Portfolio Returns

In previous modules, we calculated returns and volatility for individual securities. In this module, we extend these concepts to portfolios, which are simply collections of securities. Portfolios can contain just a few assets or thousands, but the principles of return and risk remain consistent.

A key insight here is understanding how combining assets changes risk and return, which introduces the concepts of linearity vs. non-linearity. Central to this is correlation, which determines how assets interact in a portfolio. We will also touch on exchange-traded funds (ETFs), which are securities composed of multiple underlying assets.

A portfolio's expected return is a weighted average of the expected returns of its components.

• Portfolio: a collection of securities (stocks, cryptocurrencies, etc.).
• Security: a single asset.

Example: Consider a portfolio of two stocks:

• Stock A: expected return 5%
• Stock B: expected return 11%

If both stocks are equally weighted in the portfolio, the expected return is the average:

$$\text{Portfolio Return}\ (R_p) = \frac{R_A + R_B}{2} = \frac{5\% + 11\%}{2} = 8\%$$

However, most portfolios are not equally weighted. Portfolio weights represent the proportion of total market value invested in each asset.

• If the portfolio is 60% A and 40% B:

$$R_p = 0.6 \times 5\% + 0.4 \times 11\% = 3\% + 4.4\% = 7.4\%$$

• If the portfolio is 60% B and 40% A:

$$R_p = 0.6 \times 11\% + 0.4 \times 5\% = 6.6\% + 2\% = 8.6\%$$

Key insight: The portfolio return depends on both the returns of the assets and their weights. Overweighting a higher-return asset increases portfolio return; overweighting a lower-return asset decreases it.

Advanced note: Portfolio weights can be negative (short positions) or sum to zero (dollar-neutral portfolios). These introduce leverage or hedging strategies, which can amplify returns, or losses, but are beyond the scope of this lesson.

Portfolio variance measures risk but is more complex than expected return because it depends on how assets interact.

For a two-asset portfolio:

$$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(A, B) \\ = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{A,B} \sigma_A \sigma_B$$

Where,

• $w_A, w_B$: weights of assets A and B
• $\sigma_A, \sigma_B$: standard deviations of assets A and B
• $\text{Cov}(A, B)$: covariance between returns of A and B
• $\rho_{A,B}$: correlation coefficient between A and B

Breaking it down:

1. The first two terms: weighted variance of each asset (always positive).
2. The third term: captures how assets move together (can be positive or negative) and is influenced by correlation.

Key insight: The portfolio's risk is not just a weighted average of individual risks; it also depends on how assets correlate. Combining assets with low or negative correlation can reduce overall portfolio risk, a principle known as diversification.

Portfolio standard deviation is the square root of variance:

$$\sigma_p = \sqrt{\sigma_p^2} = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{A,B} \sigma_A \sigma_B}$$

Example: For an equally weighted portfolio with two assets (50% each):

• Asset A: $\sigma_A = 10\%$
• Asset B: $\sigma_B = 20\%$
• Correlation $\rho_{A,B} = 0.5$

Calculating standard deviation:

$$\sigma_p = \sqrt{0.5^2 \times 10\%^2 + 0.5^2 \times 20\%^2 + 2 \times 0.5 \times 0.5 \times 0.5 \times 10\% \times 20\%} \\ = \sqrt{0.25 \times 0.01 + 0.25 \times 0.04 + 0.25 \times 0.1} \\ = \sqrt{0.0025 + 0.01 + 0.025} = \sqrt{0.0375} \approx 19.36\%$$

The portfolio's risk is not simply the average of individual risks; it depends on weights and correlation.

To compare risk and return, we use the coefficient of variation (CV):

$$CV_p = \frac{\sigma_p}{R_p}$$

• Lower CV = better “return per unit of risk.”
• Adding a lower-risk or negatively correlated asset can improve the portfolio's risk-return profile compared to holding a single high-return, high-risk asset.

Example: A portfolio of Stock A and Stock B may have a lower CV than Stock B alone, meaning the portfolio is more efficient in balancing risk and return.

The Sharpe ratio measures risk-adjusted return and can easily be calculated for a portfolio:

$$S_p = \frac{R_p - R_f}{\sigma_p}$$

Where,

• $R_p$: portfolio return
• $R_f$: risk-free rate
• $\sigma_p$: portfolio standard deviation (volatility)

Example: Assuming a risk-free rate of 0%, the Sharpe ratio for our portfolio of assets A and B can be computed directly if we know the portfolio return and standard deviation.

• Stock A: $S_A = \dfrac{R_A - R_f}{\sigma_A}$
• Stock B: $S_B = \dfrac{R_B - R_f}{\sigma_B}$
• Portfolio: $S_p = \dfrac{R_p - R_f}{\sigma_p}$

Insight: The portfolio can have a higher Sharpe ratio than either individual asset. By combining A and B, the portfolio achieves a better return per unit of risk, demonstrating the benefits of diversification.

The methods we’ve discussed are not limited to stocks; they apply to any assets: cryptocurrencies, bonds, real estate, or commodities.

Weights: Portfolio weights represent market value proportions, not just the number of shares. When combining assets, especially from different markets, consider:

1. Liquidity and price availability:

• Exchange-traded assets have official closing prices, making returns straightforward to calculate.

• Illiquid or OTC assets may require mark-to-model pricing, where a model estimates daily prices for return and volatility calculations.

2. Time alignment:

• Global markets operate in different time zones. To compute returns consistently, pick a standard reference time, often Greenwich Mean Time (GMT).

Once you have reliable pricing data, you can calculate returns and volatilities and then combine them using portfolio formulas.

Portfolio return is a linear combination of asset returns:

$$R_p = w_A R_A + w_B R_B$$

Properties of linearity:

• Portfolio return lies between the returns of the assets if weights are positive and sum to 1.

• If one asset’s weight is zero, the portfolio return equals the other asset’s return.

• Negative weights (short positions) can push portfolio returns beyond the range of individual assets.

Portfolio risk (standard deviation) is non-linear:

$$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{A,B} \sigma_A \sigma_B}$$

Key points:

1. The square root introduces non-linearity.
2. The correlation term $(ρ_AB)$ affects portfolio volatility:
   • Positive correlation increases risk.
   • Negative correlation reduces risk.

Example: For an equally weighted portfolio where assets A and B are negatively correlated $(ρ_AB < 0)$, the portfolio volatility can be lower than either individual asset.

Hedging insight: Assets with strong negative correlation hedge each other, reducing portfolio risk. Lower risk often comes with slightly lower expected return, but the trade-off can improve the portfolio’s risk-return efficiency.

While examples so far use two assets for simplicity, these principles scale to n-assets portfolios:

$$R_p = \sum_{i=1}^{n} w_i R_i$$

Portfolio standard deviation: involves all individual variances and pairwise correlations:

$$\sigma_p^2 = \sum_{i=1}^{n} w_i^2 \sigma_i^2 + \sum_{i=1}^{n} \sum_{j \neq i} w_i w_j \rho_{i,j} \sigma_i \sigma_j$$

As the number of assets increases, the benefits of diversification become more pronounced, especially when including assets with low or negative correlations.

In advanced portfolio management, these formulas are expressed compactly with vectors and matrices, which simplifies computations for portfolios with many assets.