As a finance professional, I often encounter investors who want to maximize returns while managing risk. One fundamental question is how to allocate capital between two risky assets efficiently. While diversification is key, the math behind optimal asset allocation is not always straightforward. In this article, I break down the principles, calculations, and real-world applications of allocating between two risky assets.
Table of Contents
Understanding Risk and Return in a Two-Asset Portfolio
Before diving into allocation strategies, I need to define the key components:
- Expected Returns – The mean return of each asset.
- Volatility (Standard Deviation) – A measure of risk.
- Correlation – How the two assets move in relation to each other.
The expected return of a portfolio with two assets is a weighted average:
E(R_p) = w_1 E(R_1) + w_2 E(R_2)Where:
- E(R_p) = Expected portfolio return
- w_1, w_2 = Weights of Asset 1 and Asset 2
- E(R_1), E(R_2) = Expected returns of Asset 1 and Asset 2
The portfolio variance, however, is not just a weighted average due to correlation:
\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{1,2} \sigma_1 \sigma_2Where:
- \sigma_p = Portfolio standard deviation
- \sigma_1, \sigma_2 = Standard deviations of Asset 1 and Asset 2
- \rho_{1,2} = Correlation coefficient between the two assets
Why Correlation Matters
If two assets are perfectly correlated (\rho = 1), diversification provides no risk reduction. However, if \rho < 1, combining them reduces overall portfolio risk.
The Efficient Frontier for Two Risky Assets
The Efficient Frontier is a set of optimal portfolios offering the highest expected return for a given level of risk. To find it, I vary the weights w_1 and w_2 (where w_2 = 1 - w_1) and plot the resulting risk-return combinations.
Example: Stocks and Bonds
Assume:
- Asset 1 (Stocks): E(R_1) = 10\%, \sigma_1 = 20\%
- Asset 2 (Bonds): E(R_2) = 5\%, \sigma_2 = 10\%
- Correlation (\rho): 0.3
Using different weights, I calculate portfolio risk and return:
| Weight Stocks (w_1) | Weight Bonds (w_2) | Portfolio Return (E(R_p)) | Portfolio Risk (\sigma_p) |
|---|---|---|---|
| 100% | 0% | 10% | 20% |
| 80% | 20% | 9% | 16.4% |
| 60% | 40% | 8% | 13.8% |
| 40% | 60% | 7% | 12.1% |
| 20% | 80% | 6% | 11.4% |
| 0% | 100% | 5% | 10% |
The table shows that a 40% stocks / 60% bonds mix reduces risk significantly while maintaining reasonable returns.
The Minimum Variance Portfolio (MVP)
The Minimum Variance Portfolio is the combination of two assets that results in the lowest possible risk. To find it, I solve for w_1 that minimizes \sigma_p^2:
w_1^{MVP} = \frac{\sigma_2^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}Using the previous example:
w_1^{MVP} = \frac{0.10^2 - 0.3 \times 0.20 \times 0.10}{0.20^2 + 0.10^2 - 2 \times 0.3 \times 0.20 \times 0.10} = \frac{0.01 - 0.006}{0.04 + 0.01 - 0.012} = \frac{0.004}{0.038} \approx 10.5\%Thus, the MVP consists of 10.5% stocks and 89.5% bonds, yielding the lowest possible risk.
The Tangency Portfolio and Optimal Risk-Return Tradeoff
If I introduce a risk-free asset (e.g., Treasury bills), the Capital Market Line (CML) becomes relevant. The Tangency Portfolio is the optimal mix of two risky assets that, when combined with the risk-free rate, provides the best risk-adjusted return.
The Sharpe Ratio (S) measures this:
S = \frac{E(R_p) - R_f}{\sigma_p}Where R_f is the risk-free rate. The goal is to maximize S.
Calculating the Tangency Portfolio Weights
The optimal weight for Asset 1 is:
w_1^{Tangency} = \frac{(E(R_1) - R_f)\sigma_2^2 - (E(R_2) - R_f)\rho \sigma_1 \sigma_2}{(E(R_1) - R_f)\sigma_2^2 + (E(R_2) - R_f)\sigma_1^2 - (E(R_1) - R_f + E(R_2) - R_f)\rho \sigma_1 \sigma_2}Assume R_f = 2\%. Plugging in the numbers:
w_1^{Tangency} = \frac{(0.10 - 0.02)(0.10)^2 - (0.05 - 0.02)(0.3)(0.20)(0.10)}{(0.10 - 0.02)(0.10)^2 + (0.05 - 0.02)(0.20)^2 - (0.08 + 0.03)(0.3)(0.20)(0.10)} = \frac{0.008 \times 0.01 - 0.03 \times 0.006}{0.008 \times 0.01 + 0.03 \times 0.04 - 0.11 \times 0.006} = \frac{0.00008 - 0.00018}{0.00008 + 0.0012 - 0.00066} = \frac{-0.0001}{0.00062} \approx -16.1\%A negative weight implies short-selling Asset 1 (stocks) and allocating more than 100% to Asset 2 (bonds). In practice, many investors avoid short-selling, so constraints may be applied.
Practical Considerations in Asset Allocation
1. Rebalancing
Markets fluctuate, causing portfolio weights to drift. I recommend periodic rebalancing (e.g., quarterly or annually) to maintain desired risk exposure.
2. Transaction Costs
Frequent trading incurs fees. I factor in costs when determining optimal rebalancing frequency.
3. Tax Implications
Capital gains taxes apply when selling assets. Tax-efficient strategies, like holding assets in tax-advantaged accounts, help.
4. Behavioral Biases
Investors often chase past performance. I stick to a disciplined strategy rather than emotional decisions.
Conclusion
Asset allocation between two risky assets involves balancing risk and return through mathematical optimization. By understanding correlation, the Efficient Frontier, and the Tangency Portfolio, I construct portfolios that align with investor goals. Real-world constraints like taxes and transaction costs must also be considered. Whether you’re an individual investor or a financial advisor, mastering these principles leads to better investment outcomes.
