Understanding Portfolio Risk and Negative Correlation

The question asks us to evaluate an Assertion (A) and a Reason (R) related to portfolio risk reduction when combining securities with less than perfect negative correlation.

Analyzing Assertion (A): Portfolio Risk Reduction through Negative Correlation

Assertion (A) states: "If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly."

Let's break this down:

• Portfolio Risk: When you invest in more than one asset, you hold a portfolio. The total risk of this portfolio is not simply the average of the individual asset risks.
• Diversification: Combining assets in a portfolio is known as diversification. The goal of diversification is often to reduce risk.
• Correlation: Correlation measures how the prices of two assets move relative to each other. It ranges from +1 (perfect positive correlation) to -1 (perfect negative correlation).
• Negative Correlation: If two assets have negative correlation, their prices tend to move in opposite directions. When one goes up, the other tends to go down (and vice versa).

If securities have perfect positive correlation ($\rho = +1$), combining them offers no diversification benefit in terms of risk reduction; the portfolio risk is just the weighted average of individual risks. If they have perfect negative correlation ($\rho = -1$), it's theoretically possible to eliminate all unsystematic risk by combining them in specific proportions. When securities have less than perfect negative correlation (meaning the correlation coefficient is between -1 and 0, i.e., $-1 < \rho < 0$), combining them still provides significant diversification benefits. When one asset performs poorly, the other is likely (though not guaranteed) to perform better, smoothing out the overall portfolio returns and reducing volatility (risk) compared to holding a single asset or positively correlated assets.

Therefore, Assertion (A) is correct.

Analyzing Reason (R): Impact of Negative Correlation on Portfolio Risk Calculation

Reason (R) states: "The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive."

To understand this, let's look at the formula for the variance of a two-asset portfolio (Variance is a common measure of risk):

The variance ($\sigma_p^2$) of a portfolio consisting of two assets, A and B, is given by:

\(\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(R_A, R_B)\)

Where:

• $w_A$ and $w_B$ are the weights of assets A and B in the portfolio ($w_A + w_B = 1$).
• $\sigma_A^2$ and $\sigma_B^2$ are the variances of assets A and B.
• $\text{Cov}(R_A, R_B)$ is the covariance between the returns of assets A and B.

Covariance is related to correlation ($\rho_{AB}$) by the formula: $\text{Cov}(R_A, R_B) = \rho_{AB} \sigma_A \sigma_B$. Substituting this into the portfolio variance formula:

\(\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho_{AB}\)

In this formula:

• $w_A^2 \sigma_A^2$ and $w_B^2 \sigma_B^2$ are always positive (as weights and variances are squared, or standard deviations are non-negative). These represent the contribution of individual asset risks to the portfolio variance.
• The third term, $2 w_A w_B \sigma_A \sigma_B \rho_{AB}$, is the covariance term. $w_A, w_B, \sigma_A, \sigma_B$ are typically positive. The sign of this entire term is determined by the sign of the correlation coefficient ($\rho_{AB}$).

If the correlation ($\rho_{AB}$) is negative (as stated in the context of Assertion A, specifically "less than perfect negative correlation", i.e., $-1 < \rho_{AB} < 0$), then the covariance term $2 w_A w_B \sigma_A \sigma_B \rho_{AB}$ becomes negative. A negative value for this term will reduce the total calculated value of the portfolio variance ($\sigma_p^2$) compared to what it would be if the correlation were zero or positive.

Therefore, Reason (R) correctly describes the mathematical effect of negative correlation on the portfolio risk calculation. Reason (R) is correct.

Relationship Between Assertion (A) and Reason (R)

Assertion (A) claims that combining securities with less than perfect negative correlation reduces portfolio risk significantly. Reason (R) explains *why* this risk reduction occurs: the negative correlation introduces a negative term into the portfolio variance formula, which reduces the overall portfolio risk value. Reason (R) provides the mathematical basis for the statement made in Assertion (A).

Thus, Reason (R) is the correct explanation of Assertion (A).

Conclusion on Assertion and Reason

Both Assertion (A) and Reason (R) are correct statements, and Reason (R) provides a valid explanation for why Assertion (A) is true regarding portfolio risk and negative correlation.

Revision Table: Key Concepts in Portfolio Management

Additional Information: Correlation and Diversification Benefits

The degree to which portfolio risk can be reduced through diversification depends critically on the correlation between the assets included in the portfolio. Here’s a quick overview:

• Perfect Positive Correlation ($\rho = +1$): Assets move in exactly the same direction. No diversification benefit in terms of risk reduction is gained by combining them. Portfolio standard deviation is the weighted average of individual standard deviations.
• Zero Correlation ($\rho = 0$): Assets' price movements are completely unrelated. Combining such assets provides substantial diversification benefits, as the ups and downs of one asset tend to cancel out the ups and downs of the other on average.
• Negative Correlation ($-1 < \rho < 0$): Assets tend to move in opposite directions. This offers even greater diversification benefits than zero correlation. When one asset performs poorly, the other is likely performing well, smoothing portfolio returns.
• Perfect Negative Correlation ($\rho = -1$): Assets move in exactly opposite directions. This offers the maximum possible diversification benefit. With a specific weighting, it is possible to construct a portfolio with zero risk if assets have perfect negative correlation (assuming non-zero risk for individual assets).

In the real world, finding assets with perfect negative correlation is rare. However, assets with low or negative correlation are sought after by portfolio managers to improve the risk-return profile of a portfolio. Combining assets with "less than perfect negative correlation" is a common and effective strategy for significant portfolio risk reduction.