# Allocating a Three-Asset Portfolio, Optimized

Our previous post showed how various allocation decisions impact an equal-weighted three-asset portfolio. However, equal-weights are not the only way to go. Every time a rebalance occurs, we can use that opportunity to re-weight the assets to minimize expected risk while maximizing expected returns. In this post, we look at two ways in which risk and returns can be optimized.

### Portfolio optimization and the efficient frontier

The intuition behind what we are going to do is quite simple: for a given set of assets, there is an ideal mix of them that perfectly balances risk with reward. Imagine a plot of risk and returns of each asset under consideration. Harry Markowitz showed back in the 1950’s that they form a parabola and at a particular tangent of the parabola lies the ideal mix. The goal of portfolio optimization is to find that point. Here are some links that explain this concept further:

For the purpose of this post, we will assume risk to either mean variance (var) or expected tail loss (ETL.) We will use portfolio optimization methods to minimize one of these risk metric and maximize expected mean returns below.

### Optimized portfolios

Like before, to keep things simple, we will go with the MIDCAP 100 index (A1), the 0-5yr TRI (A2) and the QQQ ETF (prices converted to INR, A3) as the three assets that form our portfolio.

Here is how the optimized minimum-variance portfolio performs after-tax: Asset weights after rebalance: And here is how the optimized minimum-ETL portfolio performs after-tax: Asset weights after rebalance: ### Take-away

1. All things equal, the optimized portfolios under-perform the equal-weight portfolio in terms of absolute returns.
2. Optimized portfolios show lesser risk than the equal-weight portfolio. During the 2008 carnage, for example, equal-weight drew-down ~40% whereas optimized portfolios drew-down ~20%.
3. Optimized portfolios over-weigh bonds. Hard limits were set on the maximum and minimum weights the assets can have in optimized portfolios. Toggling these will have a significant impact on portfolio risk and returns.

Code, charts and the complete result dataset are available on github.