Build 5 assets of portfolio matrix and the equation A portfolio consists of 4 securities, 1, 2, 3, and 4. The proportions of these securities are: w1=0.3, w2=0.2, w3=0.2, and w4=0.3. The standard deviations of returns on these securities (in percentage terms) are: σ1=5, σ2=6, σ3=12, and σ4=8. The correlation coefficients among security returns are: ρ12=0.2, ρ13=0.6, ρ14=0.3, ρ23=0.4, ρ24=0.6, and ρ34=0.5. What is the standard deviation of portfolio return?

Question

Asked on July 9, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-09T20:11:07.672Z

1. Building a 5-asset portfolio matrix and equation Step 1: Define the portfolio variance equation. The portfolio variance (_p^2) for N assets is given by the formula: _p^2 = _i=1^N _j=1^N w_i w_j _i _j _ij where w_i and w_j are the weights of assets i and j in the portfolio, _i and _j are their respective standard deviations, and _ij is the correlation coefficient between the returns of assets i and j. For 5 assets (N=5), the equation expands to: _p^2 = _i=1^5 w_i^2 _i^2 + _i=1^5 _j=i+1^5 2 w_i w_j _i _j _ij Step 2: Construct the portfolio covariance matrix for 5 assets. The portfolio covariance matrix () for 5 assets is a 5 × 5 matrix where the diagonal elements are the variances of individual assets (_i^2) and the off-diagonal elements are the covariances between asset returns (_ij = _i _j _ij). = _1^2 & _1 _2 _12 & _1 _3 _13 & _1 _4 _14 & _1 _5 _15 \\ _2 _1 _21 & _2^2 & _2 _3 _23 & _2 _4 _24 & _2 _5 _25 \\ _3 _1 _31 & _3 _2 _32 & _3^2 & _3 _4 _34 & _3 _5 _35 \\ _4 _1 _41 & _4 _2 _42 & _4 _3 _43 & _4^2 & _4 _5 _45 \\ _5 _1 _51 & _5 _2 _52 & _5 _3 _53 & _5 _4 _54 & _5^2 Note that _ij = _ji, so _i _j _ij = _j _i _ji. 2. Calculating the standard deviation of portfolio return for 4 securities Step 1: List the given values. • Weights: w_1=0.3, w_2=0.2, w_3=0.2, w_4=0.3 • Standard deviations: _1=5\%, _2=6\%, _3=12\%, _4=8\% • Correlation coefficients: _12=0.2, _13=0.6, _14=0.3, _23=0.4, _24=0.6, _34=0.5 Step 2: Calculate the individual variance terms (w_i^2 _i^2). • w_1^2 _1^2 = (0.3)^2 (5)^2 = 0.09 × 25 = 2.25 • w_2^2 _2^2 = (0.2)^2 (6)^2 = 0.04 × 36 = 1.44 • w_3^2 _3^2 = (0.2)^2 (12)^2 = 0.04 × 144 = 5.76 • w_4^2 _4^2 = (0.3)^2 (8)^2 = 0.09 × 64 = 5.76 Sum of variance terms: 2.25 + 1.44 + 5.76 + 5.76 = 15.21 Step 3: Calculate the covariance terms (2 w_i w_j _i _j _ij). • 2 w_1 w_2 _1 _2 _12 = 2 × 0.3 × 0.2 × 5 × 6 × 0.2 = 0.72 • 2 w_1 w_3 _1 _3 _13 = 2 × 0.3 × 0.2 × 5 × 12 × 0.6 = 4.32 • 2 w_1 w_4 _1 _4 _14 = 2 × 0.3 × 0.3 × 5 × 8 × 0.3 = 2.16 • 2 w_2 w_3 _2 _3 _23 = 2 × 0.2 × 0.2 × 6 × 12 × 0.4 = 2.304 • 2 w_2 w_4 _2 _4 _24 = 2 × 0.2 × 0.3 × 6 × 8 × 0.6 = 3.456 • 2 w_3 w_4 _3 _4 _34 = 2 × 0.2 × 0.3 × 12 × 8 × 0.5 = 5.76 Sum of covariance terms: 0.72 + 4.32 + 2.16 + 2.304 + 3.456 + 5.76 = 18.72 Step 4: Calculate the portfolio variance (_p^2). _p^2 = Sum of variance terms + Sum of covariance terms _p^2 = 15.21 + 18.72 = 33.93 Step 5: Calculate the portfolio standard deviation (_p). _p = sqrt(_p^2) = sqrt(33.93) ≈ 5.8249 Since the individual standard deviations were in percentage terms, the portfolio standard deviation is also in percentage terms. The standard deviation of portfolio return is 5.8249% What's next? Send 'em! 📸

Accepted Answer · 2026-07-09T20:11:07.672Z

1. Building a 5-asset portfolio matrix and equation Step 1: Define the portfolio variance equation. The portfolio variance (_p^2) for N assets is given by the formula: _p^2 = _i=1^N _j=1^N w_i w_j _i _j _ij where w_i and w_j are the weights of assets i and j in the portfolio, _i and _j are their respective standard deviations, and _ij is the correlation coefficient between the returns of assets i and j. For 5 assets (N=5), the equation expands to: _p^2 = _i=1^5 w_i^2 _i^2 + _i=1^5 _j=i+1^5 2 w_i w_j _i _j _ij Step 2: Construct the portfolio covariance matrix for 5 assets. The portfolio covariance matrix () for 5 assets is a 5 × 5 matrix where the diagonal elements are the variances of individual assets (_i^2) and the off-diagonal elements are the covariances between asset returns (_ij = _i _j _ij). = _1^2 & _1 _2 _12 & _1 _3 _13 & _1 _4 _14 & _1 _5 _15 \\ _2 _1 _21 & _2^2 & _2 _3 _23 & _2 _4 _24 & _2 _5 _25 \\ _3 _1 _31 & _3 _2 _32 & _3^2 & _3 _4 _34 & _3 _5 _35 \\ _4 _1 _41 & _4 _2 _42 & _4 _3 _43 & _4^2 & _4 _5 _45 \\ _5 _1 _51 & _5 _2 _52 & _5 _3 _53 & _5 _4 _54 & _5^2 Note that _ij = _ji, so _i _j _ij = _j _i _ji. 2. Calculating the standard deviation of portfolio return for 4 securities Step 1: List the given values. • Weights: w_1=0.3, w_2=0.2, w_3=0.2, w_4=0.3 • Standard deviations: _1=5\%, _2=6\%, _3=12\%, _4=8\% • Correlation coefficients: _12=0.2, _13=0.6, _14=0.3, _23=0.4, _24=0.6, _34=0.5 Step 2: Calculate the individual variance terms (w_i^2 _i^2). • w_1^2 _1^2 = (0.3)^2 (5)^2 = 0.09 × 25 = 2.25 • w_2^2 _2^2 = (0.2)^2 (6)^2 = 0.04 × 36 = 1.44 • w_3^2 _3^2 = (0.2)^2 (12)^2 = 0.04 × 144 = 5.76 • w_4^2 _4^2 = (0.3)^2 (8)^2 = 0.09 × 64 = 5.76 Sum of variance terms: 2.25 + 1.44 + 5.76 + 5.76 = 15.21 Step 3: Calculate the covariance terms (2 w_i w_j _i _j _ij). • 2 w_1 w_2 _1 _2 _12 = 2 × 0.3 × 0.2 × 5 × 6 × 0.2 = 0.72 • 2 w_1 w_3 _1 _3 _13 = 2 × 0.3 × 0.2 × 5 × 12 × 0.6 = 4.32 • 2 w_1 w_4 _1 _4 _14 = 2 × 0.3 × 0.3 × 5 × 8 × 0.3 = 2.16 • 2 w_2 w_3 _2 _3 _23 = 2 × 0.2 × 0.2 × 6 × 12 × 0.4 = 2.304 • 2 w_2 w_4 _2 _4 _24 = 2 × 0.2 × 0.3 × 6 × 8 × 0.6 = 3.456 • 2 w_3 w_4 _3 _4 _34 = 2 × 0.2 × 0.3 × 12 × 8 × 0.5 = 5.76 Sum of covariance terms: 0.72 + 4.32 + 2.16 + 2.304 + 3.456 + 5.76 = 18.72 Step 4: Calculate the portfolio variance (_p^2). _p^2 = Sum of variance terms + Sum of covariance terms _p^2 = 15.21 + 18.72 = 33.93 Step 5: Calculate the portfolio standard deviation (_p). _p = sqrt(_p^2) = sqrt(33.93) ≈ 5.8249 Since the individual standard deviations were in percentage terms, the portfolio standard deviation is also in percentage terms. The standard deviation of portfolio return is 5.8249% What's next? Send 'em! 📸

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