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106

Advanced Modelling in Finance

The Efficient Frontier

10%

12%

10%

15%

20%

25%

Risk (Standard Deviation)

Expected Return

TBills

Bonds

Constrained Points

Unconstrained Frontier

Fig1 PF

Figure 6.2 Risk–return representation of portfolios

In the following sections, we start by identifying individual points on the efﬁcient fron-

tier and then, in due course, generate the whole frontier. The ﬁnal stage is to impose

constraints on the holdings of individual asset classes (which in turn conﬁne the return

of the portfolio to a single range) and then to ﬁnd the position of the constrained frontier

points. Note that, although the scale of the chart does not allow us to conﬁrm this visu-

ally, the constrained points will always lie on or slightly to the right of the unconstrained

frontier.

If there are no additional constraints on the weights for the individual assets, frontier

portfolios can be obtained from formulas outlined by Huang and Litzenberger (1988,

section 6.4). However, generating efﬁcient portfolios using Solver is easier to explain,

so this approach is described ﬁrst, and then extended to the case when the weights on

individual asset classes are constrained. Finally, methods are suggested for generating the

entire efﬁcient frontier.

6.3 USING SOLVER TO FIND EFFICIENT POINTS

Given the same asset data set out in Figure 6.3, suppose we want to construct an efﬁcient

portfolio producing a target return of 7%. The problem is to ﬁnd the split across the

assets that achieves the target return whilst minimising the variance of return. This is

astandard optimisation problem amenable to Excel’s Solver, which contains a range of

iterative search methods for optimisation. Since portfolio variance is a quadratic function

of the weights, we will be using Solver for quadratic programming.

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Portfolio Optimisation

107

Using Solver to reproduce Unconstrained Frontier Portfolios

Asset Data

Exp Ret Std Dev

TBills

0.6%

4.3%

Target exp return

7.0%

target1

Bonds

2.1%

10.1%

9.0%

20.8%

(Ctrl + Shift + U to run Macro)

Correlation Matrix

TBills

Bonds

Frontier portfolio weights

TBills

1.00

0.63

0.09

TBills

40.0%

Bonds

0.63

1.00

0.23

Bonds

50.0%

change1

0.09

0.23

1.00

10.0%

VCV Matrix

TBills

Bonds

TBills

0.0018

0.0027

0.0008

Exp Ret

2.2%

portret1

Bonds

0.0027

0.0102

0.0048

Std Dev

7.0%

portsd1

0.0008

0.0048

0.0433

Figure 6.3 Sheet EF1 (in EQUITY1.xls) ready for optimisation

The algebraic structure of problems amenable to solution with quadratic programming

is documented fully in Eppen et al. (1998), as is the use of Excel’s Solver. Therefore,

we conﬁne ourselves to a brief explanation of how to operate Solver for this portfolio

problem using the layout in Figure 6.3. Notice the various range names attached to cells,

e.g. change1 for cells I11:I12, portret1 for I15, etc.

Solver requires ‘changing cells’, a ‘target cell’ for minimisation and the speciﬁcation

of ‘constraints’, which act as restrictions on feasible values for the changing cells. Notice

that the condition on full investment can be achieved by putting the formula:

=1SUM(I11:I12)

in cell I10. So the ‘changing cells’ for the optimisation are cells I11:I12 or, using the

range name, change1. The target cell to be minimised is the standard deviation of return

(I16), named portsd1. There is one explicit constraint, namely that the expected return

(cell I15), named portret1, must at least equal the target level (in cell I5), named target1.

(Note: The range names are particularly useful in specifying the Solver parameters and

again later in writing the VBA macros to automate the optimisation.)

The steps with Solver are:

1. Invoke Solver by choosing Tools then Options then Solver.

2. Specify in the Solver Parameter Dialog Box:

the Target cell (I16) to be optimised

specify max or min, as shown in Figure 6.4 and

the changing cells (I11:I12).

3. Choose Add to specify the constraints then OK (Figure 6.4 below on the right). This

constraint ensures that I15 must meet the target (cell I5).

4. Click on Options and ensure that Assume Linear Model is not checked.

5. Solve and get the results in the spreadsheet.

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108

Advanced Modelling in Finance

Figure 6.4 Setting up Solver to minimise variance

Applying Solver, the optimum weights suggest positive holdings of Bonds and Shares

and a negative holding of TBills in the proportions shown in the sheet in Figure 6.5.

Cell I15 conﬁrms that this efﬁcient portfolio with minimum standard deviation of 15.7%

achieves the required return of 7%. Changing the target expected return in cell I5 and

invoking Solver again with the same speciﬁcations as before produces another efﬁcient

portfolio. (The reader should conﬁrm this by changing the target return in I5 from 7%

to say 3% and then invoking Solver to get the new portfolio weights. Another exercise

is to determine the weights for the unique minimum variance portfolio. If the constraint

portret1=target1showninFigure6.4isdeleted,Solvergivestheweightsforthisspecial

case–expected return 0.7%, risk 4.1%.)

Since there are no constraints on the weights assigned to individual assets, negative

holdings are possible, which is also known as short selling. Indeed for our example in

Figure 6.5, the frontier portfolio with an expected return of 7% contains a negative holding

of 5.6% in TBills. This means that the portfolio entails borrowing a proportion of the

portfolio value, thus incurring an interest charge to offset against the expected returns

from the positive holdings in the other two assets.

Using Solver to reproduce Unconstrained Frontier Portfolios

Asset Data

Exp Ret Std Dev

TBills

0.6%

4.3%

Target exp return

7.0%

target1

Bonds

2.1%

10.1%

9.0%

20.8%

(Ctrl + Shift + U to run Macro)

Correlation Matrix

TBills

Bonds

Frontier portfolio weights

TBills

1.00

0.63

0.09

TBills

-5.6%

Bonds

0.63

1.00

0.23

Bonds

35.8%

change1

0.09

0.23

1.00

69.8%

VCV Matrix

TBills

Bonds

TBills

0.0018

0.0027

0.0008

Exp Ret

7.0%

portret1

Bonds

0.0027

0.0102

0.0048

Std Dev

15.7%

portsd1

0.0008

0.0048

0.0433

Figure 6.5 The unconstrained frontier portfolio with 7% expected return

So far, optimal portfolio weights have been obtained by minimising portfolio risk.

An alternative approach is to determine portfolio weights to maximise expected return

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Portfolio Optimisation

109

for a speciﬁed level of risk. Although these appear to be different problems for Solver,

the optimum weights correspond to portfolios lying on the efﬁcient frontier as in the

minimisation formulation.

6.4 GENERATING THE EFFICIENT FRONTIER (HUANG

AND LITZENBERGER’S APPROACH)

If there are no constraints on individual asset weights, the efﬁcient frontier can also be

produced elegantly from algebra. Although some of the more advanced textbooks (such

as Elton and Gruber, 1995) demonstrate this through the iterative solution of a set of

simultaneous equations, there is a better approach. Huang and Litzenberger (denoted HL)

have described how to ﬁnd two points on the frontier, and then to generate the whole

of the frontier from these points (by applying a result due to Black). This section builds

on their algebraic approach, the calculation sequence being explained using matrices, to

generalise the approach to portfolios with many (i.e. more than three) assets. The context

of the following description is spreadsheet implementation with Excel’s array functions.

This section is of a somewhat more advanced level and may be deferred on ﬁrst reading

until the remainder of the chapter has been understood.

Sheet EF1HL shown in Figure 6.6 includes named ranges to improve the clarity of

the formulas in the cells. The vector of expected returns (C5:C7) is named e, the vector

of weights (I5:I7) is named w and the unit vector in A24:A26 is named u. The vari-

ance–covariance matrix in C15:E17 is named V. As explained earlier, the portfolio

variance is written in matrix form as w

Vw and is evaluated with Excel’s matrix multi-

plication functions in cell I11.

The HL method for ﬁnding efﬁcient portfolios requires the inverse of the

variance–covariance matrix, which is written as V−

.Excel’s MINVERSE function for

matrix inversion does the actual calculation. The array formula to enter into the 3 by

3cell range, H15:J17, is:

DMINVERSE(C15:E17)

Note that for array functions, having selected the cell range for the formulas and entered

the appropriate formula, the keystroke combination CtrlCShiftCEnter must be pressed to

complete the entry. (If no array formula brackets fg appear in the cell, then press F2 and

try again.)

In order to ﬁnd two frontier portfolios (labelled g and gCh), Huang and Litzenberger

start by calculating four scalar quantities (A, B, C and D). The ﬁrst three, A, B and C,

are products of vectors and matrices and the fourth, D, depends on the previous three:

AD u

V−

e B D e

V−

e C D u

V−

u D D BC  A

If we deﬁne two intermediate column vectors l D V−

e and m D V−

u, shown in the

sheet in cells C24:C26 and D24:D26, the matrix multiplication expressions simplify to:

AD u

l B D e

l C D u

The single cell formula for A in cell G23: DMMULT(TRANSPOSE(u), l) involves the

MMULT array function, so it must be entered as an array formula (similarly the formulas

for B and C). Since the calculations for A, B, C and D all result in scalars, there is no

need to ‘select’ an area greater than a single cell for any of them.

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110

Advanced Modelling in Finance

Using Algebra to reproduce Unconstrained Frontier Portfolios

Asset Data

Exp Ret

Std Dev

Portfolio Weights

TBills

0.6%

4.3%

TBills

33.3%

Bonds

2.1%

10.1%

Bonds

33.3%

9.0%

20.8%

33.3%

Correlation Matrix

TBills

Bonds

via fn

TBills

1.00

0.63

0.09

Exp Ret

3.90%

Bonds

0.63

1.00

0.23

Variance

0.0080

0.09

0.23

1.00

Std Dev

8.95%

VCV Matrix

TBills

Bonds

VCV inverse

TBills

0.0018

0.0027

0.0008

901.51

-246.92

10.80

Bonds

0.0027

0.0102

0.0048

-246.92

171.13

-14.52

0.0008

0.0048

0.0433

10.80

-14.52

24.53

Finding weights, g and h, to generate points on the frontier

uvec

g+h

3.97

1.20

665.40

0.20

124.0% -1851.9%

-1727.9%

0.81

-90.30

595.91

-20.5%

805.2%

784.6%

1.97

20.82

104.15

-3.5%

1046.7%

1043.2%

Exp Ret

0.0%

100.0%

Std Dev

4.4%

237.6%

Generating Frontier Portfolios, using g and h

Target expected return

7.0%

weights

via fn

TBills

-5.6%

Exp Ret

7.0%

Bonds

35.8%

Std Dev

15.7%

69.8%

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pected return of 7% consists of holding a mixture of Bonds and Stocks whilst short

gTBills. This result from HL’s approach is identical to the result in Figure 6.5 for

roptimisation with no constraints on weights.

er points on the frontier can be evaluated using the array expression gCh

T by

ying different values for the expected return, T. In this way the efﬁcient frontier can

nerated. For example, in Excel a Data Table with a range of expected returns from

10% as inputs and portfolio risk and return as outputs can be used as the basis for

Y(Scatter) chart. In Figure 6.2, the points labelled as the ‘Unconstrained Frontier’

from such a Data Table.

is evident, the spreadsheet manipulations required to evaluate the weights for the

fﬁcient portfolios are complex. This is the ideal situation for a VBA user-deﬁned

on to simplify formula entry. In section 6.10, the code to implement Huang and

nberger’s approach is explained. The function HLPortfolioWeights has four argu-

:expret, retvec vcvmat and rf (here D 1). Argument expret is the expected return

frontier portfolio for which the weights are required, retvec is the vector of expected

sand vcvmat the variance–covariance matrix. The function returns an array of the

priate portfolio weights. In Figure 6.6, weights from this user-deﬁned function are

nin cells E36:E38 alongside the weights obtained from spreadsheet manipulation.

6.5 CONSTRAINED FRONTIER PORTFOLIOS

eare constraints on the holdings of individual assets (for example that weights must

n-negative), the analytical solution approach no longer applies. However, Solver

roduce the optimum set of weights. To distinguish these from the previous case, we

he resulting portfolio a ‘constrained’ frontier portfolio. In the ﬁrst extract of sheet

nFigure 6.7, possible portfolio weights are in cells H8:J8 (named portwts2), with

Figure 6.8 Setting up Solver with constraints on values for the weights

The frontier portfolio found by Solver has 7.4% in TBills, 20.0% in Bonds and 72.6%

in Shares. This frontier portfolio with an expected return of 7.0%, but now constrained

to have non-negative holdings, a maximum of 10% in TBills and 20% in Bonds, has a

standard deviation of 15.8%. This compares to a standard deviation of only 15.7% in the

unconstrained case, illustrating the point that typically the addition of constraints produces

asolution that is inferior to the unconstrained case.

By repeating the use of Solver for a number of different target expected returns we can

generate a series of points on the constrained frontier (see Figure 6.9). The mechanics of

the macro are described in section 6.12. With the constraints on weights detailed above,

the minimum and maximum portfolio expected returns are 6.8% and 9.0% respectively.

[The reader should check these ﬁgures by applying Solver to ﬁnd the weights to minimise

(maximise) expected return.] Assuming 11 optimisations, this range of returns is split up

into 11 target returns and Solver is used 11 times to ﬁnd frontier portfolios whose returns

lie in the feasible range (between 6.8% and 9.0%).

Constrained Frontier Portfolios

TBills

Bonds

Std Dev

Exp Ret

effwts2

10.0%

20.0%

70.0%

15.2%

6.8%

7.4%

20.0%

72.6%

15.8%

7.0%

4.7%

20.0%

75.3%

16.3%

7.2%

2.1%

20.0%

77.9%

16.8%

7.4%

0.0%

19.3%

80.7%

17.3%

7.7%

0.0%

16.1%

83.9%

17.9%

7.9%

0.0%

12.9%

87.1%

18.5%

8.1%

0.0%

9.7%

90.3%

19.0%

8.3%

0.0%

6.4%

93.6%

19.6%

8.6%

0.0%

3.2%

96.8%

20.2%

8.8%

0.0%

100.0%

20.8%

9.0%

Figure 6.9 Weights for constrained frontier portfolios

The portfolios in Figure 6.9 appear labelled Constrained Points in Figure 6.2, the last one

being identical to investing only in Shares.

Portfolio Optimisation

113

In summary, the most general technique for determining weights for risky portfolios

is to apply optimisation using Solver, essentially a numerical iterative process. However,

if there are no constraints on the values of the weights (other than that they sum to

unity), HL’s analytical approach provides formulas for the optimal weights which can be

implemented via user-deﬁned functions described in section 6.10.

6.6 COMBINING RISK-FREE AND RISKY ASSETS

This section, together with sections 6.7, 6.8 and 6.9, contains additional material pertinent

to the problem of choosing portfolios from risky assets. However, it is essentially theoret-

ical, and could be regarded as appendix material. Its main importance is that we develop

the formulas to solve the ‘three generic portfolio problems’ (as described in Taggart,

1996) and provide user-deﬁned functions. These functions provide solutions to support

ideas in Chapters 7 and 8. We introduce the concept of a risk-free asset (that is an asset

with zero risk) in order to complete the path towards the development of the capital asset

pricing model in the next chapter.

Having discussed portfolios with three (or more) risky assets and shown how to

select efﬁcient combinations which are expected to generate speciﬁed target returns with

minimum risk, we brieﬂy explore how risk and return in investment should be balanced.

This risk–return trade-off is a major factor in deciding how to split investment between

aportfolio of risky assets and a risk-free asset. The ideas are developed in the context of

three generic portfolio problems:

ž One–combining a risk-free asset with a risky asset

ž Two–combining two risky assets

ž Three–combining a risk-free asset with a risky portfolio

The spreadsheet approaches are illustrated in the sheet labelled Generic.

We require a way of measuring an individual’s trade-off between risk and return and

borrow the concept of the utility function from economics. Let us suppose that individuals

have utility functions of the form:

UD Er

 0.5A

The higher the value of the risk aversion coefﬁcient A for the investor the larger is

the penalty from portfolio risk that is subtracted from the portfolio’s expected return.

Experiments have found the typical value of A to lie between 2 and 4, and we have

chosen A D 3 for our illustrations.

Since the development in the following sections is in terms of two assets, the general

formulas for portfolio return and risk given in section 6.1 are restated here for two assets,

namely:

Portfolio return:

Er

D w

Er

C w

Er



Portfolio variance:

Varr

D 



C2w

cov1, 2

where the covariance of the two assets cov1, 2 can equally well be expressed in terms of

their correlation using cov1,2 D corr1, 2



.The risk-free asset has return denoted

with zero variance and zero covariances with the risky assets. The portfolio risk is the

square root of the variance.

114

Advanced Modelling in Finance

6.7 PROBLEM ONE–COMBINING A RISK-FREE ASSET WITH

ARISKY ASSET

The ﬁrst of the generic portfolio problems, Problem One, involves one risk-free asset

(asset0) and one risky asset (asset1) with returns assumed to be uncorrelated. The details

are shown in Figure 6.10: the risk-free rate is 1% and asset1 has risk and return somewhat

similar to the Bonds that featured in the earlier portfolio. By varying the weight invested

in asset1 (cell H14) in a range from 0% to 100% we can generate the straight line linking

the risk–return point for asset0 to the risk–return point for asset1. This line is shown in

Figure 6.11.

Generic Portfolio Problems

Problem One : risk-free asset and 1 risky asset

Asset Data

Exp Ret Std Dev

Risk aversion coefficient (A)

)

3.00

asset 0

1.0%

0.0%

asset 1

2.1%

10.1%

Corr Matrix

asset 0

asset 1

Optimal Portfolio

utility

0.0120

asset 0

1.00

0.00

asset 1

0.00

1.00

weights

asset 0

64.1%

return

1.40%

VCV Matrix

asset 0

asset 1

35.9%

std dev

3.63%

asset 0

0.0000

asset 1 via fn

35.9%

asset 1

0.0000

0.0102

Figure 6.10 Spreadsheet calculations for Problem One, sheet Generic

Problem One

10%

12%

Portfolio risk (standard deviation)

Portfolio return

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Portfolio utility

Figure 6.11 Utility and return versus risk for Problem One

Individual investors will choose a particular point on the line depending on the value

of their risk aversion coefﬁcient (in cell K6). Figure 6.11 includes the utility curve for

portfolios with different proportions of asset1 assuming a risk aversion coefﬁcient of 3

(shown as a series of separate boxes and linked to the right-hand scale of the chart). The

Portfolio Optimisation

115

utility starts at 0.01 (with 100% invested in asset0) and increases steadily as the proportion

invested in asset1 increases until the optimal point is reached, declining thereafter to a

utility value of 0.006 with 100% invested in asset1.

The optimal point on the line corresponds to that of maximum utility, which occurs

when the weight for asset1 (in cell H14) is evaluated from the formula:

D[Er

 r

]/A

In this example, the optimal portfolio will have 36% (35.9%) invested in the risky asset

to produce a utility value of 0.012 (that is, a certainty equivalent return of 1.2%). A

user-deﬁned function called Prob1OptimalRiskyWeight has been developed to provide

this weight (as in cell H15). It is described in section 6.11.

Note also that the proportion of the optimal investment put into the risky asset (as

opposed to asset0) depends on the investor’s degree of risk aversion. An investor with a

higher risk aversion coefﬁcient (of say 4) would reduce his proportion of risky assets in

his optimal portfolio to 27%.

6.8 PROBLEM TWO–COMBINING TWO RISKY ASSETS

In Problem Two both assets are risky. In our illustration, their returns are weakly positively

correlated, just as was the case for the Bonds and Shares in the earlier portfolio example.

The details are set out in Figure 6.12 with correlation 0.23 in cell D25.

Problem Two : 2 risky assets and no risk-free asset

Asset Data

Exp Ret Std Dev

Risk aversion coefficient (A)

3.00

asset 1

2.1%

10.1%

asset 2

9.0%

20.8%

Minimum Variance Portfolio

utility

0.0151

weights

Corr Matrix

asset 1

asset 2

asset 1

87.7%

return

2.95%

asset 1

1.00

0.23

asset 2

12.3%

std dev

9.77%

asset 2

0.23

1.00

Optimal Risky Portfolio

utility

0.0333

VCV Matrix

asset 1

asset 2

weights

asset 1

0.0102

0.0048

asset 1

35.2%

return

6.57%

asset 2

0.0048

0.0433

asset 2

64.8%

std dev

14.70%

asset 1 via fn

35.2%

Figure 6.12 Spreadsheet calculations for Problem Two, sheet Generic

By varying the weight invested in asset1, we can trace the outline or frontier (see the

continuous line in Figure 6.13). The resulting curve linking the risk–return points repre-

sented by combinations of the two risky assets demonstrates the beneﬁt of diversiﬁcation.

For example, if an investor starts with a portfolio wholly invested in asset1 then he can

increase the return of his portfolio by swapping part of his portfolio into asset2. The risk

of his portfolio will also be reduced by this swap despite the fact that asset2, on its own,

is a riskier asset than asset1.

This process of swapping from asset1 to asset2 reduces portfolio risk until the point

represented by the minimum variance portfolio is reached with 87.7% in asset1 and a

portfolio standard deviation of 9.77%. The formula for the minimum variance portfolio

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