Efficient frontier with a combination of Risky and Risk-free Assets

The Efficient Frontier, introduced by Harry Markowitz in Modern Portfolio Theory (MPT), represents the set of optimal portfolios that offer the highest expected return for a given level of risk. These portfolios are fully diversified, minimizing unsystematic risk while maximizing returns.

When combining risky assets (such as stocks and bonds) with risk-free assets (such as Treasury bills), investors can achieve an even more efficient allocation of assets, improving their risk-adjusted returns.

Combination of Risky and Risk-Free Assets:

Risk-Free Asset

• A risk-free asset (e.g., government Treasury bills) has zero volatility and offers a fixed return.
• It is denoted as R_f, representing the risk-free rate of return.

Risky Asset Portfolio

• A risky asset portfolio consists of stocks, bonds, and other investments with variable returns.
• The risk and return characteristics of this portfolio are determined using the mean-variance analysis.

Capital Market Line (CML) and Efficient Frontier:

1. Efficient Frontier Without a Risk-Free Asset

   • The traditional efficient frontier is a curved line showing the best possible portfolios of risky assets.
   • Investors choose a point on this frontier based on their risk tolerance.

2. Efficient Frontier With a Risk-Free Asset

   • When a risk-free asset is introduced, a straight line called the Capital Market Line (CML) is formed.
   • The CML originates from the risk-free rate (R_f) and is tangent to the efficient frontier at the tangency portfolio.
   • The equation of the CML is:

E(Rp) = [Rf + E(Rm)−Rf / σm ]*σp

• • • $)$ = Expected return of the portfolio
    • Rf​ = Risk-free rate
    • $)$ = Expected return of the market portfolio
    • σm​ = Standard deviation of the market portfolio
    • σp​ = Portfolio risk

Portfolio Selection Using Risky and Risk-Free Assets

1. Conservative Investors → Allocate more to risk-free assets.
2. Aggressive Investors → Allocate more to risky assets.
3. Leverage Investors → Borrow at the risk-free rate and invest in risky assets for higher returns.