Chapter 5
Portfolio Analysis
5.1
Covariance Matrices
For general random variables X = (x1 , . . . , xn )T , Y = (y1 , . . . , yn )T with means (µ1 , . . . , µn )T ,
(ν1 , . . . , νn )T respectively, the Covariance Matrix of X, Y is the matrix such that
(cov(X, Y ))ij = E((xi − µi )(yi − νj ))
and the Covariance Matrix of X is the matrix cov(X) such that
(cov(X))ij = (cov(X, X))ij = E((xi − µi )(xj − µj ))
Write zi = xi − µi so that
(cov(X))ij = (cov(X, X))ij = E(zi zj ) = cov(zi , zj )
The Correlation Matrix of X is the covariance matrix of the vector
(
z1
zn T
x1 − µ1
xn − µn T
,...,
) =(
,...,
)
σ1
σn
σ1
σn
where σi is the standard deviation of zi (and hence of xi ). The correlation matrix then
has 1’s along the diagonal and elements lying between -1 and 1.
Clearly, cov(X) is symmetric in that
(cov(X))ij = E(zi zj ) = (cov(X))ji
A symmetric matrix Ω is said to be positive semi-definite if for all n-vectors v we have
v T Ωv ≥ 0
and it is said to be positive definite if for all v 6= 0 we have
v T Ωv > 0
1
2
CHAPTER 5. PORTFOLIO ANALYSIS
It can be shown that a symmetric matrix is positive definite (positive semi-definite)
if and only if all its eigenvalues are > 0 (≥ 0). The number of non-zero eigenvalues
(counting multiplicity) is the same as the rank of the matrix.
It is easily seen that the covariance matrix of a vector X is positive semi-definite. For
v T cov(X)v
=
X
cov(X)ij vi vj =
ij
=
X
E(
vi zi vj zj )
=
X
X
E(
vi zi
vj zj )
=
X
E((
vi zi )2 )
≥
0
X
E(zi zj )vi vj
ij
ij
i
j
i
5.2
Portfolio Returns
We consider a portfolio of n assets with total asset values S1 , S2 , . . . , Sj , . . . , Sn , and
total value
S = S1 + S2 + · · · + Sj + · · · + Sn
(Note that in this chapter we do NOT use Si for the asset price). The value of asset j
at time i is
Sj (i)
i = 0, 1, . . . , T
In this chapter we use the notation ∆S( i) = Sj (i) − S( i − 1).
The return of asset j at time i, i = 1, . . . , T , is
rj (i) =
∆Sj (i)
Sj (i) − Sj (i − 1)
=
Sj (i − 1)
Sj (i − 1)
The return on the portfolio at time i is
r(i) =
∆S(i)
S(i − 1)
=
=
=
1
(∆S1 (i) + · · · + ∆Sn (i))
S(i − 1)
S1 (i − 1) ∆S1 (i)
Sn (i − 1) ∆Sn (i)
+ ··· +
S(i − 1) S1 (i − 1)
S(i − 1) Sn (i − 1)
x1 (i − 1)r1 (i) + · · · + xn (i − 1)rn (i)
where xj (i − 1) is the proportion by value of asset j in the portfolio at time i − 1, and
is allowed to be negative indicating a short position in this asset. We have
n
X
j=1
n
xj (i) =
S(i)
1 X
=1
Sj (i) =
S(i) j=1
S(i)
5.2. PORTFOLIO RETURNS
3
In what follows, we will drop the dependence of xj (i − 1), rj (i) and Sj (i) on i, and we
will write
r = x1 r1 + · · · + xn rn
where rj is the return at time i and xj is the proportion of asset j in the portfolio at
time i − 1. Then
n
X
Sj
xj =
,
xj = 1
S
j=1
Taking expectations, we have
E(r) =
n
X
xj E(rj )
j=1
Writing rj = E(rj ), r = E(r) this becomes
r=
n
X
xj rj
j=1
This gives an expression for the expected return of the portfolio. To calculate the
variance of the returns:
n
n
X
X
E((r − E(r))2 ) = E((
xj rj −
xj rj )2 )
=
E((
j=1
n
X
j=1
xj (rj − rj ))2 )
j=1
=
E((
n
X
n
X
xi (ri − ri ))(
xj (rj − rj )))
i=1
=
=
n
X
j=1
xi xj E((ri − ri )(rj − rj ))
i,j=1
n
X
xi xj cov(ri , rj )
i,j=1
Write
σ 2 = variance of the portfolio return r
σij = covariance of asset returns ri , rj .
So σij is a symmetric matrix since cov(ri , rj ) = cov(rj , ri ).
Hence
n
X
σ2 =
xi xj σij
i,j=1
Using vector notation, let
V = (σij ),


x1


x =  ...  ,
xn


r1


r =  ...  ,
rn

r1


R =  ... 
rn

4
Then
CHAPTER 5. PORTFOLIO ANALYSIS
σ 2 = xT Vx,
r = E(r) = xT R
and V is symmetric and positive semidefinite since σ 2 ≥ 0 for all x.
By a slight abuse of terminology, we will refer to x as a portfolio. If x, y are two
portfolios of the same assets, with returns xT r and yT r, then their covariance is, by a
similar calculation to the above,
cov(xT r, yT r) = xT Vy = yT Vx
where the last equality follows from the fact that V is symmetric.
The above figure illustrates the case n = 2. Two assets A and B with expected returns
and covariance matrix
Covariance √
Matrix√
Asset Returns
0.0025
ρ
.0025
.01
10%
√
√
ρ .0025 .01
0.01
4%
are combined in a portfolio. The correlation of the returns is ρ. ρ is a convenient
parameter to illustrate what the combinations look like. We plot σ vs E(r) for various
portfolios
(x1 , x2 ),
x1 + x2 = 1
5.2. PORTFOLIO RETURNS
5
and for various correlations ρ. The figure uses ρ = 1, −1, 0.28. Note that ρ = ±1 lead
to two straight lines joining on the r-axis. The curve joining A and B consists of 3
parts, one finite and joining A and B, and the other two infinite. The finite portion
represents portfolios with long positions in both assets. The infinite portions represent
long positions in one asset and short positions in the other. We illustrate the formulae
above by giving them explicitly in this case.
If x1 = w, x2 = 1 − w we have
σ 2 = xT V x = w2 (0.0025)2 + 2w(1 − w)(0.0025)(0.01)ρ2 + (1 − w)2 (0.01)2
E(r) = w(0.1) + (1 − w)(0.04) = 0.04 + 0.06w
For ρ = ±1 this simplifies. For example, for ρ = 1, we have
σ 2 = (0.0025w + 0.01(1 − w))2
σ = ±(0.01 − 0.0075w)
on substituting w =
1
0.06 (E(r)
− 0.04) we see that (σ, E(r)) lie on two straight lines.
5.2.1
Two-Asset Portfolios in the Spreadsheet
5.2.2
Estimating E(r), σ and Covariance from Data
We have dealt with this in Chapter 3. We can use GARCH for the variances and
historical sample means and covariances for the other parameters. These are what we
need to draw up the table that we must use.
• Place the covariance matrix with a parameter for ρ in A2:A4, and put copies in A4:B5
and A6:A8. Link the parameters to cells A9, A12, A15 respectively.
• Place the expected returns in D2:D3.
• In D5:D35 place values for x1 running from -2 to 4 in steps of 0.2. In E5:E35 place
the values of x2 so that x1 + x2 = 1.
• In F5 calculate σ for the portfolio in that row using covariance matrix A2:B3, and in
G5 calculate the return. Propagate downwards.
• Make a copy of D5:G35 starting with D36, and make another copy starting with D67,
using the other two covariance matrices that you have set up.
• Chart the three sets of (σ, r) pairs.
• Vary the 3 values of ρ, and watch the changes on the chart.
• Set up spin boxes for the three values of ρ and watch the changes.
• In particular, try the values 1,-1,0.28 for the three values of ρ.
• Add headings and improve the appearance.
6
5.3
CHAPTER 5. PORTFOLIO ANALYSIS
Optimizing Returns
• We assume that the sdv σ of a portfolio is a measure of its risk.
• We assume that each investor has an expected rate of return in mind, say r0 .
• We assume that an investor will choose a portfolio of smallest risk which has expected
return r0 .
These assumptions and others implicit in the mathematical formulation are discussed
in detail in books on portfolio theory, and are the subject of much controversy. We will
not go into this aspect here.
It is useful to plot (expected return) r on the vertical axis vs σ (sdv or volatility) on
the horizontal axis. Then the optimization problem for our investor can be written as
follows:
Problem 1:
5.3. OPTIMIZING RETURNS
7
For fixed return r0 , minimize σ over all portfolios x, or equivalently, minimize σ 2 .
σ 2 = xT Vx = min!
x
xT R = r0
xT 1 = 1
Pn
where 1 is an n-vector all of whose entries are 1, so that xT 1 = j=1 xj .
Note: There are many variations of this problem associated with different constraints
on the variable x. For example, if short sales for asset j are not allowed, the constraint
xj ≥ 0 should be added to the above formulation.
Problem 2:
We fix r0 on the r axis giving the point (0, r) . For fixed σ, we look at the points
P (r) = (σ, r) and we consider the slope of the line joining (0, r0 ) on the r-axis to P (r).
We then look for an r which maximizes the slope of the line. In this way we find the
maximum return r for a fixed level of risk σ.
r − r0
σ
xT 1
xT (R − r0 1)
√
= max!
x
xT Vx
= 1
=
Analytic Solution
It can be shown if z is the vector
z = V−1 (R − r1 1)
then
x=
1
zT 1
z
is an absolute maximum or minimum of the ratio.
We will show below that if V is invertible, then the latter is the unique critical point
(zero of the gradient or first derivative) satisfying xT 1 = 1. That all critical points are
maxima or minima follows from the convexity of the functions, but we will not go into
this point.
Let
xT (R − r0 1)
f (x) = √
,
x 6= 0
xT V x
We look for critical points of ln f , which will also be critical points of f , since 5 ln f =
5f
, where 5f is the gradient of f :
f
 ∂f 

5f = 
∂x1
..
.
∂f
∂xn


8
CHAPTER 5. PORTFOLIO ANALYSIS
Also, note that
5aT x = 5xT a = a,
5xT Ax =2x
where a is a constant column vector and A is a constant matrix. Then
ln f (x) = ln(xT (R − r0 1)) −
Hence
5 ln f (x) =
1
ln xT Vx
2
Vx
(R − r0 1)
− T
− r0 1) x Vx
xT (R
Let
λ(x) =
xT (R − r0 1)
xT Vx
Then
5 ln f (x) = 0 ⇔ λ(x)Vx = R − r0 1
Using the assumption that V is invertible, let
z = V−1 (R − r0 1)
so that
Vz = R − r0 1
Then by directly substituting for x, and using the fact that V and (hence) V−1 are
symmetric, we see that for α 6= 0 we have
λ(z) = 1,
λ(αz) =
1
α
Hence for α 6= 0
λ(αz)Vαz = λ(z)Vz = R − r0 1
which implies that for all α 6= 0, αz is a critical point (and in particular, z is a critical
point). If z 6= 0 define the vector x by
x=
z
zT 1
so that
xT 1 = 1
We have for α > 0,
f (αy) = f (y)
Hence if z is an absolute maximum of f then x = zTz 1 is also an absolute maximum
(if the denominator is positive) of f , satisfying the constraint. If the denominator is
negative then it is a minimum. A similar analysis holds for the case of an absolute
minimum.
Thus to find the critical points (which will be either maxima or minima by convexity)
we must find
z = V −1 (R − r0 1
5.4. PROPERTIES OF THE SOLUTION SET OF PROBLEM 1 OR 2
and then divide it by zT 1.
Example For two stocks:
µ
V =
σ12
ρσ1 σ2
ρσ1 σ2
σ22
9
¶
¶
σ1−2
−ρσ1−1 σ2−1
−ρσ1−1 σ2−1
σ2−2
µ
¶
E(r1 ) − rf
−1
z=V
E(r2 ) − rf
µ
¶
z
1
A1 σ22 − A2 ρσ1 σ2
x= T =
A2 σ12 − A1 ρσ1 σ2
z 1
A1 σ22 + A2 σ12 − (A1 + A2 )ρσ1 σ2
V −1 =
1
1 − ρ2
µ
A1 = E(r1 ) − rf , A2 = E(r2 ) − rf .
5.4
Properties of the Solution Set of Problem 1 or 2
The set of points (σ, r) corresponding to solutions x of Problem 1 and Problem 2 for
all real values of r0 is called the Envelope. A portfolio x with values (σ, r) lying on
the envelope is called an Envelope Portfolio. Envelope portfolios do not contain the
risk-free asset.
• We may find points on the envelope by finding z by the above formula and dividing
it by zT 1, for various values of r0 .
• For values of r0 whose tangent to the envelope intersects the envelope close to the
vertex of the hyperbola, we expect the calculation to become more unstable, as the
tangent is nearly vertical and has large slope. There are also problems with this
method when r0 is close to the value of r at the vertex.
• A set M is convex if for all 0 ≤ λ ≤ 1 and all x, y ∈ M we have λx + (1 − λ)y ∈ M ,
that is, the line segment joining any two points of the set also lies in the set. The set
of all portfolios is a convex set.
• If there is no risk-free asset in the portfolio (that is, one with σ(x) = 0), then the
envelope is a hyperbola, with axis horizontal, lying in the half-plane σ ≥ 0. If there is
a risk-free asset with return r0 then the envelope is a cone consisting of two straight
lines with vertex (0, r0 ) and lying in the right half-plane. We will analyze this further
below.
• If σ0 is the point on the hyperbola with minimum σ (that is, the vertex), then the
half of the hyperbola lying above the vertex is called the Efficient Frontier.
• The set of all envelope portfolios and the set of all frontier portfolios are convex sets.
The set of all (σ, r) corresponding to these portfolios is not convex (the points are
parts of a hyperbola or two straight lines).
10
CHAPTER 5. PORTFOLIO ANALYSIS
• Given two distinct portfolios on the envelpe, the set of all convex combinations of
the two portfolios is the set of all portfolios on the envelope. (A convex combination
of x, y is a point of the form λx + (1 − λ)y for some real λ.) The portfolios on the
envelope form a straight line in portfolio space.
• If we settle on a maximum value of risk we are willing to bear, σ1 say, then the
portfolio corresponding to the point (σ1 , r1 ) on the efficient frontier is the one that
an investor would select under the above assumptions.
• ALL portfolios correspond to values of (σ, r) which lie on or to the right of the
envelope - this is called the feasible set. This is a consequence of the fact that σ is
a minimum over all possible portfolios with the same return, and hence all possible
portfolios have (σ, r) lying to the right of the envelope point.
5.5
Portfolios with One Risk-Free Asset
We assume that there is a risk-free asset amongst the assets in the portfolio, with
return rf . Since the sdv is 0, the point (0, rf ) corresponds to an optimal portfolio.
The remaining assets generate an hyperbola corresponding to the envelope of these
5.5. PORTFOLIOS WITH ONE RISK-FREE ASSET
11
assets. By convexity, the tangent lines from (0, rf ) to the hyperbola must correspond
to portfolios, and must lie on the envelope, since if there were points on the envelope
corresponding to portfolios above or below the tangents, then there would have to be
portfolio points of the remaining assets outside the hyperbola. Hence
• The envelope is a cone whose boundary is two straight lines emanating from (0, rf )
and tangent to the hyperbola forming the envelope of the non-risk-free portfolios. The
upper tangent intersects the hyperbola at a point M = (σM , rM ) corresponding to a
portfolio xM called the Market Portfolio.
• The line through M and (0, rf ) has the equation
µ
¶
E(rM ) − rf
E(r) = rf +
σ
σM
as can be checked by substitution. It is called the Capital Market Line, CML.
We can write this, for each portfolio x, as
E(rx ) − rf
βx
= βx (E(rM ) − rf )
=
cov(xT r, xT
M r)
2
σM
Assume that this model is applied to the whole market. The market portfolio is then
12
CHAPTER 5. PORTFOLIO ANALYSIS
that of the whole market. The above equation states that
(i) There exist constants α, γ independent of the portfolio x such that
E(rx ) = α + γβx
and hence (βx , E(rx )) lie on a straight line in (β, r) space.
(ii) α = rf and γ = E(rM ) − rf for some frontier portfolio xM .
Given a set of points βx , E(rx ), we can estimate α, γ by least squares regression. We
can write the capital market line equation as:
µ
¶µ
¶
E(rM ) − rf
cov(xT r, xT
M r)
C = E(rx ) − rf =
= AB
σM
σM
where C is the excess return over the risk-free asset, A is the excess return of the market
portfolio per unit of risk, and B is the risk measure of x relative to the market portfolio
xM .
• Once we know α, γ, βx , we can use the CML to determine E(rx ) and thus value the
asset. We have remarked on the determination of α, γ. βx can be determined from
historical data. (See the chapter on Statistical Analysis of Financial Asset Data).
5.5.1
CAPM - The Capital Asset Pricing Model
This model supposes that in market equilibrium, all investors follow the conclusion
that they are best advised to invest in the market portfolio and will do this - see the
assumption below. Under this assumption, we can determine the Market Portfolio of
the whole market.
• Composition of the Market Portfolio xM
We show that the jth component of xM is
xM j =
Sj
S
where S is the total market value of all assets and Sj is the total value of the jth asset
in the market. (The market portfolio holds each asset in proportion of its total value
to the total value of the market.)
Assumption We assume that in equilibrium, every investor holds the Market Portfolio
and the risk-free asset in some proportion. If this is not the case at any time, the
situation will move towards equilibrium. (We have shown that this should be optimal
in the model we are studying.)
We can then deduce the composition of the Market portfolio.
Let S q be the total holdings of person q and Sjq the holdings of asset j by person q.
Then
X
X
Sq =
Sj
S =
q
Sj
=
Sjq
=
X
j
Sjq
q
xM j S q
5.5. PORTFOLIOS WITH ONE RISK-FREE ASSET
13
The last equality follows from the fact that the only holding of person q of asset j is
from the market portfolio, as indicated above, and must be in the proportion of the
market portfolio. So
X q
X
Sj =
Sj = xM j
S q = xM j S
q
q
Hence
xM j =
Sj
S
as required.
Actual Portfolios Approximating the Market Portfolios
In the US, S&P500 is usually taken as a surrogate for the market portfolio, since it contains 500 stocks in proportion to their market values. It is, however seldom a frontier
portfolio.
Testing CAPM
• Test 1: Calculate xM in a market with risk-free rate rf and also r = xT
M R and
σ(xM ). Check whether the tangent frontier portfolio (see below for method of calculation) with endpoint (0, rf ) has the correct proportional allocation of assets.
• Test 2: For a large collection of values of (βx , E(rx )) perform a regression to see if
there exist constants α, γ such that
E(rx ) = α + γβx
such that βx is significantly different from 0.
• Tests for CAPM are not usually positive.
To Find the CML
The CML passes through (0, rf ). We need only find the slope. We use the formulation
of Problem 2. Solve, using Solver,
xT r − rf
√
=
xT Vx
xT 1 =
max!
x
1
Alternatively, find z such that
z̃ =
z =
V−1 (r − rf 1)
1
z̃
z̃T 1
Alternative Method - Finding the Envelope by Using Convexity
14
CHAPTER 5. PORTFOLIO ANALYSIS
The set of portfolios forming the envelope is convex in the space of portfolios. (The
envelope is NOT convex in (σ, r) space.) In fact, it is a straight line in portfolio space,
as we have already remarked. Hence to find the envelope, we can proceed as follows:
• Find any two envelope portfolios x1 , x2 .
• The envelope is the set
{λx1 + (1 − λ)x2 | λ ∈ R}
5.5.2
The Security Market Line
Black has shown that the type of linear relation that holds between the E(r), E(rM ), rf
where r is the return on a frontier portfolio also holds when r is the return on a single
asset. Hence if rj is the return on asset j, we have
E(rj ) − rf = βj (E(rM ) − rf )
or
E(rj ) = α + γβj
where α, γ are constants independent of j. In order to find these two constants we must
use the values of a large number of E(rj ), βj pairs and then estimate the constants
by, for example, least squares estimation. The line is called the Security Market Line
(SML).
5.6
Finding the Efficient Frontier in the Spreadsheet
The following is the covariance matrix of four stocks, and the expected return of each.
We construct in the spreadsheet the Envelope and efficient frontier.
Covariance Matrix
0.1
0.01 0.03 0.05
0.01
0.3
0.06 -0.04
0.03 0.06
0.4
0.02
0.05 -0.04 0.02
0.5
5.6.1
Exp Return
0.06
0.08
0.1
0.15
Using Method 1 - Solving Problem 1
• In what follows, columns should be given appropriate headings.
• Place the covariance matrix in A4:D7. Call this matrix V . Place the returns in F4:F7.
Call this vector r.
• In G4:G35 place values of r0 starting from 0 and going up in steps of 0.005 ( 12 %).
For each of these values of r0 we will calculate the value of σ which puts (r0 , σ) on
the envelope.
• H4:K4 will hold values of the portfolio weights x = (x1 , x2 , x3 , x4 ). Initially, set the
values to 0.25 each, and propagate this to row 35.
5.6. FINDING THE EFFICIENT FRONTIER IN THE SPREADSHEET
15
• In L4 place xV xT (x is a row vector here). (Use an array formula). Propagate this
to row 35. Call this column σ 2 .
• In M4 place σ, and propagate to row 35.
• In N4 place the expected return on the portfolio x, that is xr (x is a row vector, r is
a column vector. Propagate.
• In O4 place the sum of the xi in that row. Propagate.
• Let 1 denote the 4-vector each component of which is 1. In P4 place xT (R − r0 1),
where r0 is the value of r0 for this row. Propagate.
P4
• Now use solver to minimize σ 2 (L4) subject to constraints i=1 xi (O4), xT (R −
r0 1) = 0 (P4) by changing the value of x (H4,I4,J4.K4). Do this for each row from 4
to 35. In the spreadsheet, we cannot propagate the use of Solver. We will be able to
automate the process using VBA.
• Chart σ (X-axis) vs E(r) (Y-axis). This charts the envelope and the efficient frontier
is the top half of the envelope.
16
CHAPTER 5. PORTFOLIO ANALYSIS
5.6.2
Automating the Use of Solver Using VBA
Inserting the array formula in column L row i and calculating the optimal value in
column L row i via Solver can be handled in two ways. We give an example of both,
one for the insertion and one for using Solver. Note that & is the string concatenation
operator.
• Write a Sub to do this.
a = CStr(i)
b = "H" & a & ":K" & a
frmla = "=MMult(" & b & ",MMult($A$4:$D$7,Transpose(" & b & ")))"
Range("L" & a).FormulaArray = frmla
SolverReset
SolverAdd CellRef:=Range("O1").Cells(i, 1), Relation:=2,
FormulaText:=0
SolverAdd CellRef:=Range("P1").Cells(i, 1), Relation:=2,
FormulaText:=0
SolverSolve UserFinish:=True
Note that Cstr(i) changes the values contained in variable i, assumed to be an integer,
into a string. The code will in fact work with i in place of a since VBA does an automatic
conversion, but it is not clear when it does this, and it is better not to rely on it.
No Short Sales
If no short sales are allowed for asset i then the constraint
xi ≥ 0
must be included.
• Write a VBA routine to do the above example for the case where no asset can be
sold short. Compare the efficient frontier for the two cases.
5.6.3
Using Method 2 - Solving Problem 2
We describe the use of Method 2 in the spreadsheet. We add the calculations below
those of Method 1.
• Below the Method 1 rows, repeat the headings of Method 1.
• Insert a range of values in the r0 column (you could use the same ones as for Method
1).
• Add a column for β to the right.
• Add 4 columns for R − r0 1 further to the right.
• Add 4 columns further to the right to hold A−1 (R − r0 1).
5.6. FINDING THE EFFICIENT FRONTIER IN THE SPREADSHEET
17
• insert formulae to calculate R − r0 1 and A−1 (R − r0 1)
• The formula for β is the ratio of two cell values
xT (R − r0 1)
σ
• The formulae for σ 2 and σ are the same as for Method 1, as is the formula for E(r).
• The formulae for the portfolio weights are the 4 elements of A−1 (R − r0 1) divided
by their sum.
• Chart σ vs E(r). Note that the values around the vertex of the hyperbola are problematic.
• In order to calculate various values of β for different r0 add a separate row which is
a copy of one of the Method 2 rows.
• On the same chart, plot the straight line with slope β and r-intercept r1 for some
chosen r1 . The line is
r = r0 + βσ
Note that it is tangent to the hyperbola. Try changing r1 . Plot the intersection point
of the tangent and the hyperbola.
• Write VBA routines to automate as much of Method 2 as you can.