Investment Management

Investment

Management UNIT 5 – INTRODUCTION TO PORTFOLIO MANAGEMENT

Source Material

 Reilly, F. K. & Brown, K.C. (2003). Investment Analysis, Portfolio

Management. 7th Ed. South Western Publishing

Portfolio Theory

 One basic assumption of portfolio theory is that as an investor

you want to maximize the returns from your investments for a given level of risk.

 First, your portfolio should include all of your assets and liabilities,

not only your stocks or even your marketable securities but also such items as your car, house, and less-marketable investments,

such as coins, stamps, art, antiques, and furniture.

 The full spectrum of investments must be considered because the returns from all these investments interact, and this

relationship between the returns for assets in the portfolio is important.

 Hence, a good portfolio is not simply a collection of individually

good investments.

Risk Defined

 Although there is a difference in the specific

definitions of risk and uncertainty, in most financial

literature, the two terms are used interchangeably.

 In fact, one way to define risk is the uncertainty of

future outcomes.

 An alternative definition might be the probability of

an adverse outcome.

Risk Aversion

 Portfolio theory also assumes that investors are basically risk averse, meaning

that, given a choice between two assets with equal rates of return, they will

select the asset with the lower level of risk.

 Evidence that most investors are risk averse is that they purchase various types of

insurance, Buying insurance basically involves an outlay of a given amount to

guard against an uncertain, possibly larger outlay in the future.

 Further evidence of risk aversion is the difference in promised yield (the required

rate of return) for different grades of bonds that supposedly have different

degrees of credit risk.

 Specifically, the promised yield on bonds increases as you go from AAA (the

lowest-risk class) to AA to A, and so on—that is, investors require a higher rate of

return to accept higher risk.

Risk Aversion

 Not every investor is risk averse…not everybody buys insurance for everything.

 Some individuals buy insurance related to some risks such as auto accidents or illness, but they also buy lottery tickets and gamble at race tracks or in casinos, where it is known that the expected returns are negative, which means that participants are willing to pay for the excitement of the risk involved.

 This combination of risk preference and risk aversion can be explained by an attitude toward risk that depends on the amount of money involved.

 Friedman and Savage speculate that this is the case for people who like to gamble for small amounts (in lotteries or slot machines) but buy insurance to protect themselves against large potential losses, such as fire or accidents.

 While recognizing this diversity of attitudes, our basic assumption is that most investors committing large sums of money to developing an investment portfolio are risk averse.

 Therefore, we expect a positive relationship between expected return and expected risk.

Markowitz Portfolio Theory

 The basic portfolio model was developed by Harry Markowitz, who derived the expected rate of return for a portfolio of assets and an expected risk measure.

 Markowitz showed that the variance of the rate of return was a meaningful measure of portfolio risk under a reasonable set of assumptions.

 Under these assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the

same (or lower) risk, or lower risk with the same (or higher) expected return.

 He derived the formula for computing the variance of a portfolio.

 This portfolio variance formula indicated the importance of diversifying your investments to reduce the total risk of a portfolio but also showed how to effectively diversify.

Markowitz Portfolio Theory Assumptions  Investors consider each investment alternative as being represented

by a probability distribution of expected returns over some holding period.

 Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.

 Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

 Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

 For a given risk level, investors prefer higher returns to lower returns.

Alternative Measures of Risk

 One of the best-known measures of risk is the variance, or standard deviation of expected returns.

 It is a statistical measure of the dispersion of returns around the expected value whereby a larger variance or standard deviation indicates greater dispersion.

 The idea is that the more disperse the expected returns, the greater the uncertainty of future returns.

 The variance or standard deviation of returns is more commonly used because (1) this measure is somewhat intuitive, (2) it is a correct and widely recognized risk measure, and (3) it has been used in most of the theoretical asset pricing models.

 Another measure of risk is the range of returns.

 It is assumed that a larger range of expected returns, from the lowest to the highest return, means greater uncertainty and risk regarding future expected returns.

Alternative Measures of Risk

 Instead of using measures that analyze all deviations from expectations, some

observers believe that when you invest you should be concerned only with

returns below expectations, which means that you only consider deviations

below the mean value.

 A measure that only considers deviations below the mean is the semi-

variance.

 Extensions of the semi-variance measure only computed expected returns

below zero (that is, negative returns), or returns below some specific asset

such as T-bills, the rate of inflation, or a benchmark.

 These measures of risk implicitly assume that investors want to minimize the

damage from returns less than some target rate.

 Assuming that investors would welcome returns above some target rate, the

returns above a target return are not considered when measuring risk.

Expected Rates of Return – Individual Investment

 The expected return for an individual risky asset with the following

set of potential returns and an assumption of equal probabilities is

shown below.

 The expected rate of return in this example is 11 percent.

Standard Deviation (Variance) of Returns – Individual Investment

 Recall that the standard deviation is the square root of the variance.

 The variance, or standard deviation, is a measure of the variation of possible rates of return, Ri, from the expected rate of return [E(Ri)] as follows:

Standard Deviation (Variance) of Returns – Individual Investment

 Sx

Expected Rates of Return – Portfolio of Investment

 The expected rate of return for a portfolio of investments is simply the weighted

average of the expected rates of return for the individual investments in the

portfolio. The weights are the proportion of total value for the investment.

 The expected return for this portfolio of investments would be 11.5 percent.

Expected Rates of Return – Portfolio of Investment

 The effect of adding or dropping any

investment from the portfolio would be easy to

determine.

You would use the new weights based on

value and the expected returns for each of

the investments.

Covariance and Correlation for Portfolio

 Before we discuss the formula for the variance of the rate of return for a portfolio, two basic concepts in statistics - covariance and correlation - must be understood.

 Covariance is a measure of the degree to which two variables “move together” relative to their individual mean values over time.

 In portfolio analysis, we usually are concerned with the covariance of rates of return rather than prices or some other variable.

 A positive covariance means that the rates of return for two investments tend to move in the same direction relative to their individual means during the same time period.

 A negative covariance indicates that the rates of return for two investments tend to move in different directions relative to their means during specified time intervals ov er time.

 The m agnitude of the cov ariance depends on the v ariances of the individual return series, as well as on the relationship between the series.

Covariance for Portfolio

 The cov ariance statistic provides an absolute measure of how prices moved together ov er time. For two assets, i and j, the cov ariance of rates of return is defined as:

 When we apply this formula to the monthly rates of return for Coca-Cola and Home Depot during 2001, it becomes:

Covariance for Portfolio

 dasc

Covariance for Portfolio

 If the rates of return for one stock are above (below) its mean rate of return

during a given period and the returns for the other stock are likewise above

(below) its mean rate of return during this same period, then the product of

these deviations from the mean is positive.

 If this happens consistently, the covariance of returns between these two stocks will be some large positive v alue.

 If, however, the rate of return for one of the securities is above its mean return

while the return on the other security is below its mean return, the product will

be negative.

 If this contrary movement happened consistently, the cov ariance between the rates of return for the two stocks would be a large negative value.

 In this example, we have a positive covariance of 6.37.

Covariance and Correlation for Portfolio  Interpretation of a number such as 6.37 is

difficult; is it high or low for covariance?

 We know the relationship between the two

stocks is generally positive, but it is not

possible to be more specific?

 Covariance and Correlation Covariance is

affected by the variability of the two

individual return series. Therefore, a number

such as the 6.37 in our example might

indicate a weak positive relationship if the

two individual series were volatile but would

reflect a strong positive relationship if the

two series were very stable.

Therefore, we need to

“standardize” this covariance

measure taking into consideration

the variability of the two individual

return series, as follows:

Covariance and Correlation for Portfolio  Standardizing the covariance by the individual standard deviations yields the

correlation coefficient (rij), which can vary only in the range –1 to +1.

 A value of +1 would indicate a perfect positive linear relationship between Ri

and Rj, meaning the returns for the two stocks move together in a completely

linear manner.

 A value of –1 indicates a perfect negative relationship between the two return

series such that when one stock’s rate of return is above its mean, the other

stock’s rate of return will be below its mean by the comparable amount.

 To calculate this standardized measure of the relationship, you need to compute

the standard deviation for the two individual return series.

 We already have the values for Rit – E(Ri) and Rjt – E(Rj) in Exhibit 7.7.

 We can square each of these values and sum them to calculate the variance of

each return series.

Covariance and Correlation for Portfolio

 sa

Covariance and Correlation for Portfolio  Thus, based on the covariance between the two series and the individual

standard deviations, we can calculate the correlation coefficient between returns for Coca-Cola and Home Depot as:

 As noted, a correlation of +1.0 would indicate perfect positive correlation, and a value of –1.0 would mean that the returns moved in a completely opposite direction. A value of zero would mean that the returns had no linear relationship, that is, they were uncorrelated statistically. That does not mean that they are independent. The value of rij =0.108 is quite low. This relatively low correlation is not unusual for stocks in diverse industries (i.e., beverages and building materials). Correlation between stocks of companies within some industries approaches 0.85.

Portfolio Standard Deviation

 Earlier, we saw where the expected rate of return of the portfolio was the

weighted average of the expected returns for the individual assets in the

portfolio; the weights were the percentage of value of the portfolio.

 In calculating portfolio standard deviation, the formula indicates that the

standard deviation for a portfolio of assets is a function of the weighted

average of the individual variances (where the weights are squared), plus the

weighted covariances between all the assets in the portfolio.

 The standard deviation for a portfolio of assets encompasses not only the

variances of the individual assets but also includes the covariances between

pairs of individual assets in the portfolio.

 Further, it can be shown that, in a portfolio with a large number of securities,

this formula reduces to the sum of the weighted covariances.

Portfolio Standard Deviation

 The general formula is represented below:

Portfolio Standard Deviation

 We can look at the computations for a three-asset portfolio but we must also consider what happens in a large portfolio with many assets.

 Specifically, what happens to the portfolio’s standard deviation when you add a new security to such a portfolio?

 The formula has two effects.

 The first is the asset’s own v ariance of returns, and

 the second is the cov ariance between the returns of this new asset and the returns of ev ery other asset that is already in the portfolio.

 The relative weight of these numerous covariances is substantially greater than the asset’s unique variance; and the more assets in the portfolio, the more this is true.

 This means that the important factor to consider when adding an investment to a portfolio that contains a number of other investments is not the investment’s own variance but its average covariance with all the other investments in the portfolio.

Portfolio Standard Deviation

 Any asset or portfolio of assets can be described by two characteristics:

 the expected rate of return and the expected standard deviation of returns.

 Therefore, the following demonstrations can be applied to two individual

assets with the indicated return–standard deviation characteristics and

correlation coefficients, two portfolios of assets, or two asset classes with the

indicated return–standard deviation characteristics and correlation

coefficients.

 Equal risk and return - changing correlation – both assets have the same risk

and standard deviation but different correlation.

• Consider the following examples where the two assets hav e equal weights in the portfolio

• (W1 = 0.50; W2 = 0.50) • Therefore, the only v alue that changes in each

example will be the correlation between the returns for the two assets.

Portfolio Standard Deviation

 Consider the following alternative correlation coefficients and

the covariances they yield.

 The covariance term in the equation will be equal to r1,2 (0.10)(0.10) because both standard deviations are 0.10. That

is, correlation coefficient x SD of assets 1 and 2.

Remember the

covariance formula is:

Portfolio Standard Deviation

 What happens to the standard deviation of the portfolio under these five

conditions?

 Recall the general formula is

 Now let us see what happens to the standard deviation of the portfolio

under these five conditions. The formula now becomes:

Portfolio Standard Deviation

 In the case of (a), a correlation coefficient of 1.00 (s.28), the portfolio SD is

0.10.

• In this case, where the returns for the two assets are perfectly positively correlated

(r1,2 = 1.0), the standard deviation for the portfolio is, in fact, the weighted

average of the individual standard deviations.

• The important point is that we get no real benefit from combining two assets that

are perfectly correlated; they are like one asset already because their returns

move together.

Portfolio Standard Deviation

 Now consider Case b, where r1,2 equals 0.50:

 The only term that changed from Case a is the last term, Cov 1,2, which changed from 0.01 to 0.005. As a result, the standard deviation of the portfolio declined by about 13 percent, from 0.10 to 0.0868. Note that the expected return did not change because it is simply the weighted average of the individual expected returns; it is equal to 0.20 in both cases.

 You should be able to confirm through your own calculations that the standard deviations for Portfolios c and d are as follows:

 c. 0.0707

 d. 0.05

Portfolio Standard Deviation

 The final case where the correlation between the two assets is –1.00 indicates

the ultimate benefits of diversification:

 Here, the negative covariance term exactly offsets the individual variance

terms, leaving an overall standard deviation of the portfolio of zero. This

would be a risk-free portfolio.

Portfolio Standard Deviation

 Perfect negative correlation gives a mean combined return for the two securities over time equal to the mean for each of them, so the returns for the portfolio show no variability.

 Any returns above and below the mean for each of the assets are completely offset by the return for the other asset, so there is no variability in total returns.

 That is, no risk, for the portfolio.

 This combination of two assets that are completely negatively correlated provides the maximum benefits of diversification

 it completely eliminates risk.

 Combining assets that are not perfectly correlated does not affect the expected return of the portfolio, but it does reduce the risk of the portfolio (as measured by its standard deviation).

 When we eventually reach the ultimate combination of perfect negative correlation, risk is eliminated.

Combining Stocks with Different Returns and Risks

 We now consider two assets (or portfolios) with different expected

rates of return and individual standard deviations as well as what

happens when we vary the correlations between them.

 The assets have the following characteristics:

Combining Stocks with Different Returns and Risks

 The new covariance calculations would be as follows:

 For example (0.50)(0.07)(0.10) = 0.0035

 Because we are assuming the same weights in all cases (0.50 – 0.50), the expected

return of the portfolio in every instance will be

Combining Stocks with Different Returns and Risks

 The Standard Deviation for case (a) is:

 Again, with perfect positive correlation, the standard deviation of the portfolio is

the weighted average of the standard deviations of the individual assets:

(0.5)(0.07) + (0.5)(0.10) = 0.085

 Changing the weights with perfect positive correlation causes the standard deviation for the portfolio to change in a linear fashion. This is an important point

to remember when we will discuss the capital asset pricing model (CAPM) in the next unit.

Combining Stocks with Different Returns and Risks

 For Cases b, c, d, and e, the standard deviation for the portfolio would be as follows:

 Note that, in this example, with perfect negative correlation the standard deviation of the portfolio is not zero. This is because the different examples have equal weights, but the individual standard deviations are not equal.

Three Asset Portfolio

 A demonstration of what occurs with a three-asset class portfolio is useful

because it shows the dynamics of the portfolio process when we add additional

assets to a portfolio.

 It also shows the rapid growth in the computations required, which is why we will

stop at three assets.

 We will assume the following characteristics for these assets:

Three Asset Portfolio

 The correlations are as follows:

rS,B = 0.25; rS,C = –0.08; rB,C = 0.15

 Given the weights specified, the E(Rp) is:

E(Rp) = (0.60)(0.12) + (0.30)(0.08) + (0.10)(0.04)

= (0.072 + 0.024 + 0.004)

= 0.10 or10%

 When we apply the generalized formula to the expected standard

deviation of a three-asset class, it is as follows:

Three Asset Portfolio

 When we apply the generalized formula to the expected standard

deviation of a three-asset class, it is as follows

Estimation Risk

 It is important to keep in mind that the results of this portfolio asset allocation

depend on the accuracy of the statistical inputs.

 In the current instance, this means that for every asset (or asset class) being

considered for inclusion in the portfolio, you must estimate its expected returns

and standard deviation.

 In addition, the correlation coefficient among the entire set of assets must also

be estimated.

 The number of correlation estimates can be significant—for example, for a

portfolio of 100 securities, the number is 4,950 (that is, 99 + 98 + 97 + . . .).

 The potential source of error that arises from these approximations is referred

to as estimation risk.

The Efficient Frontier and Investor Utility

 If we examined different two-asset combinations and derived the curves assuming all the possible weights, we would have a graph like that in Exhibit 7.14.

 The envelope curve that contains the best of all these possible combinations is referred to as the efficient frontier.

 Specifically, the efficient frontier represents that set of portfolios that has the maximum rate of return for every given level of risk, or the minimum risk for every level of return.

The Efficient Frontier and Investor Utility

 Every portfolio that lies on the efficient frontier has either a higher rate of

return for equal risk or lower risk for an equal rate of return than some

portfolio beneath the frontier.

 Thus, we would say that Portfolio A in Exhibit 7.15 dominates Portfolio C

because it has an equal rate of return but substantially less risk.

 Similarly, Portfolio B dominates Portfolio C because it has equal risk but a

higher expected rate of return.

 Because of the benefits of diversification among imperfectly correlated

assets, we would expect the efficient frontier to be made up of portfolios of

investments rather than individual securities.

 Two possible exceptions arise at the end points, which represent the asset

with the highest return and that asset with the lowest risk.

The Efficient Frontier and Investor Utility

 As an investor, you will target a point along the

efficient frontier based on your utility function

and your attitude toward risk.

 No portfolio on the efficient frontier can

dominate any other portfolio on the efficient

frontier.

All of these portfolios have different return and

risk measures, with expected rates of return

that increase with higher risk.

The Efficient Frontier and Investor Utility

 The curve in Exhibit 7.15 shows that the

slope of the efficient frontier curve

decreases steadily as you move

upward.

 This implies that adding equal

increments of risk as you move up the

efficient frontier gives you diminishing

increments of expected return.

 To evaluate this slope, we calculate

the slope of the efficient frontier as

follows:

The Efficient Frontier and Investor Utility

 An individual investor’s utility curves specify the

trade-offs he or she is willing to make between

expected return and risk.

 In conjunction with the efficient frontier, these

utility curves determine which particular portfolio

on the efficient frontier best suits an individual

investor.

 Two investors will choose the same portfolio from

the efficient set only if their utility curves are

identical.

The Efficient Frontier and Investor Utility

 Exhibit 7.16 shows two sets of utility curves along with an efficient frontier of investments.

 The curv es labeled U1 are for a strongly risk- av erse investor (with U3 U2 U1).

 These utility curves are quite steep, indicating that the investor will not tolerate much additional risk to obtain additional returns.

 The investor is equally disposed toward any E(R), σ combinations along a specific utility curve, such as U1.

 The curv es labeled U1′ (U3′ U2′ U1′) characterize a less-risk-averse investor.

 Such an investor is willing to tolerate a bit more risk to get a higher expected return.

The Efficient Frontier and Investor Utility

 The optimal portfolio is the portfolio on the

efficient frontier that has the highest utility

for a given investor.

 It lies at the point of tangency

between the efficient frontier and the

curve with the highest possible utility.

 A conservative investor’s highest utility is at

point X where the curve U2 just touches

the efficient frontier.

 A less-risk-averse investor’s highest utility

occurs at point Y, which represents a

portfolio with a higher expected return

and higher risk than the portfolio at X.