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Chapter 2

Diversification and Risky Asset Allocation

Ayşe Yüce – Ryerson University

36.pdf

VALUATION AND MANAGEMENT

Investments

JORDAN MILLER DOLVIN YÜCE

third canadian edition

fundamentals of

Ayşe Yüce – Ryerson University

Learning 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.

Diversification

Ayşe Yüce – Ryerson University

Intuitively, we all know that if you hold many investments
Through time, some will increase in value
Through time, some will decrease in value
It 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?

Ayşe Yüce – Ryerson University

Diversification and Asset Allocation

Our 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.

Ayşe Yüce – Ryerson University

Expected 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.

Ayşe Yüce – Ryerson University

Expected Return

Suppose:
There are two stocks:
Starcents
Jpod
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?

Expected Return

Ayşe Yüce – Ryerson University

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.

Ayşe Yüce – Ryerson University

Calculating Expected Returns

Ayşe Yüce – Ryerson University

Expected 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 between
The expected return on the risky asset in question and
The certain return on a risk-free investment

Ayşe Yüce – Ryerson University

Calculating the Variance of Expected Returns

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.

Ayşe Yüce – Ryerson University

Example: Calculating Expected Returns and Variances: Equal State Probabilities

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

Sheet1

Calculating Expected Returns:
		Starcents:		Jpod:
(1)	(2)	(3)	(4)	(5)	(6)
		Return if		Return if
State of	Probability of	State	Product:	State	Product:
Economy	State of Economy	Occurs	(2) x (3)	Occurs	(2) x (5)
Recession	0.50	-0.20	-0.10	0.30	0.15
Boom	0.50	0.70	0.35	0.10	0.05
Sum:	1.00	E(Ret):	0.25	E(Ret):	0.20
Calculating Variance of Expected Returns:
		Starcents:
(1)	(2)	(3)	(4)	(5)	(6)	(7)
		Return if
State of	Probability of	State	Expected	Difference:	Squared:	Product:
Economy	State of Economy	Occurs	Return:	(3) - (4)	(5) x (5)	(2) x (6)
Recession	0.50	-0.20	0.25	-0.45	0.2025	0.10125
Boom	0.50	0.70	0.25	0.45	0.2025	0.10125
Sum:	1.00			Sum = the Variance:		0.20250
				Standard Deviation:		0.45
		Jmart:
(1)	(2)	(3)	(4)	(5)	(6)	(7)
		Return if
State of	Probability of	State	Expected	Difference:	Squared:	Product:
Economy	State of Economy	Occurs	Return:	(3) - (4)	(5) x (5)	(2) x (6)
Recession	0.50	0.30	0.20	0.10	0.0100	0.00500
Boom	0.50	0.10	0.20	-0.10	0.0100	0.00500
Sum:	1.00			Sum is Variance:		0.01000
				Standard Deviation:		0.10

Ayşe Yüce – Ryerson University

Expected Returns and Variances, Starcents and Jpod

Ayşe Yüce – Ryerson University

Portfolios

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).

Ayşe Yüce – Ryerson University

Portfolios: 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 i

E(Ri) = expected return for portfolio asset i

Ayşe Yüce – Ryerson University

Example: Calculating Portfolio Expected Returns

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

Sheet1

Calculating Expected Portfolio Returns:
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
		Starcents		Starcents	Jpod		Jpod		Portfolio
		Return 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				Sum is Expected Portfolio Return:				0.225
Calculating Variance of Expected Portfolio Returns:
(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	0.01531
Sum:	1.00				Sum is Variance:	0.03063
				Standard Deviation:		0.175

Ayşe Yüce – Ryerson University

Variance 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.

Ayşe Yüce – Ryerson University

Example: Calculating Variance of Portfolio Expected Returns

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?

Sheet1

Calculating Expected Portfolio Returns:
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
		Starcents		Starcents	Jpod		Jpod		Portfolio
		Return 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	0.18	-0.04	0.30	0.82	0.25	0.21	0.105
Boom	0.50	0.70	0.18	0.13	0.10	0.82	0.08	0.21	0.105
Sum:	1.00				Sum is Expected Portfolio Return:				0.209
Calculating Variance of Expected Portfolio Returns:
(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	0.00000
Sum:	1.00				Sum is Variance:	0.00000
			Standard Deviation:			0.000

Ayşe Yüce – Ryerson University

Diversification and Risks

Ayşe Yüce – Ryerson University

Diversification and Risk

Ayşe Yüce – Ryerson University

The 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?

Ayşe Yüce – Ryerson University

The 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.

Ayşe Yüce – Ryerson University

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

Ayşe Yüce – Ryerson University

So, Should Younger Investors Put a High 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).

Ayşe Yüce – Ryerson University

Why 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.

Ayşe Yüce – Ryerson University

Why Diversification Works

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)

Ayşe Yüce – Ryerson University

Why Diversification Works

Ayşe Yüce – Ryerson University

Why Diversification Works

Ayşe Yüce – Ryerson University

Calculating Portfolio Risk

Ayşe Yüce – Ryerson University

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

Where: xA = portfolio weight of asset A

xB = portfolio weight of asset B

such that xA + xB = 1.

(Important: Recall Correlation Definition!)

The Importance of Asset Allocation

Ayşe Yüce – Ryerson University

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?

Correlation and Diversification

Ayşe Yüce – Ryerson University

Correlation and Diversification

Ayşe Yüce – Ryerson University

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.

More on Correlation and the Risk-Return Trade-Off

Ayşe Yüce – Ryerson University

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

Ayşe Yüce – Ryerson University

Sheet1

	Expected	Standard
Inputs	Return	Deviation
Risky Asset 1	14.0%	20.0%
Risky Asset 2	8.0%	15.0%
Correlation		30.0%
Percentage
in Risky	Standard	Expected
Asset 1	Deviation	Return
-60.0%	23.4%	4.4%
-50.0%	21.7%	5.0%
-40.0%	20.1%	5.6%
-30.0%	18.6%	6.2%
-20.0%	17.2%	6.8%
-10.0%	16.0%	7.4%
0.0%	15.0%	8.0%
10.0%	14.2%	8.6%
20.0%	13.7%	9.2%
30.0%	13.6%	9.8%
42.9%	13.8%	10.6%
50.0%	14.2%	11.0%
60.0%	14.9%	11.6%
70.0%	15.9%	12.2%
80.0%	17.1%	12.8%
90.0%	18.5%	13.4%
100.0%	20.0%	14.0%
110.0%	21.6%	14.6%
120.0%	23.3%	15.2%
130.0%	25.0%	15.8%
140.0%	26.8%	16.4%

Sheet1

Standard Deviation

Expected Return

Efficient Set--Two Asset Portfolio

The Importance of Asset Allocation

Ayşe Yüce – Ryerson University

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

Risk and Return with Multiple Assets

Ayşe Yüce – Ryerson University

Risk and Return with Multiple Assets

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:

Ayşe Yüce – Ryerson University

The 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 either
more return for a given level of risk

less risk for a given level of return.

Ayşe Yüce – Ryerson University

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).

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?

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.

Useful Internet Sites

Ayşe Yüce – Ryerson University

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)

Ayşe Yüce – Ryerson University

Chapter Review

Expected Returns and Variances
Expected returns
Calculating the variance
Portfolios
Portfolio weights
Portfolio expected returns
Portfolio variance

61.pdf

VALUATION AND MANAGEMENT

Investments

JORDAN MILLER DOLVIN YÜCE

third canadian edition

fundamentals of

Ayşe Yüce – Ryerson University

Chapter Review

Diversification and Portfolio Risk
The principle of diversification
The fallacy of time diversification
Correlation and Diversification
Why diversification works
Calculating portfolio risk
More on correlation and the risk-return trade-off
The Markowitz Efficient Frontier
Risk and return with multiple assets

62.pdf

VALUATION AND MANAGEMENT

Investments

JORDAN MILLER DOLVIN YÜCE

third canadian edition

fundamentals of

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Return ifReturn if

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Recession-0.20-0.100.300.15

Boom0.700.350.100.05

Sum:E(Ret):0.25E(Ret):0.20

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

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State ofStateExpectedDifference:Squared:Product:

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Recession-0.200.25-0.450.20250.10125

Boom0.700.250.450.20250.10125

Sum:0.20250

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Calculating Expected Returns:

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Calculating Variance of Expected Returns:

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State ofProb.StateWeightProduct:StateWeightProduct:Sum:Product:

Economyof State Occursin Starcents:(3) x (4)Occursin Jpod:(6) x (7)(5) + (8)(2) x (9)

Recession0.50-0.200.50-0.100.300.500.150.050.025

Boom0.500.700.500.350.100.500.050.400.200

Sum:1.000.225

Calculating Expected Portfolio Returns:

Sum is Expected Portfolio Return:

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State ofProb.StateExpectedDifference:Squared:Product:

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Recession0.500.2090.2090.000.00000.00000

Boom0.500.2090.2090.0000.00000

Sum:1.00Sum is Variance:0.00000

0.000Standard Deviation:

Calculating Variance of Expected Portfolio Returns:

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CORR(R

COV(A,

ExpectedStandard

InputsReturnDeviation

Risky Asset 114.0%20.0%

Risky Asset 28.0%15.0%

Correlation30.0%

Efficient Set--Two Asset Portfolio

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