# Surplus optimization

Surplus optimization is an approach that is used to perform liability-relative asset allocation. It is one of three approaches that can be used to perform liability-relative asset allocation. The three liability-relative asset allocation approaches are:

• Surplus optimization: this is the approach we discuss in more detail here. This is an extension of mean-variance optimization, in which we determine the efficient frontier based on the surplus with its volatility as our measure of risk, stated either in money or percentage terms.
• Two-portfolio approach: In this approach, we separate the asset portfolio into two subportfolios: a hedging portfolio and a return-seeking portfolio. More details on this approach can be found here.
• Integrated asset-liability approach: this approach integrates both the assets and the liabilities in a joint optimization method. This approach is used when the manager can determine the optimal mix of both the assets and the liabilities.

## Surplus optimization formula

To apply the approach, we first need to define the surplus return: Then the objective function to maximize is: where E(Rs,m) is the expected surplus return and Varsm is the variance of the surplus return. We apply traditional mean-variance optimization, but we also include the expected returns and variances of the liabilities. The correlations reflect the extent to which the assets are useful to hedge the liabilities.

There are a number of ways to estimate the expected returns and variances of the liabilities:

• first, we can make the assumption that the liabilities behave like corpôratae bonds and the liability inputs can be estimated using the expected return and volatility of corporate bonds
• Second, we can use a factor approach and identify the common factors that affect both the asset classes and the liabilities.

## Surplus optimization vs hedging return seeking

The main difference between surplus optimization and the hedging/return seeking portfolio is that the the surplus approach results in a single portfolio, whereas the two-portfolio approach leads to two portfolios. In particular, it yields a conservative portfolio to hedge the liabilities and a more aggressive growth portfolio to increase returns.

## Summary

We discussed surplus optimization. This approach is very similar to mean-variance optimization, but uses the surplus return and surplus variance to determine the optimal asset allocation.