The Standard Deviation Of The Market Index Portfolio Is 20 Stock A H
The problem involves assessing the impact of changes in beta and residual standard deviation on the variance of a stock (Stock A) and subsequently on the portfolio's standard deviation when an investor shifts allocations from the market index to Stock A.
Given data:
Market index standard deviation, σm = 20% (or 0.20)
Stock A’s beta, β = 1.5
Residual standard deviation of Stock A, σe = 30% (or 0.30)
Objectives:
Part (a): Determine whether increasing beta by 0.15 or increasing residual standard deviation by 3% leads to a larger increase in Stock A’s variance.
Part (b): Comparing the impact of these changes on the overall portfolio’s standard deviation when the portfolio shifts from 100% market to a 90% market and 10% Stock A allocation.
Paper For Above instruction
Calculating the Variance of Stock A:
To evaluate the impact of changes in beta and residual variance, we first need to understand how Stock A’s total variance is composed. The total variance of stock returns (σA^2) can be decomposed into systematic variance (due to market movements) and unsystematic (residual) variance (Elton & Gruber, 2014).
The variance formula for Stock A is:
σA^2 = (β^2) * (σm)^2 + (σe)^2
Where:
β is the beta of Stock A
σm is the standard deviation of the market index
σe is the residual standard deviation
Calculations before changes:
Initial variance:
σA^2 = (1.5)^2 * (0.20)^2 + (0.30)^2
= 2.25 * 0.04 + 0.09
= 0.09 + 0.09
= 0.18
The initial variance of Stock A is 0.18 (or 18%).
Part (a): Impact of increasing beta vs. residual standard deviation
First, determine the change in variance when beta increases by 0.15:
New beta = 1.5 + 0.15 = 1.65
σA_new^2_beta = (1.65)^2 * (0.20)^2 + (0.30)^2
= 2.7225 * 0.04 + 0.09
= 0.1089 + 0.09
= 0.1989
Change in variance due to beta increase:
∆σA^2_beta = 0.1989 - 0.18 = 0.0189
Next, assess the impact of increasing residual standard deviation by 3%:
New residual std dev = 0.30 + 0.03 = 0.33
σA_new^2_residual = (1.5)^2 * (0.20)^2 + (0.33)^2
= 2.25 * 0.04 + 0.1089
= 0.09 + 0.1089
= 0.1989
Change in variance due to residual standard deviation increase:
∆σA^2_residual = 0.1989 - 0.18 = 0.0189
Observation: Both increases lead to an identical increase in variance (0.0189). Therefore, from the perspective of variance, both changes have the same magnitude of impact.
Part (b): Impact of changes on portfolio standard deviation with a shift in allocation
Initial scenario: The portfolio is entirely invested in the market index; thus, the initial portfolio standard deviation is 20%.
New portfolio: 90% in the market (σm = 0.20) and 10% in Stock A (with its variance), changing the overall portfolio standard deviation.
We will examine two cases:
Increase beta by 0.15, resulting in higher variance of Stock A, and therefore, a higher stock contribution to the portfolio risk.
Increase residual deviation by 3%, similarly increasing the stock’s variance.
Calculating the combined portfolio variance:
The variance of a two-asset portfolio is:
σp^2 = w_m^2 * σm^2 + w_A^2 * σA^2 + 2 * w_m * w_A * Cov(m, A)
Since the covariance between the market and Stock A is:
Cov(m, A) = β * (σm)^2 = 1.5 * 0.04 = 0.06
Assuming the initial weights are:
w_m = 0.90
w_A = 0.10
Initial variance of the portfolio (before any change):
σp_initial^2 = (0.9)^2 * 0.04 + (0.1)^2 * 0.18 + 2 * 0.9 * 0.1 * 0.06 = 0.81 * 0.04 + 0.01 * 0.18 + 2 * 0.9 * 0.1 * 0.06
= 0.0324 + 0.0018 + 0.0108 = 0.045
Initial portfolio standard deviation:
σp_initial = √0.045 ≈ 0.212 or 21.2%
Impact of increased beta:
If beta increases to 1.65, then the variance of Stock A becomes 0.1989, as previously calculated. The covariance between market and Stock A now is:
Cov(m, A) = 1.65 * 0.04 = 0.066
Variance of the portfolio in this case:
σp_beta_change^2 = 0.81 * 0.04 + 0.01 * 0.1989 + 2 * 0.9 * 0.1 * 0.066
= 0.0324 + 0.001989 + 0.01188
= 0.0463
The corresponding standard deviation:
σp_beta_change ≈ √0.0463 ≈ 0.215 or 21.5%
Difference from initial:
∆σp_beta = 21.5% - 21.2% ≈ 0.3%
Impact of increased residual deviation:
Residual standard deviation increases from 0.30 to 0.33, so Stock A’s variance becomes 0.1989.
The covariance remains based on the original beta and market variance, that is, 0.06.
σp_residual_change^2 = 0.81 * 0.04 + 0.01 * 0.1989 + 2 * 0.9 * 0.1 * 0.06
= 0.0324 + 0.001989 + 0.0108 = 0.0452
The new portfolio standard deviation is:
σp_residual_change ≈ √0.0452 ≈ 0.213 or 21.3%
The change in portfolio risk here is:
Comparison: Increasing beta results in a larger increase in the overall portfolio standard deviation (~0.3%) compared to increasing residual deviation (~0.1%). This suggests that changes in beta have a more significant influence on the portfolio’s risk profile when the investor’s allocation shifts from fully market to a diversified mix with Stock A.
Conclusion
Analyses indicate that both increasing beta by 0.15 and residual standard deviation by 3% lead to identical increases in Stock A’s variance. However, when these changes are applied within the context of an actual portfolio—particularly one shifting from full market exposure to a mixed allocation—the impact of an increased beta on overall portfolio risk is more pronounced than an equivalent residual increase. This underscores the importance of beta as a measure of systematic risk, especially when constructing diversified portfolios. Investors should consider how shifts in market-related risk factors impact their overall risk exposure beyond individual stock variances.
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