Title: Asset Allocation
1Asset Allocation
2Asset Allocation The Fundamental Question
- How do you allocate your assets amongst different
assets? - There are literally thousands of assets available
to you for investment purposes. Which ones will
you invest in, and how much will you invest in
each of these? - In this class, we will limit our discussion only
to the universe of stocks and one risk-free asset
(the Treasury). However, the same ideas can be
extended to any set of risky assets, like real
estate, hedge fund investments, etc. - We can divide this question into two separate
decisions.
3The Two Decisions
- How do you allocate your assets amongst different
assets? There are two decisions that you have to
make - A. How will you allocate between the risk-free
asset and the portfolio of risky assets (stocks)?
- To figure this out, ask yourself this question
Of all the money you have available to you for
investment, how much do you want to keep in
cash? - We shall see that there is no best way to
allocate your allocation will depend on your
preferences and risk tolerance. Thus, your
decision will depend on criteria like your
current age, total wealth, current financial
commitments, etc. - B. How will you allocate between different risky
assets within the portfolio of risky assets. - We shall see that there is an optimal allocation
between risky assets one best way to divide
all your cash between the risky stocks. - This is the primary question we shall deal with
over here.
4Assumptions
- Traditionally, when we decide asset allocation,
we will assume that all the assets are fairly
priced. - If, instead, one asset is not fairly priced (it
is under- or over-valued), it may be optimal for
you to simply allocate all your money into that
one asset! - Moreover, we will assume that we know, or can
estimate from past history, all that we need to
know about the expected returns, volatilities,
and correlations of our stocks.
5The Objective of Allocation
- What should be our objective when we decide to
allocate between different assets? - For example, why should we not invest in only
one asset? We may not wish to invest in one asset
as one of our goals is to diversify (and, thus,
reduce) risk. - Specifically, we will set our objective as Earn
the highest return per unit of risk. - We will measure our return as excess returns R
Rf - We will measure our risk by the volatility of the
return. The volatility is the standard deviation
of the return.
6The Sharpe Ratio
- Given our objective of maximizing the return per
unit of risk, we will use a metric based on the
expected return and volatility of the asset that
is commonly known as the Sharpe Ratio. - The Sharpe ratio measures the tradeoff between
risk and return for each portfolio. - Sharpe Ratio (R-Rf)/(Vol).
- We will use the Sharpe ratio as our criteria for
choosing between different allocations.
7Maximizing the Sharpe Ratio
- It is important to note that maximizing the
Sharpe ratio, i.e., maximizing the excess return
per unit of risk is not equivalent to either (a)
maximizing the return, or (b) minimizing the
risk. - Example
- Between 1/1994 and 9/2004, the average return
earned on a stock of KO was 11.85/year. Over the
same period, the return earned on PEP was
14.69/year. The volatility of KOs return was
25.35/year, and the volatility of PEPs return
was 24.28/year. Thus, PEP earned a higher return
with a lower risk than KO over this period. - Qt Suppose you expect KO and PEP to perform the
same over the next 10 years. Does this mean that
you should invest all your money in Pepsi, and
nothing in Coke? Answer No.
8Notations and Useful Formulae
- Let there be two assets, Asset 1 and Asset 2.
- R1, R2 expected returns on Asset 1 and Asset 2,
respectively. - Vol1, Vol2 volatilities of the returns on Asset
1 and Asset 2, respectively. - The volatility is the standard deviation of the
returns. - Rho12 correlation between returns on Asset 1
and Asset 2 - W1 proportion in Asset 1.
- W2 proportion in Asset 2.
- Rp expected return on portfolio of the two
assets w1 R1 w2 R2 - Volp Volatility of portfolio of the two assets
(w1)2 (Vol1)2 (w2)2 (Vol2)2 2 x Rho12 x
w1 x w2 x Vol1 x Vol2
9Asset Allocation A. Risky vs. Riskless Asset
- First, consider the allocation between the risky
and riskless asset. - Rf expected return on riskfree asset.
- Rp expected return on risky 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
individual-specific. Thus, there is no one best
portfolio allocation.
10Portfolio 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 risky asset is 15.
11Riskfree Return5, Risky Return12, Vol of
Risky Asset0.15
12Portfolio Return vs. Portfolio Volatility
13How 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
individual-specific 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
risk-return tradeoff.
14B. Portfolio of Risky Assets
- 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 investor
allocate between two or more risky stocks? - Once again we will assume that investors want to
maximize the Sharpe ratio (so that investors want
the best tradeoff between return and volatility).
15Determining 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 portfolio). - When we only have two risky assets, 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 minimum
variance 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.
16Frontier 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.
- 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 over the 10-year sample
period 1994-2004, with monthly data. - As can be seen, the optimal weight for a
portfolio (to get the maximum Sharpe ratio) is
about 28 for KO. - If the exact answer is required, we can easily
solve for it using the solver in Excel..
17Volatility-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, drawn such that it has
the risk-free rate as its intercept. - This is because the slope of the line that passes
connects the risk-free 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 also called the capital
allocation line. All investments represented on
this line are optimal (and will comprise of
combination of the riskfree asset and risky stock
portfolio).
18Portfolio Return-Volatility Frontier KO PEP,
1994-2004