FINANCE 495 (Rev’d Sp15)
ADVANCED INVESTMENT ANALYSIS
LECTURE NOTE 5
I. Mean-Variance Portfolio Analysis, Cont’d.
A. Using SOLVER to find the Efficient Frontier: The
efficient frontier represents the means and standard
deviations of the portfolios which are mean-variance
(MV) efficient. Generally each mean—standard
deviation point on the boundary is achieved with a
unique portfolio of assets. The Excel SOLVER
function can be used to identify asset combinations and
weighting schemes (especially) that (beyond being
merely feasible) lie on the Efficient Frontier.
“Annual income twenty pounds, annual expenditure nineteen nineteen six, result
happiness. Annual income twenty pounds, annual expenditure twenty pounds ought
and six, result misery.” - Charles Dickens, ‘David Copperfield, Chap. 12.’
B. SOLVER can be used to find the allocation of money
to be invested in the portfolio assets that either: 1)
maximise the portfolio return for a given level of risk,
or, 2) minimise the portfolio risk for a specified
portfolio return (sound familiar?).
C. How SOLVER works: The following information
about SOLVER is obtained from the Excel Help
function.
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About Solver
Solver is part of a suite of commands sometimes called what-if analysis tools. With
Solver, you can find an optimal value for a formula in one cell — called the target cell —
on a worksheet. Solver works with a group of cells that are related, either directly or
indirectly, to the formula in the target cell. Solver adjusts the values in the changing cells
you specify — called the adjustable cells — to produce the result you specify from the
target cell formula. You can apply constraints to restrict the values Solver can use in the
model, and the constraints can refer to other cells that affect the target cell formula.
Use Solver to determine the maximum or minimum value of one cell by changing other
cells — for example, you can change the amount of your projected advertising budget
and see the affect on your projected profit amount.
Define and solve a problem
1. On the Data tab, in the Analysis group, click Solver.
If the Solver command or the Analysis group is not available, you need to
activate the Solver add-in.
How to activate the Solver add-in
a. Click the File tab, click Options, and then click the Add-Ins category.
b. In the Manage box, click Excel Add-ins, and then click Go.
c. In the Add-ins available box, select the Solver Add-in check box, and
then click OK.
1. In the Set Objective box, enter a cell reference or name for the objective cell.
The objective cell must contain a formula.
2. Do one of the following:
o If you want the value of the objective cell to be as large as possible,
click Max.
o If you want the value of the objective cell to be as small as possible,
click Min.
o If you want the objective cell to be a certain value, click Value of, and
then type the value in the box.
3. In the By Changing Variable Cells box, enter a name or reference for each
decision variable cell range. Separate the nonadjacent references with
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commas. The variable cells must be related directly or indirectly to the
objective cell. You can specify up to 200 variable cells.
4. In the Subject to the Constraints box, enter any constraints that you want to
apply by doing the following:
a. In the Solver Parameters dialog box, click Add.
b. In the Cell Reference box, enter the cell reference or name of the cell range for
which you want to constrain the value.
c. Click the relationship ( <=, =, >=, int, bin, or dif ) that you want between the
referenced cell and the constraint.If you click int, integer appears in the Constraint
box. If you click bin, binary appears in the Constraint box. If you click dif, alldifferent
appears in the Constraint box.
d. If you choose <=, =, or >= for the relationship in the Constraint box, type a
number, a cell reference or name, or a formula.
e. Do one of the following:
 To accept the constraint and add another, click Add.
 To accept the constraint and return to the Solver Parameters
dialog box, click OK.
Note You can apply the int, bin, and dif relationships only in
constraints on decision variable cells.
You can change or delete an existing constraint by doing the
following:
f. In the Solver Parameters dialog box, click the constraint that you want to change
or delete.
g. Click Change and then make your changes, or click Delete.
5. Click Solve and do one of the following:
o To keep the solution values on the worksheet, in the Solver Results
dialog box, click Keep Solver Solution.
o To restore the original values before you clicked Solve, click Restore
Original Values.
6. You can interrupt the solution process by pressing Esc. Excel recalculates the
worksheet with the last values that are found for the decision variable cells.
7. To create a report that is based on your solution after Solver finds a solution,
you can click a report type in the Reports box and then click OK. The report is
created on a new worksheet in your workbook. If Solver doesn't find a
solution, only certain reports or no reports are available.
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8. To save your decision variable cell values as a scenario that you can display
later, click Save Scenario in the Solver Results dialog box, and then type a
name for the scenario in the Scenario Name box.
D. SOLVER as applied to Developing Efficient Portfolios
(adapted from the “Portfolio of Securities” ‘Tab’ from
the SOLVER Sample problems)
1. One of the basic principles of investment management is diversification. By holding a portfolio of
several stocks, for example, you can earn a rate of
return that represents the average of the returns from
the individual stocks, while reducing the risk that any
one stock will perform poorly.
2. Using this model, you can use Solver to find the
allocation of funds to stocks that minimises the
portfolio risk for a given rate of return, or that
maximises the rate of return for a given level of risk.
3. This worksheet contains monthly stock prices and
annualized, monthly returns for a 10 stock portfolio.
Additionally, arithmetic means (AVERAGE fn),
population standard deviations (STDEVP fn),
coefficients of variation and the variance/covariance matrix have been calculated. Initially equal
amounts (10 percent of the portfolio) are invested in
each security.
4. In the PROJECT analysis students will be required
to base their 10-asset portfolios on minimum and
maximum asset allocations. Specifically, invest-
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ments will be constrained as shown in the following
figure.
Constraint
Minimum
Maximum
Percentage
3.0%
25.0%
5. Use Solver to try different percentage allocations
(weights) of funds in stocks to either maximise the
portfolio rate of return for a specified level of risk or
minimise the risk for a given rate of return. With the
initial allocation of 10% across the board, the
portfolio return is 17.8801% and the variance is
0.022820 (Std Devn = 15.1062%).
Problem Specifications
Target cell(s)
$T$25 (Max) or
$U$25 (Min)
Changing cells
Constraints
$S$27:$S$36
$S$27:$S$36>=0.03
$S$27:$S$36<=0.25
$S$37=1
Goal is to maximise
portfolio return or
minimise variance.
Weight of each stock.
Weights must be no less
than 3.0%.
Weights must be no more
than 25.0%.
Weights must sum to 1.0.
E. Steps/Procedure to Use SOLVER in Spreadsheet 5
1. In the Spreadsheet, the following cell references/
equations need to be set up before you use SOLVER.
The cell references/equations for the required cells in
this spreadsheet are as follows.
 Expected Asset Returns: P27:P36
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



Asset Standard Deviations: Q27:Q36
Asset Coefficients of Variation: R27:R36
Variance/Covariance Matrix: N9:W18
Matrix of Weights: N44:W53 (Note: These need to
be referenced to the Weight cell array in the
SOLVER problem, i.e., S27:S36).
Note: If you set up the ‘below diagonal’ weights
using the equations and the ‘above diagonal’
elements as cell references by cleverly anchoring
the cells you should be able to use copying (since
the weights are set up vertically in the SOLVER
problem).
 Calculated Portfolio Return: S39 → $T$25 needs to
reference this cell.
 Calculated Markowitz Variance: S41 (Using the
Matrix Multiplication calculation) → $U$25 needs
to reference this cell.
2. Click the Tools menu.
3. Choose Solver
4. Set Solver Parameters as shown above in the Problem
Specifications.
 Set Target Cell
 By Changing Cells
 Subject to the Constraints [Add]
5. Click Solve
Ex. 5.1: Project Example III
Monthly data for the PROJECT SPREADSHEET have
been collected over a 61-month period from 4/01/2010 to
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4/01/2015 on ten assets. These assets include the eight
assets from Ex. 2.4 plus two additional stocks (HSP and
SIMO). The actual annualised monthly returns, mean
returns, population standard deviations, coefficients of
variations and the variance/covariance matrix are provided
in Spreadsheet 5. Additionally, you will find a template
provided to set up the SOLVER problems.
a) Enter the required values to get the SOLVER problem
set up.
b) Starting with equal weights use SOLVER to find the
optimal weights to maximise the portfolio return.
c) Having solved the First SOLVER problem, skip one
column (i.e., col. X) and Copy the first solution directly
to the right in the adjacent columns. Verify that this
second solution template can be Cut (and pasted) and
moved five rows higher into cells (Y2:AI48) without
adversely affecting the template.
d) In this Second SOLVER problem, use the maximum
return weights and choose to find the weights that
minimise the variance. Note: because of the cell anchors
in the First Solver problem, you will need to adjust the
Target, Changing and Constraint cells accordingly.
A5.1. A facsimile excerpt from Spreadsheet 5 is shown
below after running SOLVER in part d).
20
O
P
Q
Efficient Stock Portfolio - Portfolio 2
21
22
23
Find the weightings of stocks in an efficient portfolio that maximise the
portfolio rate of return for a given level of risk or minimise portfolio risk
for a given level of return.
Max Retn
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S
T
U
Min Risk
24
25
26
Asset
27
28
29
30
31
32
33
Asset 1
Asset 2
Asset 3
Asset 4
Asset 5
Asset 6
Asset 7
34
35
36
37
38
39
40
41
Expected
Return
Stnd
Devn
Coef
Varn
0.16437
0.14476
0.8807
0.25000
0.14093
0.47723
0.29715
0.55175
2.1085
1.1561
0.08193
0.03000
0.15835
0.15365
0.16947
0.14634
1.0702
0.9524
0.03000
0.03000
0.09335
0.14446
0.19065
0.13325
2.0423
0.9224
0.03000
0.25000
Asset 8
Asset 9
0.15511
0.15218
0.14629
0.16691
0.9431
1.0968
0.03000
0.03000
Asset 10
0.14837
0.15780
1.0636
0.23807
1.0000
Portfolio Totals:
Return
StndDevn
MrkVar
Weight
0.159773
10
0.015984
10
15.9773%
12.6429%
0.015984
Note: In forming the matrix of weights you want to enter the
formulas starting with the diagonal VARIANCE
calculations. Then enter the covariance weight calculations
in the first column. Anchor the column of the first weight so
that you will be able to copy the columns below diagonal.
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Matrix of Weights
M
43
44
45
46
47
48
49
50
51
52
53
w1.
w2.
w3.
w4.
w5.
w6.
w7.
w8.
w9.
w10.
N
O
P
Q
R
S
T
U
V
W
w. 1
w. 2
w. 3
w. 4
w. 5
w. 6
w. 7
w. 8
w. 9
w. 10
0.06250
0.02048
0.00750
0.00750
0.00750
0.00750
0.06250
0.00750
0.00750
0.05952
0.02048
0.00671
0.00246
0.00246
0.00246
0.00246
0.02048
0.00246
0.00246
0.01951
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.06250
0.02048
0.00750
0.00750
0.00750
0.00750
0.06250
0.00750
0.00750
0.05952
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.00750
0.00246
0.00090
0.00090
0.00090
0.00090
0.00750
0.00090
0.00090
0.00714
0.05952
0.01951
0.00714
0.00714
0.00714
0.00714
0.05952
0.00714
0.00714
0.05668
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