1 chapter 2 diversification and risky asset allocation ayşe yüce – ryerson university copyright...

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Chapter 2Diversification and Risky Asset

AllocationAyşe Yüce – Ryerson UniversityAyşe Yüce – Ryerson University Copyright © 2012 McGraw-Hill RyersonCopyright © 2012 McGraw-Hill Ryerson

222Ayşe Yüce – Ryerson UniversityAyşe Yüce – Ryerson University Copyright © 2012 McGraw-Hill RyersonCopyright © 2012 McGraw-Hill Ryerson

Learning ObjectivesLearning Objectives

To get the most out of this chapter, spread your study time across:

1. How to calculate expected returns and variances for a security.

2. How to calculate expected returns and variances for a portfolio.

3. The importance of portfolio diversification.

4. The efficient frontier and the importance of asset allocation.

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DiversificationDiversification

Ayşe Yüce – Ryerson UniversityAyşe Yüce – Ryerson University Copyright © 2012 McGraw-Hill RyersonCopyright © 2012 McGraw-Hill Ryerson

Intuitively, we all know that if you hold many investments

Through time, some will increase in valueThrough time, some will decrease in valueIt is unlikely that their values will all change in the

same way

Diversification has a profound effect on portfolio return and portfolio risk.

But, exactly how does diversification work?


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

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

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

the highest expected return, given its risk.You must be aware that diversification concerns

expected returns.


Expected ReturnsExpected Returns Expected return is the “weighted average” return on a

risky asset, from today to some future date. The formula is:

To calculate an expected return, you must first: Decide on the number of possible economic scenarios

that might occur. Estimate how well the security will perform in each

scenario, and Assign a probability, ps, to each scenario. (BTW, finance professors call these economic scenarios,

“states.”)

The upcoming slides show how the expected return formula is used when there are two states. � Note that the “states” are equally likely to occur in this

example. � BUT! They do not have to be equally likely--they can

have different probabilities of occurring.

n

1ssi,si returnpreturn expected


Expected ReturnExpected Return

Suppose:There are two stocks:

StarcentsJpod

We are looking at a period of one year.

Investors agree that the expected return: for Starcents is 25 percent for Jpod is 20 percent

Why would anyone want to hold Jpod shares when Starcents is expected to have a higher return?

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Expected ReturnExpected Return


The answer depends on risk

Starcents is expected to return 25 percent

But the realized return on Starcents could be significantly higher or lower than 25 percent

Similarly, the realized return on Jpod could be significantly higher or lower than 20 percent.


Calculating Expected ReturnsCalculating Expected Returns


Expected Risk PremiumExpected Risk Premium

Recall:

Suppose risk free investments have an 8% return. If so,The expected risk premium on Jpod is 12%The expected risk premium on Starcents is

17%

This expected risk premium is simply the difference betweenThe expected return on the risky asset in

question andThe certain return on a risk-free investment

Rate RiskfreeReturn Expected Premium Risk Expected


Calculating the Variance of Expected Calculating the Variance of Expected ReturnsReturns The variance of expected returns is calculated using this formula:

This formula is not as difficult as it appears. This formula says:

� add up the squared deviations of each return from its expected return

� after it has been multiplied by the probability of observing a particular economic state (denoted by “s”).

The standard deviation is simply the square root of the variance.

n

1s

2ss

2 return expectedreturnpσVariance

Varianceσ Deviation Standard


Example: Calculating Expected Returns Example: Calculating Expected Returns and Variances: Equal State and Variances: Equal State ProbabilitiesProbabilities

(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

Calculating Expected Returns:Starcents: Jpod:

(2)

1.00

Calculating Variance of Expected Returns:Starcents:

(2)

Probability ofState of Economy

0.500.50

1.00 Sum = the Variance:

Standard Deviation:

Probability ofState of Economy

0.500.50

Note that the second spreadsheet is only for Starcents. What would you get for Jpod?


Expected Returns and Variances, Starcents and Expected Returns and Variances, Starcents and JpodJpod


PortfoliosPortfolios

Portfolios are groups of assets, such as stocks and bonds, that are held by an investor.

One convenient way to describe a portfolio is by listing the proportion of the total value of the portfolio that is invested into each asset.

These proportions are called portfolio weights.Portfolio weights are sometimes expressed in

percentages.However, in calculations, make sure you use

proportions (i.e., decimals).


Portfolios: Expected ReturnsPortfolios: Expected Returns

The expected return on a portfolio is a linear combination, or weighted average, of the expected returns on the assets in that portfolio.

The formula, for “n” assets, is:

In the formula: E(RP) = expected portfolio return wi = portfolio weight for portfolio

asset iE(Ri) = expected return for portfolio

asset i

n

1iiiP REwRE


Example: Calculating Portfolio Expected Example: Calculating Portfolio Expected ReturnsReturns

Note that the portfolio weight in Jpod = 1 – portfolio weight in Starcents.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)Starcents Starcents Jpod Jpod PortfolioReturn if Portfolio Contribution Return if Portfolio Contribution Return

State of Prob. State Weight Product: State Weight Product: Sum: Product:Economy of State Occurs in Starcents: (3) x (4) Occurs in Jpod: (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 not a simple weighted average of the variances of the assets in the portfolio.

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

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

ps = probability of state of economy, s E(Rp,s) = expected portfolio return in state s

E(Rp) = portfolio expected return

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

n

1s

2Psp,sP REREpRVAR


Example: Calculating Variance of Portfolio Expected Example: Calculating Variance of Portfolio Expected ReturnsReturns

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

What happens when you use .40 as the weight in Starcents?

(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 0.209 0.209 0.00 0.0000 0.00000

Boom 0.50 0.209 0.209 0.00 0 0.00000Sum: 1.00 Sum is Variance: 0.00000

0.000Standard Deviation:

Calculating Variance of Expected Portfolio Returns:


Diversification and RisksDiversification and Risks


Diversification and Risk Diversification and Risk


The Fallacy of Time DiversificationThe Fallacy of Time Diversification

Young people are often told that they should hold a large percent of their portfolio in stocks.

The advice could be correct, but often the typical argument used to support this advice is incorrect.The Typical Argument: Even though stocks are more

volatile, over time, the volatility “cancels out.”Sounds logical, but the typical argument is

incorrect.

This argument is the fallacy of time diversification fallacy

How can such a plausible argument be incorrect?


The Fallacy of Time DiversificationThe Fallacy of Time Diversification

How can such logical-sounding advice be bad? You might remember from your statistics class that we can add

variances. This fact means that an annual variance grows each year by

multiplying the annual variance by the number of years.

Standard deviations cannot be added together: An annual standard deviation grows each year by the square root of the number of years.

As we showed earlier in the chapter, a randomly selected portfolio of large-cap stocks has an annual standard deviation of about 20%. If we held this portfolio for 16 years, the standard deviation

would be about 80 percent, which is 20 percent multiplied by the square root of 16.

Bottom line: Volatility increases over time—volatility does not “cancel out” over time.

Investing in equity has a greater chance of having an extremely large value AND increases the probability of ending with a really low value.


The Very Definition of Risk—A Wider Range The Very Definition of Risk—A Wider Range of Possible Outcomes from Holding Equityof Possible Outcomes from Holding Equity


So, Should Younger Investors Put a High So, Should Younger Investors Put a High Percent of Their Money into Equity?Percent of Their Money into Equity?

The answer is probably still yes, but for logically sound reasons that differ from the reasoning underlying the fallacy of time diversification.

If you are young and your portfolio suffers a steep decline in a particular year, what could you do?

You could make up for this loss by changing your work habits (e.g., your type of job, hours, second job).

People approaching retirement have little future earning power, so a major loss in their portfolio will have a much greater impact on their wealth.

Thus, the portfolios of young people should contain relatively more equity (i.e., risk).


Why Diversification WorksWhy Diversification Works

Correlation: The tendency of the returns on two assets to move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the portfolio standard deviation.

Positively correlated assets tend to move up and down together.

Negatively correlated assets tend to move in opposite directions.

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



The correlation coefficient is denoted by Corr(RA, RB) or simply, A,B.

The correlation coefficient measures correlation and ranges from -1 to 1:

-1 (perfect negative correlation)

0 (uncorrelated)

+1 (perfect positive correlation)

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Calculating Portfolio RiskCalculating Portfolio Risk


For a portfolio of two assets, A and B, the variance of the return on the portfolio is:

)RCORR(Rσσx2xσxσxσ

B)COV(A,x2xσxσxσ

BABABA2B

2B

2A

2A

2p

BA2B

2B

2A

2A

2p

Where: xA = portfolio weight of asset A

xB = portfolio weight of asset B

such that xA + xB = 1.

(Important: Recall Correlation Definition!)

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The Importance of Asset AllocationThe Importance of Asset Allocation


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

Uh, is this decision a wise one?

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Correlation and DiversificationCorrelation and Diversification


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Correlation and DiversificationCorrelation and Diversification


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

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

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

The undesirable portfolios are said to be dominated or inefficient.

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More on Correlation and the Risk-Return More on Correlation and the Risk-Return Trade-OffTrade-Off


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Example: Correlation and the Example: Correlation and the Risk-Return Trade-Off, Two Risky AssetsRisk-Return Trade-Off, Two Risky Assets


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

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The Importance of Asset The Importance of Asset AllocationAllocation


We can illustrate the importance of asset allocation with 3 assets.

How? Suppose we invest in three mutual funds: One that contains Foreign Stocks, F One that contains U.S. Stocks, S One that contains U.S. Bonds, B

Figure 11.6 shows the results of calculating various expected returns and portfolio standard deviations with these three assets.

Expected Return Standard Deviation

Foreign Stocks, F

18% 35%

U.S. Stocks, S 12 22

U.S. Bonds, B 8 14

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Risk and Return with Multiple Risk and Return with Multiple AssetsAssets


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Risk and Return with Multiple Risk and Return with Multiple AssetsAssets

We used these formulas for portfolio return and variance:

But, we made a simplifying assumption. We assumed that the assets are all uncorrelated.

If so, the portfolio variance becomes:

2B

2B

2S

2S

2F

2F

2p

BSBSBS

BFBFBF

SFSFSF

2B

2B

2S

2S

2F

2F

2p

BBSSFFp

σxσxσxσ

)RCORR(Rσσx2x

)RCORR(Rσσx2x

)RCORR(Rσσx2x

σxσxσxσ

RxRxRxr


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The Markowitz Efficient FrontierThe Markowitz Efficient Frontier

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

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

All the other possible combinations are inefficient. That is, investors would not hold these portfolios because they could get eithermore return for a given level of risk orless risk for a given level of return.


Investors face two problems with they form portfolios of multiple securities from different asset classes.

These are as follows: an asset allocation problem & a security selection problem.

The asset allocation problem involves a decision regarding what percentage should be allocated among different asset classes ( stocks, bonds, derivatives, foreign securities).

The security selection problem involves deciding which to pick in each class and what percentage to allocate to these securities (RIM, Royal Bank, Molson).

39Ayşe Yüce Copyright © 2012 McGraw-Hill Ryerson

Professional Investors have traditionally used modern portfolio theory to help make investment decisions.

This approach examines past returns, volatility and correlation to determine the optimum percentage of a portfolio to invest in to achieve an expected rate of return for a given level of risk.

“Modern Portfolio theory focuses on diversifying your risk away, but the crisis has shown the limits of the approach.”

What are the alternatives?How should investors be looking to construct their

portfolios?

40Ayşe Yüce Copyright © 2012 McGraw-Hill Ryerson

Investors should make asset allocations that give the best chance of meeting their own unique future financial commitments, rather then simply trying to maximize risk-adjusted returns.

Life cycle investing, takes into account the investor’s specific time horizons, something that modern portfolio theory does not take into account.

Controlling the overall risk level of investments to make sure it is in line with risk appetite.

41Ayşe Yüce Copyright © 2012

McGraw-Hill Ryerson

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Useful Internet SitesUseful Internet Sites


www.investopedia.com (for more on risk measures)

www.teachmefinance.com (also contains more on risk measure)

www.morningstar.com (measure diversification using “instant x-ray”)

www.moneychimp.com (review modern portfolio theory)

www.efficientfrontier.com (check out the reading list)

http://www.investopedia.com/

http://www.teachmefinance.com/

http://www.morningstar.com/

http://www.moneychimp.com/

http://www.efficientfrontier.com/


Chapter Review Chapter Review

Expected Returns and VariancesExpected returnsCalculating the variance

PortfoliosPortfolio weightsPortfolio expected returnsPortfolio variance


Chapter Review Chapter Review

Diversification and Portfolio RiskThe principle of diversificationThe fallacy of time diversification

Correlation and DiversificationWhy diversification worksCalculating portfolio riskMore on correlation and the risk-return trade-

off

The Markowitz Efficient FrontierRisk and return with multiple assets

1 chapter 2 diversification and risky asset allocation ayşe yüce – ryerson university copyright...

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