Port Risk

Return to Return to Return to Return toPeriod VFINX SPY DGAGX SPX2004 10.74 10.70 5.57 10.88 -0.14 -0.18 -5.312003 28.50 28.18 20.39 28.68 -0.18 -0.50 -8.292002 -22.15 -21.59 -17.14 -22.10 -0.05 0.51 4.962001 -12.02 -11.75 -10.75 -11.89 -0.13 0.14 1.142000 -9.06 -9.73 1.80 -9.11 0.05 -0.62 10.911999 21.07 20.39 9.97 21.04 0.03 -0.65 -11.071998 28.61 28.28 30.85 28.58 0.03 -0.30 2.271997 33.21 33.48 27.85 33.36 -0.15 0.12 -5.511996 22.86 22.55 25.67 22.96 -0.10 -0.41 2.711995 37.44 38.05 37.88 37.58 -0.14 0.47 0.30

Average (%): 13.92 13.86 13.21 14.00 -0.08 -0.14 -0.79

Periodic Tracking Error (%): 0.09 0.43 6.68

Period/Yr: 1.00 1.00 1.00

Annual Tracking Error (%): 0.09 0.43 6.68

Dvfinx Dspy Ddgagx

Five-Asset Class Portfolio Risk Decomposition

***Input: Capital Market Variables***

R1 = 1.00% 4.00% 0.50 0.30

R2 = 2.00% 8.00% 0.20 0.10

R3 = 7.50% 18.00% 0.30 0.70

R4 = 10.00% 25.00% 0.50 0.15

R5 = 4.00% 12.00% 0.40 0.20

*** Input: Portfolio Weights*** Portfolio Std. Deviation:

w1 = 3.10% 9.67%

w2 = 21.48%

w3 = 23.77% Portfolio Return:

w4 = 11.52%

w5 = 40.13% Rp = 5.00%

______

Sum = 100%

1. Total Portfolio Risk Decomposition:

Total Contribution

Asset Class Marginal Risk to Portfolio Risk

1 2.11% 0.07%

2 4.00% 0.86%

3 14.39% 3.42%

4 19.11% 2.20%

5 7.78% 3.12%

Weighted Average: 9.67%

2. "Roll-up" Portfolio Risk Decomposition:

Marginal Volatility

Asset Class Include (yes/no) of Asset Roll-Up

1 yes 3.76%

2 yes

3 no

4 no

5 no

Variance-Covariance Matrix (Note: This is calculated by the program)

0.0016 0.0016 0.00144 0.003 0.0024

0.0016 0.0064 0.00576 0.006 0.00096

0.00144 0.00576 0.0324 0.0315 0.00324

0.003 0.006 0.0315 0.0625 0.006

0.0024 0.00096 0.00324 0.006 0.0144

s1 = r12 = r24 =

s2 = r13 = r25 =

s3 = r14 = r34 =

s4 = r15 = r35 =

s5 = r23 = r45 =

sp =

This spreadsheet calculates the asset-specific marginal risk contributions for a five-asset class portfolio. The user must specify the expected return and standard deviations for each asset class, as well as the correlation coefficients between the asset classes.

The spreadsheet automatically computes: (i) total portfolio risk, (ii) marginal risk of each asset, and (iii) each asset total contribution to risk. The user can also roll-up the risk calculations for various subsets of the asset classes.

Calculation of the Mean-Variance Efficient Weights for a Three Asset Portfolio

This program calculates the optimal portfolio allocations that define the mean-variance efficient frontier for a three-asset class portfolio. The user needs to input: expected returns and standard deviations for the asset classes; correlations between the asset classes; and the expected return goal. The program calculates the analyticalsolution to the constrained portfolio variance minimization problem with short sales allowed.


R1 = 12.00% s1 = 21.20% 0.18R2 = 5.10% s2 = 8.30% -0.07R3 = 3.60% s3 = 3.30% 0.22

***Input: Return Goal***

R* = 8.00%

******** ******** ********

***Calculation of Optimal Weights (Note: Calculation is Automatic)***

w1* = 45.99% w2* = 35.80% w3* = 18.21%

Test: Sum of Weights = 1Test: R* Constraint = 0.08

***Calculation of Portfolio Std. Deviation***

10.71%

******* ******* *******

(Note: The following calculations are the intermediate steps in the analytical solution presented above.)

Calculation of V Matrix and Inverse of V

V Matrix: cofactor V:0.044944 0.00316728 -0.00049 7.139E-06 -3.74E-06 5.2822E-06

0.0031673 0.006889 0.0006026 -3.744E-06 4.87E-05 -2.863E-05-0.00049 0.00060258 0.001089 5.2822E-06 -2.86E-05 0.00029959

Det V: adj V:7.139E-06 -3.74E-06 5.2822E-06

3.0641E-07 -3.744E-06 4.87E-05 -2.863E-055.2822E-06 -2.86E-05 0.00029959

r12 = r13 = r23 =

sp =

Calculation of the Mean-Variance Efficient Weights for a Three Asset Portfolio

Inv V: Test: (V)x(Inv V)23.2989 -12.219778 17.239055 1 0 0

-12.21978 158.950995 -93.44808 0 1 017.239055 -93.448081 977.73393 0 0 1

Calculation of Efficient Set Constants

R'(Inv V): i'(Inv V):2.79326533 28.318177 A = 38.57051793.27599647 53.283136 B = 1.6723128832.5012562 901.52491 C = 983.126222

Calculation of M Matrix and Inverse of M

M Matrix: cofactor M:1.67231288 38.570518 983.126222 -38.5705238.5705179 983.12622 -38.570518 1.6723129

Det M: adj M:983.126222 -38.57052

156.40979 -38.570518 1.6723129

Inv M: Test: (M)x(Inv M):6.28557982 -0.246599 1 0-0.2465991 0.0106919 0 1

Intermediate Calculation for Weights

(Inv V)[R i]: (Inv M)[R* 1]':2.79326533 28.318177 0.256247263.27599647 53.283136 -0.009036132.5012562 901.52491

Mean-Variance Efficient Portfolio Weights With No Short Sales


R1 = 12.00% 21.20% 0.18

R2 = 5.10% 8.30% -0.07

R3 = 3.60% 3.30% 0.22

Optimal Portfolio Weights: ***Input: Return Goal***

w1 = 85.51%

w2 = 14.49% R* = 11.00%

w3 = 0.00%

______

Sum = 100%

Portfolio Return: Portfolio Std. Deviation:

Rp = 11.00% 18.38%


0.044944 0.00316728 -0.00048972

0.00316728 0.006889 0.00060258

-0.00048972 0.00060258 0.001089

s1 = r12 =

s2 = r13 =

s3 = r23 =

sp =

When run with the Solver add-in on MS Excel for Windows, this spreadsheet is capable of calculating the mean-variance efficient weights for a three asset portfolio. The user must provide the following inputs: (i) returns, standard deviations, and correlations for the three asset classes; and (ii) a return goal to be acheived by the portfolio.

The Solver uses a numerical approximation procedure to select the new weight values that minimize the portfolio variance subject the following constraints: (i) the weights sum to one; (ii) the return is at least as great as the goal; and (iii) no short selling is allowed.

Mean-Variance Efficient Portfolio Weights With Limited Short Sale Restrictions


R1 = 12.00% 21.20% 0.18

R2 = 5.10% 8.30% -0.07

R3 = 3.60% 3.30% 0.22


w1 = 77.71%

w2 = 58.16% R* = 11.00%

w3 = -35.87%

______

Sum = 100%


Rp = 11.00% 18.03%


0.044944 0.00316728 -0.00048972

0.00316728 0.006889 0.00060258

-0.00048972 0.00060258 0.001089

s1 = r12 =

s2 = r13 =

s3 = r23 =

sp =


The Solver uses a numerical approximation procedure to select the new weight values that minimize the portfolio variance subject the following constraints: (i) the weights sum to one; (ii) the return is at least as great as the goal; and (iii) limited short selling is allowed.


R1 = 1.00% 4.00% 0.50 0.30

R2 = 2.00% 8.00% 0.20 0.10

R3 = 7.50% 18.00% 0.30 0.70

R4 = 10.00% 25.00% 0.50 0.15

R5 = 4.00% 12.00% 0.40 0.20


w1 = 3.10%

w2 = 21.48% R* = 5.00%

w3 = 23.77%

w4 = 11.52%

w5 = 40.13%

______

Sum = 100%


Rp = 5.00% 9.67%


0.0016 0.0016 0.00144 0.003 0.0024

0.0016 0.0064 0.00576 0.006 0.00096

0.00144 0.00576 0.0324 0.0315 0.00324

0.003 0.006 0.0315 0.0625 0.006

0.0024 0.00096 0.00324 0.006 0.0144

s1 = r12 = r24 =

s2 = r13 = r25 =

s3 = r14 = r34 =

s4 = r15 = r35 =

s5 = r23 = r45 =

sp =



When run with the Solver add-in on MS Excel for Windows, this spreadsheet is capable of calculating the mean-variance efficient weights for a five asset class portfolio. The user must provide the following inputs: (i) returns, standard deviations, and correlations for the five asset classes; and (ii) a return goal to be acheived by the portfolio.



R1 = 4.35% 4.02% 0.41 0.21

R2 = 5.69% 8.81% -0.14 0.00

R3 = 9.79% 15.62% -0.16 0.65

R4 = 9.94% 15.35% -0.20 0.69

R5 = 10.89% 21.29% -0.02 0.73


w1 = 13.96%

w2 = 26.96% R* = 8.00%

w3 = 29.68%

w4 = 26.19%

w5 = 3.21%

______

Sum = 100%


Rp = 8.00% 8.93%


0.00161604 0.0014520642 -0.000879094 -0.000987312 -0.001711716

0.0014520642 0.00776161 -0.000275224 0.0028399035 -7.5026E-05

-0.000879094 -0.000275224 0.02439844 0.015584855 0.0229459362

-0.000987312 0.0028399035 0.015584855 0.02356225 0.0238565095

-0.001711716 -7.5026E-05 0.0229459362 0.0238565095 0.04532641

s1 = r12 = r24 =

s2 = r13 = r25 =

s3 = r14 = r34 =

s4 = r15 = r35 =

s5 = r23 = r45 =

sp =



When run with the Solver add-in on MS Excel for Windows, this spreadsheet is capable of calculating the mean-variance efficient weights for a five asset class portfolio. The user must provide the following inputs: (i) returns, standard deviations, and correlations for the five asset classes; and (ii) a return goal to be acheived by the portfolio.


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Port Risk