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
As an investor (or investment advisor), the
ultimate goal is to choose the portfolio that is
“optimal.”
◦ How do you know what is optimal for a particular
investor? (Preferences)
◦ How do we evaluate risky assets? (Risk and return
measurement)
◦ Given preferences and asset characteristics, how do
we select a portfolio? (Optimization)
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Prior to the 1950’s, investment analysis was
largely non-quantitative and unscientific.
◦ “The wise man puts all his eggs in one basket and
watches the basket.” Andrew Carnegie\Mark Twain

Harry Markowitz’s path-breaking work in the
1950’s revolutionized portfolio theory.
◦ Why would any rational person take on greater risk
unless compensated by greater expected return?
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Obviously, they range from very sophisticated
(particularly for large institutions) to some
very silly approaches (some financial
planners, some online firms)
Today’s approach will (hopefully) fit with the
former group.
Let’s briefly look at the latter group.
The following is constructed from several
online “optimizers.”
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Current Age
50
Current Savings
$100,000
Annual Savings (today’s $)
$12,000
Retirement Age
65
Annual Retirement Spending
(today’s $)
$70,000
Years in Retirement
20
Investment Return
10.5%
Inflation (%)
3.5%
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
What’s with the “years in retirement” input? Is
this a polite way of asking someone when
they plan on dying?
The user inputs the all-important investment
return number?
◦ This is like going to the doctor with a headache,
and the doctor asking you if it is a tumor.

Is there any notion of risk?
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Congratulations! Your plan works.
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Warning! Your plan does not work.
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Save more
Retire later
Spend less
Increase investment return
1.
2.
3.
4.
The first 3 seem kind of unpleasant. Why
not simply choose a higher return?

◦
Note that a fifth potential solution is to die
sooner.
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You typically get pie charts of the following
variety:
Where did this come from?
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

In a later module, we will look more deeply at
“life-cycle” or dynamic investing.
In this module, we’ll focus on a quantitative
approach to trading-off risk and return, in a
manner that best suits an investor’s tastes.
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A risky asset (or portfolio) is simply one
where its return is not known with certainty.
Mathematically, we say that the return on a
risky asset is a random variable.
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You can’t describe a risky return with simply
one number. In general, one uses a
probability distribution.
We shall find that 2 metrics are most helpful
for describing risky returns: expected returns
and variances.
We denote asset i’s expected return and
variance as:
2
E (r ) and 
i
i
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The expected return is the mean, or average
return.
Since we don’t know the true E(ri), we must
estimate it.
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
One naïve method is to simply use the
historical average return.
That is, if you look at asset i’s historical
returns over periods 1, 2, … , T, its historical
average return is:
1
r  r
T
T
i
t 1
i ,t
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Historical average returns are notoriously
unreliable. For example, if you compute an
average return over one 5-year period, it will
provide little guidance for what will take place
over the following 5-year period.
Typically, we use equilibrium models or
economic judgment to compute expected
returns. We will see some of this today.
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Variance measures the degree of spread of a
distribution around its expected value.
Again, we do not know  , so we must estimate
it.
The standard estimator using historical data is:
2
i
S
2
i
1

 r  r 
T 1
2
T
t 1
i ,t
i
and the estimator of standard deviation is simply
its square root, Si.
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Historical estimations of variances are also
imperfect. There are a wide variety of
estimation techniques that improve upon the
simple one just described (GARCH, factor
models, implied volatility)
That being said, the historical estimate of
variance is much more predictive than the
historical estimate of expected return.
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Finally, we also need a measure of the degree
to which assets’ returns co-move.
We denote the covariance between assets i
and j as  .
A positive (negative) covariance implies that
when one asset’s return is above its expected
return, the other asset’s return is likely to be
above (below) it expected return.
The correlation between assets i and j is
i,j



 

i,j
i,j
i
j
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
The standard covariance estimator using
historical data is:
1
C 
 r  r r  r 
T 1
T
i,j

i ,t
t 1
i
j ,t
j
The estimate of correlation (Corri,j) is then:
C
Corr 
SS
i,j
i,j
i
j
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
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A portfolio is a set of weights w1, w2, … , wn on
individual assets. It is a vector. The weights must
sum to 1.
For each selection of weights, the portfolio’s
expected return and variance are determined.
The expected return on a portfolio is equal to:
E (r )  w E (r )  w E (r )  ...  w E (r )
p
1
1
2
2
n
n
  w E (r )
n
i 1

i
i
Thus, the portfolio’s expected return is simply
the weighted average of the individual asset’s
returns.
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
The variance of a portfolio is equal to:


2
p
   ww
n
n
i 1
j 1
i
j
i,j
The variance of a portfolio is complicated,
since it must take into account covariances
across all assets.
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If you have a negative weight in the risky
portfolio, you are shorting that asset.
You are essentially “borrowing” the risky
asset.
◦ What is the most you can make when short-selling?
◦ What is the most you can lose when short-selling?

While some investors engage in extensive
shorting (hedge funds), many others abstain
(most mutual funds)
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How do you measure someone’s preferences
over portfolio returns?
This is actually complicated.
In microeconomics, we typically think of
utility over consumption goods such as
apples and cars. Mathematically, it would
simply be a function such as u(apples, cars).
Now, since portfolio returns are random
variables, the mathematical mapping is
considerably more complicated.
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For the most part, we will make a simplifying
assumption about preferences that will permit us
to use a very simple utility structure.
We assume mean-variance preferences.
1. Individuals care only about expected return and
variance.
2. They like higher expected return, and like lower
variance.

Note that if all asset returns are normally
distributed, and investors are risk averse, mean
variance utility will hold. However, in reality,
return distributions deviate from normality.
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While any M-V utility function that is increasing in the
mean and decreasing in the variance will work, we
shall find it useful to use a simple, linear function.
A very common mean-variance utility function is the
following linear formulation:
A
u  E (r )  
2
p



2
p
The parameter A represents the investor’s degree of
risk aversion. (Why the ½?)
One starts with the mean, and then subtracts out a
“penalty” for risk.
The higher the risk aversion, the higher is the risk
penalty.
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If A>0, the investor dislikes risk, and is risk
averse.
If A=0, the investor doesn’t care about risk,
and is risk neutral.
If A<0, the investor likes risk, and is risk
seeking.
We will almost always assume risk aversion.
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Consider the following thought experiment.
◦ Imagine a risky portfolio with a mean of 8% and a
variance of 4%.
◦ Would you rather have that portfolio or a riskless
asset paying 2%? What about 5%?
◦ Determine the riskless rate that would leave you
indifferent.
◦ Suppose that riskless rate is 2.8%. Then we have:
A
.028  .08  .04
2
A  2.6
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
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The individual’s optimization problem is to
select the portfolio (weights) so as to
maximize utility.
Optimization (in general) involves an
objective function, control variable(s), and
constraints.
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Here the objective function is the meanvariance utility:
A
u  E (r )  
2
p
2
p
which now can be written as:
A
u   w E (r )    w w 
2
n
i 1
n
i
i
n
i 1 j 1
i
j
i,j
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Here the control variables are the n weights.
That is, all other aspects of the problem are
beyond the investor’s control.
◦ Can you think of any potential real-world violation
of this assumption?
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
The standard constraint is that the weights
sum to 1:
n
w
i 1

i
 1.
The set of constraints can be expanded to
account for short-sell constraints or the
inclusion of upper and lower bounds.
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We write the investor’s portfolio problem as:
A
max  w E (r )    w w 
2
subject to
n
w1 , w2 ,...,wn
n
w
i 1


i
i 1
i
i
n
n
i 1
j 1
i
j
i,j
 1.
For this particular problem, one can solve it
completely using basic calculus (Lagrange).
However, as constraints are added, numerical
optimization (Solver) is the way to go.
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
You are currently deciding how to allocate
your portfolio across the following 8 asset
classes:
1.
2.
3.
4.
5.
6.
7.
8.
US Bonds
Foreign Bonds
US Large Cap Growth
US Large Cap Value
US Small Cap Growth
US Small Cap Value
Foreign Equity (Developed)
Emerging equity
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Expected return
US Bonds
Foreign Bonds
US Large Cap Growth
US Large Cap Value
US Small Cap Growth
US Small Cap Value
Foreign Equity
Emerging Equity
Standard Deviation
4.00%
2.75%
10.76%
8.17%
10.91%
9.91%
4.70%
4.95%
3.32%
8.35%
15.40%
15.92%
20.31%
20.04%
18.64%
22.70%
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Foreign
US Bonds Bonds
US Bonds
Foreign
Bonds
US Large
Cap
Growth
US Large
Cap Value
US Small
Cap
Growth
US Small
Cap Value
Foreign
Equity
Emerging
Equity
US Large
US Small
Cap
US Large Cap
US Small Foreign
Growth Cap Value Growth Cap Value Equity
Emerging
Equity
1.00
0.58
0.05
0.03
-0.07
-0.05
0.15
0.20
0.58
1.00
0.34
0.35
0.23
0.25
0.53
0.51
0.05
0.34
1.00
0.93
0.91
0.84
0.90
0.82
0.03
0.35
0.93
1.00
0.89
0.92
0.87
0.76
-0.07
0.23
0.91
0.89
1.00
0.94
0.81
0.73
-0.05
0.25
0.84
0.92
0.94
1.00
0.76
0.67
0.15
0.53
0.90
0.87
0.81
0.76
1.00
0.91
0.20
0.51
0.82
0.76
0.73
0.67
0.91
1.00
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

Suppose your Risk Aversion parameter A =
4.0.
Here are the optimal weights:
US Bonds
-18.03%
Foreign
Bonds
26.61%
US LargeCap
Growth
US LargeCap Value
US SmallCap
Growth
Small-Cap
Value
Foreign
Equity
Emerging
Markets
536.67% -204.07% -139.81% 140.81% -237.84% -4.34%
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Holy cow, is the what “state-of-the-art”
advice gives you?
Would you put 536% in large cap growth?
Would you short foreign stock 238%? Large
cap value 204%?
What is driving these results?
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It seems that the optimized results are largely
driven by the historical experience in average
returns.
But, the advice is wildly inconsistent with any
equilibrium notion.
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A very common adjustment is to plug in
market-based expected returns.
Essentially, we “reverse-optimize” and find
the expected returns that are consistent with
the existing market weights. (Note analogy to
using implied volatilities)
An easy way of doing this is to simply use
CAPM expected returns. (Comment on why
we need 2 inputs)
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
As of January 1, 2018, these are the relative
market weights of the 8 asset classes:
US Bonds
19.86%
Foreign
Bonds
25.97%
US LargeCap
Growth
US LargeCap Value
US SmallCap
Growth
Small-Cap
Value
12.69%
12.46%
1.07%
1.00%
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Foreign
Equity
21.23%
Emerging
Markets
5.72%
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
Using the market weights (keep them fixed),
we construct a data series of monthly market
returns. We then use the CAPM SML equation
to derive expected returns:
E (r )  r   ( E (r )  r )

We need 2 inputs:
f
m
f
◦ For the riskless rate, I choose the current 5 year TBill yield of 2.33% per annum.
◦ For the market risk premium, E ( r )  r , I use the 25
year historical average (for these 8 asset classes) of
4.83% per annum.
◦ Make sure you divide by 12 for monthly inputs
m
f
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US Bonds
2.75%


Foreign
Bonds
4.78%
US LargeCap
Growth
US LargeCap
Value
US SmallCap
Growth
8.73%
8.83%
9.88%
SmallCap
Value
Foreign
Equity
Emerging
Markets
10.60%
11.75%
9.55%
These expected returns make much more
sense. They are broadly consistent with
relative risk.
Note again that these are annualized. Use the
monthly returns for optimization.
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
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Suppose your Risk Aversion parameter A =
4.0.
Here are the optimal weights:
US Bonds
11.70%
Foreign
Bonds
29.29%
US LargeCap
Growth
US LargeCap Value
US SmallCap
Growth
Small-Cap
Value
14.08%
12.87%
0.11%
1.72%
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Foreign
Equity
23.40%
Emerging
Markets
6.83%
44
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The positions now seem reasonable.
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
Consider the advice to an investor with risk
aversion parameter A= 4.3875 .
US Bonds
Foreign
Bonds
US LargeCap
Growth
US LargeCap
Value
US SmallCap
Growth
19.86%
25.97%
12.69%
12.46%
1.07%


SmallCap
Value
1.00%
Foreign
Equity
21.23%
Emerging
Markets
5.72%
These are precisely the market weights. (Is
there a simple formula for deriving this? Try
to figure it out.)
Easy to use “guess & check”. Don’t use Solver
for this.
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
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A very common approach for quantitative
asset allocation is to begin with the marketimplied expected returns.
Then, you add to the model your own “views”
on the relative expected returns. For
example, you might be bullish on US large
cap growth, and believe a 15% expected
return is reasonable.
Finally, you weight the market’s views and
your own views, based on the degree of
confidence you have in your views.
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