Modern Portfolio Theory on US ETFs

After reading @doctorpot1 's post about the Modern Portfolio Theory (READ FIRST), some research was done to have a deeper understanding on the theory, and I've applied the theory to 3 paperfolios.

2. Due to limited data from $Invesco NASDAQ 100 ETF(QQQM.US)$ , efficient frontier is tested with $Invesco QQQ Trust(QQQ.US)$ historical performance. Both ETFs are moving close to, if not identical to each other. Proven by the correlation value of 1.00.

3. Due to lack of data from $Vanguard S&P 500 ETF(VOO.US)$ , efficient frontier is tested with $SPDR S&P 500 ETF(SPY.US)$ , with the same reason as 2.

4. The efficient frontier is calculated using the time frame of 2005 Jan 1 - 2024 April 30

5. Allocations are done with the closing price of the ETFs as at 3rd May 2024. Since there is no fractional trading in Moomoo Malaysia at the moment, the ratios are calculated with at least 1 share of each ETF, and based on their market value as at 3rd May 2024.

1st Portfolio - The Base Portfolio
We start off by having the base portfolio consisting of 100% $SPDR S&P 500 ETF(SPY.US)$, which has an expected return of 10.45% p.a. and a standard deviation (risk) of 15.13%.

By following the optimal portfolio's allocation of 62.73% $Invesco NASDAQ 100 ETF(QQQM.US)$ and 37.27% $SPDR Gold ETF(GLD.US)$ (2 QQQM shares : 1 GLD share), we could gain an additional 2.4% potential return, totalling 12.85% p.a., while reducing risk by 1.56% to 13.57%.

This portfolio is built on the basis that satisfies my preferred exposure to gold (<20% of the portfolio). The portfolio consists of 80.80% $Invesco NASDAQ 100 ETF(QQQM.US)$ and 19.20% $SPDR Gold ETF(GLD.US)$ (5 QQQM shares : 1 GLD share). It is still considered an optimal portfolio for investors with a higher risk appetite, providing an additional of 0.94% of potential return, totalling 13.79% p.a. by compensating with 1.94% of additional risk to 15.51% when compared to Portfolio 2.

This portfolio is built on the 3rd portfolio, with an attempt to include a S&P 500 ETF into the mix. It consists of 58.34% $Invesco NASDAQ 100 ETF(QQQM.US)$ , 19.81% $SPDR Gold ETF(GLD.US)$ , and 21.86% $Vanguard S&P 500 ETF(VOO.US)$ (7 QQQM shares : 2 GLD shares : 1 VOO share).
The portfolio is sub-optimal as it does not land on the efficient frontier, meaning that it does not provide enough return for the level of risk. While the expected return of 12.81% for 14.51% of risk beats Portfolio 1, Portfolio 2 and 3 outperforms this portfolio.