© 2009 McGraw-Hill Ryerson Limited 2-1 Chapter 2 Diversification and Asset Allocation  Expected Return and Variances  Portfolios  Diversification and Portfolio Risk  Correlation and Diversification  Markowitz Efficient Frontier  Asset Allocation and Security Selection Decisions of Portfolio Formation

© 2009 McGraw-Hill Ryerson Limited 2- 2 “Put all your eggs in the one basket and WATCH THAT BASKET!” -- Mark Twain As we shall see in this chapter, this is probably not the best advice! Intuitively, we all know that diversification is important when we are managing investments. Intuitively, we all know that diversification is important when we are managing investments. In fact, diversification has a profound effect on portfolio return and portfolio risk. In fact, diversification has a profound effect on portfolio return and portfolio risk. But, how does diversification work, exactly? But, how does diversification work, exactly?

© 2009 McGraw-Hill Ryerson Limited 2- 3 Diversification and Asset Allocation Our goal in this chapter is to examine the role of diversification and asset allocation in investing. 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. 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.” 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. An efficiently diversified portfolio is one that has the highest expected return, given its risk. You must be aware of the difference between historic return and expected return! You must be aware of the difference between historic return and expected return!

© 2009 McGraw-Hill Ryerson Limited 2- 4 Example: Historical Returns From you can download historical prices, and then calculate historical returns. From you can download historical 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? Although it is important to be able to calculate historical returns, we really care about expected returns, because we want to know how our portfolio will perform from today forward. Although it is important to be able to calculate historical returns, we really care about expected returns, because we want to know how our portfolio will perform from today forward.

© 2009 McGraw-Hill Ryerson Limited 2- 5 Expected Return Expected return is the “weighted average” return on a risky asset, from today to some future date. The formula is: 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: 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, and Estimate how well the security will perform in each scenario, and Assign a probability to each scenario Assign 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 when there are two states. The next slide shows 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. They can have different probabilities of occurring. 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 occurring.

© 2009 McGraw-Hill Ryerson Limited 2- 6 Expected Return II Suppose there are two stocks: NetCap and JMart Suppose 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 percent for NetCap is 25 percent and for JMart is 20 percent Why would anyone want to hold JMart shares when NetCap is expected to have a higher return? Why would anyone want to hold JMart shares when NetCap is expected to have a higher return? The answer depends on risk Jmart is expected to return 20 percent The answer depends on risk Jmart is expected to return 20 percent But the realized return on Jmart could be significantly higher or lower than 20 percent But the realized return on Jmart could be significantly higher or lower than 20 percent Similarly, the realized return on NetCap could be significantly higher or lower than 25 percent. Similarly, the realized return on NetCap could be significantly higher or lower than 25 percent.

© 2009 McGraw-Hill Ryerson Limited 2- 7 Expected Return

© 2009 McGraw-Hill Ryerson Limited 2- 8 Expected Risk Premium Recall: Recall: Suppose riskfree investments have an 8% return. So, in the previous slide, the risk premium expected on Jmart is 12%, and 17% for Netcap. So, in the previous slide, the risk premium expected on Jmart is 12%, and 17% for Netcap. This expected risk premium is the difference between the expected return on the risky asset in question and the certain return on a risk-free investment. This expected risk premium is the difference between the expected return on the risky asset in question and the certain return on a risk-free investment.

© 2009 McGraw-Hill Ryerson Limited 2- 9 Calculating the Variance of Expected Returns The variance of expected returns is calculated as the sum of the squared deviations of each return from the expected return, multiplied by the probability of the state. Here is the formula: The variance of expected returns is calculated as the sum of the squared deviations of each return from the expected return, multiplied by the probability of the state. Here is the formula:

© 2009 McGraw-Hill Ryerson Limited Standard Deviation The standard deviation is simply the square root of the variance. The standard deviation is simply the square root of the variance. The following slide contains an example that shows how to use these formulas in a spreadsheet. The following slide contains an example that shows how to use these formulas in a spreadsheet.

© 2009 McGraw-Hill Ryerson Limited Expected Returns and Variances Equal State Probabilities Note that this is for the individual assets, Netcap and Jmart.

© 2009 McGraw-Hill Ryerson Limited Expected Returns and Variances, Netcap and Jmart

© 2009 McGraw-Hill Ryerson Limited Portfolios Portfolios are groups of assets, such as stocks and bonds, that are held by an investor. 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. 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. These proportions are called portfolio 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.

© 2009 McGraw-Hill Ryerson Limited Portfolios: Expected Returns The expected return on a portfolio is the weighted average, of the expected returns on the assets in that portfolio. The formula, for “n” assets, is: The expected return on a portfolio is the weighted average, of the expected returns on the assets in that portfolio. The formula, for “n” assets, is: In the formula: E(R P ) = expected portfolio return w i = portfolio weight in portfolio asset i w i = portfolio weight in portfolio asset i E(R i ) = expected return for portfolio asset i

© 2009 McGraw-Hill Ryerson Limited Example: Calculating Portfolio Expected Returns Note that the portfolio weight in Jmart = 1 – portfolio weight in Netcap.

© 2009 McGraw-Hill Ryerson Limited 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. 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: If there are “n” states, the formula is:

© 2009 McGraw-Hill Ryerson Limited Portfolio Variance In the formula, In the formula, VAR(RP) = variance of portfolio expected return VAR(RP) = variance of portfolio expected return p s = probability of state of economy, s p s = probability of state of economy, s E(R p,s ) = expected portfolio return in state s E(R p,s ) = expected portfolio return in state s E(R p ) = portfolio expected return E(R p ) = portfolio expected return Note that the formula is like the formula for the variance of the expected return of a single asset. Note that the formula is like the formula for the variance of the expected return of a single asset.

© 2009 McGraw-Hill Ryerson Limited Calculating Variance of Portfolio Returns It is possible to construct a portfolio of risky assets with zero portfolio variance! What? How? (Open the spreadsheet and set the weight in Netcap to 2/11ths.) It is possible to construct a portfolio of risky assets with zero portfolio variance! What? How? (Open the spreadsheet and set the weight in Netcap to 2/11ths.)

© 2009 McGraw-Hill Ryerson Limited Diversification and Risk, I. Table 2.7 Portfolio Standard Deviations Table 2.7 Portfolio Standard Deviations

© 2009 McGraw-Hill Ryerson Limited Diversification and Risk, II. Figure 2.1 Portfolio Diversification

© 2009 McGraw-Hill Ryerson Limited Why Diversification Works, I. 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. 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. Negatively correlated assets tend to move in opposite directions. Imperfect correlation, positive or negative, is why diversification reduces portfolio risk. Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.

© 2009 McGraw-Hill Ryerson Limited Why Diversification Works, II. The correlation coefficient is denoted by Corr(R A, R B ) or simply,  A,B. The correlation coefficient is denoted by Corr(R A, R B ) or simply,  A,B. The correlation coefficient measures correlation and ranges from: The correlation coefficient measures correlation and ranges from: From: From: (perfect negative correlation) Through:0(uncorrelated) To:+1 (perfect positive correlation)

© 2009 McGraw-Hill Ryerson Limited Why Diversification Works, III.

© 2009 McGraw-Hill Ryerson Limited Why Diversification Works, IV.

© 2009 McGraw-Hill Ryerson Limited Why Diversification Works, V.

© 2009 McGraw-Hill Ryerson Limited Calculating Portfolio Risk For a portfolio of two assets, A and B, the variance of the return on the portfolio is: For a portfolio of two assets, A and B, the variance of the return on the portfolio is: Where: w A = portfolio weight of asset A w B = portfolio weight of asset B w B = portfolio weight of asset B such that w A + w B = 1.

© 2009 McGraw-Hill Ryerson Limited Correlation and Diversification 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? 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? Uh, is this decision a wise one?

© 2009 McGraw-Hill Ryerson Limited Correlation and Diversification Note: A correlation of 0.10 is assumed in calculations

© 2009 McGraw-Hill Ryerson Limited Correlation and Diversification

© 2009 McGraw-Hill Ryerson Limited Correlation and Diversification The various combinations of risk and return available all fall on a smooth curve. 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. 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. 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. The undesirable portfolios are said to be dominated or inefficient.

© 2009 McGraw-Hill Ryerson Limited Correlation and Risk &Return Trade off

© 2009 McGraw-Hill Ryerson Limited Correlation and the Risk-Return Trade-Off

© 2009 McGraw-Hill Ryerson Limited Risk and Return with Multiple Assets

© 2009 McGraw-Hill Ryerson Limited 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 for a given a return. The Markowitz Efficient frontier is the set of portfolios with the maximum return for a given risk AND the minimum risk for a given a return. For the plot, the upper left-hand boundary is the Markowitz efficient frontier. For the plot, the upper left-hand boundary is the Markowitz efficient frontier. Efficient Frontier All the other possible combinations are inefficient. That is, investors would not hold these portfolios because they could get either 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, or more return for a given level of risk, or less risk for a given level of return. less risk for a given level of return.

© 2009 McGraw-Hill Ryerson Limited Example of Efficient Frontier

© 2009 McGraw-Hill Ryerson Limited Portfolio Formation Decisions Investors can form their using the following financial assets : Investors can form their using the following financial assets : 1- Domestic Stocks, 2- Domestic Bonds (government, municipal, provincial bonds, corporate bonds) 3- Derivative Instruments (options, futures, swaps), 4- Money market instruments (T-Bills, certificate of deposits, repurchase agreements, etc), 5- Foreign Stocks and Bonds. 1- Domestic Stocks, 2- Domestic Bonds (government, municipal, provincial bonds, corporate bonds) 3- Derivative Instruments (options, futures, swaps), 4- Money market instruments (T-Bills, certificate of deposits, repurchase agreements, etc), 5- Foreign Stocks and Bonds.

© 2009 McGraw-Hill Ryerson Limited Asset Allocation and Security Selection Investors face with two problems when they form portfolios of multiple securities from different asset classes. Investors face with two problems when they form portfolios of multiple securities from different asset classes. 1. Asset Allocation Decision Asset allocation problem involves a decision of what percentage of investor ’ s portfolio should be allocated among different asset classes (stocks, bonds, derivatives, money market instruments, foreign securities). Asset allocation problem involves a decision of what percentage of investor ’ s portfolio should be allocated among different asset classes (stocks, bonds, derivatives, money market instruments, foreign securities). 2. Security Selection Decision Security selection is deciding which securities to pick in each class and what percentage of funds to allocate to these securities (for example choosing different stocks and their percentages within the asset class). Security selection is deciding which securities to pick in each class and what percentage of funds to allocate to these securities (for example choosing different stocks and their percentages within the asset class).

© 2009 McGraw-Hill Ryerson Limited Portfolio Decisions Suppose you have $100,000 for investment. First you have to decide which assets you want to hold. For example, you may decide to invest $20,000 in money market instruments, $30,000 in bonds and $50,000 in domestic bonds. Then, your portfolio consists of 20% of money market instruments, 30% of bonds and 50% in stocks. This is your asset allocation decision. Suppose you have $100,000 for investment. First you have to decide which assets you want to hold. For example, you may decide to invest $20,000 in money market instruments, $30,000 in bonds and $50,000 in domestic bonds. Then, your portfolio consists of 20% of money market instruments, 30% of bonds and 50% in stocks. This is your asset allocation decision. Then you may decide within the stock part of your portfolio, to allocate $50,000 equally among the following five stocks: Nortel, Research In Motion, Royal Bank, Bombardier, and Molson. That decision, your security selection decision, indicates that you will buy $10,000 worth of stocks of each company. Then you may decide within the stock part of your portfolio, to allocate $50,000 equally among the following five stocks: Nortel, Research In Motion, Royal Bank, Bombardier, and Molson. That decision, your security selection decision, indicates that you will buy $10,000 worth of stocks of each company.

© 2009 McGraw-Hill Ryerson Limited Useful Internet Sites (to find expected returns) (to find expected returns) (for more on risk measures) (for more on risk measures) (also contains more on risk measure) (also contains more on risk measure) (measure diversification using “instant x-ray”) (measure diversification using “instant x-ray”) (review modern portfolio theory) (review modern portfolio theory) (check out the online journal) (check out the online journal)