Notesale: Turn your study into money

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1

Portfolio Theory

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

2

Diversification and Portfolio Risk
• What is the best way to put together a portfolio of stocks?
• Before Harry Markowitz (Nobel prize winner and the father of
portfolio theory) wrote his seminal article on portfolio selection in
1952, investors were using ad hoc methods and relying mostly on
their intuition to pick stocks and form their portfolios
• Markowitz’s contribution was to think about investing in a more
structured, “scientific” way
...
Diversification makes sense as well as
being common practice
...
”

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

3

Diversification and Portfolio Risk
•

Diversification reduces risk
•
•

The old adage “Don’t put all your eggs in one basket” is a good starting point but
not specific enough to apply to investment decisions

•

•

This, I presume, is known by every person and does not even require any
background in finance

Portfolio Theory is basically about how to distribute your proverbial “eggs” into
those “baskets”

Think of risk as having two components
•

Market or Systematic risk: risk related to the macro economic factors

•

Firm-specific or Nonsystematic risk: risk related to the individual firm

•

Total risk = Systematic + Nonsystematic

•

Firm-specific risk can be reduced or even eliminated by holding a portfolio of
stocks because the reasons behind the fortunes of one company are likely to
be different from the reasons of another

•

Market risk cannot be eliminated because it affects firms all the same (i
...
, "a
rising tide lifts all boats“)

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

4

Diversification and Portfolio Risk
• In the graphs below, the x-axis shows the number of securities in a
portfolio and the y-axis shows the risk of the portfolio (as measured
by the standard deviation of the portfolio)
• Look at what happens to the risk of the portfolio as more and more
securities are included in the portfolio
• At the end of this lecture we will be able to quantify the reduction in
risk that can be achieved by investing in multiple securities

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

5

Two Risky Assets
•

We start simple, with only two risky assets
...
2%
Gold
E(r)= 1/3(45%+10%-25%) = 10%
Variance = 1/3(45% – 10%)2 + 1/3(10% – 10%)2 + 1/3(-25% – 10%)2 = 817
Standard deviation = √817 = 28
...
Prof
...
2
28
...
dev
...
2

C

0
...
2

–2
...
6

25
...
6

11
...
2

0
...
2

10
...
4

10
...
6

F

0

1

45

10

–25

10

28
...
Prof
...
0 - - - - - - - - - - - - - - - - - - -  A (S&P 500)
11
...
4 - - - - - - - - - - - - - - -D 
10
...
2

18
...
2

28
...
Prof
...
Prof
...
We
can now do the same using the statistics of the two assets and the portfolio
variance formula
...

S&P 500
Gold

Recession
–14%
45%

Normal
12%
10%

High Growth
38%
–25%

Expected returns and standard deviations of the individual assets (as previously calculated):
S&P 500: E(r) = 12%, σ = 21
...
6%
Cov(rSP , rG) = 1/3[(-14-12)*(45-10) + 0 + (38-12)*(–25-10)]=-607
Var(PC) = 0
...
2)2 + 0
...
6)2 + 2*0
...
2*(–607)=126
Std
...
= √126 = 11
...

With the portfolio variance formula it is easy to see that the significant reduction in risk is
due to the negative covariance term
...
Prof
...
52*202)+(0
...
14%

•

•

Try it: Suppose there are 2 stocks with standard deviations of 20% each, zero covariance between
the two and that you are equally invested in both
...
5*20% +0
...
Prof
...
Prof
...
Prof
...
Dev
...
Dev
...
Prof
...
Prof
...
The two
risky assets act as a perfect hedge in
this scenario
...
Investors who seek higher
expected returns will choose portfolios
to the northeast, and investors who
seek lower standard deviations will
choose portfolios to the southwest
...
Prof
...
Prof
...
3)
A and B are two feasible portfolios
which can be constructed by mixing
the two assets
•
•

•
•
•

E(rA)= 8
...
45%
E(rB)= 9
...
70%

Two CALs are drawn from the riskfree rate (5%) to portfolios A and B
Would you rather invest in A or B?
Let’s calculate the Sharpe Ratios
• SA = (8
...
45 = 0
...
5 – 5)/11
...
38

•
•

Portfolio B dominates A because it
offers more return per unit of risk
Can you do better?

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

18

Two Risky Assets and a Risk-Free Asset
•

We want the portfolio with the highest
Sharpe Ratio - in graphical terms, the CAL
with the steepest slope

•

This will be the line originating from the
risk-free rate and tangent to the portfolio
opportunity set

•

The point of tangency determines the
optimal risky portfolio

•

The solution method is to pose it as an
optimization problem and use the tools of
calculus
Max(wi) = [E(rp)-rf]/σp
s
...
Σwi = 1

•

When dealing with more than two assets the
analytical solution becomes impractical, so
in general we will use Excel’s “Solver” (or
some other software optimizer)

•

Note: The formula for the two asset case is given in
the book and if you plug in this set of values the
optimal portfolio turns out to be: wE = 0
...
4 and the corresponding E(rp) = 11% and σp=
14
...


Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

19

Two Risky Assets and a Risk-Free Asset
•
•

Notice that the optimal risky
portfolio does not depend on the
risk aversion of the investor!
Does that mean that the investor
will put all his money in Portfolio P?
•

•

•

No
...


The investor can set his desired
level of expected return or the
standard deviation he is
comfortable with, which will pin
down the actual allocation
Alternatively, we can assign a risk
aversion parameter and a utility
function to calculate the amount he
should be investing in the risky
asset

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

20

Two Risky Assets and a Risk-Free Asset
•

By using Excel’s Solver (or the formula
given in the book) we solved for the
optimal weights of the risky portfolio, P:
wE = 0
...
4
E(rp) = 11% , σp= 14
...
11 – 0
...
1422)
y* = 0
...
40*0
...
2976
...
60*0
...
4463

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

21

Markowitz Model
•
•
•

The Markowitz model is the generalization of the two risky asset problem to multiple
risky assets
It takes the security characteristics as inputs (means, variances, covariances) and
calculates the minimum variance frontier
This is done by minimizing the variance of the portfolio (by changing the weights)
for a target level of return, r, and constraining the weights to sum up to 1

1 n
minimize
 wi w j ij
2 i , j 1
n

subject to

 w E (r )  r
i

i

w

1

i 1
n

i 1

•

i

If you have taken an optimization course you will recognize this as a quadratic
program
...


Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

22

Efficient Frontier
•

•

•

•

To determine the minimum variance
frontier, run the optimizer for a series
of target returns and graph the returns
against the portfolio standard
deviations calculated using the
optimum weights
The first insight is that the individual
assets lie on the inside of the frontier
...
Hence investing in a single
asset is inefficient
...
Hence the bottom
part of the frontier is also inefficient
...
This curve is
called the efficient frontier
...
Prof
...


•

Once the optimal portfolio P is found,
there is no further need to change the
composition of the assets in the
portfolio, which means that a portfolio
manager will offer this portfolio, and
this portfolio only, to all his clients!

•

Clients’ risk preferences only come
into play in terms of how much they
want to invest in this portfolio

•

Reality check: This theoretical solution
is based on assuming away real-world
frictions such as taxes, liquidity,
borrowing/lending restrictions, etc
...
Prof
...
Assume the simplest case:
all stocks have the same standard deviation and zero correlation with each
other
...
In a big way
...
Prof
...
What happens when assets are positively
correlated? For simplicity let’s assume all pairwise correlations between the
stocks are equal to some constant ρ
...

n

   wi w j ij
2
p

i , j 1
n

n

   w    wi w j ij
2
p

 
2
p

i 1

2
n

2
i

2
i

i , j 1
j i

n

1
 ij i j
n2
i , j 1


j i

2

n2  n
 

 2
n
n2
 2 (1   )
2
p 
  2
n
2
p

•
•

What happens as n increases? Goes to infinity?
What happens when ρ is zero? One? Less than one?

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University

26

The Power of Diversification
•

•
•
•

We don’t really have to assume an identical
correlation structure among the stocks to
derive this result
...

When n goes to infinity (n-1)/n approaches
1, hence portfolio variance approaches the
average covariance
This means that diversification eliminates
all risk, except the average covariance of
the stocks
Since investors can do this easily (and
happily!) we’ll see in our next set of
lectures that:
•
•

The market only rewards people for taking
“market risk”
There is no reward for taking “firm-specific
risk”

1 n 2
   i
n i 1
2

n
1
 ij  2
  ij
n  n i , j 1
j i

n

   wi w j ij
2
p

i , j 1
n

n

   w    wi w j ij
2
p

2
i

i 1

2
i

i , j 1
j i

n

1 2 n 1
   2  i   2  ij
i 1 n
i , j 1 n
2
p

j i

1
n

2

2
p   

(n  1)
 ij
n

Koç Üniversitesi – Mühendislik Fakültesi

Assist
...
Umut Gökçen / Koç University



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