(Pension) risk attribution

For portfolio construction, and monitoring

Last time I did a lot of throat-clearing about the ultimate goal of pension investing.

Replacement ratio

Making future real income explicit as a goal—either as an individual investor or on behalf of the collective of a pension fund—accomplishes two things:

• It provides a measure of adequacy. Knowing the price of your envisaged annuity P tells you how much of it you could buy with your current savings A (namely A/P), and so how close you are to your pension goal. It can also illustrate the cost (resp. benefit) δPₜ of retiring earlier (resp. later), and separately how that price is sensitive to the prevailing interest rate ∂P/∂r or other factors.

• It provides an indication of risk. Your savings could be adequate in themselves, but if their exposure to risk factors is dissimilar to the target, they could still fall short in certain scenarios (e.g. low interest rates). Since we’re targeting A/P we should consider risks to that ratio, which behave as ≈ σ(a - p)¹.

The last comment suggests that the risks to manage in our investment portfolio are the tracking error (TE) of the portfolio with respect to the targeted (real) annuities². It is this quantity that we have been calling PVAR (Pension Value At Risk) at the office.

For portfolio construction you want to calculate this relative risk and see how the individual elements of the portfolio contribute. For this we need a model of asset categories i and their covariance matrix 𝛴. For a portfolio with allocation weights wᵢ and a benchmark bᵢ in these assets (weights summing to one so |w|=|b|=1), we denote the relative risk σ (a scalar) and its sensitivity (gradient vector∇σ ):

\(\sigma = \sqrt{(w-b)^T \Sigma (w-b)}, \qquad (\nabla \sigma)_i=\frac{\partial \sigma}{\partial w_i}=\frac{\left(\Sigma(w-b)\right)_i}{\sqrt{(w-b)^T \Sigma (w-b) }}\)

That marginal risk is useful because it provides a measure of contributions:

\(\sum_i (w_i-b_i) \frac{\partial \sigma}{\partial w_i}=(w-b)^T \nabla \sigma= \frac{\sigma^2}{\sigma}=\sigma\)

This weighted sum of all positions adds up to total risk σ! In that sense, ∂σ/∂wᵢ is a good measure of the risk contribution of position i with size (wᵢ-bᵢ). This lets you spot which deviations from the strategic portfolio are contributing to risk, and which might be reducing it.

This also holds if, as in the original case, the benchmark b is a distinguished, univestible³ category. Identify this special category by index 0, and let w₀=0 and b₀=1.

\(\sigma = \sqrt{w_i \Sigma_{ij} w_j + \Sigma_{00} - 2w_i\Sigma_{0i}}, \quad \frac{\partial \sigma}{\partial w_i}=\frac{\Sigma_{ij} w_j-\Sigma_{0i}}{\sigma}, \quad \frac{\partial \sigma}{\partial w_0}=\frac{\Sigma_{0j} w_j-\Sigma_{00}}{\sigma} \)

(where we let i and j run over indices ≠0 in the implied summations). In the expression for σ, recognize w𝛴w as the variance of the portfolio and 𝛴₀₀ as the variance of the benchmark. Remark that the marginal risk of the benchmark ∂σ/∂w₀ may be quite a large—depending on how much the portfolio manages to hedge it—givens its weight |b|=1≫wᵢ for any other i. Explaining this contribution in a risk report is a challenge.

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A postscript to the physics-inclined: some of the subscripts should really be superscripts in the Einstein summation convention (Wiki). Also, in that convention using indices (i,j) rather than Greek letters (μ,ν) already implies excluding index 0 from the summation! Now the formulas look even fancier:

\(\sigma = \sqrt{w_j \Sigma_i^j w^i + \Sigma_0^0 - 2w_j\Sigma_0^j}, \quad \frac{\partial \sigma}{\partial w_i}=\frac{1}{\sigma}\left(\Sigma_j^i w^j-\Sigma_0^i\right), \quad \frac{\partial \sigma}{\partial w_0}=\frac{1}{\sigma}\left(\Sigma_j^0 w^j-\Sigma_0^0\right)\)

1. When considered as returns, convention is write lower case for those (log) variables and so log(A/P) = log(A)-log(P) =: a-p.

2. There is more to say about how to measure this—the frequency and horizon (and confidence interval) should align with your investment process, for instance—and I might revisit it elsewhere. But this TE is what to measure.

3. For instance a hypothesized entirely risk-avoiding investment policy (DNB). Of course it is much better if such products are available for purchase (like Renda+ in Brazil, thank you Arun Muralidhar!) then you don’t have to be a sophisticated institutional investor to manage this risk.