© 2009 mcgraw-hill ryerson limited 2-1 chapter 2 diversification and asset allocation expected...

© 2009 McGraw-Hill Ryerson © 2009 McGraw-Hill Ryerson LimitedLimited 2-2-11

Chapter 2Chapter 2Diversification and Diversification and

Asset Allocation Asset Allocation

Expected Return and VariancesExpected Return and Variances PortfoliosPortfolios Diversification and Portfolio RiskDiversification and Portfolio Risk Correlation and DiversificationCorrelation and Diversification Markowitz Efficient FrontierMarkowitz Efficient Frontier Asset Allocation and Security Asset Allocation and Security

Selection Decisions of Portfolio Selection Decisions of Portfolio FormationFormation

© 2009 McGraw-Hill Ryerson © 2009 McGraw-Hill Ryerson LimitedLimited 2- 2- 22

““Put all your eggs in the one basket and Put all your eggs in the one basket and WATCH THAT BASKET!” -- Mark TwainWATCH THAT BASKET!” -- Mark Twain

As we shall see in this chapter, this is probably not As we shall see in this chapter, this is probably not the best advice!the best advice!

Intuitively, we all know that diversification is Intuitively, we all know that diversification is important when we are managing investments.important when we are managing investments.

In fact, diversification has a profound effect on In fact, diversification has a profound effect on portfolio return and portfolio risk.portfolio return and portfolio risk.

But, how does diversification work, exactly?But, how does diversification work, exactly?


Diversification and Asset AllocationDiversification and Asset Allocation

Our goal in this chapter is to examine the role of Our goal in this chapter is to examine the role of diversification and asset allocation in investing.diversification and asset allocation in investing.

In the early 1950s, professor Harry Markowitz was the first to In the early 1950s, professor Harry Markowitz was the first to examine the role and impact of diversification.examine the role and impact of diversification.

Based on his work, we will see how diversification works, and Based on his work, we will see how diversification works, and we can be sure that we have “efficiently diversified we can be sure that we have “efficiently diversified portfolios.” portfolios.” An efficiently diversified portfolio is one that has the An efficiently diversified portfolio is one that has the

highest expected return, given its risk.highest expected return, given its risk. You must be aware of the difference between historic You must be aware of the difference between historic

return and expected return!return and expected return!


Example: Historical ReturnsExample: Historical Returns

From From http://finance.yahoo.com, you can download historical , you can download historical prices, and then calculate historical returns.prices, and then calculate historical returns.

How about the historical return on eBay from 1998 to 2002?How about the historical return on eBay from 1998 to 2002?

Year: Start: End: Return1999 $20.10 $31.30 55.7%2000 $31.30 $16.50 -47.3%2001 $16.50 $33.45 102.7%2002 $33.45 $33.91 1.4%

Average: 28.1%

eBay Prices:

Although it is important to be able to calculate historical returns, Although it is important to be able to calculate historical returns, we really care about expected returns, because we want to know we really care about expected returns, because we want to know how our portfolio will perform from today forward.how our portfolio will perform from today forward.


Expected ReturnExpected Return Expected return Expected return is the “weighted average” return on a risky is the “weighted average” return on a risky

asset, from today to some future date. The formula is:asset, from today to some future date. The formula is: To calculate an expected return, you must first:To calculate an expected return, you must first:

Decide on the number of possible economic scenarios that might occur,Decide on the number of possible economic scenarios that might occur, Estimate how well the security will perform in each scenario, andEstimate how well the security will perform in each scenario, and Assign a probability to each scenarioAssign a probability to each scenario (BTW, finance professors call these economic scenarios, “states.”)(BTW, finance professors call these economic scenarios, “states.”)

The next slide shows how the expected return formula is used The next slide shows how the expected return formula is used when there are two states. when there are two states. Note that the “states” are equally likely to occur in this example. BUT! Note that the “states” are equally likely to occur in this example. BUT!

They do not have to be. They can have different probabilities of They do not have to be. They can have different probabilities of occurring.occurring.

n

1ssi,si returnpreturn expected


Expected Return IIExpected Return II Suppose there are two stocks: NetCap and JMartSuppose there are two stocks: NetCap and JMart We are looking at a period of one year.We are looking at a period of one year. Investors agree that the expected return:Investors agree that the expected return: for NetCap is 25 percent and for JMart is 20 percentfor NetCap is 25 percent and for JMart is 20 percent Why would anyone want to hold JMart shares when NetCap is Why would anyone want to hold JMart shares when NetCap is

expected to have a higher return?expected to have a higher return? The answer depends on riskThe answer depends on risk

Jmart is expected to return 20 percentJmart is expected to return 20 percent But the realized return on Jmart could be significantly higher But the realized return on Jmart could be significantly higher

or lower than 20 percentor lower than 20 percent Similarly, the realized return on NetCap could be significantly Similarly, the realized return on NetCap could be significantly

higher or lower than 25 percent.higher or lower than 25 percent.


Expected ReturnExpected Return


Expected Risk PremiumExpected Risk Premium

Recall:Recall:

Suppose riskfree investments have an 8% return.Suppose riskfree investments have an 8% return. So, in the previous slide, the risk premium expected on Jmart So, in the previous slide, the risk premium expected on Jmart

is 12%, and 17% for Netcap.is 12%, and 17% for Netcap. This expected risk premium is the difference between the This expected risk premium is the difference between the

expected return on the risky asset in question and the certain expected return on the risky asset in question and the certain return on a risk-free investment.return on a risk-free investment.

rate riskfreeireturn expected ipremium risk expected


Calculating the Variance of Expected ReturnsCalculating the Variance of Expected Returns

The vThe variance of expected returns ariance of expected returns is calculated as the sum of is calculated as the sum of the squared deviations of each return from the expected the squared deviations of each return from the expected return, multiplied by the probability of the state. Here is the return, multiplied by the probability of the state. Here is the formula:formula:

n

1s

2isi,si return expectedreturnpVariance


Standard DeviationStandard Deviation The standard deviation is simply the square root of the The standard deviation is simply the square root of the

variance.variance.

The following slide contains an example that shows how The following slide contains an example that shows how to use these formulas in a spreadsheet.to use these formulas in a spreadsheet.

Variance Deviation Standard


Expected Returns and VariancesExpected Returns and Variances Equal State Probabilities Equal State Probabilities

(1) (3) (4) (5) (6)Return if Return if

State of State Product: State Product:Economy Occurs (2) x (3) Occurs (2) x (5)Recession -0.20 -0.10 0.30 0.15

Boom 0.70 0.35 0.10 0.05Sum: E(Ret): 0.25 E(Ret): 0.20

(1) (3) (4) (5) (6) (7)Return if

State of State Expected Difference: Squared: Product:Economy Occurs Return: (3) - (4) (5) x (5) (2) x (6)Recession -0.20 0.25 -0.45 0.2025 0.10125

Boom 0.70 0.25 0.45 0.2025 0.10125Sum: 0.20250

0.45

1.00 Sum is Variance:

Standard Deviation:

Probability ofState of Economy

0.500.50

1.00

Calculating Variance of Expected Returns:Netcap:

(2)

Probability ofState of Economy

0.500.50

Calculating Expected Returns:Netcap: Jmart:

(2) Note that this is for the individual assets, Netcap and Jmart.


Expected Returns and Variances, Expected Returns and Variances, Netcap and JmartNetcap and Jmart


PortfoliosPortfolios PortfoliosPortfolios are groups of assets, such as stocks and are groups of assets, such as stocks and

bonds, that are held by an investor.bonds, that are held by an investor. One convenient way to describe a portfolio is by One convenient way to describe a portfolio is by

listing the proportion of the total value of the portfolio listing the proportion of the total value of the portfolio that is invested into each asset.that is invested into each asset.

These proportions are called These proportions are called portfolio weightsportfolio weights.. Portfolio weights are sometimes expressed in percentages.Portfolio weights are sometimes expressed in percentages. However, in calculations, make sure you use proportions.However, in calculations, make sure you use proportions.


Portfolios: Expected ReturnsPortfolios: Expected Returns The expected return on a portfolio is the weighted average, of The expected return on a portfolio is the weighted average, of

the expected returns on the assets in that portfolio. The the expected returns on the assets in that portfolio. The formula, for “n” assets, is:formula, for “n” assets, is:

In the formula: In the formula: EE((RRPP) =) = expected portfolio return expected portfolio return

wwi i = portfolio weight in portfolio asset = portfolio weight in portfolio asset ii

EE((RRii) =) = expected return for portfolio asset expected return for portfolio asset ii

n

1iiiP REwRE


Example: Calculating Portfolio Expected Example: Calculating Portfolio Expected ReturnsReturns

Note that the portfolio weight in Jmart = 1 – portfolio Note that the portfolio weight in Jmart = 1 – portfolio weight in Netcap.weight in Netcap.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)Netcap Netcap Jmart Jmart Portfolio

Return if Portfolio Contribution Return if Portfolio Contribution ReturnState of Prob. State Weight Product: State Weight Product: Sum: Product:

Economy of State Occurs in Netcap: (3) x (4) Occurs in Jmart: (6) x (7) (5) + (8) (2) x (9)Recession 0.50 -0.20 0.50 -0.10 0.30 0.50 0.15 0.05 0.025

Boom 0.50 0.70 0.50 0.35 0.10 0.50 0.05 0.40 0.200

Sum: 1.00 0.225

Calculating Expected Portfolio Returns:

Sum is Expected Portfolio Return:


Variance of Portfolio Expected ReturnsVariance of Portfolio Expected Returns

Note: Unlike returns, portfolio variance is generally Note: Unlike returns, portfolio variance is generally notnot a simple weighted average of the variances of the a simple weighted average of the variances of the assets in the portfolio.assets in the portfolio.

If there are “n” states, the formula is:If there are “n” states, the formula is:

n

1s

2Psp,sP REREpRVAR


Portfolio VariancePortfolio Variance

In the formula, In the formula, VAR(RP) = variance of portfolio expected returnVAR(RP) = variance of portfolio expected return

pps s = probability of state of economy, s= probability of state of economy, s E(RE(Rp,sp,s) = expected portfolio return in state s) = expected portfolio return in state s E(RE(Rpp) = portfolio expected return) = portfolio expected return

Note that the formula is like the formula for the Note that the formula is like the formula for the variance of the expected return of a single asset.variance of the expected return of a single asset.


Calculating Variance of Portfolio ReturnsCalculating Variance of Portfolio Returns

It is possible to construct a portfolio of risky assets It is possible to construct a portfolio of risky assets with zero portfolio variance! What? How? (Open the with zero portfolio variance! What? How? (Open the spreadsheet and set the weight in Netcap to 2/11ths.) spreadsheet and set the weight in Netcap to 2/11ths.)

(1) (2) (3) (4) (5) (6) (7)Return if

State of Prob. State Expected Difference: Squared: Product:Economy of State Occurs: Return: (3) - (4) (5) x (5) (2) x (6)Recession 0.50 0.05 0.225 -0.18 0.0306 0.01531

Boom 0.50 0.40 0.225 0.18 0.030625 0.01531Sum: 1.00 Sum is Variance: 0.03063

0.175Standard Deviation:

Calculating Variance of Expected Portfolio Returns:


Diversification and Risk, I.Diversification and Risk, I. Table 2.7 Portfolio Standard Deviations


Diversification and Risk, II.Diversification and Risk, II. Figure 2.1 Portfolio Diversification


Why Diversification Works, I.Why Diversification Works, I. Correlation:Correlation: The tendency of the returns on two assets to The tendency of the returns on two assets to

move together. Imperfect correlation is the key reason why move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the diversification reduces portfolio risk as measured by the portfolio standard deviation.portfolio standard deviation. PositivelyPositively correlated assets tend to move up and down correlated assets tend to move up and down together.together.

NegativelyNegatively correlated assets tend to move in opposite correlated assets tend to move in opposite directions.directions.

Imperfect correlation, positive or negative, is why Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.diversification reduces portfolio risk.


Why Diversification Works, II.Why Diversification Works, II. The The correlation coefficientcorrelation coefficient is denoted by Corr( is denoted by Corr(RRAA, R, RBB) )

or simply, or simply, A,BA,B. . The correlation coefficient measures correlation and The correlation coefficient measures correlation and

ranges from:ranges from:

From: From: -1 -1 (perfect negative correlation)(perfect negative correlation)

Through:Through: 00 (uncorrelated)(uncorrelated)

To:To: +1+1 (perfect positive correlation)(perfect positive correlation)


Why Diversification Works, III.Why Diversification Works, III.


Why Diversification Works, IV.Why Diversification Works, IV.


Why Diversification Works, V.Why Diversification Works, V.


Calculating Portfolio RiskCalculating Portfolio Risk For a portfolio of two assets, A and B, the variance of For a portfolio of two assets, A and B, the variance of

the return on the portfolio is:the return on the portfolio is:

Where: wWhere: wA A = portfolio weight of asset A= portfolio weight of asset A

wwBB = portfolio weight of asset B = portfolio weight of asset B

such that such that wwAA + + wwBB = 1. = 1.

)(

),(

BABABABBAAp

BABBAAp

RRCORR

BACOV

2

222222

22222


Correlation and DiversificationCorrelation and Diversification

Suppose that as a Suppose that as a very conservativevery conservative, risk-averse , risk-averse investor, you decide to invest all of your money in a bond investor, you decide to invest all of your money in a bond mutual fund. Very conservative, indeed? mutual fund. Very conservative, indeed?

Uh,Uh, is this decision a wise one?is this decision a wise one?



Note: A correlation of 0.10 is assumed in calculations



The various combinations of risk and return available The various combinations of risk and return available all fall on a smooth curve.all fall on a smooth curve.

This curve is called an This curve is called an investment opportunity setinvestment opportunity set, , because it shows the possible combinations of risk because it shows the possible combinations of risk and return available from portfolios of these two and return available from portfolios of these two assets.assets.

A portfolio that offers the highest return for its level A portfolio that offers the highest return for its level of risk is said to be an of risk is said to be an efficient portfolioefficient portfolio..

The undesirable portfolios are said to be The undesirable portfolios are said to be dominateddominated oror inefficientinefficient..


Correlation and Risk &Return Trade offCorrelation and Risk &Return Trade off


Correlation and the Risk-Return Trade-OffCorrelation and the Risk-Return Trade-Off

Expected StandardInputs Return Deviation

Risky Asset 1 14.0% 20.0%Risky Asset 2 8.0% 15.0%Correlation 30.0%

Efficient Set--Two Asset Portfolio

0%

2%4%

6%8%

10%12%

14%16%

18%

0% 5% 10% 15% 20% 25% 30%

Standard Deviation

Exp

ecte

d R

etu

rn


Risk and Return with Multiple AssetsRisk and Return with Multiple Assets


The The MarkowitzMarkowitz Efficient Frontier Efficient Frontier

The Markowitz Efficient frontier is the set of portfolios with the The Markowitz Efficient frontier is the set of portfolios with the maximum return for a given risk AND the minimum risk for a maximum return for a given risk AND the minimum risk for a given a return.given a return.

For the plot, the upper left-hand boundary is the Markowitz For the plot, the upper left-hand boundary is the Markowitz efficient frontier. efficient frontier.

Efficient FrontierEfficient Frontier All the other possible combinations are inefficient. That All the other possible combinations are inefficient. That

is, investors would not hold these portfolios because is, investors would not hold these portfolios because they could get eitherthey could get either more return for a given level of risk, ormore return for a given level of risk, or less risk for a given level of return.less risk for a given level of return.


Example of Efficient FrontierExample of Efficient Frontier


Portfolio Formation DecisionsPortfolio Formation Decisions

Investors can form their using the following financial Investors can form their using the following financial assets :assets :

1- Domestic Stocks, 2- Domestic Bonds 1- Domestic Stocks, 2- Domestic Bonds (government, municipal, provincial bonds, corporate (government, municipal, provincial bonds, corporate bonds) 3- Derivative Instruments (options, futures, bonds) 3- Derivative Instruments (options, futures, swaps), 4- Money market instruments (T-Bills, swaps), 4- Money market instruments (T-Bills, certificate of deposits, repurchase agreements, etc), 5- certificate of deposits, repurchase agreements, etc), 5- Foreign Stocks and Bonds. Foreign Stocks and Bonds.


Asset Allocation and Security SelectionAsset Allocation and Security Selection Investors face with two problems when they form portfolios Investors face with two problems when they form portfolios

of multiple securities from different asset classes.of multiple securities from different asset classes.

1. Asset Allocation Decision1. Asset Allocation Decision Asset allocation problem involves a decision of what Asset allocation problem involves a decision of what

percentage of investorpercentage of investor’’s portfolio should be allocated among s portfolio should be allocated among different asset classes (stocks, bonds, derivatives, money different asset classes (stocks, bonds, derivatives, money market instruments, foreign securities). market instruments, foreign securities).

2. Security Selection Decision2. Security Selection Decision Security selection is deciding which securities to pick in Security selection is deciding which securities to pick in

each class and what percentage of funds to allocate to these each class and what percentage of funds to allocate to these securities (for example choosing different stocks and their securities (for example choosing different stocks and their percentages within the asset class).percentages within the asset class).


Portfolio DecisionsPortfolio Decisions Suppose you have $100,000 for investment. First you have to Suppose you have $100,000 for investment. First you have to

decide which assets you want to hold. For example, you may decide which assets you want to hold. For example, you may decide to invest $20,000 in money market instruments, $30,000 decide to invest $20,000 in money market instruments, $30,000 in bonds and $50,000 in domestic bonds. Then, your portfolio in bonds and $50,000 in domestic bonds. Then, your portfolio consists of 20% of money market instruments, 30% of bonds consists of 20% of money market instruments, 30% of bonds and 50% in stocks. This is your asset allocation decision. and 50% in stocks. This is your asset allocation decision.

Then you may decide within the stock part of your portfolio, to Then you may decide within the stock part of your portfolio, to allocate $50,000 equally among the following five stocks: allocate $50,000 equally among the following five stocks: Nortel, Research In Motion, Royal Bank, Bombardier, and Nortel, Research In Motion, Royal Bank, Bombardier, and Molson. That decision, your security selection decision, Molson. That decision, your security selection decision, indicates that you will buy $10,000 worth of stocks of each indicates that you will buy $10,000 worth of stocks of each company. company.


Useful Internet SitesUseful Internet Sites

www.411stocks.com (to find expected returns) (to find expected returns) www.investopedia.com (for more on risk measures) (for more on risk measures) www.teachmefinance.com (also contains more on risk (also contains more on risk

measure)measure) www.morningstar.com (measure diversification using “instant (measure diversification using “instant

x-ray”)x-ray”) www.datachimp.com (review modern portfolio theory) (review modern portfolio theory) www.efficentfrontier.com (check out the online journal) (check out the online journal)

© 2009 mcgraw-hill ryerson limited 2-1 chapter 2 diversification and asset allocation expected...

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