Download Portfolio Theory - University of Toronto

Portfolio Theory
An Investor’s Preference
Return
4
2
3
1
Risk
• 2 dominates 1 - same risk, but higher return
• 2 dominates 3 - same return, but lower risk
• 4 dominates 3 – same risk, but higher return
Indifference Curves
Indifference Curve
Return
-
Represents individual’s
willingness to trade-off
return and risk
-
Assumptions:
1) 5 Axioms
2) Prefer more to less (Greedy)
=> Max E[U(W)]
3) Risk aversion
Increasing Expected Utility
Risk
Indifference Curves
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)
=> Max E[U(W)]
3) Risk aversion
4) Assets jointly normally
distributed
Increasing Expected Utility
Standard Deviation σR
The transition
Indifference Curve
-
-
Represents individual’s
willingness to trade-off
return and risk
Assumptions:
From U(RETURN, RISK)
To U(expected return, standard deviation of return)
•
This transition means the objects of choice for an
investor is normally distributed.
•
Because only if a distribution is normally
distributed can it be described only by its mean
and standard deviation.
•
This poses a question:
“Does that mean the risky assets investors invest
in have to be always normally distributed in terms
of their returns?”
1) 5 Axioms
2) Prefer more to less (Greedy)
=> Max E[U(W)]
3) Risk aversion
4) Assets jointly normally
distributed
The answer: NO! Individual asset’s return does not have normal distribution.
So why should we assume normally distributed returns for the objects of choice?
The transition
Indifference Curve
-
-
Represents individual’s
willingness to trade-off
return and risk
Assumptions:
From U(RETURN, RISK)
To U(expected return, standard deviation of return)
•
This transition means the objects of choice for an
investor is normally distributed.
•
Because only if a distribution is normally
distributed can it be described only by its mean
and standard deviation.
•
This poses a question:
“Does that mean the risky assets investors invest
in have to be always normally distributed in terms
of their returns?”
1) 5 Axioms
2) Prefer more to less (Greedy)
=> Max E[U(W)]
3) Risk aversion
4) Assets jointly normally
distributed
The answer: NO! Individual asset’s return does not have normal distribution.
So why should we assume normally distributed returns for the objects of choice?
BECAUSE RATIONAL INDIVIDUAL INVESTORS CHOOSE TO INVEST IN A
PORTFOLIO BUT NOT A SINGLE ASSET!
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.
NOTE: (N = No. of stocks in the portfolio)
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.
3 reasons to support mean-variance criteria
1) As the table shows, a portfolio consisting of large number of risky assets
tend have a return that is very 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).
3) Cost-effective consideration: data required to compute 3rd and 4th
moments are very demanding. Both computational and cost inefficient.
Conclude
•
Thus we are convinced that expected returns and variances are good
enough to describe the objects of choice.
•
From the investor’s point of view, it is cost-effective, and more intuitive.
•
From the objects of choice, if the unit of choice is an investment portfolio
with large number of risky assets, it is close to normally distributed.
•
So we are cool with the indifference curves.
•
Now, we’ll work on the computations of expected return and standard
deviation (i.e., the square root of variance) of an investment portfolio.
•
We make our lives easy by assuming individual asset’s return is
normally distributed to illustrate. We don’t buy this assumption, but for
the purpose of math, we’ll stick with it.
Math Review I
• Asset j’s return in State s:
Rjs = (Ws – W0) / W0
• Expected return on asset j:
E(Rj) = ∑sProb(State=s)xRjs
• Asset j’s variance:
σ2j = ∑sProb(State=s)x[Rjs- E(Rj)]2
• Asset j’s standard deviation:
σj = √σ2j
• Thus, Rj ~ N(E(Rj), σ2j)
Math Review I
• Covariance of asset i’s return & j’s return:
Cov(Ri, Rj)= E[(Ris- E(Ri))x(Rjs- E(Rj))]
=∑sProb(State=s)x[Ris- E(Ri)]x[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:
$100,000
40%
Expected Return:
rbad = ($80,000 – $100,000) / $100,000 = -20%
(Notation: Let αs denote the Prob(State=s)
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: A case of 2 risky assets
• Assume you have 2 risky assets (x & y) to
invest, both are normally distributed.
Rx ~ N(E(Rx), σ2x) & Ry ~ N(E(Ry), σ2y)
In your investment portfolio, you put a in x, b in y.
• a + b = 1 (a and b in %)
Your Portfolio’s Expected Return E(Rp) is:
E(Rp) = E[aRx + bRy]
=aE(Rx)+ bE(Ry)
Illustration: A case of 2 risky assets
• Rx ~ N(E(Rx), σ2x) & Ry ~ N(E(Ry), σ2y)
To calculate your Portfolio Variance:
σ 2p
= 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)
Var σ2p
= a2 σ2x + b2 σ2y + 2abσxσyρxy
s.d. σp
= √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
• Thus, the portfolio’s return is normally distributed too
Rp ~ N(E(Rp), σ2p)
Illustration: A case of 2 risky assets
σp
= √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
σp increases as ρxy increase.
Implication: given a (and thus b, because they add up to 1),
if ρxy is smaller, the portfolio’s variance is smaller (i.e.,
the risk is lower)
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.
Qs.: Can you link this concept to the reason why we focus
on expected return and s.d. of return as the objects of
choice?
Diversification
Proposition: portfolio of less than perfectly
correlated assets always offer better risk-return
opportunities than the individual component assets
on their own.
Proof:
If ρxy = 1 (perfectly positively correlated)
then, σp = a σx + b σy
=> portfolio s.d. is merely the weighted avg. s.d.
If < 1 (less than perfectly correlated)
then, σp < a σx + b σy
=> portfolio s.d. is less than the weighted avg. s.d.
Varying the portfolio weight
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 = portfolio weight on Asset x (in %), a+b=100%
a
Varying the portfolio weight
Suppose:
σp
rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)
σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
20%
ρxy=1
12%
0%
100%
a = portfolio weight on Asset x (in %), a+b=100%
a
Min-Variance opportunity set with
the 2 risky assets
E(Rp)
13%
=1
%8
σp
12%
20%
Varying the portfolio weight
Suppose:
rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)
σp
σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
20%
ρxy=0.3
12%
0%
100%
a = portfolio weight on Asset x (in %), a+b=100%
a
Min-Variance opportunity set with
the 2 risky assets
E(Rp)
13%
 = .3
%8
σp
12%
20%
Varying the portfolio weight
Suppose:
σp
rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)
σp = √(a2 σ2x + b2 σ2y + 2abσxσyρxy)
20%
ρxy=-1
12%
0%
100%
a = portfolio weight on Asset x (in %), a+b=100%
a
Min-Variance opportunity set with
the 2 risky assets
E(Rp)
13%
 = -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
- Given a level of expected return, the
lowest possible risky is?
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
Investors’ choices with many 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
Investors’ choices with many
risky assets, no risk-free asset
CML
E(Rp)
B
Q
M
A
rf
σp
Investors’ choices with many
risky assets, no risk-free asset
CML
E(Rp)
B
Q
M
This guy’s investment
portfolio consist of
around -50% in riskfree asset, 150% on
the market portfolio
A
rf
This guy’s investment
portfolio consist of
around 50% in riskfree asset, 50% on the
market portfolio
σp
2-Fund Separation
• 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.
2-Fund Separation:
• “Each investor will have a utility-maximizing
portfolio that is a combination of the risk-free
asset and a portfolio (or fund) of risky assets that
is determined by the line drawn from the riskfree rate of return tangent to the investor’s
efficient set of risky assets.”