Managed Funds


Attached is a template for this execise. There are numerous passively and actively managed funds (e.g. mutual funds, ETFs, and hedge funds). Visit to explore various ETF strategies. The objective of this exercise is to 1) practice programming numerical optimization and 2) learn about popular active management fund strategies.

Passively Managed Broad Stock Market Index

The Vanguard Total Stock Market ETF (VTI) is an example of a passively managed broad stock market index. It seeks to replicate the performance of a benchmark, here US Total Market Index. Notice the performance of the fund tracks it's benchmark very closely. Also notice that the expense ratio is 0.04%. For every $100 you invest in the fund, you pay a management fee of 4 cents. This is an industry low and it's low because the investment strategy of this fund is easy to implement - essentially buy all the securities in the market and hold them. We will see other strategies that require more active management to implement.

Leveraged ETFs

Leveraged ETFs use leverage to enhance their returns. The ProShares UltraPro QQQ seeks to have a 3x daily return of the Nasdaq-100 Index. It accomplishes this by borrowing $2 of leverage for every $1 of equity invested. Recall that financial leverage will magnify both your gains and losses. Below is plot of the TQQQ fund vs the broad market. Notice that when the market declines, TQQQ declines even further; and when the market rises, TQQQ rises even more.

In Solver, we can represent leverage as a negative investment weight in the risk-free rate. First, assuming a fund can use no leverage, find the optimal investment weights that maximizes the portfolio's expected return subject to having a Sharpe ratio at least equal to 0.18. What is the expected return of the fund? Next, assume the fund can use up to 3x leverage. Repeat the exercise and find the optimal investment weights and fund's expected return. What's the relationship between the two expected returns?

130 / 30 Funds

130/30 funds are funds that seek to have a 130% fund investment in long positions and 30% in short positions. The JPMorgan U.S. Large Cap Core Plus Fund is an example of a fund that utilizes this strategy to try to outperform large cap US stocks. Attached is a summary by Charles Schwab on long/short strategies.

In Solver, we can modify our standard setup to include maximum long and short positions amongst a group of securities. Specifically, we can relax our standard no-short selling constraint and allow securities to have short positions by allowing Solver to have negative investment weights. Find the optimal investment weights for a 130/30 fund that is maximizing its Sharpe ratio. Assume the total amount of short positions across all assets (i.e. the absolute value of the sum of the negative investment weights) cannot exceed 30%.

Minimum Volatility ETFs

Minimum volatility ETFs seek to track the performance of a benchmark but acheive less volatility compared to the benchmark. The Vangaurd Global Minimum Volatility Fund (VMNVX) is a fund employing this strategy with respect to the FTSE Global All Cap Index. Notice in the plot below that VMNVX has similar returns compared to the board market but less volatility.

In Solver, we can set our objective function to target the performance of benchmark subject to constraining the volatility of the portfolio. Find the optimal investment weights that maximize the expected return of the portfolio subject to having no more than 5% volatility.