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Context

This is a technical note with on the idea of waiting to invest as discussed on a recent 80k hours podcast.

The idea of expected value of a portfolio is raised a number of times. Some nuance is missed which is of substantial import to decision making. The point which piqued my interest was when a guest said (around 1hr in) something like volatility was beneficial.

Models

Now, here is the starting place for how people think about compound interest describing the value of a portfolio, V, at time T,

V(T) = exp{ r * T }

Where r is the appropriate interest rate. The figure of 7% annual growth is oft mentioned, implying r=log(1.07).

Do you always get 7% by investing in the stock market? No. We all know it goes up and down.

The next natural model, undoubtedly terrible in many ways, is:

V(T) = exp{ q * T + X(T)}

Where X is normally distributed with mean 0 and variance T * s^2. Here, s is the volatility previously mentioned. If we want to talk volatility, there must be some randomness in the model, and this is as simple as it gets.

Why q and not r? Because if q=r then E[V]>1.07. For E[V]=1.07 one must set q = r -
s^2/2. Check out wikipedia on the lognormal distribution for details.

Well, the 7% figure comes from looking at 'long' run index returns. Now the annual volatility on such indices could be on the order of 10-20%. A figure of s = 2r would not be out of order.

Punchline

So what? Well we still have

E(T) = E[V(T)] = exp{rT} = exp{T*(q+ 0.5 s^2)}

But the median is

Q(T) = exp{qT}

While the mode is

M(T) = exp{T*(q - s^2)}

To put this in perspective, for T=279, r=log(1.07) and s=2r:

E(T) / M(T) = 2128

While:

E(T) / Q(T) = 12

Or to put it another way P(V(T) > E[V(T)]) = 13%. The chance of attaining the expected value is the same as the chance as blowing your head off in one round of Russian roulette.

The expected value of returns is enormous compared to the most likely scenario. If returns were deterministic, this ratio would be 1. Suppose (waving hands) we think about uncertain returns as though passing them through the expectation operator makes them behave deterministically. Then we're most likely to find ourselves in a world 1-3 orders of magnitude more disappointing world than the one we had imagined. Volatility is not our friend here.

The general purpose of this pedantic sermon is that expected value is an inadequate guide to decision making. Instead, serious consideration should be given to minimising the probability of certain bad things happening.

The interaction with catastrophic risks makes this worth caring about. A naive expectation maximizer will tend to be content with distributions over future worlds which involve, with substantial probability, everyone being dead.

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New Comment

Hey, I know that episode : )

Thanks for these numbers. Yes: holding expected returns equal, our propensity to invest should be decreasing in volatility.

But symmetric uncertainty about the long-run average rate of return—or to a lesser extent, as in your example, time-independent symmetric uncertainty about short-run returns at every period—increases expected returns. (I think this is the point I made that you’re referring to.) This is just the converse of your observation that, to keep expected returns equal upon introducing volatility, we have to lower the long-run rate from r to q = r – s^2/2.

Whether these increased expected returns mean that patient philanthropists should invest more or less than they would under certainty is in principle sensitive to (a) the shape of the function from resources to philanthropic impact and (b) the behavior of other funders of the things we care about; but on balance, on the current margin, I’d argue it implies that patient philanthropists should invest more. I’ll try writing more on this at some point, and apologies if you would have liked a deeper discussion about this on the podcast.

Thanks for the response. To clarify: in the second model both the drift and the diffusion term impact on the expected returns. If you substitute in a model return e^{q + sz}, with z a standard normal:

E[V(1)] = E[e^{q + s z}] = E[e^{sz}]e^q = e^{s^2/2} e^q > e^q

So, if we have fixed from some source that E[V(1)]=1.07=e^r then we cannot set q=r in the model with randomness while maintaining the equality. Where the equality cashes out as 'the expected rate of return a year from now is 7%'.

Empirically estimated long run rates already take into account the effects of randomness since they are typically some sort of mean of observed returns. If this were not the case one would always have to, at least, quote the parameters in pairs (drift=such and such, vol=such and such) and perform a calculation in order to get out the expected returns.

Yup, no disagreement here. You're looking at what happens when we introduce uncertainty holding the absolute expected return constant, and I was discussing what happens when we introduce uncertainty holding the expected annual rate of return constant.

So, what do you think of the idea that aiming for high expected returns in long term investments might not be the best thing to do, given the skewed distribution? This is, we want to ensure that most futures are 'good'; not just a few that are 'excellent' lost in a mass of 'meh' or worse.

BTW, I did like the podcast - it does take something to make me tap out forum posts :)