Capital Allocation Between Risky and Risk-Free Assets

Introduction

Investors face a fundamental decision: how to allocate their capital between risky assets (stocks, corporate bonds, real estate) and risk-free assets (Treasury bills, government bonds, cash equivalents). This decision balances expected return against risk, shaping the overall performance and volatility of an investment portfolio. Capital allocation is central to portfolio theory and forms the basis of strategies like the capital market line (CML) and modern portfolio theory (MPT).

Understanding Risk-Free and Risky Assets

1. Risk-Free Assets

• Defined as investments with known returns and virtually zero default risk.
• Common examples:
  • U.S. Treasury bills (short-term)
  • Government savings bonds
  • Cash equivalents (money market funds)
• Key characteristics: stability, liquidity, low return.

2. Risky Assets

• Investments with uncertain returns, including potential gains or losses.
• Examples:
  • Stocks
  • Corporate bonds
  • Real estate
  • Commodities
• Key characteristics: higher expected returns, higher volatility.

The Risk-Return Tradeoff

• Investors seek to maximize returns for a given level of risk.
• Risky assets offer higher expected returns, compensating investors for uncertainty.
• Risk-free assets provide security and liquidity, but lower returns.

Expected Return and Portfolio Risk

Let:

• R_f = risk-free rate
• E[R_p] = expected portfolio return
• w = proportion invested in risky asset
• R_m = expected return of risky asset

Then expected portfolio return:

E[R_p] = w \cdot E[R_m] + (1-w) \cdot R_f

Portfolio standard deviation (risk):
\sigma_p = w \cdot \sigma_m
Where \sigma_m is the standard deviation of the risky asset.

Insight: Increasing allocation to risky assets increases both expected return and portfolio volatility.

Capital Allocation Line (CAL)

• Graphical representation of risk-return combinations for different allocations between risk-free and risky assets.
• Equation:

E[R_p] = R_f + \frac{E[R_m] - R_f}{\sigma_m} \cdot \sigma_p

The slope, known as the Sharpe ratio, measures excess return per unit of risk.

Example Scenario

Investor has $100,000:

• Risk-free asset: Treasury bills, 3% return
• Risky asset: Stock portfolio, 10% expected return, 15% standard deviation

Allocation (w)	Portfolio Return E[R_p]	Portfolio Risk \sigma_p
0% risky	3%	0%
50% risky	6.5%	7.5%
100% risky	10%	15%

• Interpretation: Higher allocation to risky assets increases expected return but also risk.

Optimal Capital Allocation

1. Based on Risk Tolerance

• Conservative: Higher allocation to risk-free assets
• Moderate: Balanced allocation
• Aggressive: Majority in risky assets

2. Using Utility Functions

• Investors maximize expected utility considering risk aversion:
  U = E[R_p] - \frac{1}{2} A \cdot \sigma_p^2
  Where A = risk aversion coefficient
• Optimal risky asset proportion:

w^* = \frac{E[R_m] - R_f}{A \cdot \sigma_m^2}

3. Incorporating Multiple Risky Assets

• Use mean-variance optimization to combine multiple risky assets for diversification.
• Allocation to risk-free asset is then determined along the Capital Market Line (CML).

Leveraging and Borrowing

• Investors can borrow at the risk-free rate to invest more in risky assets, increasing expected return and risk.
• This extends the capital allocation line beyond 100% investment in risky assets.

Practical Considerations

1. Time Horizon: Longer horizons can tolerate more risk due to the potential for compounding and recovery from downturns.
2. Liquidity Needs: Investors needing short-term access may favor risk-free allocation.
3. Diversification: Even within risky assets, diversification reduces overall portfolio risk.
4. Market Conditions: Expected returns and volatility estimates affect the optimal allocation.

Example Calculation

Investor risk aversion coefficient A = 3, risky asset expected return E[R_m] = 10%, standard deviation \sigma_m = 15%, risk-free rate R_f = 3%

Optimal risky weight:

w^* = \frac{0.10 - 0.03}{3 \cdot 0.15^2} = \frac{0.07}{0.0675} \approx 1.04

• Interpretation: Slightly over 100%, indicating the investor could borrow at the risk-free rate to leverage the investment.

Conclusion

Capital allocation between risky and risk-free assets is a fundamental component of portfolio management. The balance determines expected return, volatility, and the potential for achieving investment goals. Using concepts like the Capital Allocation Line, Sharpe ratio, and utility optimization, investors can structure portfolios that reflect their risk tolerance, time horizon, and financial objectives. Strategic allocation, combined with diversification and monitoring, ensures investors maximize expected return for the level of risk they are willing to assume.