Math 53 (Financial Math)
March 30, 2007
Suppose that we want to invest B dollars in some mix of securities, in such a way as to maximize expected return and minimize risk. These are competing objectives, and we will need to compromise between them.
We will construct a model involving only one time step. We choose a set of investments at time zero, and observe the results at time one.
1. Feasible portfolios and the feasible set
Let’s start by defining the choices that are available to us. We are given a list of N securities, indexed by i = 1, …, N . For each i , let x i
be the number of dollars we invest in security i . We are free to choose the values of x
1
, …, x n
subject to a budget constraint , x
1
 x
2
  x
N

B (1)
Note that equality is required. The x i
’s need not be integers, but they may be subject to various other linear constraints. To begin with, we assume that the only other linear constraints are nonnegativity constraints : x i

0 (2) for each i .
A vector x = ( x
1
, … , x
N
) satisfying constraints (1) and (2) is called a portfolio, or a feasible portfolio.
The feasible portfolios form a compact, convex subset of R
N
called the feasible set.
From this set, we want to choose a portfolio that (in some sense) maximizes expected return and minimizes risk.
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2. Assumptions about the securities
By the return on a security we mean the profit we make from investing in the security, per dollar invested. Let R i
denote the return on the i -th security, treated as a random variable. It can be positive or negative. (It follows that our profit from investing x i
dollars in the i -th security is also a random variable, given by x i
R i
.)
We assume that the mean returns from the securities and all variances and covariances are known. That means that we know all of the following values.
 i
= E ( R i
) = expected return from security i
 i
= standard deviation of R i
 i
2 = Var ( R i
) = variance of R i
 ij
= covariance of R i
and R j
( note that
 ii
is the same as
 i
2
. )
 ij
= correlation of R i
and R j
( so that
 ij
=
 ij
 i
 j
. )
With this notation, we can write the return from the portfolio x = ( x
1
, …, x
N
) as a random variable:
 x
 x R
1 1
  x R
N N
(3)
(The symbol

is an upper-case Greek pi, standing for “
 rofit.”) We want to maximize the mean of

( x ) and minimize its variance. We denote the mean of

( x ) by

( x ). We write
Var( x ) for Var(

( x ) ) and

( x ) for the standard deviation of

( x ), both as an abbreviation and in order to emphasize the dependence on the portfolio x . These statistics are given by
  x

1 1
  x

N N
(4) and
Var
  x x
 i j ij
.
(5)
We want to maximize

( x ) and minimize Var( x ), in some combination. It would be just as good to minimize the standard deviation

( x ) = ( ) . For short, we’ll refer to

( x ) as the
“reward,” and to Var( x ) or

( x
) as the “risk.”
Let’s see which risk-reward combinations ( Var( x ),

( x ) ) are possible as x ranges over the set of feasible portfolios. Think of the pair (Var,

) as a function from R
N
to two-dimensional
(Var,

) space:
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R
N x
1-
,…
,x
B
N
,
B x
1-
,
… x

B x
(Var(x),

(x))
,
 x
1
…
,x
N
,
F x
1
…
,x
N Var x
11-
The yellow image of this map is a compact set --- that is, closed and bounded --- because it is a continuous image of a compact set. It usually isn’t convex. The yellow image set is also often called the feasible set; we’ll call it
F .
(Fair warning: For the first few sections we need for the coordinates of risk-reward space be Var and

. But at some point later on, we’ll change the coordinates to become

and

. Risk is still represented on the horizontal axis, and still increases to the right, but in a different way. Each form of the picture has is uses and both appear commonly in the literature.)
The key to understanding the shape of F is this lemma.
Lemma: Any line segment in the feasible set is mapped to a (finite segment of a) parabola in
(Var,

) space, opening to the right, or to a line segment.
Proof: Let x and y be any two feasible portfolios. The line segment between them is given by the portfolios
( 1 –

) x +
 y for values of

in [ 0, 1 ]. Each of these is a feasible portfolio (which is to say that the feasible set is convex). It is easy to check that the return from the combined portfolio is

((1
 
) x
  y ) (1
 x
 y (6) so that its mean is

((1
 
) x
  y ) (1
  x
    
  y
  x
 
.
(7)
If

( y ) =

( y ), this means that

is constant on the line segment, so the entire line segment is mapped into a connected part of a horizontal line in ( Var,

) space --- that is, a horizontal line segment, as claimed. In all other cases we can assume that

( y ) –

( y )

0, which means that the linear relationship in (7) is reversible, and

is itself a linear function of the

coordinate

((1-

) x +
 y ).
The variance of the return from the combined portfolio is
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Var ((1
 
) x
  y )
  
)
2

[ Var x
Var

Var
 
2
Var y

2(1
  
Cov x y y

2 Cov x y

2  
Var x

2 Cov x y
 
[ Var x
(8) which is a quadratic function of

. Its leading coefficient is positive (or at least nonnegative) by the Cauchy-Schwarz inequality. (Or: note that Var ( )

Var y

2 Cov x y

Var ( x
 y ) is nonnegative.) Here, Cov( x , y ) means Cov(

( x ),

( y)).
Since this is a quadratic function of

, it is also a quadratic function of the expected return from the same portfolio. That is, Var is a quadratic function of

in (Var,

) space. It is not hard to see that the leading coefficient of the quadratic is positive, which means that the image of the line segment is a portion of a parabola opening to the right.
(end of proof)
It follows that the left edge of F is convex (meaning, concave to the right). That is the only convexity property that we need. (If we assume that there are no linear relationships among the securities --- that is, no nonzero combination of securities yields a return with variance zero --- and that the securities do not all have the same expected return, then the left edge of F is strictly concave to the right.)
The upper-left edge of the yellow image is called the efficient frontier.
Each point on the efficient frontier represents a portfolio that…
(a) Maximizes

for a given value of the variance Var, and at the same time,
(b) Minimizes Var for a given reward

.
We call these efficient portfolios.
Our model tells us that we should choose an efficient portfolio, but it offers no guidance as to which efficient portfolio we should choose. That depends on the investor’s taste for risk.
Therefore, a reasonable statement of our portfolio selection problem is to identify all of the points of the efficient frontier, and to find portfolios corresponding to each of these points.
Tangent lines.
Let’s make the assumptions needed to guarantee that the efficient frontier is strictly convex. Then for each point on the frontier, there is a line tangent to the frontier at that point, and at only that point. Further, for each nonnegative of the slope (including, with the obvious meaning, slope +∞) there is a unique line of that slope that is tangent to the frontier. In this way we establish a one-to-one correspondence between slopes in [0, +∞] and points of the frontier.
For technical reasons, we denote the slope by 1/(2

). Now

also ranges over the interval
[0, +∞], and values of

are also in one-to-one correspondence with the points of the frontier.
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The topmost point of F represents the portfolio with the highest reward, regardless of risk. It also corresponds to the tangent line with slope 0, or

= +∞. The leftmost point of F represents the portfolio of least risk, regardless of reward. It corresponds to the tangent line with slope +∞, or

= 0.
In between,

represents the tradeoff between reward and risk. At the point corresponding to

,
1 unit of reward (measured by

) is being traded for 2

units of risk (measured by Var).
Put another way: The point corresponding to

is the point of F that maximizes the linear function
 
- (1/2) Var.
That’s the function you want to maximize if you think one unit of reward is worth 2 
units of risk.
3. Matrix formulation
We can restate the problem in the language of vectors and matrices.
We introduce column vectors to represent a portfolio, x , the vector of mean returns,

, and the vector of all 1’s, e : x

 x
1

 
 
 x
N

,
μ 
 

1

 

N

, e 
 
  .
 
Also, we define the covariance matrix ,

, as follows:



11

1 N


.

N 1

NN
Now the constraints can be written as x
T e = B x ≥ 0
(budget constraint) and
(non-negativity constraints).
The various objective functions become
Mean:
Variance:

( x ) = x
T r ;
Var( x ) = x
T  x ;
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Standard deviation:

( x ) = Var( ) .
4. Formal Statements
We could state the problem formally in either of two ways. Either we want to maximize the reward for each possible choice K of the variance…
For each K ,
Maximize by choice of x

subject to the constraints x x
T
T
=
 x x
T r

K , e = B , x

0.
…or we want to minimize the variance for each possible choice
L of the reward:
For each L ,
Minimize by choice of x
Var =
subject to x
T  x x
T r

L , x T e = B , x

0.
But there’s a better way to state the problem, based on finding all of the points of tangency. This method also gives all of the point of the efficient frontier:
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For each

in [0, +

],
Maximize by choice of x

subject to x
T
=
 x
T r e = B , x

0.
– (1/2) x
T  x
Each value of

corresponds to one point on the efficient frontier.
For each

, this is a quadratic programming problem (“an instance of a quadratic program”).
The only sense in which it is not entirely routine (as a quadratic programming problem) is that we are to solve the problem for a family of
 ’s. It turns out that it is much more efficient to solve the family together than to apply standard quadratic programming algorithms separately for different values of

. Therefore there are specialized algorithms for this problem. The construct the efficient frontier as a sequence of portions of parabolas, and provide an explicit portfolio to represent each point of the frontier.
5. What if there is cash?
At this point it is convenient to introduce a simplification. Since the entire problem scales with B , we might as well assume that B = 1. The budget constraint becomes x
1
 x
2
  x
N

1 . (9)
We can imagine that we actually have only one dollar to invest, or, perhaps more naturally, we can interpret each x i
as the fraction of the portfolio that is invested in the i -th security.
Also, we will take this occasion to switch from Var( x ) to

( x ) in our graphs of the efficient frontier. Clearly it makes no difference whether we minimize Var( x ) or

( x ). The graph looks almost the same --- at least, at the cartoonish level at which we have been drawing them --- but all of those parabolas become hyperbolas. That is, the efficient frontier is now composed of segments of hyperbolas, but it is still concave to the right. We make this change because

( x ) is more appropriate for discussion of portfolios that contain cash.
The cash asset.
In this section we introduce a new asset, called cash and indexed by i = 0, with a guaranteed return of

0
. We call it a guaranteed return because we set

0
= 0, and also
 i 0
= 0 for each i = 1, …, N .
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Sometimes the return

0
is called the risk-free rate , although that terminology is more appropriate for a model that is cast in terms of return per unit time, which we have not done.
Assets other than cash are called risk assets .
We assume that we can borrow or lend as much cash we want with the level of return specified.
That is, there is no non-negativity constraint on x
0
--- the fraction of our portfolio that is in cash can be positive or negative. If N = 2, for example, the portfolio
( x
0
, x
1
, x
2
) = ( -0.20, +0.60, +0.60 ) is a feasible portfolio. We are keeping the non-negativity constraints on the risky assets, however, so it is impossible to have more than 100% of the portfolio in cash.
What does this do the feasible set F in (

,

)-space, and to the efficient frontier?
First, we add to F the point (

,

) = ( 0,

0
), corresponding to the all-cash portfolio.
Now, suppose we form a portfolio x by having x
0
in cash, and allocating the remaining fraction ( 1 – x
0
) to risky assets in proportion to M , where M was a feasible portfolio under the old formulation. Then:

( x ) = x
0


( x
0
+ (1– x
0
)

( M )
) = (1– x
0
)

( M ).
That is, the reward is a mix of the reward for M and the reward for holding cash, and the risk is reduced in proportion to the amount of the portfolio that is in risky assets. (Well, we should put
“reduced” in quotes, because 1 – x
0
can be negative.)
In (

,

)-space, the point (

( x ),

( x )) lies on the line joining the all-cash point ( 0,

0
) to the point (

( M ),

( M )). More particularly, as ranges along the half-line starting at ( 0,

0 x
0
ranges from +1 to –

, the point (

( x ),

( x ))
) and passing through (

( M ),

( M )).
If we repeat this procedure with all portfolios M in the original feasible set, we get (in (

,

)space) the cone, from the origin, generated by our previous yellow image set F
. Let’s call the result F’ . The new efficient frontier is the ray from ( 0,

0
) that is tangent to F at the point realized by a particular portfolio which we will call M* .
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
, x
1
…
M*
,x
N

Note that there is now a unique optimal portfolio M* of risky assets. Every optimal portfolio consists of some amount of cash and some amount of M* . We exercise our risk preference only by deciding how much cash to hold, and how much of M* .
M* is usually called the market portfolio , for the reason that we will see in a moment. M* happens to be the portfolio that maximizes the Sharpe Ratio,
Sharpe Ratio =

( M )


( M )

0
.
6. The Capital Asset Pricing Model (CAPM)
Now assume that everyone agrees on the inputs

and

. Then everybody calculates the same
M* . Everybody holds the same mix of non-cash securities. Investors differ only in how much cash they hold, and how much of M* .
M* must necessarily be a proportional share of all the securities that exist. We call it the market portfolio.
Since every security is in M* , it follows from the solution of the quadratic program above (and lots of algebra) that every security satisfies this equation:
 i
=

0
+
 i
(

M*
–

0
) , where

M*
=

( M* ).
The coefficient
 i
is the same as the regression coefficient in the equation
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That is,
 i

R( i ) =
 i
R( M* ) +

.
 j  1..
N

 N

 ij x j j

.
This is a theory of what expected returns must necessarily be, if we accept all of the assumptions made so far.
(end)
Walter Stromquist wstromq1@swarthmore.edu
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