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Asset Allocation

- Week 4

Asset 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.

The 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.

Assumptions

- 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.

The 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.

The 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.

Maximizing 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.

Notations 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

Asset 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.

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 risky asset is 15.

Riskfree Return5, Risky Return12, Vol of

Risky Asset0.15

Portfolio Return vs. Portfolio Volatility

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

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.

B. 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).

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 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.

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.
- 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..

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, 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).

Portfolio Return-Volatility Frontier KO PEP,

1994-2004