chapter 7: portfolio theory

Question 1

expected return

standard deviation of the return

Answer 1

• the expected return which we denote r, and

* the standard deviation of the return which we denote σ

Question 2

given a set of investments whose returns have known expected values
and standard deviations, which one is “optimal”

Answer 2

“highest expected returns”is not =“optimal”: the whole distribution of the returns needs to be taken
into account

Question 3

investments of 1000

Portfolio A: Will be worth £1,100 with probability 1.

Portfolio B: Will be worth £1,000 with probability 1/2 and £1,300 with probability 1/2.

Portfolio C: Will be worth £500 with probability 1/10, £1,200 with probability 8/10 and
£3,000 with probability 1/10

Expected returns:

Answer 3

Expected returns:

r_A = (1100 − 1000)/1000 = 0.1

r_B = (0.5)(1000 − 1000)/1000
\+ 0.5(1300 − 1000)/1000 = 0.15
r_C =
(1/10)(500 − 1000)/1000
\+ (8/10)(1200 − 1000)/1000
\+
(1/10)(3000 − 1000)/1000
= 0.31

percentage of payout/initial payment

Question 4

Mr. X: Wants to sail around the world on a cruise costing £1,200 a year from now.
Ms. Y: Must repay her mortgage in a year and must have £1,100 to do so.
Dr. Z: Needs to buy a rare book worth £1,300.

Portfolio A: Will be worth £1,100 with probability 1.

Portfolio B: Will be worth £1,000 with probability 1/2 and £1,300 with probability 1/2.

Portfolio C: Will be worth £500 with probability 1/10, £1,200 with probability 8/10 and
£3,000 with probability 1/10

Answer 4

The chances of Mr X. sailing around the world if he invests in investments A,B or C are 0, 1/2
and 9/10 respectively, so he should be advised to invest in C. Only investment A guarantees a return
sufficient for Ms. Y to pay her mortgage and she should choose it. The probability of the investments
being worth at least £1,300 after a year are 0, 1/2 and 1/10 respectively, so Dr. Z should invest in
portfolio B.
So different investors prefer different investments!

Question 5

Consider two investments A and B with expected returns r_A and r_B and standard deviation of
returns σ_A and σ_B

Answer 5

A1: Investors are greedy:
If σ_A = σ_B and rA > rB investors prefer A to B.

A2: Investors are risk averse:
If rA = rB and σ_A > σ_B investors prefer B to A.

A3: Transitivity of preferences: If investment B is preferable to A and if investment C is
preferable to B then investment C is preferable to A.

Question 6

We are introducing a partial ordering ≺ on the points of the σ-r plane

considering diagram of σ against r

Answer 6

r
|   *B           *  C
|   *A
|      *D
\_\_\_\_\_\_\_\_\_σ 
B at (σB ,rB ) etc

A is directly under B, C on same horizontal as B

B is preferable to both A and C. Investments A and C are
incomparable.

We are introducing a partial ordering ≺ on the points of the σ-r plane

** ie those on a horizontal line have same r (r_A=r_B), expected return ,and lower (right ) σ preferred

** ie those on a vertical line have same σ, deviation, so preferred higher expected return r (towards top)

Question 7

INDIFFERENCE CURVES

Answer 7

describe preferences of investor by specifying sets of
investments which are equally attractive to the given investor

DEF indifference curve of an investor: this is the curve
consisting of points (σ,r) for which investments with these
expected returns and standard deviation of returns are all equally attractive to our investor.

Question 8

INDIFFERENCE CURVES and assumptions of risk aversion etc

Answer 8

A1, A2 and A3 imply that these curves must be non-decreasing

Question 9

EXAMPLE: indifference curves for investors

r
|  |  |  |  |
|  |  |  |  |
|  |  |  |  |
L |_|_|_|\_\_\_σ

Answer 9

Investor X cannot tolerate uncertainty at all: this investor
prefers a certain zero return rather a very large expected
return with a small degree of uncertainty.

less risk

Question 10

EXAMPLE: indifference curves for investor Y

r    |      |   |      |
|    /     /   /     /
|   /     /   /     /
|_/    /    /    /
L\_\_/\_\_/\_\_/\_\_σ

Answer 10

Investor Y is willing to take some risks

slightly curved

less risk would be straighter

Question 11

EXAMPLE: indifference curves for investor Z
               -  -  -  -
r               -  -   -
|                 - - -
|      /  /  /       ---
|   /  /  /     /
|/  /  /     /
|/   /    /
L/\_\_/\_\_\_\_\_σ

Answer 11

Investor Y is willing to take some risks but not as much as investor Z
(his indifference curves are steeper.)

curves hard to show

Question 12

EXAMPLE: indifference curves for investor

r
|\_\_\_\_\_\_\_
|\_\_\_\_\_\_\_
|\_\_\_\_\_\_\_
|\_\_\_\_\_\_\_
L\_\_\_\_\_\_\_σ

Answer 12

Investor W is risk neutral.

Question 13

Consider two investments A and B with expected returns r_A and r_B and standard deviation of returns σ_A and σ_B .
We split an investment of £1 between the two investments:
consider portfolio Πt consisting of t units of investment A and
1 − t units of investment B.

call this Πt #1

risky investments A and B ie we have risks/ standard dev non zero

Answer 13

We split an investment of £1 between the two investments:
consider portfolio Πt consisting of t units of investment A and
1 − t units of investment B.

We can do this for any t and not just 0 ≤ t ≤ 1.
For example, to construct portfolio Π2 we short sell £1 worth of B
and buy £2 worth of A, for a total investment of £1.

Question 14

portfolio Πt #1

VARIANCE of

Answer 14

Let A and B be the random variables representing the annual return of investments A and B.

The variance of Πt is given by Var(Πt) = Var(tA + (1 − t)B)
= Var(tA) + Var((1 − t)B) + 2Covar(tA,(1 − t)B)
= t^2Var(A) + (1 − t)^2Var(B) + 2t(1 − t)Covar(A,B)
= t^2Var(A) + (1 − t)^2Var(B)+ 2t(1−t)ρ(A,B)√[Var(A)Var(B)]

= (tσ_A)^2 + 2t(1 −t)ρ(A,B)σ_Aσ_B + ((1 − t)σ_B)^2

Question 15

portfolio Πt #1
The shapes of these curves
Proposition 7.1: T

Answer 15

The shapes of these curves are concave:

The curve in the σ-r plane given parametrically
by √[(tσ_A)^2 + 2t(1−t)ρ(A,B)σ_Aσ_B + ((1 − t)σ_B^2],

tr_A + (1 − t)rB for 0 ≤ t ≤ 1 lies to the left of the line segment
connecting the points (σ_A,r_A) and (σ_B ,r_B ).

Question 16

proof of
portfolio Πt #1
The shapes of these curves
Proposition 7.1:

by looking as all points on curve as investments

Answer 16

Since ρ(A,B) ≤ 1,
√[(tσA)^2 + 2t(1 − t)ρ(A,B)σ_Aσ_B + ((1 − t)σ_B )^2
≤ √[(tσA)^2 + 2t(1 − t)σ_Aσ_B + ((1 − t)σ_B)^2]
= √[(tσ_A + (1 − t)σ_B)^2]
= tσ_A + (1 − t)σ_B.

The result follows from the fact that the parametric equation of
the line segment connecting the points (σA,rA) and (σB ,rB ) is
{(tσ_A + (1 − t)σ_B, tr_A + (1 − t)r_B ) | 0 ≤ t ≤ 1}

Question 17

The feasible set

as#1 was 2 investments

Answer 17

Suppose now that there are many different investments A1,… . ,An
available.
We can invest our one unit of currency by investing ti
in Ai for each 1 ≤ i ≤ n as long as Σ_[i=1,n]of t_i = 1.

What are all possible pairs (σ,r) corresponding to these portfolios?
This set of points in the σ-r plane is called the feasible set.

Question 18

example diagram

consider an indifference curve similar to a trajectory of a ball with u middle at max r and w at end where risk larger than that at u and a point v below u

this is a decreasing curve

Answer 18

cant say
u≺v≺w as on same indifference curve

but
can say v ≺u
can say w ≺u

so violated axiom?

thus indifference curves are non decreasing

u is more preferable than w
and u is more preferable than v

Question 19

feasible set diagram

efficient frontier

Answer 19

by considering all possible investments between those invested

shapes moons, “comb”

efficient frontier is the top of it feasible set is all of it

Question 20

EFFICIENT PORTFOLIOS

Answer 20

An efficient portfolio is a feasible portfolio that provides the greatest expected return for a given level of risk, or equivalently,
the lowest risk for a given expected return. This is also called an optimal portfolio.

(Which portfolios among all possible ones should an investor satisfying axioms A1,A2 and A3 choose?)

Which portfolio along the efficient frontier will our investor choose? This is where risk preferences
start playing a role.

Question 21

efficient frontier

Answer 21

The efficient frontier is the set of all efficient portfolios. Obviously, our investor should choose a portfolio along
the efficient frontier!
Joining all investments st concave

• Feasible sets are convex along efficient frontier
• efficient frontier is the top (non jagged edge of “comb”)

ie diagram comb A to B on left outside follow this:
The feasible set is convex along the efficient frontier, in the sense that for any two portfolios A and B in the feasible set, there exist feasible portfolios above the portfolios in the segment connecting A and B.

Question 22

considering the different investors and their indifference curves (non decreasing) we can ask which are those points/investments which are equally attractive

Answer 22

Consider investors X (with no risk tolerance at all) and W (risk neutral) discussed before together with investor U whose indifference curves are given in a diagram (not shown)

indifference curves are level curves of some smooth function F(σ,r)

Question 23

Optimal portfolios when?

Answer 23

Optimal portfolios occur where indifference curves are
tangent to the efficient frontier.

**Otherwise, if it occurs at a point where the indifference curve intersects the efficient frontier transversally, find an almost parallel
indifference curve very close to the original one and to its left

**diagram of “comb”

A good portfolio shown as straight line through efficient frontier
A BETTER is shown to cross at a point on efficient frontier st somewhere (further back) it has the same gradient ie would be tangent at that point

one that isnt a tangent is some kind of curve

Question 24

examples of curves drawn

Answer 24

• drawing those of investor Y above our feasible set until FIRST one which intersects with the feasible sets

ie it might have a “lower attractiveness” (not first curve we draw) but we have found an optimal investment at intersection. This point is a point of tangency between the efficient frontier and the indifference curves

*eg if curve passing through intersection and non tangent to the efficient frontier will cross the efficient frontier again, this wouldn’t be our optimal because could find ones with higher attractiveness (this time drawing curves higher up) which still cross the efficient frontier -our investment wasnt optimal

Question 25

We now add a risk-free investment (only one)

this is on the returns axis ie 0 risk called B

what happens when investing in B and in some risky ones in feasible set

Answer 25

Let rB be its (expected)
return. Since rB is constant, its covariance with the returns of any other portfolio Π is zero so the portfolio Πt consisting of t units of currency invested in B and (1 − t) units of currency invested in Π has expected return

E (tB + (1 − t)Π) = tr_B + (1 − t)rΠ

(where we used B and Π to denote also the returns of the
investments B and Π) and standard deviation of return
√[Var(tB + (1 − t)Π)] = √[Var((1−t)Π)]
=√[(1 − t)^2Var(Π)] = |1 − t|σ_Π

The curve t → (σΠt,rΠt) for t ≤ 1 is a straight line passing
through the points (0,rB ) and (σΠ,rΠ) and all the points on or
below such a line will be part of the feasible set.

ie t → ((1-t)σA,tr_B +(1-t)r_A)

line from B to point A in feasible set eg -2 pounds in B means borrow 2 pounds and new collection of feasible set

Question 26

What happens to the efficient frontier? Consider the set S
consisting of all the slopes s of lines ℓs
in the σ-r plane which pass
through the point (0,rB ) and intersect the feasible set. Let m be
the supremum of S. Consider now the line ℓm which is above all
the others

Answer 26

The line ℓm will either be tangent to the efficient frontier or
asymptotic to it.
We will see in Chapter 8 that, if we impose additional conditions
on markets and investors, ℓm cannot be an asymptote of the
efficient frontier and so it is tangent to it.

Question 27

The line ℓm will either be tangent to the efficient frontier or
asymptotic to it.

Answer 27

This point of tangency is called the market portfolio and we shall denote the corresponding portfolio with M.
The new efficient frontier, ℓm is called
the capital market line.

• the new efficient frontier is line from B to top of efficient frontier , its tangent to it and the point corresponds to the market portfolio
• the investment will be the market portfolio- the new point of tangency on the straight line from B and an investment of risk free in B
• ie straight line intersecting at top of comb

Question 28

Theorem 7.2:

Answer 28

In the presence of a risk-free investment there exists an
(essentially) unique investment choice consisting entirely of risky investments which is efficient, namely, the market portfolio.

Any other efficient investment is a combination of an investment in the market portfolio and in the risk-free investment.

Question 29

line from B to point A in feasible set eg -2 pounds in B means borrow 2 pounds and new collection of feasible set

Answer 29

ignoring the risk free investment and consider some risk investments to give some feasible set

drawing line from B to each point and so new feasible set is formed

eg triangle coming outwards
THE FEASIBLE SET CONSISTING OF INVESTMENTS IN BOTH RISKY AND NON RISKY

the new efficient frontier is upper left boundary of the feasible set (straight line above original concave)

Question 30

SUMMARISE:

Answer 30

• if it’s on the capital market line its an efficient investment and its optimal
• if it’s on the old feasible set then it it consists of risky investments
• this is a unique point because otherwise it would not be optimal (diagrams in lectures show this)