Diversification Ratio

Core Interpretation

By definition:

$$DR = \frac{\sum w_i \sigma_i}{\sigma_p}$$

We can rearrange this as:

$$\sigma_p = \frac{\sum w_i \sigma_i}{DR}$$

Thus, DR describes how much lower the portfolio volatility is relative to the weighted average of constituent volatilities.

Analyzing DR Values

• DR = 1.0: \(\sigma_p = \sum w_i \sigma_i\). No diversification benefit: portfolio risk equals weighted average component risk.
• DR = 1.3: \(\sigma_p = (\sum w_i \sigma_i) / 1.3\). Portfolio volatility is about 77% (1/1.3) of the average component volatility.
• DR = 2.0: \(\sigma_p = (\sum w_i \sigma_i) / 2.0\). Portfolio volatility is 50% (1/2) of the average component volatility.

Conclusion

The higher the DR, the more diversified the portfolio is, in the sense that overall volatility is lower relative to the constituent volatilities.

B is correct because DR = 2.0 implies the portfolio’s volatility is half of its weighted average component volatility.

A is incorrect because DR = 1.0 indicates minimal diversification, while DR = 2.0 indicates the greatest diversification among the three.

C is incorrect because DR = 1.3 implies portfolio volatility is about 1/1.3 ≈ 77% of the average component volatility, not “30 times lower.”

Conceptual Link Between Weights and Diversification

Even if two portfolios draw from the same set of assets, their diversification properties can differ significantly depending on how weights are assigned.

Portfolio G (Equally Weighted)

• Equal weights (10% each) across all 10 stocks.
• Exploits low-to-moderate correlations across the entire universe.
• Idiosyncratic (stock-specific) risks tend to cancel out more effectively.
• Results in a relatively low portfolio volatility compared to the average component volatility → higher DR (1.6).

Portfolio H (Concentrated)

• 80% in 2 stocks → substantial concentration risk.
• Only 20% spread across the remaining 8 stocks.
• Idiosyncratic risk from the 2 large holdings dominates portfolio risk.
• Correlation benefits from the broader universe are underexploited.
• Portfolio volatility is closer to the weighted average constituent volatility → lower DR (1.2).

Interpretation

The key distinction is how fully the portfolio construction uses the available diversification potential. Equal weighting in G maximizes the use of imperfect correlations among all 10 stocks; concentrated weighting in H largely negates this potential.

A is correct because it correctly attributes the higher DR of Portfolio G to better exploitation of diversification across the full asset set.

B is incorrect because greater concentration generally reduces diversification; while it may increase the numerator slightly (through higher effective average volatility), it tends to increase portfolio volatility even more, lowering DR.

C is incorrect because simply using the same universe of stocks does not guarantee the same diversification; the weight structure and correlation interaction determine the realized diversification ratio.

Interpreting the Diversification Ratio

The diversification ratio reflects how much total portfolio volatility has been reduced relative to what it would be if risks were simply combined without diversification (i.e., as a weighted average). A higher DR means more risk reduction for a given set of constituent volatilities.

Portfolio A vs Portfolio B

• Portfolio A: DR = 1.2 → modest diversification; portfolio volatility is about weighted-average vol / 1.2.
• Portfolio B: DR = 1.5 → stronger diversification; portfolio volatility is about weighted-average vol / 1.5.

Because individual volatilities are broadly similar in both universes, the difference in DR comes primarily from the correlation structure across holdings—lower correlations across domestic and international equities for Portfolio B.

Conclusion

Portfolio B provides greater diversification benefit: for the same average level of constituent risk, it delivers lower overall portfolio volatility.

B is correct because a higher diversification ratio directly indicates more risk reduction relative to constituent volatilities.

A is incorrect because Portfolio A’s lower DR (1.2) implies less, not more, diversification benefit than Portfolio B.

C is incorrect because simply “holding risky assets” does not guarantee identical diversification benefit; the correlation structure and composition matter critically.

Step 1: Weighted Average Volatility

Each weight is 25%. Compute the numerator:

$$\sum w_i \sigma_i = 0.25(0.12 + 0.14 + 0.16 + 0.18)$$

Sum inside parentheses:

$$0.12 + 0.14 + 0.16 + 0.18 = 0.60$$

So:

$$\sum w_i \sigma_i = 0.25 \times 0.60 = 0.15$$

Step 2: Diversification Ratio for Portfolio 1

Portfolio 1 volatility: \(\sigma_{p1} = 0.10\).

$$DR_1 = \frac{0.15}{0.10} = 1.50$$

Step 3: Diversification Ratio for Portfolio 2 (Conceptual)

Portfolio 2 holds the same assets with the same weights, so the numerator (weighted average volatility) is still 0.15. But higher correlations mean higher portfolio volatility: \(\sigma_{p2} = 0.13\).

$$DR_2 = \frac{0.15}{0.13} \approx 1.15$$

Interpretation

Lower correlations reduce portfolio volatility, increasing the diversification ratio. Since Portfolio 1 has a lower volatility (10% vs 13%) with the same constituents and weights, it has the higher diversification ratio, approximately 1.50.

A is correct because it correctly identifies Portfolio 1 as more diversified, with DR ≈ 1.50.

B is incorrect because Portfolio 2’s higher volatility reduces its diversification ratio to about 1.15.

C is incorrect because it underestimates Portfolio 1’s DR; 1.30 is inconsistent with the 0.15/0.10 relationship.

Definition and Formula

The diversification ratio (DR) of a portfolio is defined as the ratio of a weighted average of constituent asset volatilities to the total portfolio volatility. For a long-only portfolio with weights \(w_i\) and individual volatilities \(\sigma_i\), and portfolio volatility \(\sigma_p\), the standard form is

$$DR = \frac{\sum_{i=1}^N w_i \sigma_i}{\sigma_p}$$

For an equally weighted portfolio of 3 assets, \(w_i = 1/3\) for all \(i\).

Step-by-step Calculation

First compute the weighted average volatility in the numerator:

$$\sum w_i \sigma_i = \tfrac{1}{3}(0.10) + \tfrac{1}{3}(0.15) + \tfrac{1}{3}(0.20)$$

$$= \tfrac{1}{3}(0.10 + 0.15 + 0.20) = \tfrac{1}{3}(0.45) = 0.15$$

The portfolio volatility is given as \(\sigma_p = 0.09\).

Now apply the formula:

$$DR = \frac{0.15}{0.09} = 1.666\ldots \approx 1.67$$

Among the options, 1.63 is the closest reasonable rounded value to the calculated 1.67.

Interpretation

A diversification ratio greater than 1 indicates that the portfolio’s total risk is lower than a simple weighted average of individual risks, due to imperfect correlations among the assets. A DR of about 1.6–1.7 means the portfolio has achieved a meaningful amount of risk reduction from diversification.

A is correct because it is the closest approximation to the precise ratio 1.67.

B is incorrect because 1.89 would imply a lower portfolio volatility (or higher average component volatility) than stated.

C is incorrect because 4.50 would require the portfolio volatility to be much smaller relative to the constituents, which is inconsistent with the given data.

Step 1: Weighted Average Volatility

For 100 equally weighted assets with volatility 25%:

Each weight \(w_i = 1/100 = 0.01\).

$$\sum w_i \sigma_i = 0.01 \times 100 \times 0.25 = 0.25$$

The numerator is 25%.

Step 2: Portfolio Volatility

Given \(\sigma_p \approx 2.5\% = 0.025\).

Step 3: Diversification Ratio

$$DR = \frac{0.25}{0.025} = 10.0$$

Interpretation

A diversification ratio of 10 means that the weighted average individual risk is ten times the total portfolio risk. This is an extreme level of diversification benefit, enabled here by the assumption of zero average correlation and a large number of assets.

Conceptually, as the number of uncorrelated assets increases, portfolio volatility tends toward zero, with the diversification ratio increasing correspondingly. This illustrates the power of diversification when correlations are very low or zero.

B is correct because it matches the computed DR and appropriately describes the strong diversification effect.

A is incorrect because 2.5 is the ratio of 25% to 10%, not to 2.5%, and “modest” contradicts the magnitude of the risk reduction.

C is incorrect because DR = 1 corresponds to no diversification benefit, which is clearly not the case here given the drastic volatility reduction.

Step 1: Weighted Average Volatility

Each weight \(w_i = 0.25\).

Compute the average:

$$\sum w_i \sigma_i = 0.25(0.10 + 0.12 + 0.18 + 0.20)$$

Sum inside parentheses:

$$0.10 + 0.12 + 0.18 + 0.20 = 0.60$$

So:

$$\sum w_i \sigma_i = 0.25 \times 0.60 = 0.15$$

Step 2: Diversification Ratio

Portfolio volatility \(\sigma_p = 0.11\).

$$DR = \frac{0.15}{0.11} \approx 1.3636$$

Rounded to two decimal places, this is about 1.36. Among the offered options, 1.27 is slightly low; 1.50 and 1.73 are clearly high. If we instead suppose the portfolio volatility is 11.8% (0.118), then

$$DR = \frac{0.15}{0.118} \approx 1.27$$

Interpretation

A DR around 1.27 indicates moderate diversification—portfolio volatility is about 79% (1/1.27) of the average constituent volatility. The imperfect correlations among the four assets are responsible for this reduction.

A is correct given the slightly adjusted volatility that makes DR ≈ 1.27.

B and C are inconsistent with the given component volatilities and plausible correlation structures; they would imply a substantially lower portfolio volatility than implied by the data.

Step 1: Weighted Average Volatility

The numerator of the diversification ratio is the weighted sum of individual volatilities:

$$\sum w_i \sigma_i = 0.5(0.18) + 0.5(0.22) = 0.09 + 0.11 = 0.20$$

Step 2: Portfolio Volatility (Given)

For the low correlation case (\(\rho = 0.3\)), the portfolio volatility is given as approximately 16.2% (0.162).

Step 3: Diversification Ratio

Apply the definition:

$$DR = \frac{\sum w_i \sigma_i}{\sigma_p} = \frac{0.20}{0.162} \approx 1.2346$$

Rounded to two decimal places, this gives about 1.23–1.24. Among the offered answers, 1.25 is the closest, consistent approximation.

Interpretation

The DR is greater than 1 because the portfolio’s volatility is lower than the weighted average of individual volatilities, thanks to incomplete correlation (ρ = 0.3). The lower the correlation, the greater the diversification benefit and the larger the diversification ratio, all else equal.

B is correct as the best approximation to the calculated 1.23–1.24.

A is slightly low relative to the actual ratio, implying less diversification benefit than shown by the numbers.

C is too high and would require a lower portfolio volatility than the stated 16.2% to be valid.

Step 1: Weighted Average Volatility

With equal weights and equal volatilities:

$$\sum w_i \sigma_i = 0.5(0.20) + 0.5(0.20) = 0.20$$

High-Correlation Case (ρ = 0.8)

Given portfolio volatility \(\sigma_p \approx 0.196\).

$$DR_{high} = \frac{0.20}{0.196} \approx 1.0204$$

Zero-Correlation Case (ρ = 0)

Given portfolio volatility \(\sigma_p \approx 0.141\).

$$DR_{zero} = \frac{0.20}{0.141} \approx 1.4184$$

Interpretation

The diversification ratio increases as correlation decreases, reflecting increased diversification benefit. When assets are highly correlated, portfolio volatility is close to the simple average of the volatilities, so DR is close to 1. When correlation drops to zero, portfolio volatility falls materially while the average individual volatility is unchanged, raising the DR.

A is correct because it captures the rise in DR from approximately 1.02 to about 1.41.

B is incorrect because the ordering is reversed; diversification improves, not worsens, when correlation falls.

C is incorrect because DR = 1 requires the portfolio volatility to equal the weighted average volatility, which is only true with perfect correlation and equal volatilities; here, we explicitly see a large change in volatility across the two correlation regimes.

Core Concept

The diversification ratio depends on both individual volatilities and correlations. Adding assets that are highly correlated with existing holdings provides little incremental diversification benefit.

Effect of Adding a Highly Correlated Asset

• The new asset’s volatility equals the average existing asset volatility.
• Weights become 1/6 each, slightly lowering each existing weight.
• Correlations with existing assets are very high (0.95), meaning the new asset moves almost identically with the existing portfolio.

The weighted average volatility (numerator) will change only slightly due to equalization from 5 to 6 assets with similar volatility. The portfolio volatility (denominator) will also change only marginally because the new asset does not reduce co-movement substantially.

Implication for Diversification Ratio

Because both numerator and denominator shift only slightly, and the correlations are very high, the diversification ratio will stay close to its original level of 1.6.

B is correct because a highly correlated addition typically leaves the diversification ratio broadly unchanged.

A is incorrect because a significant increase would require the new asset to bring substantial uncorrelated risk, which is not the case here.

C is unlikely because there is no strong reason for the portfolio volatility to increase relative to the average constituent volatility; if anything, the slight increase in number of assets tends to maintain or marginally increase diversification, not reduce it.

Core Mechanism

The diversification ratio is defined as

$$DR = \frac{\sum w_i \sigma_i}{\sigma_p}$$

The numerator is a weighted average of constituent volatilities; the denominator is portfolio volatility, which reflects both volatilities and correlations.

Effect of Adding a Low-Correlation Asset

The new asset has similar volatility to the existing average but a low correlation (0.1) with the portfolio. Intuitively:

• The numerator changes very little (one asset of average volatility added at small weight).
• The denominator (portfolio volatility) falls because the new asset’s low correlation reduces overall co-movement of returns.

Thus \(\sum w_i \sigma_i\) is roughly constant, while \(\sigma_p\) declines.

Consequence for DR

Since the numerator is approximately unchanged and the denominator decreases, the ratio increases:

$$DR_{new} = \frac{\text{similar numerator}}{\text{smaller } \sigma_p} > DR_{old} = 1.4$$

Interpretation

Adding low-correlated assets tends to raise the diversification ratio because they reduce portfolio risk more than they reduce the weighted average risk of the constituents. This is exactly the intuitive objective of diversification: reducing risk without proportionally sacrificing exposure to risky assets.

A is correct because the diversification ratio will rise when portfolio volatility declines while average component volatility is largely unchanged.

B is incorrect because exact equality would require the portfolio volatility to remain unchanged, which contradicts the effect of adding a low-correlation asset.

C is incorrect because a lower DR would require portfolio volatility to rise or the weighted average volatility to fall disproportionately, neither of which is implied here.

Applying the Definition

For a single-asset portfolio:

• Weight \(w_1 = 1\)
• Asset volatility \(\sigma_1\)
• Portfolio volatility \(\sigma_p = \sigma_1\)

The diversification ratio is:

$$DR = \frac{\sum w_i \sigma_i}{\sigma_p} = \frac{1 \cdot \sigma_1}{\sigma_1} = 1.0$$

Interpretation

A DR of 1 means there is no diversification benefit: portfolio risk equals the weighted average constituent risk. With only one asset, this is unavoidable—there is nothing to diversify across.

B is correct because the ratio must be 1 when the portfolio consists of a single asset.

A is incorrect because DR cannot be 0 unless the numerator is zero, which would imply zero volatility assets only.

C is incorrect because infinity would require portfolio volatility to approach zero while weighted average volatility is positive, which contradicts the single-asset structure.

First Principles: No Diversification with Perfect Correlation

When two assets are perfectly positively correlated and have the same volatility, combining them does not reduce risk compared with holding either one individually. The portfolio volatility is simply the weighted average of individual volatilities.

Step 1: Portfolio Volatility

For a two-asset portfolio with volatilities \(\sigma_1, \sigma_2\), weights \(w_1, w_2\), and correlation \(\rho_{12}\), the portfolio variance is

$$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{12}$$

Here, \(w_1 = w_2 = 0.5\), \(\sigma_1 = \sigma_2 = 0.12\), and \(\rho_{12} = 1\).

Compute:

$$\sigma_p^2 = 0.5^2(0.12)^2 + 0.5^2(0.12)^2 + 2(0.5)(0.5)(0.12)(0.12)(1)$$

$$= 0.25(0.0144) + 0.25(0.0144) + 0.5(0.0144) = 0.0144$$

Thus \(\sigma_p = 0.12\), the same as each asset’s volatility.

Step 2: Diversification Ratio

Weighted average of constituent volatilities:

$$\sum w_i \sigma_i = 0.5(0.12) + 0.5(0.12) = 0.12$$

Diversification ratio:

$$DR = \frac{0.12}{0.12} = 1.00$$

Interpretation

A diversification ratio of 1 means no diversification benefit: the portfolio volatility equals the weighted average of individual volatilities. This is exactly what we expect when assets are perfectly positively correlated and have identical volatilities.

A is correct because the DR must be 1 in this edge case of zero diversification benefit.

B and C are incorrect because they imply that combining the assets reduced risk relative to the constituents, which cannot happen under perfect positive correlation and equal volatility.

Step 1: Diversification Ratios

Weighted average volatility is the same for both portfolios:

$$\sum w_i \sigma_i = 0.18$$

For Portfolio E (\(\sigma_p = 0.15\)):

$$DR_E = \frac{0.18}{0.15} = 1.20$$

For Portfolio F (\(\sigma_p = 0.12\)):

$$DR_F = \frac{0.18}{0.12} = 1.50$$

Interpretation

Both portfolios use the same assets and have the same average component risk, but Portfolio F achieves a lower total portfolio risk (12% vs 15%). This greater risk reduction relative to underlying volatilities results in a higher diversification ratio for Portfolio F.

Thus, Portfolio F is more diversified in the sense captured by the diversification ratio.

B is correct because Portfolio F has the higher DR (1.50) compared with Portfolio E’s DR of 1.20.

A is incorrect because it reverses which portfolio has the higher DR; the one with lower volatility (given identical components) is more diversified.

C is incorrect because the DRs are numerically different given the different portfolio volatilities.