Visiting the Asset Liability Efficient Frontier

This post is a bad idea, a little like that last round of shots after a long night of drinking: probably doesn’t need to be done and likely to leave a bad impression somewhere. In this case, though, the harm is from posting something that is only interesting to a handful of people on Earth rather than getting lost on the way home from the bar.

That said, I am interested in managing asset-liability portfolios and I recently re-read William Sharpe and Lawrence Tint’s 1990 paper called “Liabilities- A New Approach” where they lay out their formulas for the efficient frontier incorporating liabilities and discuss liability hedging credits, which are an adjustment to the efficient frontier due to the correlation of assets and liabilities. In addition, they include a formula for other assets beyond the control of the asset allocator, which is interesting to me since nearly everything seems so focused on the efficient frontier assets only. I hope to come back to this a few times in the future, so I’m going to show where it comes from.

In the case where there are no liabilities, this simply becomes the asset-only efficient frontier that we’re all familiar with. But as liabilities are included in the analysis, there is an additional factor the modifies the efficient frontier.

The analysis starts with the basics: \(S = A - L\) or surplus equals assets minus liabilities, and \(S_0 = A_0 - L_0\) are those as of today (time 0). Sharpe and Tint include a factor \(k\) for how important the surplus optimization is, which I will set to one since I’m obviously interested in surplus optimization. They denote next year’s surplus, which is unknown, as \(\dot{S_1} = \dot{A_1} - \dot{L_1}\). The objective of the optimization is to maximize surplus, but to get there you divide by \(A_0\) and multiplying by \(L_0/L_0\), the objective becomes:

$$ \frac {\dot{A_1}}{A_0} - \frac{L_0}{A_0} \frac {\dot{L_1}}{L_0} \ $$

or

$$ \left( 1 - \frac{L_0}{A_0} \right) \quad + \quad \left( \dot{R_A} - \frac{L_0}{A_0} \dot{R_L} \right) $$

Where \(\dot{R_A}\) is the return on assets from time 0 to 1. As they point out, the first term has no uncertainty, so that leaves the second expression which they label \( \dot{Z}\). The objective is to maximize utility, which is defined as:

$$ \mathrm{U = Exp( } \dot{Z} \mathrm{) - Variance(} \dot{Z} \mathrm{)}/t $$

Where \(t\) is the risk aversion parameter. The authors point out that you can’t change the return on the liabilities or its risk, so these drop out of the equation. The objective becomes to maximize:

$$ \mathrm{U = Exp( } \dot{R_A} \mathrm{)} \: - \frac{\mathrm{Variance(} \dot{R_A} \mathrm{)}}{t} - \frac{2}{t} \frac{L_0}{A_0} \: \mathrm{Covariance (} \dot{R_A}, \dot{R_L} \mathrm{)} $$

So, if you want to program this, what are the portfolio weights? I’ve had enough, you’re thinking, this is a good place to stop. But like the guy at the bar we have to go one deeper. I made a couple of variable changes and will use the risk aversion parameter \(\lambda = 2/t\) and \(\xi = \frac{L_0}{A_0} Cov(\dot{R_A}, \dot{R_L})\). The asset weights in the portfolio are represented by the column vector \(w\), and \(\Sigma\) is the variance-covariance matrix. Here \( \mathbf{1} \) represents a column vector of 1’s.

Our goal is to maximize the objective:

$$ \mathrm{Max} \quad w’ \mu \quad - \quad \frac{\lambda}{2} w’ \Sigma w \quad + \quad \lambda \frac{L_0}{A_0} \mathrm{Cov}(\dot{R_A}, \dot{R_L}) \quad \mathrm{Subject \: to \:} \quad w’\mathbf{1} = 1 $$

We do this by the method of Lagrange multipliers, so that:

$$ \begin{align*} \mathcal{L} \quad &= \quad w’ \mu \: - \frac{\lambda}{2} \: w’\Sigma w \: + \: \lambda w’ \xi + \gamma (1 - w’ \mathbf{1})\\ \frac{\partial \mathcal{L}}{\partial w} \quad &= \quad \mu \qquad - \quad \lambda \: \Sigma \: w \:+\: \lambda \: \xi \: - \: \gamma \: \mathbf{1} = 0\\ \frac{\partial \mathcal{L}}{\partial \gamma} \quad &= \quad 1 \quad - \quad w’ \: \mathbf{1} = 0 \end{align*} $$

The last row is just the constraint that \(w’ \: \mathbf{1} \:= \: 1\). Substituting gamma

$$ \gamma \quad = \quad \frac{ \mathbf{1}’ \: \Sigma^{-1} \: \mu + \lambda \: \mathbf{1}’ \: \Sigma^{-1} \: \xi \: - \: \lambda}{\mathbf{1}’ \Sigma^{-1} \mathbf{1}} $$

into the middle equation and doing some algebra gives the final answer for portfolio weights \(w\):

$$ w \quad = \quad \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}’ \: \Sigma^{-1} \: \mathbf{1}} + \frac{1}{\lambda} \left[ \Sigma^{-1} \mu \: - \: \frac{\mathbf{1}’ \: \Sigma^{-1} \mu \; \Sigma^{-1} \mathbf{1}} {\mathbf{1}’ \Sigma^{-1} \mathbf{1}} \right] + \left[ \Sigma^{-1} \xi \: - \: \frac{\mathbf{1}’ \: \Sigma^{-1} \xi \; \Sigma^{-1} \mathbf{1}} {\mathbf{1}’ \Sigma^{-1} \mathbf{1}} \right] $$

I like this portrayal of the efficient frontier where you have the Global Minimum Variance portfolio (the first term) plus a risk aversion parameter times a self-financing portfolio of risky assets (the second term). Adding a liability component adds the third term, which is not multiplied by the risk aversion parameter, so it shifts the location of the efficient frontier but not it’s response to the risk tolerance. The amount of the shift depends on the covariance between assets and liabilities and the value of liabilities compared to assets. A pretty cool paper from 1990.

There’s our math bender for today. We’ve overdone it, so we’ll recover for a while before we come back.