best improves the aggregate portfolio on the margin.

However, it makes more sense to start by modeling what philanthropists should do in the aggregate, and then later use that to figure out what to do on the margin. This post focuses on that first step.

The risk-free rate is 0%.

A more sophisticated model should include a term for the risk-free rate, but it’s simpler to assume that the rate is 0%, and this simplification doesn’t change much.

All random variables are lognormally distributed.

It’s common to assume that asset prices follow lognormal distributions. I also assume that the quantity of mission target is lognormally distributed, which makes sense if it’s something that tends to grow exponentially (CO2 emissions, AI progress, and numbers of farmed animals probably all behave this way).

The MVO asset has an arithmetic mean return of 8% and a standard deviation of 18%.

I could spend all day talking about market expectations, but in short, I believe these numbers appropriately represent an aggressive but appropriately-diversified portfolio (e.g., a global equity index fund).

The optimal degree of mission hedging isn’t determined by the expected return or standard deviation alone, but by the ratio between the two.

(I wrote more about market expectations in a previous essay.)

The hedge asset has an arithmetic mean return of 0% and a standard deviation of 18%.

By assumption, the MVO asset is the optimal selfish portfolio, and the hedge asset has zero correlation with the MVO asset. That means the optimal selfish portfolio contains no allocation to the hedge, and therefore the hedge asset cannot have an expected return greater than 0%.

If expected return equals 0%, then you can add leverage or mix in cash to get whatever standard deviation you want. I set the standard deviation at 18% so it’s the same as the MVO asset.

The MVO asset has zero correlation to the mission target.

This might not actually be true, but it keeps things simple. To the extent that the MVO asset is positively correlated with the mission target, that makes investing in the MVO asset look more compelling. It does not reduce the optimal allocation to the hedge, but it does reduce the optimal relative allocation. For example, increasing the correlation might change the optimal allocation of (MVO, hedge) from (200%, 30%) to (210%, 30%).

Table of variables

The model uses the following variables. m, h, b follow lognormal distributions (and thus their logarithms follow normal distributions), and x can refer to any one of m, h, b.

m	MVO asset price
h	hedge asset price
b	mission target
W	wealth
\(\alpha_x\)	log(mean of x)
\(\mu_x\)	mean of log(x) = log(geometric mean of x)
\(\sigma_x\)	standard deviation of log(x)
\(\sigma_{xy}\)	covariance between log(x) and log(y)
\(\rho\)	correlation between log(h) and log(b)
\(\gamma\)	coefficient of risk aversion
\(\omega_x\)	proportion of wealth allocated to asset x

Results

We now have a model of the expected utility of mission hedging. What does this model say a philanthropist should do?

That depends on the inputs. This model has four free variables:

1. \(\rho\): correlation between the hedge asset and the mission target
2. \(\gamma\): risk aversion
3. \(\alpha_b\): arithmetic mean growth rate of the mission target
4. \(\sigma_b\): standard deviation of the growth rate of the mission target

The model outputs are \(\omega_m\) and \(\omega_h\): the optimal allocations to the MVO asset and the hedge asset, respectively.

Unconstrained results

In this section, I provide a formula for the solution, then I offer some concrete answers for particular numeric inputs.

General solution

The MVO asset’s optimal portfolio allocation \(\omega_m\) is given by

\begin{align} \omega_m = \displaystyle\frac{\alpha_m}{\sigma_m^2 \gamma} \end{align}

where \(\alpha_m\) and \(\sigma_m\) are the arithmetic mean and standard deviation of the MVO asset, respectively.

(This is identical to the solution to Merton’s portfolio problem for an ordinary investor.)

And optimal hedge allocation \(\omega_h\) equals

\begin{align} \omega_h = \displaystyle\frac{\sigma_{bh}}{\sigma_h^2 \gamma} \end{align}

where \(\sigma_{bh}\) is the covariance of the mission target and the hedge, and \(\sigma_h\) is the standard deviation of the hedge.

We could also write this as

\begin{align} \omega_h = \displaystyle\frac{\rho \sigma_b}{\sigma_h \gamma} \end{align}

where \(\sigma_b\) is the standard deviation of the mission target.

That means the optimal allocation to the hedge asset is proportional to its correlation with the mission target, proportional to the mission target’s standard deviation, and inversely proportional to the standard deviation of the hedge and to the philanthropist’s degree of risk aversion.

(See Appendix C for proof.)

Some qualitative observations:

• \(\omega_m\) and \(\omega_h\) are independent, except that they both depend on risk aversion \(\gamma\).
• \(\omega_h\) increases with \(\rho\) and \(\sigma_b\), decreases with \(\sigma_h\) and \(\gamma\), and does not change with \(\alpha_b\). This makes intuitive sense:
  • We want to mission hedge more when our hedge is more effective (higher covariance with the mission target), including when the mission target is more volatile.
  • We want to mission hedge less when the hedge is more volatile. We get more “bang for our buck” with each dollar we put in the hedge, so we don’t need as much.
  • We want to mission hedge less when we’re more risk-averse, because we prefer a safer (less volatile) portfolio.
• It’s perhaps not immediately obvious why \(\omega_h\) varies with \(\sigma_b\) but not \(\alpha_b\). The explanation is that we’re trying to hedge against future worlds where the mission target is unexpectedly prevalent, which is more likely to happen when \(\sigma_b\) is large.⁶ \(\alpha_b\) doesn’t affect how the hedge covaries with the mission target.
• The optimal relative allocation \(\displaystyle\frac{\omega_h}{\omega_m + \omega_h}\) does not depend on risk aversion \(\gamma\). In other words, risk aversion doesn’t change our relative preference for hedging vs. traditional investing.

Specific numeric results

When provided reasonable inputs, what output does this model give?

Within a reasonable range of inputs, the optimal relative allocation to the hedge asset \(\displaystyle\frac{\omega_h}{\omega_m + \omega_h}\) falls between 1.7% and 29%. So, at least given the model assumptions described previously, we know that 50% is too much and 0.1% is not enough.

Specifically, I tested input values within these ranges:

• \(\rho\) (correlation) from 0.25 (weak hedge) to 0.9 (very strong hedge)
• \(\gamma\) (risk aversion) from 1.1 (approximately logarithmic) to 2 (substantially risk-averse)
• \(\sigma_b\) (mission target volatility) from 3% (highly stable thing, e.g., CO2 emissions⁷) to 20% (volatile or hard-to-predict thing, e.g., AI progress⁸)

Mission hedging looks most favorable when \(\rho = 0.9, \sigma_b = 20\%\). With those parameters, the optimal portfolio allocates 71% to the MVO asset and 29% to the hedge in relative terms. The absolute allocations depend on \(\gamma\). With \(\gamma = 1.1\), the optimal allocation is 227% MVO, 91% hedge (giving 3.18:1 leverage). \(\gamma = 2\) gives 124% MVO, 50% hedge.

(I do think it’s plausible that \(\sigma_b > 20\%\) for some causes, so the optimal relative allocation to mission hedging could perhaps be higher than 29%. \(\sigma_b = 30\%\) gives a relative allocation of up to 38%.)

Mission hedging looks least favorable when \(\rho = 0.25, \sigma_b = 3\%\). In that case, the optimal relative allocation is 98.3% MVO, 1.7% hedge.

Mission hedging boosts expected utility by 0.01% on the low end and 0.3% on the high end.

It’s still an open question as to whether we should invest closer to 1.7% or closer to 29% in mission hedging. That depends on (a) the correlation between the hedge and the mission target and (b) the volatility of the mission target, which I will leave as questions for future research.

For bigger tables of results, see Appendix D.

Results with a leverage constraint

All the optimal portfolios so far have required us to use leverage—sometimes a lot of leverage. But many investors can’t, at least not easily. What happens if we constrain the solution to disallow leverage?

This changes the results in one key way: Unless we’re particularly risk-averse, we can’t take on as much risk as we want to. If we want to mission hedge, we have to give up some expected return. That doesn’t mean we should never mission hedge, but it does change the tradeoffs.

Specifically, it means we should allocate 0% to mission hedging unless (a) mission hedging looks particularly compelling (high volatility of the mission target, high correlation) or (b) we’re very risk-averse, to the point that we wouldn’t want to use leverage anyway (given the return/risk expectations I used, this happens when \(\gamma > 3\)). At \(\sigma_b = 20\%, \gamma = 1.5\), we only want to mission hedge if \(r \ge 0.88\), which is probably unattainable.

Results with a legacy investment

One common scenario: You have a lot of money in some position. It’s not well-diversified, but you still hold it for historical reasons. Maybe you don’t want to sell all of your legacy asset right away, but you can sell a little bit and invest the proceeds in something else. Should you prioritize mission hedging, or is it better to invest in the mean-variance optimal asset?

Under this model, you should prefer the MVO asset, even given parameters that heavily lean toward mission hedging.⁹ Even if we already invest half our portfolio in the MVO asset (and the other half in the legacy asset), it’s still (slightly) better on the margin to move money into MVO.¹⁰

This is a much stronger result than what we got from the unconstrained model—the input parameters don’t change the outcome except at the extreme high end.

How accurate is this model?

Getting results out of a model always requires making a lot of assumptions. And every time you make an assumption, that’s another opportunity for the model to diverge from reality.

My mission hedging framework includes a lot of parameters, which are easy to modify. It also builds in some less-easily-fixable properties, including:

1. This model treats investment returns as lognormal. In reality, investments are more likely to show big declines than a lognormal distribution would suggest. A more accurate model would invest more conservatively, but it’s not clear how this would affect mission hedging. Intuitively, I’d expect an improved model to relatively favor whichever asset has a smaller left skew.
2. This model uses a particular class of utility functions. These utility functions have certain nice features, for example, the optimal asset allocation doesn’t depend on your time horizon or on how much money you start with. A philanthropist’s real-life utility function probably doesn’t look like the one I used, but real-life utility functions are notoriously difficult to ascertain. (We could perhaps improve the utility function by further investigating how our ability to do good varies with the target we’re trying to hedge.)
3. This model treats leverage, short positions, and trading as free. But even for investors who can use leverage and shorts, they have to pay some extra cost for those.
4. This model ignores the direct effects of investing. Buying stock in a company might cause the company to become more successful. And philanthropists might want to prioritize impact investing over mission hedging or profit maximization.

The most robust output of this model is the result with a legacy investment: if you currently hold most of your money in a legacy investment, then it’s better on the margin to shift toward the mean-variance optimal portfolio than to shift toward mission hedging. This result held true regardless of how I changed the model parameters (within reason). However, this result still depends on the choice of utility function, and it’s possible to construct a utility function where it’s better on the margin to put funds toward mission hedging.

For an unconstrained portfolio, it’s unclear how much to mission hedge. This model can produce a wide range of numbers depending on input parameters. The choice of risk aversion has a fairly small impact, choice of correlation has a moderate impact, and the standard deviation of the mission target matters a lot.¹¹ This suggests that, if we want to know more about what to do, we should prioritize figuring out the mission target’s volatility.

Future work

Six ideas for projects that could help philanthropists decide how to mission hedge:

1. What changes would make mission hedging worthwhile on the margin?

The strongest conclusion of my model is that, on the margin, mission hedging is worse than improving the risk-adjusted return of an investment portfolio. This result held across a range of input parameters. What assumptions could break this result? Are those changes reasonable?

If no such changes exist, we can conclude that mission hedging isn’t worth doing in practice.

2. Mission hedging vs. impact investing.

Many philanthropists want to directly do good with their investments. But impact investing often has the exact opposite prescriptions as mission hedging. Does divestment do enough good to damper—or even negate—the benefits of mission hedging? Should we over-weight investments in bad things, or under-weight them? How do we decide?

Jonathan Harris, founder of the Total Portfolio Project, has done some relevant work¹² on this.

3. Find the specific assets that provide the strongest mission correlation.

What can we invest in to mission hedge various causes? Which ones will work best? Or, in cases where mission leverage makes more sense than mission hedging, which assets provide the most effective mission leverage? And how can we tell?

For some causes, like climate change (and maybe farm animal welfare), plenty of data exists, and it’s just a matter of putting it together. For others, there’s not much existing data, so we’d have to get more creative.

4. Optimization over real-world historical data.

Pick some mission target, then get historical data on the growth rate/volatility of that target and find an investment asset that hedged it well. What would the historically optimal portfolio allocation have been?

Using real-world data alleviates certain concerns with using a model. The model assumes asset prices follow lognormal distributions and maintain consistent correlations over time, but those things aren’t true. Optimizing over historical data would show us how asset prices really behave (or, at least, how they used to behave).

Getting this data shouldn’t be too hard to do for some causes. For climate change, we could look at historical annual CO2 emissions, market returns, and returns for some plausibly good hedges like oil company stocks or oil futures.

5. Shape of the utility function.

Does the marginal utility of wealth really increase linearly with the mission target? Maybe it increases with some power function of the mission target, or maybe the relationship is more complicated. What if the mission target is a good thing (see here for examples)?

My current thinking on this subject is pretty primitive, so I expect there’s room to come up with something a lot better.

6. Alternative approaches for evaluating mission-correlated investing.

Could we look at the value of mission-correlated investing in an entirely different way? What other methods might we use to decide how much of our portfolio (if any) to dedicate to it? Could we decide on a percentage allocation using a qualitative, rather than quantitative, approach?

Acknowledgments

Thank you to Jonathan Harris for providing feedback on drafts of this essay.

Appendices

Appendix A: Mission leveraging

Mission leveraging is the opposite of mission hedging. With mission hedging, we invest in a way that gives us more money when the world gets worse. With mission leveraging, we get more money when the world gets better, doubling down on our mission instead of hedging it.

I won’t discuss when to hedge vs. leverage in full generality. (See here for a discussion plus some real-world examples.) But when should we hedge vs. leverage when we’re working with the class of utility functions discussed in this essay?

With any utility function that matches my stated assumptions, we always prefer to mission hedge. But there are similar utility functions under which we’d prefer to leverage.

We can take a utility function with the same form as before:

\begin{align} U(W, b) = b \displaystyle\frac{W^{1 - \gamma}}{1 - \gamma} \end{align}

But instead of \(\gamma > 1\), let \(\gamma < 1\). This changes two of the six stated assumptions:

1. Utility is now bounded below by 0, and has no upper bound.
2. \(b\) is now a good thing: utility is positive instead of negative, which means a larger \(b\) increases utility rather than decreasing it.

Because \(b\) is now a good thing, by investing in an asset that’s correlated with \(b\), we are leveraging rather than hedging.

If we want \(b\) to be a bad thing, we can write the utility function as \(\displaystyle\frac{1}{b} \frac{W^{1 - \gamma}}{1 - \gamma}\). The solution with this new utility function becomes:

\begin{align} \omega_h = -\displaystyle\frac{\sigma_{bh}}{\sigma_h^2 \gamma} \end{align}

So now, rather than buying the hedge asset, we want to short it. And the higher its covariance with \(b\), the larger our short position should be. In other words, we want to mission leverage.

In broad terms, when utility of wealth grows quickly (\(\gamma < 1\)), we should leverage. And when utility of wealth grows slowly (\(\gamma > 1\)), we should hedge.

The exact conditions are:

1. When \(\frac{dU}{db}\) is negative, \(b\) is a bad thing. When it’s positive, \(b\) is a good thing.
2. When \(\frac{d^2U}{db dW}\) has the same sign as \(\frac{dU}{db}\), we want to leverage. When it has an opposite sign, we want to hedge.

Appendix B: Why use this approach?

The model I use in this post is an extension of Harry Markowitz’s mean-variance optimization model (see Markowitz (1952)¹³). Markowitz won a Nobel Prize for this model, and it has been explored extensively in academic finance. At the same time, few investing practitioners use it.

Mean-variance optimization finds the portfolio with the best risk-adjusted return (a.k.a. Sharpe ratio) when given a set of assets with known means, standard deviations, and correlations with each other. This approach has some serious downsides:

1. If you feed the model historical data for return/volatility/correlation, the model only tells you what portfolio would have been optimal historically. As William Bernstein put it: “If you can predict the optimizer inputs well enough to come close to the future [optimal portfolio], then you don’t need an optimizer in the first place.”¹⁴
2. The Sharpe-ratio-maximizing portfolio is only optimal if standard deviation fully captures what people mean by “risk” (spoiler: it doesn’t).
3. Often, the resulting portfolio requires using so much leverage that it’s impossible to invest in in practice.

Most practitioners prefer other portfolio construction approaches. A simple equal-weight strategy—where you divide your money evenly across every asset class—works pretty well.¹⁵¹⁶ However, a few firms (such as RHS Financial) explicitly use portfolio optimization, but instead of naively inputting historical data, they take a sophisticated approach to generate more reasonable model results.

So mean-variance optimization doesn’t produce good results (at least if applied naively). Why, then, am I extending this model to mission-correlated investing?

One reason is that it’s a good starting point—this model is well-understood, and so a simple extension of it is easy to reason about.

A second reason is that, just as recent research has found more practical variations on the mean-variance approach, so too might we find a modification of my model that fixes the biggest issues. (It might be as simple as taking a published extension to the Markowitz model and incorporating mission-correlated investing.)

But most importantly, it’s because I don’t see how a non-quantitative approach could work. With ordinary investing, you can simply invest in an equal-weighted portfolio and you won’t go too wrong. Or you can use popular rules of thumb like 60% stocks, 40% bonds. But there’s no fallback approach for allocating to mission-correlated investing. Equal-weighting would tell us to put 50% in mission hedging/leveraging, which seems excessive. And we don’t have any convenient rules of thumb, either. Without any strong guiding heuristics, I believe the best approach is to make a bunch of reasonable(ish) assumptions and use those to explicitly calculate the portfolio that maximizes expected utility.

Appendix C: Proof of analytic solution

Thanks to Gordon Irlam’s Lifetime Portfolio Selection: A Simple Derivation, which provides a simple proof of the asset allocation result from Merton (1969)¹⁷, and which I found indispensable in proving the result in this section.¹⁸

Recall that we defined

\begin{align} U(W, b) = b \displaystyle\frac{W^{1 - \gamma}}{1 - \gamma} \end{align}

For this proof, let’s generalize this to

\begin{align} U(W, b) = b^\lambda \displaystyle\frac{W^{1 - \gamma}}{1 - \gamma} \end{align}

where \(\lambda\) determines the rate at which it becomes easier to affect the mission target as the mission target grows.

Wealth is determined by the proportional allocation to the MVO asset and to the hedge asset and the returns of each. Each variable follows a lognormal distribution.

A single-variable lognormal distribution is given by \(e^{\mu + \sigma N(0,1)}\) where \(N(0,1)\) is a standard normal distribution (with mean 0 and standard deviation 1).

\(b\) is a lognormal distribution given by \(\exp(\mu_b + \sigma_b N(0,1))\).

Our objective is to maximize expected utility with respect to allocations \(\omega_m\) and \(\omega_h\):

\begin{align} \arg \max\limits_{\omega_m, \omega_h} E\left[b^\lambda \displaystyle\frac{W^{1 - \gamma}}{1 - \gamma}\right] \end{align}

Expanding \(b\) and \(W\), we get

\begin{align} \arg \max\limits_{\omega_m, \omega_h} E\left[\exp(\lambda \mu_b + \lambda \sigma_b N_b(0,1)) (\exp(\omega_m \mu_m + \omega_m \sigma_m N_m(0,1) + \omega_h \mu_h + \omega_h \sigma_h N_h(0,1)))^{1 - \gamma}\right] \end{align}

where \(N_m(0,1), N_h(0,1), N_b(0,1)\) are all standard normal distributions.

(ignoring the \(1 - \gamma\) in the denominator, because scaling by a constant does not change the argmax)

When taking an expected value, we can separate out any independent variables—\(E[X Y] = E[X] E[Y]\). By assumption, \(m\) is independent of both \(b\) and \(h\). So we can separate this into two maximization problems:

\begin{align} \arg \max\limits_{\omega_m} E\left[\exp(\omega_m \mu_m + \omega_m \sigma_m N_m(0,1)) \right] \end{align}

\begin{align} \arg \max\limits_{\omega_h} E\left[\exp(\lambda \mu_b + \omega_h \mu_h (1 - \gamma) + \lambda \sigma_b N_b(0,1) + \omega_h \sigma_h (1 - \gamma) N_h(0,1)) \right] \end{align}

The first problem is simply the traditional portfolio optimization problem, which has the solution

\begin{align} \omega_m = \displaystyle\frac{\mu_m + \frac{1}{2} \sigma_m^2}{\sigma_m^2 \gamma} \end{align}

If we let \(\alpha = \mu_m + \frac{1}{2} \sigma_m^2\), we can write this as \(\displaystyle\frac{\alpha_m}{\sigma_m^2 \gamma}\). This is convenient in some cases because \(e^\alpha\) is the arithmetic mean of a lognormal distribution.

The second maximization problem does not have a pre-existing solution (to my knowledge), so let’s solve it.

If \(N_b(0,1)\) and \(N_h(0,1)\) have correlation \(\rho\), then the part inside the exponent is a sum of dependent normally distributed random variables. The mean of the sum is simply the sum of the means. The standard deviation of the sum equals

\begin{align} \sigma_{b + h} = \sqrt{\sigma_b^2 + (\omega_h \sigma_h)^2 + 2 \rho \sigma_b (\omega_h \sigma_h)} \end{align}

Note that \(\rho \sigma_b \sigma_h\) is the covariance between \(b\) and \(h\), so we can replace this with \(\sigma_{bh}\).

The expected value of a lognormal distribution parameterized by \(\mu\) and \(\sigma\) is \(e^{\mu + \sigma^2/2}\). Maximizing this quantity is equivalent to maximizing its logarithm, so we want to find

\begin{align} \arg \max\limits_{\omega_h} \left[ \lambda \mu_b + \omega_h \mu_h (1 - \gamma) + \frac{1}{2} \lambda^2 \sigma_b^2 + \frac{1}{2} \omega_h^2 \sigma_h^2 (1 - \gamma)^2 + \omega_h \sigma_{bh} (1 - \gamma) \right] \end{align}

At this point, it is convenient to replace \(\mu\) with \(\alpha - \sigma^2/2\) (recall that \(\alpha\) is the log of the arithmetic mean). In an efficient market, an uncorrelated asset such as a mission hedge has \(\alpha = 0\), which allows us to ignore the \(\alpha_h\) term if we want to.

Setting the derivative to 0,

\begin{align} \displaystyle\frac{dE(U)}{d\omega_h} = \alpha_h (1 - \gamma) - \omega_h \sigma_h^2(1 - \gamma) + \omega_h \sigma_h^2 (1 - \gamma)^2 + \lambda \sigma_{bh} (1 - \gamma) = 0 \end{align}

Solving for \(\omega_h\) and simplifying gives

\begin{align} \omega_h = \displaystyle\frac{\lambda \sigma_{bh} + \alpha_h}{\sigma_h^2 \gamma} \end{align}

And naturally, for \(\lambda = 1, \alpha_h = 0\), this further simplifies to

\begin{align} \omega_h = \displaystyle\frac{\sigma_{bh}}{\sigma_h^2 \gamma} \end{align}

Jonathan Harris wrote an alternative proof here, starting from the model he developed in A Framework for Investing in Altruism—which is more general than my model—and plugging in some numbers. His model provides an approximation, not an exact general solution, but the solution happens to be exact in this case.

Appendix D: Tables of optimization results

Results for \(\sigma_b = 3\%\)

\(\rho\)	\(\gamma\)	market	hedge	ratio
0.25	1.1	2.245	0.038	1.7%
	1.5	1.646	0.028	1.7%
	2.0	1.235	0.021	1.7%
0.5	1.1	2.245	0.076	3.3%
	1.5	1.646	0.056	3.3%
	2.0	1.235	0.042	3.3%
0.9	1.1	2.245	0.136	5.7%
	1.5	1.646	0.100	5.7%
	2.0	1.235	0.075	5.7%

Results for \(\sigma_b = 20\%, \gamma = 1.5\) (note: optimal absolute market allocation for a given \(\gamma is the same as for\)\sigma_b = 3\%$$)

\(\rho\)	market	hedge	ratio
0.25	1.646	0.185	10.1%
0.5	1.646	0.370	18.4%
0.9	1.646	0.667	28.8%

Results for \(\sigma_b = 30\%, \gamma = 1.5\)

\(\rho\)	market	hedge	ratio
0.25	1.646	0.278	14.4%
0.5	1.646	0.556	25.2%
0.9	1.646	1.000	37.8%

Results for \(\sigma_b = 20\%\), no leverage allowed (\(\omega_m + \omega_h = 1\)), computed by numeric approximation:

\(\rho\)	\(\gamma\)	market	hedge
0.25	1.1	1.497	-0.497
	1.5	1.230	-0.230
	2	1.048	-0.048
0.5	1.1	1.370	-0.370
	1.5	1.138	-0.138
	2	0.978	0.022
0.9	1.1	1.177	-0.177
	1.5	0.992	0.008
	2	0.869	0.131

Appendix E: Numeric approximation of the optimal solution

It is possible to numerically approximate the optimal allocation for any utility function. In this section, I describe how I did this.

Epistemic status: Before I started working on this problem, I didn’t understand most of the required math, and I picked it up as I went along. So there’s a reasonable chance that I’m making some mistakes.

We want to find the utility-maximizing asset allocation, where:

1. Utility is a two-dimensional function of wealth and the mission target.
2. There are three lognormally-distributed random variables: price of the MVO asset \(m\), price of the hedge asset \(h\), and quantity of the mission target \(b\).

The three variables are parameterized by a length-3 mean vector \(\mu\) and a covariance matrix \(\Sigma\), which together describe a three-dimensional multivariate normal distribution. The random variables we care about—\(m, h, b\)—are defined as the exponentials of three normally-distributed random variables \(x_1, x_2, x_3\), which together form a vector \(x\). That is, \(m = e^{x_1}, h = e^{x_2}, b = e^{x_3}\).

The probability density function for a three-variable multivariate normal distribution is

\begin{align} f(x) = \displaystyle\frac{\exp(-\frac{1}{2}(x - \mu)^T \Sigma^{-1} (x - \mu))}{\sqrt{(2\pi)^3 \det(\Sigma)}} \end{align}

where \(\Sigma^{-1}\) is the inverse of the covariance matrix \(\Sigma\) and \(\det(\Sigma)\) is its determinant.

Transforming this normal density function into lognormal space according to the multivariate change of variables formula, letting \(y = [m, h, b] = \exp(x)\), we have

\begin{align} f(y) = \displaystyle\frac{\exp(-\frac{1}{2}(\log(y) - \mu)^T \Sigma^{-1} (\log(y) - \mu))}{m h b \sqrt{(2 \pi)^3 \det{\Sigma}}} \end{align}

where \(\frac{1}{m h b}\) is the determinant of the Jacobian matrix of \(y\). (This Stack Exchange post gives an intuitive explanation of why the change-of-variables formula works this way.)

Let \(\omega_m, \omega_h\) be the asset proportions in the MVO asset and the hedge asset, respectively (recall that these can sum to greater or less than 1, and they can be negative). Wealth is then calculated as \(w = \exp(\omega_m m + \omega_h h)\). The mission target \(b\) is simply the third value of the random vector \(y\).

Expected utility is given by

\begin{align} E[U(W, b)] = \int_{-\infty}^\infty U(W, b) f(y) dy \end{align}

Our goal is to maximize this function.

To solve this problem, I wrote a program to compute expected utility using numerical integration and then perform gradient descent to find the optimal asset proportions.

My numerical integration uses Richardson’s extrapolation formula¹⁹ with 12 and 24 trapezoids per dimension (up to 24³ = 13,824 four-dimensional trapezoids per integral). According to my tests, this method is accurate to within 0.02% (e.g., if the true optimal allocation is 10%, this method will give an answer between 9.998% and 10.002%).

I define the trapezoid bounds by first generating evenly-spaced 3D bases for a three-variable i.i.d. standard normal distribution. Then I transform the bounds of these squares according to \(g(x) = e^{\mu + \sigma x}\), where \(\sigma\) is the vector of standard deviations for the three variables in \(x\). Then I use these new bounds given by \(g(x)\) as the base of each trapezoid. This transformation function assumes the three variables are independent, which they’re not, but I found that this method was accurate enough.

My source code is available here.

Notes