Portfolio Theory
Indifference Curve
Indifference Curve
Expected Return E(r)
-
Represents individual’s
willingness to trade-off
return and risk
-
Assumptions:
1) 5 Axioms
2) Prefer more to less (Greedy)
3) Risk aversion
4) Assets jointly normally
distributed
Increasing Utility
Standard Deviation σ(r)
Dominance
Expected Return
4
2
3
1
Standard Deviation
• 2 dominates 1; has a higher return
• 2 dominates 3; has a lower risk
• 4 dominates 3; has a higher return
Jointly normally distributed?
Individual stock return may not be normally distributed, but a portfolio
consists of more and more stocks would have its return increasingly close
to being normally distributed.
Jointly normally distributed?
1st moment: Mean = Expected return of portfolio
2nd moment: Variance = Variance of the return of portfolio (RISKNESS)
Mean and Var as sole choice variables => Distribution of return can be
adequately described by mean and variance only
That means, distribution has to be normally distributed.
2 reasons to support mean-variance criteria
1) As the table shows, a portfolio with large number of risky assets tend to
be close to normally distributed.
2) The fact that investors rebalance their own portfolios frequently will act
so as to make higher moments (3rd, 4th, etc) unimportant (Samuelson
1970)
Math Review I
• Asset j’s return in State s:
rjs = (Ws – W0) / W0
• Expected return on asset j:
E(rj) = ∑sαsrjs
• Asset j’s variance:
σ2j = ∑sαs[rjs- E(rj)]2
• Asset j’s standard deviation:
σj = √σ2j
Math Review I
• Covariance of asset i’s return & j’s return:
Cov(ri, rj)= E[(ris- E(ri)) (rjs- E(rj))]
=∑sαs[ris- E(ri)] [rjs- E(rj)]
• Correlation of asset i’s return & j’s return:
ρij = Cov(ri, rj) / (σiσj)
-1 ≤ ρij ≤ 1
When ρij = 1 => i and j are perfectly positively
correlated. They move together all the time.
When ρij = -1 => i and j are perfectly negatively
correlated. They move opposite to each other all
the time.
A simple example: Asset j
60%
$150,000
Good State: rgood = ($150,000 – $100,000) / $100,000 = 50%
$80,000
Bad State:
$10,000
40%
rbad = ($80,000 – $100,000) / $100,000 = -20%
Expected Return:
E(rj) = ∑sαsrjs = 60%(50%) + 40%(-20%) = 22%
Variance:
σ2j = ∑sαs[rjs- E(rj)]2 = 60%(50%-22%)2 + 40%(-20%-22%)2 = 11.76%
Standard Deviation:
σj = √σ2j = √11.76% = 34.293%
Math Review II
• 4 properties concerning Mean and Var
Let ũ be random variable, a be a constant
1) E(ũ+a) = a + E(ũ)
2) E(aũ) = aE(ũ)
3) Var(ũ+a) = Var(ũ)
4) Var(aũ) = a2Var(ũ)
Portfolio Theory – a bit of history
•
Modern portfolio theory (MPT)—or portfolio theory—was introduced by
Harry Markowitz with his paper "Portfolio Selection," which appeared in
the 1952 Journal of Finance. 38 years later, he shared a Nobel Prize with
Merton Miller and William Sharpe for what has become a broad theory for
portfolio selection.
•
Prior to Markowitz's work, investors focused on assessing the risks and
rewards of individual securities in constructing their portfolios. Standard
investment advice was to identify those securities that offered the best
opportunities for gain with the least risk and then construct a portfolio from
these. Following this advice, an investor might conclude that railroad
stocks all offered good risk-reward characteristics and compile a portfolio
entirely from these. Intuitively, this would be foolish. Markowitz formalized
this intuition. Detailing a mathematics of diversification, he proposed that
investors focus on selecting portfolios based on their overall risk-reward
characteristics instead of merely compiling portfolios from securities that
each individually have attractive risk-reward characteristics. In a nutshell,
inventors should select portfolios not individual securities. (Source:
riskglossary.com)
Link to his Nobel Prize lecture if you are interested:
• http://nobelprize.org/economics/laureates/1990/markowitz-lecture.pdf
Illustration: 2 risky assets
• Assume you have 2 risky assets (x & y) to
choose from, both are normally distributed.
rx ~ N(E(rx), σ2x) & ry ~ N(E(ry), σ2y)
• You put a of your money in x, b in y.
• a+b=1
• Portfolio Expected Return:
E(rp) = E[arx + bry]=aE(rx)+ bE(ry)
Illustration: 2 risky assets
• rx ~ N(E(rx), σ2x) & ry ~ N(E(ry), σ2y)
• Portfolio Variance:
σ2p = E[rp - E(rp)]2
= E[(arx + bry)-E[arx + bry]]2
= E[(arx - aE[rx])+(bry - bE[bry])]2
= E[a2(rx - E[rx])2 + b2(ry - E[ry])2 + 2ab(rx
- E[rx])(ry - E[ry])]
= a2 σ2x + b2 σ2y + 2abCov(rx, ry)
= a2 σ2x + b2 σ2y + 2abCov(rx, ry)
σ2p = a2 σ2x + b2 σ2y + 2abσxσyρxy
σp
= √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
Illustration: 2 risky assets
σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
σp increases as ρxy increase.
Implication: given a (and thus b), if ρxy is smaller,
variance of portfolio is smaller.
Diversification: you want to maintain the expected
return at a definite level but lower the risk you
expose. Ideally, you hedge by including another
asset of similar expected return but highly
negatively correlated with your original asset.
Diversification
Proposition: portfolio of less than perfectly
correlated assets always offer better riskreturn opportunities than the individual
component assets on their own.
Proof:
If ρxy = 1 (perfectly positively correlated)
then, σp = a σx + b σy
If < 1 (less than perfectly correlated)
then, σp < a σx + b σy
Varying the portion on X & Y
Suppose:
E(rp)
rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)
E(rp) = E[arx + bry]=aE(rx)+ bE(ry)
13%
%8
0%
100%
a
Varying the portion on X & Y
Suppose:
rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)
σp
σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
20%
ρxy=1
ρxy=-1
ρxy=0.3
12%
0%
100%
a
Min-Variance opportunity set with
the 2 risky assets
E(rp)
13%
 = -1
 = .3
 = -1
=1
σp
12%
20%
%8
Min-Variance opportunity set with
the Many risky assets
E(rp)
Efficient
frontier
Individual risky assets
Min-variance opp. set
σp
Min-Variance opportunity set
E(rp)
Min-Variance Opportunity set – the locus of risk & return
combinations offered by portfolios of risky assets that yields
the minimum variance for a given rate of return
σp
Efficient set
E(rp)
Efficient set – the set of mean-variance choices from the
investment opportunity set where for a given variance (or
standard deviation) no other investment opportunity offers a
higher mean return.
σp
Individual’s decision making with 2
risky assets, no risk-free asset
E(rp)
U’’’ U’’ U’
Efficient set
S
P
Q
More
risk-averse
investor
Less
risk-averse
investor
σp
Introducing risk-free assets
• Assume borrowing rate = lending rate
• Then the investment opp. set will involve any
straight line from the point of risk-free assets to
any risky portfolio on the min-variance opp. set
• However, only one line will be chosen because it
dominates all the other possible lines.
• The dominating line = linear efficient set
• Which is the line through risk-free asset point
tangent to the min-variance opp. set.
• The tangency point = portfolio M (the market)
Capital market line = the linear
efficient set
E(rp)
E(Rm)
M
5%=Rf
σm
σp
Individual’s decision making with 2
risky assets, with risk-free asset
CML
E(rp)
B
Q
M
A
rf
σp
Implication
• All an investor needs to know is the
combination of assets that makes up
portfolio M as well as risk-free asset. This
is true for any investor, regardless of his
degree of risk aversion.