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The expected return of a portfolio is like predicting what you might earn on average from an investment. To calculate it, you look at the potential returns from each asset, or piece of the portfolio, and weigh them based on how much of each asset you own. For example, if you have a combination of stocks and bonds, and you know what each might return, you can find a weighted average to see the expected return of the entire portfolio.
This is calculated using the formula:\[ E[R_p] = \pi \times E[R_{S\&P500}] + (1-\pi) \times E[R_{T-bills}] \]Here, \(\pi\) stands for the percentage of your portfolio invested in a higher-risk asset like the S&P 500. The remainder, \(1-\pi\), goes into a safer asset like T-bills. With this approach, you balance the potential gains from riskier investments against the reliability of safer ones.

Beta is a measure of how sensitive a stock or portfolio is to market movements. Think of it as the heartbeat of market risk. A beta of 1 means the asset tends to move with the market, while a beta greater than 1 indicates more sensitivity – it might rise more than the market in good times and fall more in bad.
For a mixed portfolio, you calculate beta by taking the beta of each asset, multiplied by their weight in the portfolio. This method helps in understanding how much market risk your entire portfolio carries:\[ \text{Beta}_p = \pi \times \text{Beta}_{S\&P500} \]Since treasury bills, being risk-free, have a beta of 0, the portfolio beta effectively depends only on the S&P 500 portion’s beta. This helps investors gauge potential volatility and adjust accordingly.

The risk-return tradeoff is a fundamental concept that tells us that to achieve higher returns, one must be willing to accept more risk. It’s a balancing act every investor considers: the potential of higher profits against the possibility of losses.
When you look at different portfolios with varying proportions of risk-free and risky assets, you find that as the percentage in a risky asset increases, so does both the expected return and the risk (beta). This reveals how increased risk leads to the potential for higher rewards, following the risk-return tradeoff principle. It underscores the decision-making process in the financial world, where more risk should theoretically bring about higher expected returns.

The Security Market Line (SML) graphically demonstrates the expected return for various levels of market risk (beta). It's like a financial roadmap showing where each investment might stand in terms of risk versus expected return.
On the SML, each asset is plotted with its beta against its expected return. The line itself represents market equilibrium – meaning the expected returns are consistent with the level of risk, based on the market. From the exercise, the portfolios line up along this line, reaffirming that higher risk (beta) is associated with higher returns. Investors use the SML to assess whether an investment is fairly priced compared to its risk.

A weighted portfolio involves assigning different weights, or percentages, to different assets in an investment mix. These weights are crucial because they determine how much influence each asset has on the overall portfolio performance.
By adjusting the weights, investors can customize their risk and expected return. For instance, having more weight in a high-return stock may increase potential earnings but also brings more risk. Conversely, increasing the weight in safer assets like T-bills may result in lower volatility but also less profit potential. This flexibility in structuring a portfolio makes it a powerful tool for investors to align their investments with their personal risk tolerance and financial goals. Understanding weighted portfolios helps in crafting a well-balanced investment strategy.