Asset Allocation
Week 4
1
Asset Allocation: The Fundamental
Question
• How do you allocate your assets amongst different assets?
• Traditionally, we divide the discussion here into two parts:
• A. The allocation between riskfree 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
• Thus, there are two 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?
• Typically, the first objective of a financial advisor is to
determine for her clients the appropriate allocation
between the risky and riskless assets, and then to choose
how the risky portfolio should be constructed.
3
The Sharpe Ratio
• To compare one portfolio with another, we will use
a metric called the “Sharpe Ratio”.
• The Sharpe ratio measures the tradeoff between
risk and return for each portfolio.
–
–
–
–
R = Expected Portfolio Return.
Rf = Riskfree 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).
• We may use the Sharpe ratio as a criteria for
determining the right portfolio.
4
Asset Allocation: Risky vs. Riskless
Asset
• Consider the allocation between the risky and riskless
asset.
• Rf = expected return on riskfree asset.
• Rp= expected return on risky asset portfolio.
• Volatility of riskfree asset = 0.
• W1 = proportion in riskfree asset.
• W2 = proportion in risky asset.
• Is there an optimal w1, w2?
• We shall show that the choice of w1, w2 is individualspecific, and will depend on the individual’s risk aversion
and objectives. Thus, there is no one optimal portfolio.
5
Portfolio of Risky + Riskless Asset
• To calculate the portfolio return and portfolio variance
when we combine the risky asset and riskless asset, we can
use the usual formulas, noting that the volatility of the
riskfree rate is zero.
• 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).
• This simplification in the formula for the portfolio
volatility occurs because the vol of the riskfree asset is
zero.
• To understand the tradeoff between risk and return, we can
graph the portfolio return vs the the portfolio volatility.
• The following graph shows this graph for the case when
the mean return for the riskfree asset is 5%, the mean
return for the risky asset is 12%, and the volatility of the 6
risky asset is 15%.
Riskfree 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 graph from the previous slide is called the capital
allocation line (CAL).
• For the special case when one of the two assets is the
riskfree asset, the CAL is a straight line, with a slope of
(Rp-Rf)/(Vol of risky portfolio).
• This slope equals the increase in return of the portfolio for
a unit increase in volatility. Therefore, it is also called the
reward-to-variability ratio. We will also refer to this ratio
as the Sharpe ratio.
• The greater the slope the greater the reward for taking risk.
Ideally, you want to achieve the highest return per unit
risk, so that you choose a risky portfolio that gives you the
steepest slope.
• Note that this tradeoff will be essentially determined by the
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mean return and volatility of the risky portfolio.
How to allocate between the riskfree asset and
the risky stock portfolio.
• The conclusion we draw from the straight-line graph is that: when we
combine a riskfree asset with the risky stock portfolio, all portfolios
have the same Sharpe ratio.
• Therefore, it is not possible to make a decision on allocation between
the riskfree asset and the risky stock portfolio based solely on the
Sharpe ratio. Instead, we will have to take into account individualspecific considerations. There is no single allocation here that is best
for all investors.
• Your decision to allocate between the risky asset and the riskfree asset
will be determined by your level of risk aversion and your objectives,
depending on factors like your age, wealth, horizon, etc. The more risk
averse you are, the less you will invest in the risky asset.
• Although different investors may differ in the level of risk they take,
they are also alike in that each investor faces exactly the same riskreturn tradeoff.
10
A Digression into “Market Timing”
• Why not actively manage the allocation between the
riskfree asset and the risky stock portfolio?
• 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.
• Such a trading strategy is also called market timing. The
objective of market timing is to be invested in stocks in a
bull market, and to be invested in bonds/cash in a bear
market.
11
Returns to Market Timing
• Here is an example that illustrates how you could do if you
were a good/bad market timer. 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%).
• Moral of the story: time the market only if you have a good
crystal ball.
– But its tempting to keep trying even when one doesn’t have a
crystal ball.
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?
• Once again we will assume that investors want to
maximize the Sharpe ratio (so that investors want the best
tradeoff between return and volatility).
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).
• When we only have two risky assets, as in this case, it is easy to
construct this graph by simply calculating the portfolio returns for all
possible weights.
• When we have more than 2 assets, it becomes more difficult to
represent all possible portfolios, and instead we will only graph only a
subset of portfolios. Here, we will choose only those portfolios that
have the minimum volatility for a given return. We will call this graph
the variance-return frontier.
• Once we solve for this minimum variance frontier, we will show that
there exists one portfolio on this frontier that has the highest Sharpe
ratio, and thus is the optimal stock 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
Frontier with KO and PEP
• As an example, consider a portfolio of KO and PEP. What should be
the optimal combination of KO and PEP?
– Refer to excel file on web page.
• As we only have two assets here, we can easily tabulate the Sharpe
ratio for a range of portfolio weights, and check which portfolio has
the highest Sharpe ratio.
• The next slide shows the results. In the calculation of the Sharpe ratio,
it is assumed that the riskfree rate is constant (which is not strictly
true). The portfolio mean and portfolio return are calculated with the
usual formulae over the 10-year sample period 1993-2002, with
monthly data.
• As can be seen, the optimal weight for a portfolio (to get the maximum
Sharpe ratio) appears to be in the range of 0.6 in KO. If the exact
answer is required, we can easily solve for it using the excel solver.
• It can also be observed that, amongst these 11 portfolios, the portfolio
with the minimum volatility is one that invests 50% in each of the two
stocks. This is the minimum variance portfolio. The minimum variance
portfolio may be different from the portfolio with the highest Sharpe 15
ratio.
The Sharpe Ratio: KO + PEP
Weight
Vol
Return
Sharpe Ratio Constant Rf
100.00%
25.40%
12.54% 0.375790522
3.00%
90.00%
24.18%
12.40% 0.388925896
80.00%
23.19%
12.27% 0.399591614
70.00%
22.45%
12.13% 0.406607569
60.00%
21.98%
11.99% 0.408887809
50.00%
21.81%
11.85% 0.405711573
40.00%
21.94%
11.71% 0.396954454
30.00%
22.37%
11.57% 0.383158416
20.00%
23.07%
11.43% 0.365393678
10.00%
24.03%
11.29% 0.344983145
0.00%
25.22%
11.15%
0.323221
Max Sharpe Ratio
0.408887809
16
Volatility-Return Frontier
• Consider the graph of the portfolio return vs. Portfolio
volatility.
• 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 riskfree rate as its
intercept.
• This is because the slope of the line that passes connects
the riskfree asset and the risky portfolio is equal to the
Sharpe ratio. Thus, the steeper the line, the higher the
Sharpe ratio. The tangent to the graph has the steepest
slope, and thus the portfolio that lies on this tangent is the
optimal portfolio (having the highest Sharpe ratio).
• This tangent is now the capital allocation line. All
investments represented on this line are optimal (and will
comprise of combination of the riskfree asset and risky
17
stock portfolio).
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%
18
Creating the mean variance frontier
How to use a spreadsheet to calculate
the frontier when there are more than
2 assets
19
The Minimum Variance Frontier
• With two assets, as we saw, we can construct the frontier
by brute force - by listing almost all possible portfolios.
• When we have more than 2 assets, its 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.
• We will use Excel’s “Solver” for these calculations (look
under Tools. If it is not there, then add it into the menu
through Add-in).
20
The Steps
• We will implement the procedure in three 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. Your objective here is to
determine the weights of the portfolio that will allow you
to achieve a specified required rate of return with the
lowest possible volatility.
• 3. Repeat 2 for a range of returns, and plot the frontier
(return vs. volatility).
21
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.
22
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.
23
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.
24
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.
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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].
26
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.
.
27
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.
28
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
29
weighted.
Factors that affect 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.
30
(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.
31
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%
32
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.
33
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.
34
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).
35
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.
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