Asset Allocation Methods
Lecture 10
1
The Fundamental Question

How do you allocate your assets amongst
different assets?

A. The allocation between risk free and a
portfolio of risky assets.

B. The allocation between different risky asset
within the portfolio of risky assets.
2
The Decisions That an Investor
Must Make

1. Which is the risky stock portfolio that
results in the best risk-return tradeoff?

2. After making the choice of the risky stock
portfolio, how should you allocate your assets
between this risky portfolio and the riskfree
asset?
3
The Sharpe Ratio

The Sharpe ratio measures the tradeoff
between risk and return for each
portfolio.
◦
◦
◦
◦

R = Expected Portfolio Return.
Rf = Risk free Rate.
Vol = Portfolio Vol.
Portfolio Vol = (w1)^2 (vol of asset 1)^2 + (w2 )^2
(vol of asset 2)^2 + 2 (correlation) (w1 )(w2 ) (vol of
asset 1)(vol of asset 2) + …+ ….(additional terms of
volatilities and correlations).
Sharpe Ratio = (R-Rf)/(Vol).
4
Asset Allocation: Risky vs.
Riskless Asset
Rf = expected return on risk free asset.
 Rp= expected return on risky asset portfolio.
 Volatility of risk free asset = 0.
 W1 = proportion in risk free asset.
 W2 = proportion in risky asset.


Is there an optimal w1, w2?
5
Portfolio of Risky + Riskless Asset

Portfolio Return = w1 Rf + w2 Rp.

Portfolio Variance = (w1)^2 (0) + (w2 )^2 (vol
of risky asset)^2 + 2 (correlation) (w1 )(w2 )
(0)(vol of risky asset).

Portfolio Volatility = w2 *(vol of risky asset).

The following graph shows for the case when
the mean return for the risk free asset is 5%,
the mean return for the risky asset is 12%,
and the volatility of the risky asset is 15%.
6
Risk free Return=0.05, Risky
Return=0.12, Vol of Risky
Asset=0.15
w_1 (weight of riskfree) w_2 (weight of risky) Portfolio Return Portfolio Vol
0
1
0.12
0.15
0.1
0.9
0.113
0.135
0.2
0.8
0.106
0.12
0.3
0.7
0.099
0.105
0.4
0.6
0.092
0.09
0.5
0.5
0.085
0.075
0.6
0.4
0.078
0.06
0.7
0.3
0.071
0.045
0.8
0.2
0.064
0.03
0.9
0.1
0.057
0.015
1
0
0.05
0
7
Portfolio Return
Portfolio Return vs. Portfolio
Volatility
0.15
0.1
CAL
0.05
0
0
0.05
0.1
0.15
0.2
Portfolio Volatility
8
Capital Allocation Line (CAL)

The slope equals the increase in return of the
portfolio for a unit increase in volatility.
Therefore, it is also called the reward-tovariability ratio. We will also refer to this ratio
as the Sharpe ratio.

The greater the slope the greater the reward
for taking risk.
9
How to allocate between the risk
free asset and the risky stock
portfolio.

It is not possible to make a decision on allocation
between the risk free asset and the risky stock portfolio
based solely on the Sharpe ratio.

Your decision to allocate between the risky asset and
the risk free asset will be determined by your level of
risk aversion and your objectives, depending on factors
like your age, wealth, horizon, etc.

Although different investors may differ in the level of
risk they take, they are also alike in that each investor
faces exactly the same risk-return tradeoff.
10
A Digression into “Market
Timing”

There are funds that actively manage the
decision to allocate between the risky/riskless
asset for the investor: these funds are
typically called “market allocation” funds.

Typically, the funds actively manage a mix of
stocks, bonds and money market securities,
and they may change the fraction of their
holding in each of these assets, depending on
what they think is “optimal” at that time.
11
Returns to Market Timing
If you could time the market, using the S&P
500, what would your returns be over the
period Jan 1950- Dec 2002?
 We start with $1 on January 1, 1950, and ask
how much we would have on December 31,
2002.
 1. Buy and hold strategy: $51.60 (average
return=7.72%).
 2. Perfect timer: $238,203 (26.31%) (!!).
 3. Occasional timer (miss the worst 10
months): $200 (10.52%).
 4. Mis-timer (miss the best 10 months):
$16.87 (5.48%).
 5. Miss both best/worst 10 months: $65.49
(8.21%).

12
The Optimal Risky Stock
Portfolio

We discussed the allocation between the risky
(stock) portfolio and the riskless (cash)
portfolio.

Now we will consider the other decision that
an investor must make: how should the risky
stock portfolio be constructed?
13
Determining the Optimal Portfolio

If we can plot the portfolio return vs. Portfolio volatility
for all possible allocations (weights), then we can easily
locate the optimal portfolio with the highest Sharpe
ratio of (Rp - Rf)/(Vol of risky portfolio).

Because there exists one specific portfolio with the
highest Sharpe ratio, all investors will want to invest in
that portfolio. Thus, the weights that make up this
portfolio determines the optimal allocation between the
risky assets for all investors.
14
Volatility-Return Frontier

Graphically, the optimal portfolio (with the
highest Sharpe ratio) is the portfolio that lies
on a tangent to the graph. This tangent is
drawn so that it has the risk free rate as its
intercept.

This tangent is now the capital allocation line.
All investments represented on this line are
optimal (and will comprise of combination of
the risk free asset and risky stock portfolio).
15
Portfolio Return-Volatility Frontier
14.90%
14.80%
14.70%
14.60%
14.50%
Series1
14.40%
14.30%
14.20%
14.10%
21.00%
22.00%
23.00%
24.00%
25.00%
26.00%
16
Creating the mean variance
frontier
How to use a spreadsheet to
calculate the frontier when
there are more than 2 assets
17
The Minimum Variance Frontier

When we have more than 2 assets, it gets
difficult to consider all possible portfolio
combinations. Instead, we will make the
process simpler by considering only a subset
of portfolios: those portfolios that have the
minimum volatility for a given return.

When we plot the return and volatilities of
these portfolios, the resultant graph will be
known as the minimum variance (or volatility)
frontier.
18
The Steps

1. For each asset (and for the time period that
you have chosen), calculate the mean return,
volatility and the correlation matrix.

2. Set up the spreadsheet so that the Solver
can be used. See the sample spreadsheet.

3. Repeat 2 for a range of returns, and plot
the frontier (return vs. volatility).
19
Step 1: Assembling the Data


A. Fix the time period for the analysis. You
want a sufficiently long period so that your
estimates of the mean return, volatility and
correlation are accurate. But you don’t want a
period too long, because the data may not be
valid.
B. Estimate the mean return and volatility for
each of your assets. Next, calculate the
correlation between each pair of assets. If
there are N assets, you will have to calculate
N(N-1) correlations.
20
Step 2: Setting up the
spreadsheet to use the Solver
(1/4)



The objective here is to set up the
spreadsheet in a manner that is easy to use
with the solver.
The estimates of the return, volatility and the
correlation matrix are used to set up a matrix
for covariances, which is then used to
calculate the portfolio volatility for a given set
of weights.
To create the frontier, you will ask the solver
to find you the weights that gives you the
minium volatility for a required return.
21
Step 2: Using the Solver (2/4)


1.Target Cell: When you call the solver, it will
ask you to specify the objective or the “target
cell”. Your objective is to minimize the
volatility - so in this case, you will specify the
cell that calculates the portfolio volatility
[$B$25]. As you want to minimize the
volatility, you click the “Min”.
2. Constraints: You will have to specify the
constraints under which the optimization must
work. There are two constraints that hold, and
a third which will usually also apply.
22
Using the Solver: Constraints on
the Optimization (3/4)



1. First, the sum of the weights must add up
to 1.
2. Second, you have to specify the required
rate of return for which you want the portfolio
of least volatility. For each level of return, you
will solve for the weights that give you the
minimum volatility. To construct the frontier,
you will vary this required return over a range.
Thus, you will have to change this constraint
every time you change the required return.
Third, if there are constraints to short-selling,
you will have to specify that each portfolio
weight is positive.
23
Step 2: (4/4)

Finally, you specify the arguments that need
to be optimized. In this case, you are
searching for the optimal weights, so you will
have to specify the range in the spreadsheet
where the portfolio weights used [A20, A21,
A22].
24
Step 3

The final step is to simply repeat step
2, until you have a sufficiently large
data set so that the minimum variance
frontier can be plotted.
.
25
The Optimal Allocation
We can now use the graph of the minimum
variance frontier to figure out the portfolio
with the highest Sharpe Ratio. This portfolio
will be the portfolio such that the CAL passing
through it is tangent to the minimum variance
frontier.
 The weights of this portfolio determines the
optimal allocation within the assets that make
up the “risky portfolio”. All investors should
opt for this allocation.
 The portfolio will always be on the upper
portion of the frontier, above the portfolio with
the lowest volatility - this portion is called the
efficient frontier.

26
Diversification (1/6)



We have observed that by combining
stocks into portfolios, we can create an
asset with a better risk-return tradeoff.
The reduction of risk in a portfolio occurs
because of diversification. By combining
different assets into a portfolio, we can
diversify risk and reduce the overall
volatility of the portfolio.
Let us review the factors that affect how
risk can be diversified. Here we will ignore
the issue of allocation (as we have already
considered it), and instead assume that
our portfolio is equally weighted.
27
diversification in an equally
weighted portfolio (2/6)
There are two main factors that affect
the extent to which volatility can be
reduced: the number of assets in the
portfolio, and the correlation between
the assets.
 Increasing the number of assets
reduces the volatility of the portfolio.
 Adding an asset with a low correlation
with the existing assets of a portfolio
also helps to reduce the volatility of the
portfolio.

28
(3/6)
To examine the effect of correlation and
the number of assets, lets assume, for
simplicity, that each of the assets have
the same volatility (say, 40%) and the
same average correlation with each
other.
 The portfolio volatility can then be
calculated by the usual formula, and we
can examine the reduction in volatility
of the portfolio as we change the
number of assets, or the correlation.

29
Sample spreadsheet (4/6)
How m any stocks does it take to diversify for a given correlation
Average Vol Avg Correlation
40.00%
0.6
50.00%
40.00%
30.00%
Series1
20.00%
10.00%
0.00%
0
20
40
60
N
1
3
5
10
20
30
40
50
100000
Port Vol
40.00%
34.25%
32.98%
32.00%
31.50%
31.33%
31.24%
31.19%
30.98%
30
Some Conclusions (5/6)



By changing N=number of stocks in
portfolio, and the correlation, we can
examine how the portfolio volatility
decreases.
We can make the following observations:
1. For all positive correlation, there is a threshold
beyond which we cannot reduce the portfolio
volatility. This threshold depends on the magnitude
of the correlation. If the correlation is zero or less
than zero, then it is possible to bring down the
portfolio volatility to zero by having a large
number of assets. This threshold represents the
undiversifiable or the systematic risk of the
portfolio.
31
Some Conclusions (6/6)
2. As the correlation decreases, the more we can
reduce the portfolio volatility. However, it takes
more assets to bring down the portfolio volatility to
its theoretical minimum.
 Example: if the correlation is 0.9 and the average
volatility of each stock in the portfolio is 40%, then
the lowest portfolio volatility that is possible is
about 37.95%. We can reach within 0.5% of this
minimum volatility by creating a portfolio of only 4
assets. Suppose instead that the average
correlation is 0.5. Then the lowest possible
portfolio volatility is 28.28%; however, to reach
within 0.5% of this value, we need as many as 30
stocks.

32
In Summary (1/2)
1. The optimal allocation is determined in two steps. First,
we decide the allocation between the risky portfolio, and
the riskless asset. Second, we determine the allocation
between the assets that comprise the risky portfolio.
 2. As every portfolio of the risky assets and the riskless
asset has the same Sharpe ratio, there is not one optimal
portfolio for all investors. Instead, the allocation will be
determined by individual-specific factors like risk aversion
and the objectives of the investor, taking into account
factors like the investor’s horizon, wealth, etc.
 3. When we are considering the allocation between
different classes of risky assets, it is possible to create a
portfolio that has the highest Sharpe Ratio. The weights of
the risky assets in this portfolio will determine the optimal
allocation between various risky assets. This portfolio can
be determined graphically by drawing the capital allocation
line (CAL) such that it is tangent to the minimum variance
frontier. This portfolio will always lie on the upper part of
the frontier (or on the efficient part of the frontier).

33
In Summary (2/2)


4. The extent to which you can decrease the volatility of
the portfolio depends also on the correlation. The lower
the average correlation of the stocks in your portfolio,
the lower you can decrease the volatility of your
portfolio.
5. The homework provides you with an exercise to
determine the optimal allocations.
34