There are 100 stocks, all of which have expected return 5% and volatility 10%. The correlation between each pair of stocks is 0.5. What is the minimum volatility that any portfolio of these 100 stocks can have?

Question

BlackTom AI · Accepted Answer

We have 100 stocks each with expected return 5% and volatility 10%, and every pair has correlation 0.5. The minimum possible portfolio volatility occurs for a portfolio that minimizes w^T Σ w subject to sum w = 1, where Σ is the covariance matrix.

For identically distributed assets with σ = 0.10 and pairwise correlation ρ = 0.5, the covariance matrix Σ has diagonal entries σ^2 and off-diagonal entries ρ σ^2.

A standard result for equi-correlated assets is that the minimum-variance portfolio weights are found via w* ∝ Σ^{-1} 1, and the minimum variance is Var_min = 1 / (1^T Σ^{-1} 1).

Compute Σ^{-1} for this structure: Σ = σ^2[(1−ρ)I + ρ J], where J is the all-ones matrix. The inverse has the form Σ^{-1} = (1/σ^2)[a I + b J], with a = 1/(1−ρ) and b = −ρ / [(1−ρ)(1−ρ + ρ N)]. Here N = 100.

Plugging in ρ = 0.5, 1−ρ = 0.5, and N = 100:
- a = 1/0.5 = 2
- b = −0.5 / [0.5 × (0.5 + 0.5×100)] = −0.5 / [0.5 × 50.5] = −0.5 / 25.25 ≈ −0.0198

Now compute 1^T Σ^{-1} 1 = (1/σ^2)[a N + b N^2] = (1/0.01)[2×100 + (−0.0198)×10000] = 100×(200 − 198) = 100×2 = 200.

Therefore Var_min = 1 / (1^T Σ^{-1} 1) = 1 / 200 = 0.005.

The minimum portfolio standard deviation is sqrt(0.005) ≈ 0.07071, i.e., about 7.07%.

Thus, the smallest possible volatility for any portfolio of these 100 stocks is approximately 7%, rounding to the nearest percent. Since all assets are identical in distribution and perfectly symmetric, any deviation from the equal-weighting structure would not yield a lower variance than this minimum, given the fixed pairwise correlation.

In summary, the calculation shows a minimum volatility around 7.07%, which matches the given answer choice of 7%.

There are 100 stocks, all of which have expected return 5% and volatility 10%. The correlation between each pair of stocks is 0.5. What is the minimum volatility that any portfolio of these 100 stocks can have?单项选择题

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