Better portfolios with higher moments


A toolset beyond mean–variance portfolio optimization is appropriate for those instances where higher return moments might need to be taken into account, either for individual decisions or for pricing studies. Maximizing expected log surplus utility is superior for compounding returns in excess of financial obligations. Here, it is matched with a more flexible scenario representation of the investor’s joint probability distribution of returns and with an agnostic optimization engine. We show simple examples based on extrapolating historical stock and bond returns and then extended using hypothetical option prices. We clarify how Black–Scholes implied volatility anomalies can arise in a portfolio context.

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Appendix 1

Why mean–variance is not generally consistent with expected utility?

Consider both U(x), a von Neumann–Morgenstern utility function of investment outcome x, and also P(x), an outcome probability distribution. Expected utility then is the sum of P(x)U(x). For mean–variance optimization to be precisely equivalent, maximizing mean(P(x)) − L*variance(P(x))/2 where L is a risk aversion parameter would have to give the same allocation result as maximizing ΣP(x)U(x). This is much more difficult requirement than apparent at first glance.

Consider the components P(x) and U(x) individually. For each, we have arguments that they alone could justify the mean–variance approach as fully compatible with expected utility maximization. Though the contrary suggestion is often repeated in the finance literature, one cannot take for granted that the approximation succeeds perfectly if P(x) is a normal distribution, because that eliminates the third return moment but not the fourth moment. The other justification, a quadratic utility U(x), would imply that utility goes down increasingly rapidly with increasing wealth once U(x) reaches a maximum, contrary to rational utility assumptions unless we don a blindfold to the possibility that utility might exceed that amount. These observations do not make the task impossible, but they narrow the possibility to non-obvious special cases.

Maximizing medians in the long term

For long-term compounded risky results, the eventual wealth distribution approximates a log normal distribution, with a tiny probability of huge returns. For a finite-lived investor, this is less relevant than median results. Fortunately, growth optimal policies, if the generalized central limit theorem (GCLT) can be applied, eventually maximize median terminal wealth. The GCLT covers cases where variance varies and where simple forms of autocorrelation are permitted. Then:

  1. 1.

    Maximizing the expected log return each period maximizes its sum over multiple periods,

  2. 2.

    The sum of log returns converges to a normal distribution,

  3. 3.

    Which is symmetric, with maximized mean equal to maximized median.

  4. 4.

    Taking the antilog preserves rank order,

  5. 5.

    Maximizing median terminal wealth.

In the text, this applies to terminal surplus, which, if financial obligations were to remain constant, maximizes terminal wealth.

Taylor series analysis behind Eq. 1

In mathematics, many smooth functions can be expressed as an infinite series of polynomials of increasing powers of distances from some local point. We can often get a good approximation from the first few terms.

A Taylor series around the point r = E, where r is return and E is the expected return, looks like:

$$f\left( r \right) = \sum f^{n} \left( E \right)\left( {r - E} \right)^{n} /n!$$

where fn(E) is the nth-order derivative of the function evaluated at E and n! is 1 times 2 … times n.

We consider 0! = 1 and f0 (E) = f(E).

Let f(r) = ln(1 + r). We want to estimate it around the location E.

Then the zeroth term is ln(1 + E).

The first-order term is (r − E)/(1 + E).

Then deriving the successive-order derivatives, and canceling offsetting values, we get the following.

$$\ln \left( {1 + r} \right) \cong \ln \left( {1 + E} \right) + \frac{(r - E)}{(1 + E)} - \frac{{(r - E)^{2} }}{{2(1 + E)^{2} }} + \frac{{(r - E)^{3} }}{{3(1 + E)^{3} }} - \frac{{(r - E)^{4} }}{{4(1 + E)^{4} }} \cdots$$

Taking the expected value of ln(1 + r), the first-order term is zeroed out, and we get

$${\text{Expected}}\,\,\,\ln \left( {1 + r} \right) \cong \ln \left( {1 + E} \right) - \frac{V\left( r \right)}{{2\left( {1 + E} \right)^{2} }} + \frac{{S\left( r \right)V\left( r \right)^{{\frac{3}{2}}} }}{{3\left( {1 + E} \right)^{3} }} - \frac{{K\left( r \right)V\left( r \right)^{2} }}{{4\left( {1 + E} \right)^{4} }} \cdots$$

where r is return, E is expected r, V(r) is variance of r, S(r) is skewness of r, and V(r) is kurtosis of r.

Now, since we actually want expected return on surplus rather than expected return on investments, substitute LE for E, where L is the risk aversion parameter, and substitute LV1/2 for V1/2. Noticing that a common factor in all but the zeroth term is Q = LV1/2/(1 + LE), we finally arrive at

$${\text{Expected}}\,\,\,\ln (1 + Lr) = \ln \left( {1 + LE} \right){-}Q^{2} /2 + SQ^{3} /3{-}KQ^{4} /4 + \cdots .$$

Note that Q determines the relative size of the impact of skewness and kurtosis. If it is too large, the series will not converge to a finite sum. A portfolio characterized by such a Taylor series should be avoided!

Working Python code

Appendix 2, available for download here, contains a working Python 3.7 program for calculating optimal allocations starting with a comma delimited (CSV) file with its top row containing ticker labels and successive rows showing returns for each ticker symbol.

Appendix 2


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Wilcox, J. Better portfolios with higher moments. J Asset Manag (2020).

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  • Asset allocation
  • Portfolio optimization
  • Higher moments
  • Skewness
  • Kurtosis
  • Scenarios