203
Chapter 6. Covariance, Correlation
Problem PP169
To illustrate graphically the meaning of correlation, generate 10 values of a random variable X
uniformly distributed between 1 and 12 in cells A2:A11: = 1 + 𝑅𝐴𝑁𝐷( ) ∗ 11.
Generate 10 values of a random variable Y in cells B2:B11. The conditional pdf of Y given 𝑋 = 𝑥
1
is 𝑓(𝑦|𝑥) = 12−𝑥 , 1 < 𝑦 < 13 − 𝑥. Example for cell B2: = 1 + (12 − 𝐴2) ∗ 𝑅𝐴𝑁𝐷( ) and
similarly for cells B3:B11.
To prepare for a scatter plot of the values (𝑋 𝑌), input the values 0 and 13 in cells D2:D3, the
values 0 and 12 in cells D5:D6, compute the mean of the ten X values in both cells E2 and E3, the
mean of the 10 Y values in both cells E5 and E6.
1. Use Insert\Scatter\Scatter with only markers to graph the X (on the horizontal axis) and Y
values (on the vertical axis). Select the horizontal axis from 0 to 12, the vertical axis from
0 to 13. Add a vertical line at the mean of the X values (use cells D2:E3, connect the
points by a linear trend line), a horizontal line at the mean of the Y values (use cells
D5:E6). Consider both trend lines as new coordinate axes. Compute the correlation
coefficient between the X and the Y values in cell E8: = 𝐶𝑂𝑅𝑅𝐸𝐿(𝐴2: 𝐴11; 𝐵2: 𝐵11).
Click on one of the data points in the graph and add a linear trend line.
Check the sign of the correlation and the slope of the trend line.
Repeat generating the values of X and Y (key F9), check the correlation coefficient and
the position of the points with respect to the new axes. Also compute the sample
covariance in cell D9: = 𝐶𝑂𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸. 𝑆(𝐴2: 𝐴11; 𝐵2: 𝐵11);
2. On a new worksheet generate 10 values of X as in Step 1. Generate 10 values of Y in cells
B2:B11 as follows (example for cell B2): = 16 − 2 ∗ 𝐴2 and likewise for cells B3:B11.
Use a scatter plot to graph the 10 values of X and Y.
Compute the correlation coefficient between the X and the Y values. Notice the perfect
(negative) correlation for points on a straight line (with negative slope);
3. On a new worksheet we generate 50 points on the circumference of a circle with radius 1
as follows: 50 values of X, randomly distributed between -1 and +1, in the range A2:A51:
= −1 + 2 ∗ 𝑅𝐴𝑁𝐷( ), 50 values of Y in cells B2:B51. Example for cell B2:
= 𝐼𝐹(𝑅𝐴𝑁𝐷( ) < .5; 𝑆𝑄𝑅𝑇(1 − 𝐴2 ∗ 𝐴2); −𝑆𝑄𝑅𝑇(1 − 𝐴2 ∗ 𝐴2)) and similarly for cells
B3:B51.
Use a scatter plot to graph the 50 values of (𝑋 𝑌).
Compute the correlation coefficient between the X and the Y values. Notice the poor
correlation although the points are ‘perfectly’ related since 𝑋 2 + 𝑌 2 = 1;
4. On a new worksheet generate the points as in Step 1. Add a point in row 12: the value 12
in cell A12, the value 13 in cell B12.
Compute again the correlation between X and Y. The added point can be considered an
outlier in the data. Notice the strong effect an outlier can have on the correlation.
When repeatedly generating new data the correlation coefficient will be just mildly
204
negative or even positive.
Assignment PA169
Repeat Step 1 but generate the values of Y as (for cell B2): = 1 + 𝐴2 ∗ 𝑅𝐴𝑁𝐷( ).
Repeat Step 2 but generate the values of Y as (for cell B2): = 6 + 2 ∗ 𝐴2.
Repeat Step 3 but generate the values of Y (for cell B2): = 𝑃𝑂𝑊𝐸𝑅(𝐴2; 4).
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Problem PP170
Consider the random variables (X, Y) in Problem PP135.
1. Compute the expected values, the variances and standard deviations of X and Y.
Answer: 𝐸(𝑋) = 0.8, 𝑣𝑎𝑟(𝑋) = 0.76, 𝐸(𝑌) = 0.4, 𝑣𝑎𝑟(𝑌) = 0.24;
2. Compute the expected value of X*Y and the covariance and correlation between X and Y.
Answer: 𝐸(𝑋 ∗ 𝑌) = 0.5, 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 0.18, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = 0.4215;
3. Derive the conditional pmf of Y given X for all possible values of X.
Answer:
𝑝𝑦|𝑥 (0|0) = 0.8, 𝑝𝑦|𝑥 (1|0) = 0.2,
𝑝𝑦|𝑥 (0|1) = 0.5, 𝑝𝑦|𝑥 (1|1) = 0.5,
𝑝𝑦|𝑥 (0|2) = 1⁄3, 𝑝𝑦|𝑥 (1|2) = 2⁄3;
4. Generate 1000 values of X using the marginal pmf of X as follows: input random numbers
in the range A2:A1001 and generate random values of X in cells B2:B1001. Example for
cell B2:
= 𝐼𝐹(𝐴2 < .5,0; 𝐼𝐹(𝐴2 < .7; 1; 2)) and similarly for cells B3:B1001.
Use the conditional pmf of Y given X to generate 1000 values of Y in cells C2:C1001.
Example for cell C2: = 𝐼𝐹 (𝐵2 = 0; 𝐼𝐹(𝑅𝐴𝑁𝐷( ) < .8; 0; 1); 𝐼𝐹(𝐵2 = 1; 𝐼𝐹(𝑅𝐴𝑁𝐷( ) <
.5; 0; 1); 𝐼𝐹(𝑅𝐴𝑁𝐷( ) < 1⁄3; 0; ,1))) and similarly for cells C3:C1001.
In cell D11, compute the (sample) covariance of the first ten observations:
= 𝐶𝑂𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸. 𝑆(𝐵$2: 𝐵11; 𝐶$2: 𝐶11). Similarly for cells D12:D1001 for the first 11,
12, … observations.
In cell E11 compute the correlation coefficient of the first 10 observations:
= 𝐶𝑂𝑅𝑅𝐸𝐿(𝐵$2: 𝐵11; 𝐶$2: 𝐶11) and similarly for cells E12:E1001 for the first 11, 12,
… observations.
Report the values 10, 11, 12, …, 1000 in cells F11:F1001. Use columns F and D to graph
the evolution of the sample covariance as a function of the sample size (use the option
Scatter with Straight Lines and Markers). Add a straight line for the value of 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌)
as computed in Step 2.
In a similar way use columns F and E to graph the evolution of the sample correlation.
Assignment PA170
Consider the Assignment in Problem PP135. Compute the expected values, the variances and the
standard deviations of X and Y.
Compute the covariance and the correlation between X and Y.
Generate 1000 values of X and Y.
As in Step 4 compute the sample covariance and correlation of the first 10, 11, 12, …,1000
observations. Graph the evolution of the sample covariance and correlation as in Step 4.
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Problem PP171
The sides of two coins are numbered 1 and 2. Both coins are tossed. Let X be the sum of the
numbers on both coins and Y the maximum.
1. Derive the joint pmf of X and Y.
Answer:
Y: Maximum
Marginal X
1
2
2
.25
.25
X: Sum
3
.5
.5
Marginal Y
4
.25
.25
.25
.75
2. Derive the marginal pmf of X and Y.
Answer: see Table in Step 1;
3. Compute the expected values, the variances and the standard deviations of X and Y.
Answer: 𝐸(𝑋) = 3, 𝜎𝑋2 = 0.5, 𝜎𝑋 = √. 5 , 𝐸(𝑌) = 1.75, 𝜎𝑌2 = .1875, 𝜎𝑌 = √. 1875;
4. Compute the covariance and the correlation between X and Y.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 1⁄4, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = √2⁄3 = 0.8165;
5. Compute 𝑉𝐴𝑅(𝑋 − 𝑌) from the results in Step 3 and 4.
Answer: 𝑉𝐴𝑅(𝑋 − 𝑌) = 3⁄16;
6. Generate 1000 values of the two coins in cells A2:B1001, the values of the sum X in cells
C2:C1001, the values of the maximum Y in cells D2:D1001.
Derive the sample estimates of the results in Step 1 to Step 5 and compare with the exact
results.
Assignment PA171
Two sides of a die are colored blue, two sides are colored green and two sides are colored red.
When rolling the die, define the random variables B (𝐵 = 1 when outcome is blue, 0 otherwise)
and R (𝑅 = 1 when the outcome is red, 0 otherwise).
Compute the covariance and the correlation between B and R. Compute the variance of the sum
𝐵 + 𝑅.
Simulate the problem over 1000 lines and compute the sample covariance and sample correlation
between B and R. Compute the sample variance of 𝐵 + 𝑅.
Compare the sample values with the exact results.
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Problem PP172
Consider Problem PP136.
1. Compute the covariance and the correlation between X and Y. Does the sign of the
correlation make intuitive sense? Why?
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 0.9452, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = 0.4795;
2. Compute 𝐸(1⁄𝑌), 𝑣𝑎𝑟(1⁄𝑌) 𝑎𝑛𝑑 𝑠𝑡𝑑𝑒𝑣(1⁄𝑌).
Answer: 𝐸(1⁄𝑌) = 0.5514, 𝑣𝑎𝑟(1⁄𝑌) = .0984, 𝑠𝑡𝑑𝑒𝑣(1⁄𝑌) = .3137;
3. Verify that 𝐸(1⁄𝑌) = .5514 ≠ 1⁄𝐸(𝑌) = .3956;
4. Compute 𝐸(𝑋⁄𝑌), 𝑣𝑎𝑟(𝑋⁄𝑌). Verify that 𝐸(𝑋⁄𝑌) = 2.275 ≠ 𝐸(𝑋)⁄𝐸(𝑌) = 1.7692
and that 𝐸(𝑋⁄𝑌) = 2.275 ≠ 𝐸(𝑋) ∗ 𝐸(1⁄𝑌) = 2.4659;
5. Compute 𝑐𝑜𝑣𝑎𝑟(𝑋, 1⁄𝑌 ) and 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 1⁄𝑌 ) using the results in Steps 2 and 4.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 1⁄𝑌) = −.1909, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 1⁄𝑌) = −.4335;
6. Compute 𝐸(𝑌⁄𝑋) and 𝑣𝑎𝑟(𝑌⁄𝑋).
Answer: 𝐸(𝑌⁄𝑋) = .5833, 𝑣𝑎𝑟(𝑌⁄𝑋) = .0713;
7. Compute 𝑐𝑜𝑣𝑎𝑟(𝑋⁄𝑌 , 𝑌⁄𝑋) en 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋⁄𝑌 , 𝑌⁄𝑋) using the results in Steps 4 and 6.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋⁄𝑌 , 𝑌⁄𝑋) = −.3271, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋⁄𝑌 , 𝑌⁄𝑋) = −.8766;
8. Generate values of X and Y as in Step 3 of Problem PP136. Compute the values of
𝑋⁄𝑌, 𝑌⁄𝑋 𝑎𝑛𝑑 1⁄𝑌 in columns E, F and G.
Compute all sample analogues of the results in Steps 1 to 7.
Compare the sample values with the exact theoretical results.
Assignment PA172
Consider Assignment PA136.
Compute 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌), 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌), 𝑐𝑜𝑣𝑎𝑟(𝑋⁄𝑌 , 𝑌⁄𝑋), 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋⁄𝑌 , 𝑌⁄𝑋).
Generate 1000 values of X and Y. Compute 𝑋⁄𝑌 and 𝑌⁄𝑋 in columns C and D.
Compute estimates of the four parameters from the sample data.
Repeat the simulation a number of times and compare the sample values with the exact
theoretical values.
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Problem PP173
Consider the random variables in Problem PP135 and Problem PP170.
1. Compute the variance of the sum 𝑋 + 𝑌 from the pmf as derived in Step 5 of Problem
PP135.
Answer: 𝑣𝑎𝑟(𝑋 + 𝑌) = 1.36;
2. Compute the variances of X and Y and the covariance between X and Y.
Answer: 𝑣𝑎𝑟(𝑋) = 0.76, 𝑣𝑎𝑟(𝑌) = 0.24, 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 0.18;
3. Compute the variance of the sum 𝑋 + 𝑌 from the results in Step 2;
4. Generate 1000 values of X and Y as in Step 4 of Problem PP170.
Compute the sum of the values of X and Y in cells D2:D101.
Compute the sample variances of X and Y and the sample covariance between the data in
columns B and C:
= 𝑉𝐴𝑅. 𝑆(𝐵2: 𝐵1001), = 𝑉𝐴𝑅. 𝑆(𝐶2: 𝐶1001), = 𝐶𝑂𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸. 𝑆(𝐵2: 𝐵1001; 𝐶2: 𝐶1001).
Compute the sample variance of the sum 𝑋 + 𝑌 in two ways: directly from the data in
column D: = 𝑉𝐴𝑅. 𝑆(𝐷2: 𝐷1001) and indirectly from the sample variances and the
sample covariance:
= 𝑉𝐴𝑅. 𝑆(𝐵2: 𝐵1001) + 𝑉𝐴𝑅. 𝑆(𝐶2: 𝐶1001) + 2 ∗ 𝐶𝑂𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸. 𝑆(𝐵2: 𝐵1001; 𝐶2: 𝐶1001).
Compare both results.
Compare the sample variance of the sum with the exact value computed in Step 3.
Assignment PA173
Consider the random variables X and Y in Assignments PA135 and PA170.
In PA170 you have computed 𝑣𝑎𝑟(𝑋) = 0.5, 𝑣𝑎𝑟(𝑌) = 1, 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 0.5.
Derive 𝑣𝑎𝑟(𝑌 − 𝑋) from these results. What is the meaning of the random variable 𝑌 − 𝑋 in this
problem? Derive the variance of 𝑌 − 𝑋 from this new interpretation.
Simulate 1000 values of X and Y, compute the differences 𝑌 − 𝑋 and the sample variance of the
differences.
Compute the sample variances of X and Y and the sample covariance between X and Y.
Compute again the sample variance of the differences 𝑌 − 𝑋 from these results.
Compare the results of both approaches.
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Problem PP174
Consider Problem PP136 and Problem PP172.
1. Compute 𝑣𝑎𝑟(𝑋 − 𝑌) using the pmf of 𝑋 − 𝑌 derived in Step 8 of Problem PP136.
Answer: 2.0525;
2. From the marginal pmf of X and Y derived in Step 2 of Problem PP136, compute 𝑣𝑎𝑟(𝑋)
and 𝑣𝑎𝑟(𝑌).
Answer: 𝑣𝑎𝑟(𝑋) = 𝑣𝑎𝑟(𝑌) = 1.9715;
3. Compute 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) using the joint pmf of X and Y derived in Step 1 of Problem PP136.
Answer: 0.9452;
4. Derive 𝑣𝑎𝑟(𝑋 − 𝑌) from the results in Step 2 and 3.
Answer: see Step 1;
5. Start from the simulated values in Step 3 of Problem 136 or Step 2 in Problem PP172.
Compute the sample variance of the values 𝑋 − 𝑌 generated in column E;
6. Compute the sample variances and the sample covariance of the values of X and Y
generated in columns D and C;
7. Derive the sample variance of the values of 𝑋 − 𝑌 from the results in Step 6;
8. Compute 𝑣𝑎𝑟(𝑋 + 𝑌) using the pmf of 𝑋 + 𝑌 derived in Step 8 of Problem PP136.
Answer: 5.8333;
9. From the results in Step 2 and 3 again compute the variance of 𝑋 + 𝑌.
Answer: see Step 8;
10. Call the result of rolling a die T. What is the value of 2 ∗ 𝑣𝑎𝑟(𝑇)? Compare with the
results in Step 8 and 9. Explain;
11. Start from the simulated values in Step 3 of Problem PP136 or Step 2 in Problem PP172.
Compute the sample variance of the values 𝑋 + 𝑌 generated in column F;
12. Derive the sample variance of the values of 𝑋 + 𝑌 generated in column F from the results
in Step 6;
13. Let 𝑇1 and 𝑇2 be the outcome on the first and the second die in columns A and B.
Compute the sample variances and the sample covariance of the data generated in
columns A and B. From these values derive the sample variance of the values 𝑇1 + 𝑇2 and
compare with the values in Steps 11 and 12. Explain.
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Assignment PA174
Consider the random variables X and Y defined in PA136 and PA172.
Derive the pmf of 𝑋 − 𝑌 from the joint pmf of X and Y.
Derive 𝑣𝑎𝑟(𝑋 − 𝑌) from the pmf of 𝑋 − 𝑌.
Compute the expected values and variances of X and Y.
Compute the covariance between X and Y.
Derive 𝑣𝑎𝑟(𝑋 − 𝑌) using 𝑣𝑎𝑟(𝑋), 𝑣𝑎𝑟(𝑌) and 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌). Compare both results.
Generate 1000 values of X, Y and 𝑋 − 𝑌.
Compute the sample variance of 𝑋 − 𝑌 in two ways: firstly, from the 1000 differences and
secondly from the sample variance of X, the sample variance of Y and the sample covariance
between X and Y.
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Problem PP175
Random variables X and Y have a variance equal to 1 and 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = 𝜌. For any constant c
let 𝑉 = 𝑌 + 𝑐𝑋 and 𝑊 = 𝑋 + 𝑐𝑌.
1. Derive 𝑐𝑜𝑣𝑎𝑟(𝑉, 𝑊) and 𝑐𝑜𝑟𝑟𝑒𝑙(𝑉, 𝑊).
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑉, 𝑊) = 𝜌 + 𝜌𝑐 2 + 2𝑐, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑉, 𝑊) =
𝜌+𝜌𝑐 2 +2𝑐
1+𝑐 2 +2𝜌𝑐
;
2. Check the correlation when 𝑐 = 1. Explain;
3. Assume X and Y are bivariate normal with
𝐸(𝑋) = 10, 𝐸(𝑌) = 20, 𝑣𝑎𝑟(𝑋) = 𝑣𝑎𝑟(𝑌) = 1, 𝜌(𝑋, 𝑌) = 𝜌.
Generate 1000 values of X in cells A2:A1001: = 𝑁𝑂𝑅𝑀. 𝐼𝑁𝑉(𝑅𝐴𝑁𝐷( ); 10; 1), generate
1000 values of Y in cells B2:B1001 for 𝜌 = 0.5 . Example for cell B2:
= 𝑁𝑂𝑅𝑀. 𝐼𝑁𝑉 (𝑅𝐴𝑁𝐷( ); 20 ∗ .5 ∗ (𝐴2 − 10); 𝑆𝑄𝑅𝑇(1 − 𝑃𝑂𝑊𝐸𝑅(. 5; 2))).
In columns C and D report values of Y for 𝜌 = 0 and 𝜌 = −0.5.
Let 𝑐 = 4. In columns E and F report values of V and W for 𝜌 = 0.5, in columns G and H
report values of V and W for 𝜌 = 0, in columns I and J report values of V and W for 𝜌 =
−0.5.
Based on the results in Step 1 compute the covariance and the correlation between V and
W for the three different correlations between X and Y.
Compute the sample covariances and sample correlations and compare with the exact
theoretical values.
Assignment PA175
Two analysts measure a same variable Y. The measurements involve errors. Let 𝑋𝑘 be the value
measured by analyst 𝑘, 𝑘 = 1, 2 and assume 𝑋𝑘 = 𝑌 + 𝜀𝑘 , 𝑘 = 1,2. Assume that 𝜀1 and 𝜀2 are
independent and that both are also independent of Y. Let 𝐸(𝜀1 ) = 𝐸(𝜀2 ) = 0 (both analysts
expect to measure Y correctly), 𝑣𝑎𝑟(𝜀1 ) = 𝑣𝑎𝑟(𝜀2 ) = 𝜏 2 and 𝑣𝑎𝑟(𝑌) = 𝜎 2 .
Compute the covariance and the correlation between both measurements as a function of 𝜎 2 and
𝜏 2 . When is the correlation minimal, when maximal?
Generate 1000 values of Y in cells A2:A1001 (assume Y to be normally distributed with expected
value 100 and standard deviation 2), generate 1000 values of 𝜀1 (𝜀2 ) in cells B2:B1001
(C2:C1001) (assume both to be standard normally distributed).
Compute the values of 𝑋1 (𝑋2) in column D (E). Compute the sample covariance and the sample
correlation between 𝑋1 and 𝑋2 and compare with the exact values computed in Step 1.
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Problem PP176
Random variable Z is standard normally distributed. Let 𝑋 = 2 ∗ 𝑍 and 𝑌 = 𝑍 2 .
1. Compute the correlation between Z and X.
Answer: 𝜌(𝑍, 𝑋) = 1. Explain;
2. Compute the correlation between Z and Y. Are Z and Y independent? Explain.
Answer: 𝜌(𝑍, 𝑋) = 0. Z and Y are not independent;
3. Generate 1000 values of Z in A2:A1001: = 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑅𝐴𝑁𝐷( )).
Generate the corresponding values of X and Y in columns B and C.
Compute the sample correlation between Z and X and between Z and Y.
Compare the sample correlations with the exact correlations.
Assignment PA176
Random variable Z is uniformly distributed on the interval (-1 1).
Let 𝑋 = 2 ∗ 𝑍 and 𝑌 = 𝑍 2 .
Compute the covariance and the correlation between Z and X. Compute the covariance and the
correlation between Z and Y.
Generate 1000 values of Z, X and Y.
Compute the sample covariances and correlations between Z and X and between Z and Y.
Compare the sample estimates with the exact theoretical values.
213
Problem PP177
For random variables X and Y it holds that 𝐸(𝑋) = 𝐸(𝑌) = 𝜇, 𝑣𝑎𝑟(𝑋) = 𝜎𝑋2 , 𝑣𝑎𝑟(𝑌) = 𝜎𝑌2 and
𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 𝜎𝑋𝑌 . Assume that the expected value 𝜇 is not known but that the variances and
the covariance are known. We estimate the expected value by 𝑊 = 𝑎𝑋 + (1 − 𝑎)𝑌 where a is a
constant to be determined.
1. Show that 𝐸(𝑊) = 𝜇 for any value of a (the expected value of the estimate is “correct”);
2. Determine the value of a for which 𝑣𝑎𝑟(𝑊) is minimal.
Answer: 𝑎 = (𝜎𝑌2 − 𝜎𝑋𝑌 )⁄(𝜎𝑋2 + 𝜎𝑌2 − 2 ∗ 𝜎𝑋𝑌 );
3. Assume the joint density of (𝑋, 𝑌) is bivariate normal with parameters 𝜇 = 10, 𝜎𝑋2 = 4,
𝜎𝑌2 = 1, 𝜎𝑋𝑌 = −1. It follows that the correlation 𝜌(𝑋, 𝑌) = −0.5.
Report the value of the correlation in cell A1, the value of the covariance in cell B1.
Generate 1000 values of X in cells A2:A1001: = 𝑁𝑂𝑅𝑀. 𝐼𝑁𝑉(𝑅𝐴𝑁𝐷( ); 10; 2).
Generate 1000 values of Y in cells B2:B1001. Example for cell B2: =
𝑁𝑂𝑅𝑀. 𝐼𝑁𝑉 (𝑅𝐴𝑁𝐷( ); 10 + $𝐴$1 ∗
(𝐴2−10)
2
; 𝑆𝑄𝑅𝑇(1 − $𝐴$1 ∗ $𝐴$1)).
Input different values of a in cells C1:K1: -2, -1.5, -1, -.5, 0, .5, 1, 1.5, 2. In cell L1 input
the value of a derived in Step 2: = (1 − $𝐵$1)⁄(4 + 1 − 2 ∗ $𝐵$1).
Compute the values of W for the different values of a in columns D to M. Example for
cell D2: = 𝐷$1 ∗ $𝐴2 + (1 − 𝐷$1) ∗ $𝐵2 and likewise for cells D3:D1001 and columns
E to M.
Repeat the values of a in cells O1:X1. Compute the averages of the values of W in cells
O2:X2 for the different values of a. Compare the averages with the result in Step 1.
Compute the sample variances of the values of W in cells O3:X3 for the different values
of a. For which value of a is the sample variance minimal?
Compare this result to the conclusion drawn in Step 2;
4. Change the value of the correlation (and the covariance) in cell A1 (B1) to different
values and always check the results in O2:X2 and O3:X3.
Assignment PA177
Random variables S and V are derived from random variables X and Y: 𝑆 = 𝑋 + 𝑌, 𝑉 = 𝑋 − 𝑌.
Show that 𝑐𝑜𝑣𝑎𝑟(𝑆, 𝑉) = 𝑣𝑎𝑟(𝑋) − 𝑣𝑎𝑟(𝑌).
Generate 1000 values of X, Y, S and V. Assume that (𝑋, 𝑌) are bivariate normal with parameters
𝐸(𝑋) = 10, 𝑣𝑎𝑟(𝑋) = 1, 𝐸(𝑌) = 20, 𝑣𝑎𝑟(𝑌) = 4, 𝜌(𝑋, 𝑌) = 0.5 and compute the sample
covariance between S and V. Compare with 𝑐𝑜𝑣𝑎𝑟(𝑆, 𝑉).
214
Problem PP178
Random variable X is uniformly distributed over the interval (0 1).
1. Compute 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑋 𝑘 ) for any constant 𝑘 > 0.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑋 𝑘 ) = 𝑘⁄(2 ∗ (𝑘 + 1) ∗ (𝑘 + 2));
2. Derive the value of k for which the covariance is maximal.
Answer: 𝑘 = √2;
3. Graph the covariance as a function of k for k in the interval (0 12);
4. Compute 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑋 𝑘 ) for any 𝑘 > 0.
Answer: 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑋 𝑘 ) = √3 ∗ (2𝑘 + 1)⁄(𝑘 + 2);
5. Derive the value of k for which the correlation is maximal.
Answer: 𝑘 = 1;
6. Graph the correlation as a function of k for k in the interval (0 12);
7. Generate values of X in the range A2:A1001. Compute the values of 𝑋 𝑘 for
𝑘 = 0.5, 2, 3, 4, 5 in columns B, C, D, E, F.
Compute the sample covariances and correlations between X and 𝑋 𝑘 and compare with
the exact values derived in Steps 1 and 4.
Assignment PA178
A random variable X is uniformly distributed over the interval (1 2). A square has a side of
length X.
Compute the covariance and the correlation between the circumference and the area of the
square.
Generate 1000 values of X. Compute the circumference and the area of the 1000 squares.
Compute the sample covariance and the sample correlation and compare to the exact values.
215
Problem PP179
For any constant a, −1 < 𝑎 < 1, and 𝑍1 and 𝑍2 independent standard normal random variables,
1+𝑎
consider random variables X and Y: 𝑋 = √
2
1−𝑎
∗ 𝑍1 + √
2
1+𝑎
∗ 𝑍2 , 𝑌 = √
2
1−𝑎
∗ 𝑍1 − √
2
∗ 𝑍2 .
1. Compute the covariance and the correlation between X and Y.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 𝑎, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = 𝑎;
2. Derive the joint pdf of X and Y.
Answer: bivariate normal with 𝐸(𝑋) = 𝐸(𝑌) = 0, 𝑣𝑎𝑟(𝑋) = 𝑣𝑎𝑟(𝑌) = 1, 𝜌(𝑋, 𝑌) = 𝑎;
3. Report a value of a in cell A1, e.g. 𝑎 = 0.8.
Generate 1000 values of 𝑍1 and 𝑍2 in cells A3:A1002 and B3:B1002. Example for cell
A3: = 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑅𝐴𝑁𝐷( )) and likewise for cells A4:A1002 and column B.
Compute the values of X and Y in columns C and D for the value of a in cell A1. Example
for cell C3: = 𝑆𝑄𝑅𝑇((1 + $𝐴$1)⁄2) ∗ 𝐴3 + 𝑆𝑄𝑅𝑇((1 − $𝐴$1)⁄2) ∗ 𝐵3 and likewise
for cells C4:C1002 and column D.
Compute the sample covariance between X and Y and compare with the value in cell A1.
Change the value in cell A1 to .2, to 0, to -.5.
Assignment PA179
Let 𝑍1 and 𝑍2 be independent standard normal random variables.
For constants 𝑚1 , 𝑚2 , 𝑐1 , 𝑐2 𝑎𝑛𝑑 0 < 𝜌 < 1, let
𝑋 = 𝑚1 + 𝑐1 ∗ 𝑍1 and 𝑌 = 𝑚2 + 𝑐2 ∗ (𝜌 ∗ 𝑍1 + √1 − 𝜌2 ∗ 𝑍2 ).
Compute the covariance and the correlation between X and Y.
Derive the joint pdf of X and Y.
Generate 1000 values of 𝑍1 , 𝑍2 , X and Y for 𝑚1 = 2, 𝑚2 = 4, 𝑐1 = −3, 𝑐2 = 2, 𝜌 = 0.5.
Compute the sample covariance and the sample correlation between X and Y.
Compare to the exact theoretical values.
216
Problem PP180
Consider the random variables X and Y in Problem PP167.
1. Compute the covariance between X and Y.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑋, 𝑌) = 1⁄𝜆2;
2. Compute the correlation between X and Y.
Answer: 𝑐𝑜𝑟𝑟𝑒𝑙(𝑋, 𝑌) = √2⁄2;
3. Compute 𝐸(𝜆𝑋 − 𝑌) and 𝑣𝑎𝑟(𝜆𝑋 − 𝑌).
Answer: 𝐸(𝜆𝑋 − 𝑌) = (𝜆 − 2)⁄𝜆, 𝑣𝑎𝑟(𝜆𝑋 − 𝑌) = 1 + 2⁄𝜆2 − 2⁄𝜆 ;
4. To simulate the above results, report a value of λ in cell A1, say 𝜆 = 2.
Generate 1000 values of X (Y) in cells A3:A1002 (B3:B1002).
Example for cell A3: = 𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉(𝑅𝐴𝑁𝐷( ); 1; 1⁄$𝐴$1).
Example for cell B3:= 𝐴3 − (1⁄$𝐴$1) ∗ 𝑙𝑛(𝑅𝐴𝑁𝐷( )).
Compute the values of 𝜆𝑋 − 𝑌 in cells C3:C1002.
Compute the sample covariance and sample correlation between X and Y. Compare with
the values in Step 1 and 2.
Compute the sample mean and the sample variance of 𝜆𝑋 − 𝑌 in column C and compare
with the values derived in Step 3.
Assignment PA180
Consider the random variables in Problem PA167.
Compute the covariance and the correlation between X and Y.
Generate 1000 values of Y and X, compute the sample covariance and sample correlation and
compare with the exact theoretical values.
217
Problem PP181
A machine produces parts which can be one of four different kinds: the part satisfies all
requirements (probability is 0.8), it can be reworked and then satisfies the requirements
(probability is 0.05), it has insufficient length and is total loss (probability is 0.1) or it has
insufficient width and is total loss (probability is 0.05).
The selling price of the part is €18, the unit production cost is €10, the cost of reworking a part is
€5.
A lot of 1000 parts will be produced today. Let S be the number of parts that satisfy all
requirements, R the number of parts to be reworked, L the number of parts of insufficient length
and W the number of parts of insufficient width.
1. Compute the expected value and the variance of S and R.
Answer: 𝐸(𝑆) = 800, 𝐸(𝑅) = 50, 𝑣𝑎𝑟(𝑆) = 160, 𝑣𝑎𝑟(𝑅) = 47.5;
2. Compute the covariance and correlation between S and R.
Answer: 𝑐𝑜𝑣𝑎𝑟(𝑆, 𝑅) = −40, 𝑐𝑜𝑟𝑟𝑒𝑙(𝑆, 𝑅) = −.4588;
3. Express the total profit of the lot P as a function of S and R.
Answer: 𝑃 = 18 ∗ 𝑆 + 13 ∗ 𝑅 − 10000;
4. Compute the expected value of P.
Answer: 𝐸(𝑃) = 5050;
5. Compute the standard deviation of P.
Answer: 𝜎𝑃 = 202.8485;
6. To simulate the problem, generate 1000 random numbers in the range A2:A1001.
Generate the quality of 1000 parts in cells B2:B1001. Example for cell B2:
= 𝐼𝐹 (𝐴2 < .8; "𝑆";𝐼𝐹(𝐴2 < .85; "𝑅";𝐼𝐹(𝐴2 < .95; "𝐿";"𝑊"))) and similarly for cells
B3:B1001.
Compute the profit made on the parts in cells C2:C1001. Example for cell C2:
= 𝐼𝐹(𝐵2 = "𝑆";8; 𝐼𝐹(𝐵2 = "𝑅";3; −10)) and similarly for cells C3:C1001.
Compute the total number of S-parts in cell E2, the total number of R-parts in cell F2, the
total number of L-parts in cell G2, the total number of W-parts in cell H2 and the total
profit of the 1000 parts in cell I2.
Use the TABLE-option of Excel to generate these values 1000 times.
Compute the sample averages and sample variances of S and R and compare with the
values computed in Step 1.
Compute the sample covariance and correlation between S and R and compare with the
values in Step 2.
Compute the sample mean of profits and compare with the expected value computed in
Step 4.
Compute the sample standard deviation of profit and compare with the standard deviation
computed in Step 5.
218
Assignment PA181
A die is rolled 60 times. Let X be the number of times an even number is rolled and Y the number
of times the outcome 1 is obtained.
Compute the expected values and variances of X and Y, compute the covariance and the
correlation between X and Y.
Compute the expected value and the variance of 𝑋 + 𝑌 using the previous results.
Compute again the expected value and the variance of 𝑋 + 𝑌 using results from a binomial
process.
Generate 60 rolls of a die.
Compute the value of X, Y and 𝑋 + 𝑌 and use the TABLE-option to repeat this process 1000
times.
Compute sample estimates of the expected values, variances, covariance and correlation
computed above and compare the sample estimates to the exact values.
219
Problem PP182
Two risky assets have random returns 𝑅1 and 𝑅2 over the next period.
Let 𝜇1 = 𝐸(𝑅1 ), 𝜇2 = 𝐸(𝑅2 ) be the expected returns, 𝜎1 = 𝑠𝑡𝑑𝑒𝑣(𝑅1 ), 𝜎2 = 𝑠𝑡𝑑𝑒𝑣(𝑅2 ) be the
standard deviations of the returns and 𝜌12 = 𝑐𝑜𝑟𝑟𝑒𝑙(𝑅1 , 𝑅2 ), the correlation between both
returns.
A portfolio P is built by investing a capital of 1 for a proportion 𝑎1 in asset 1 and a proportion
𝑎2 = 1 − 𝑎1 in asset 2 (no short selling, so 0 ≤ 𝑎1 ≤ 1).
1. Derive the expected return and the variance of the portfolio.
Answer:
𝜇(𝑃) = 𝑎1 ∗ 𝜇1 + (1 − 𝑎1 ) ∗ 𝜇2 ,
𝜎 2 (𝑃) = 𝑎12 ∗ 𝜎12 + (1 − 𝑎1 )2 ∗ 𝜎22 + 2 ∗ 𝑎1 ∗ (1 − 𝑎1 ) ∗ 𝜌12 ∗ 𝜎1 ∗ 𝜎2 ;
2. Derive the proportions 𝑎1𝑚 and 𝑎2𝑚 to be invested in both assets when minimizing the risk
(risk being measured by the variance of the portfolio).
𝜎2 −𝜌
∗𝜎1 ∗𝜎2
12
Answer: 𝑎1𝑚 = 𝜎2 +𝜎2 2 −2∗𝜌
1
2
12 ∗𝜎1 ∗𝜎2
, possibly corrected to 0 when the ratio is negative and to
1 when the ratio is larger than 1 (this correction is not required when short selling is
allowed), 𝑎2𝑚 = 1 − 𝑎1𝑚 ;
3. Assume 𝜇1 = 0.14, 𝜇2 = 0.08, 𝜎1 = 0.2, 𝜎2 = 0.15, 𝜌12 = 0. Report these values in
cells A3:E3. For all values of 𝑎1 between 0 and 1 in steps of 0.05 (cells A5:A25),
compute the expected return of the portfolio in cells B5:B25, the standard deviation of the
portfolio in cells C5:C25 using the results in Step 1 (use the values in cells A3:E3).
Use a scatter plot to graph the expected value (on the vertical axis) versus the standard
deviation (on the horizontal axis);
4. For the parameter values in Step 3, compute the portfolio 𝑃𝑚 with minimal variance (also
minimal standard deviation) using the result in Step 2.
Answer: 𝑎1𝑚 = 0.36, 𝑎2𝑚 = 0.64, 𝐸(𝑃𝑚 ) = 0.1016, 𝜎 2 (𝑃𝑚 ) = 0.0144, 𝜎(𝑃𝑚 ) = 0.12.
5. Assume a risk averse investor (preferring less risk for the same expected return and more
expected return for the same risk), indicate in the graph in Step 3 the portfolios that are
not dominated by other possible portfolios (the so called efficient frontier);
6. Use the Solver to compute the minimal variance portfolio as already derived in Step 4;
7. Compute the portfolio with minimal risk when requiring an expected profit of at least
0.11.
Answer: 𝑎1 = .5, 𝑎2 = .5, 𝐸(𝑃) = 0.11, 𝜎 2 (𝑃) = 0.125, 𝜎(𝑃) = 0.3535;
8. Compute the portfolio with minimal risk when requiring an expected profit of at least
0.10.
Answer: 𝑎1 = .36, 𝑎2 = .64, 𝐸(𝑃) = 0.1016, 𝜎 2 (𝑃) = 0.0144, 𝜎(𝑃) = 0.12;
220
9. Use the Solver to derive the portfolios computed in Steps 7 and 8;
10. Change the value of the correlation in cell E3 to .25, .5, 1, -.25, -.5, -1 and check how the
graph derived in Step 3 changes;
11. For values of the correlation between -1 and +1, steps of 0.1 (cells A32:A52), use Excel
to compute the value of 𝑎1 and the standard deviation of the minimal variance portfolio.
Graph the standard deviation as a function of the correlation. Justify intuitively the shape
of this graph;
12. Assume both returns are normally distributed. Compute the portfolio of maximal expected
return but for which the probability of a loss of at least 0.08 is limited to 0.1. Use both
hand computation and the Solver.
Answer (rounded): 𝑎1 = .796, 𝐸(𝑃) = .1278, 𝑠𝑡𝑑𝑒𝑣(𝑃) = 0.1621.
Assignment PA182
Perform Steps 1, 2, 3, 5, 7 and 8 for the case of one risky asset (𝜇1 = .15, 𝜎1 = .25) and one riskfree asset (𝜇2 = .06, 𝜎2 = 0). In Step 12 assume a loss of at least .15 limited to a probability of
0.01.
221
Problem PP183
Three assets have random returns 𝑅1 , 𝑅2 and 𝑅3 over the next period. Expected returns are 0.1,
0.2 and -0.02. Variances are 0.015, 0.02 and 0.005. Correlations are: 𝜌12 = 0.5, 𝜌13 = −0.4 and
𝜌23 = −0.2.
Report the expected values in cells B2:D2, the variances in cells B3:D3, the correlation matrix in
the range B7:D9.
1. Compute the covariance matrix in the range G7:I9.
Answer:
.015
.00866
-.00346
.02
-.002
.005
2. Consider all possible portfolios of the three assets (steps of .05) in cells A13:C242 (no
short selling). For example start with 0, 0, 1 in cells A13:C13 (all funds invested in asset
3), 0, 0.05, 0.95 in cells A14:C14 (5% invested in asset 2, 95% in asset 3), etc. Compute
the expected returns of all portfolios in column D. Example for cell D13:
= 𝐴13 ∗ $𝐵$2 + 𝐵13 ∗ $𝐶$2 + 𝐶13 ∗ $𝐷$2.
Compute the standard deviation of all portfolios in column E. Example for cell E13:
= 𝑆𝑄𝑅𝑇(𝐴13 ∗ 𝐴13 ∗ $𝐵$3 + 𝐵13 ∗ 𝐵13 ∗ $𝐶$3 + 𝐶13 ∗ 𝐶13 ∗ $𝐷$3 + 2 ∗ 𝐴13 ∗
𝐵13 ∗ $𝐺$8 + 2 ∗ 𝐴13 ∗ 𝐶13 ∗ $𝐺$9 + 2 ∗ 𝐵13 ∗ 𝐶13 ∗ $𝐻$9);
3. Graph a Scatter Plot (Scatter with only markers) of the standard deviation (horizontal
axis) versus the expected value (vertical axis). The efficient portfolios can be clearly seen
on the graph;
4. Underneath the graph put three fractions adding up to 1 in cells H38:J38, e.g. .7, .1, .2.
Compute the expected value and the standard deviation of the portfolio in cells K38:L38
by copying any of the cells in column D in cell K38 and any of the cells in column E in
cell L38. Compute the sum of the fractions in cell M38: = 𝑆𝑈𝑀(𝐻38: 𝐽38).
Construct a horizontal line segment from the point (0 =K38) to the point (=L38 =K38) as
follows: 0 in cell H40, =K38 in cell I40, =L38 in cell H41, =K38 in cell I41.
Add the two series in cells H40:I41 to the graph (as Scatter with straight lines).
Construct a vertical line segment from the point (=L38 0) to the point (=L38 K38)
similar to the construction of the horizontal segment.
Notice that the line segments join in the point that represents the standard deviation and
expected value of the portfolio with fractions in H38:J38.
Change the fractions and check how the graph changes;
5. Use the Solver to compute the portfolio with minimal standard deviation (minimal risk) in
cell L38. Allow cells H38:J38 to change. Check the graph.
Answer: 𝜎(𝑃) = 0.0477 (cell L38); 𝜇(𝑃) = 0.0264 (cell K38); fractions to be invested
(cells H38:J38): 0.2688, 0.0641, 0.6671;
222
6. Use the Solver to minimize the standard deviation of the portfolio (cell L38) under the
restriction that the expected return (cell K38) be at least 0.12. Check the graph.
Answer: : 𝜎(𝑃) = 0.086; 𝜇(𝑃) = 0.12, fractions to be invested (cells H38:J38): 0.173,
0.542, 0.285;
7. Change the expected return required in Step 6 to different values. Notice that when an
expected return smaller than 0.02636 (expected return corresponding to the portfolio with
minimal risk) is required, then the solution is the minimal risk portfolio.
Assignment PA183
Keep the data above with the following changes: the variance of asset 1 is 0 (asset with no risk),
the correlations between the return of asset 1 with both other returns are 0.
Repeat Steps 1 to 7 above. What does the efficient set of non-dominated portfolios look like?
223
Problem PP184
N assets have random returns 𝑅1 , 𝑅2 , … , 𝑅𝑁 over the next period. Expected returns are
𝝁𝑇 = [𝜇1 𝜇2 … 𝜇𝑁 ]. Standard deviations are 𝜎1 , 𝜎2 , … , 𝜎𝑁 . Denote the covariance matrix by 𝛀.
A capital of 1 is available to invest in the assets. Let the proportions invested in the assets be
denoted as 𝒂𝑇 = [𝑎1 𝑎2 … 𝑎𝑁 ] with ∑𝑁
1 𝑎𝑖 = 1.
1. Compute the expected return and the variance of the portfolio (use matrix notation).
Answer: 𝐸(𝑃) = 𝒂𝑇 𝝁, 𝜎 2 (𝑃) = 𝒂𝑇 𝛀𝒂;
2. Use the data and the set-up of the Excel sheet in Problem PP183, first 9 lines. Assume
fractions 𝑎𝑇 = [. 1 .6 .3] in cells H12:J12. Compute the expected value of the portfolio in
cell G14 using the array instruction: = 𝑀𝑀𝑈𝐿𝑇(𝐻12: 𝐽12; 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐵2: 𝐷2)).
Compute the variance in cell G15 using the array instruction:
= 𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻12: 𝐽12; 𝐺7: 𝐼9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻12: 𝐽12)).
Compare expected value and variance with the values computed in Step 2 of Problem
PP183 (note: the standard deviation was computed there, not the variance);
3. Derive the weights a for the portfolio with minimal variance (short selling allowed).
Compute its expected value and variance.
Answer: 𝒂 = 𝛀−1 𝟏⁄1𝑇 𝛀−1 𝟏
(1 is a N dimensional vector of ones),
𝐸(𝑃) = 𝟏𝑇 𝛀−1 𝝁⁄𝟏𝑇 𝛀−1 𝟏, 𝜎 2 (𝑃) = 1⁄𝟏𝑇 𝛀−1 𝟏 ;
4. To compute the minimal variance portfolio, its expected value and variance, use the
results of Step 3 as follows: first compute the inverse covariance matrix in the range
L7:N9. To do so, select cells L7:N9 and use the array instruction: = 𝑀𝐼𝑁𝑉𝐸𝑅𝑆𝐸(𝐺7: 𝐼9).
Report the values 1, 1 and 1 in cells H17:J17. Select cells G18:G20 and compute the
weights of the minimal variance portfolio using the array instruction: =
𝑀𝑀𝑈𝐿𝑇(𝐿7: 𝑁9; 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻17: 𝐽17))⁄𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻17: 𝐽17; 𝐿7: 𝑁9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻17: 𝐽17))
Compute the expected value of the minimal variance portfolio in cell G22 (array
instruction):
= 𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻17: 𝐽17; 𝐿7: 𝑁9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐵2: 𝐷2))⁄𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻17: 𝐽17; 𝐿7: 𝑁9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻17: 𝐽17))
Use a similar instruction to compute the variance in cell G23:
= 1⁄𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻17: 𝐽17; 𝐿7: 𝑁9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻17: 𝐽17));
5. Use the Solver to compute the minimal variance portfolio. Work as follows: any weights
in H25:J25. Cell K25: = 𝑆𝑈𝑀(𝐻25: 𝐽25).
Cell G27: = 𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝐻25: 𝐽25; 𝐺7: 𝐼9); 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐻25: 𝐽25)), the
variance of the portfolio. Use the Solver to minimize cell G27, restriction that cell K25
equals 1. Compute the expected return of the minimal variance portfolio in cell G28.
Compare expected value and variance with the same values computed in Step 4;
6. Derive the weights a for the portfolio with minimal variance among all portfolio’s with an
expected return of at least 𝜇𝑚𝑖𝑛 (short selling allowed).
Answer: Let 𝜶 = 𝛀−1 𝝁, 𝜷 = 𝛀−1 𝟏, 𝑇1 = 𝝁𝑇 𝜶, 𝑇2 = 𝟏𝑇 𝜷, 𝑇3 = 𝝁𝑇 𝜷.
224
Then,
1
𝒂 = 𝑇 𝑇 −𝑇 2 [𝑇1 𝜷 − 𝑇3 𝜶 + 𝜇𝑚𝑖𝑛 (𝑇2 𝜶 − 𝑇3 𝜷)];
1 2
3
7. On the Excel sheet compute the vector values of 𝜶 and 𝜷 in cells G31:G33 and H31:H33,
the values of the constants 𝑇1 , 𝑇2 𝑎𝑛𝑑 𝑇3 in cells G36:I36. In cells G40:G58 input values
of 𝜇𝑚𝑖𝑛 : -.02, -.01, 0, .0125, .025, .0375, …, 0.2. Select cells H40:J40 and compute the
value of the vector 𝒂𝑇 for 𝜇𝑚𝑖𝑛 = −.02 (array instruction):
= (1⁄($𝐺$36 ∗ $𝐻$36 − $𝐼$36 ∗ $𝐼$36)) ∗ ($𝐺$36 ∗ 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸($𝐻$31: $𝐻$33) − $𝐼$36 ∗
𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸($𝐺$31: $𝐺$33) + 𝐺40 ∗ ($𝐻$36 ∗ 𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸($𝐺$31: $𝐺$33) − $𝐼$36 ∗
value of 𝒂𝑇 compute the expected value of the
portfolio in cell K40 (should be -.02), its variance in cell L40 (.003531) and its standard
deviation in cell M40. Drag cells H40:M40 to line 58. For minimal expected values as
given in cells G40:G58 we have available the efficient portfolios (with minimal standard
deviation). Make a scatter plot (with full lines) of the expected value (vertical axis) versus
the standard deviation (horizontal axis);
𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸($𝐻$31: $𝐻$33))). Using this
8. Use the Solver to compute the efficient portfolio for a minimal expected return of 0.125.
Report fractions adding up to 1 in cells G61:G63. In cell G64: = 𝑆𝑈𝑀(𝐺61: 𝐺64).
Compute the expected return in cell I61: = 𝑀𝑀𝑈𝐿𝑇(𝐵2: 𝐷2; 𝐺61: 𝐺63), the variance in
cell I62: = 𝑀𝑀𝑈𝐿𝑇(𝑀𝑀𝑈𝐿𝑇(𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸(𝐺61: 𝐺63; 𝐺7: 𝐼9); 𝐺61: 𝐺63)). Minimize
cell I62, requiring cell G64 to be 1 and cell I61 at least 0.125. Change the value of the
minimal expected return to 0.02 and solve the problem again (notice some short selling).
Using the Solver you can forbid short selling by requiring all fractions to be nonnegative.
Assignment PA184
Consider 4 assets with expected returns 𝝁 = [. 1 .3 .4 .2] and standard deviations
𝝈 = [0 .05 .1 .07] (asset 1 is a non-risk asset). Correlations are: 𝜌23 = .5, 𝜌24 = −.4, 𝜌34 =
−.2. Compute the covariance matrix of the returns of assets 2, 3 and 4. Compute the expected
return, the variance and the standard deviation of a portfolio with fractions .2, .3, .4 and .1.
Use the Solver to compute the optimal portfolio (minimal variance) for expected values of at least
.1, .125, .15, …, .4 (short selling allowed). Compute the expected value, the variance and the
standard deviation for each portfolio.
Next compute optimal portfolio’s using only the risky assets 2, 3 and 4 for the same expected
returns .1, .125, …, .4. For these portfolios assume it is required that the restriction on the
expected value is exactly equal to (rather than at least equal to). Construct in the same graph
(Scatter plot with full lines) the expected values (vertical axis) as a function of the standard
deviations (horizontal axis) for both types of portfolios (with and without the no-risk asset).
Notice how the efficient set of the four asset model is a straight line tangent to the efficient set of
the three asset model). Denote the vector of expected returns of the risky assets by 𝝁𝑅 and the
return of the non-risky asset by 𝜇𝑁𝑅 . Then it can be shown that the weights of the tangency
portfolio are given by 𝛀−1 (𝝁𝑅 − 𝜇𝑁𝑅 𝟏)⁄1𝑇 𝛀−1 (𝝁𝑅 − 𝜇𝑁𝑅 𝟏). Compute these weights and derive
the expected return, the variance and the standard deviation of the tangent portfolio.