FIN 377: Investments Topic 4: An Introduction to Portfolio Management

FIN 377: Investments Topic 4: An Introduction to Portfolio Management
Larry Schrenk, Instructor

Overview 6.1 Some Background Assumptions
6.2 Markowitz Portfolio Theory 6.3 The Efficient Frontier 6.4 Capital Market Theory

Learning Objectives @

Readings Reilley, et al., Investment Analysis and Portfolio Management, Chap. 6

6.1 Some Background Assumptions

6.1 Some Background Assumptions
Investors want to maximize return from the total set of investments for a given level of risk Portfolio includes all assets and liabilities Relationship between the returns for assets in the portfolio is important A good portfolio is not simply a collection of individually good investments

6.1.1 Risk Aversion Given a choice between two assets with equal rates of return, risk-averse investors will select the asset with the lower level of risk Evidence Many investors purchase insurance purchase various types of insurance, including life insurance, car insurance, and health insurance Yield on bonds increases with risk classifications, which indicates that investors require a higher rate of return to accept higher risk Not all investors are risk averse It may depend on the amount of money involved Risking small amounts, but insuring large losses

6.1.2 Definition of Risk For most investors, risk means the uncertainty of future outcomes An alternative definition may be the probability of an adverse outcome

Diversification: An Example
We bounce a rubber ball and record the height of each bounce. The average bounce height is very volatile As we add more balls… Average bounce height less volatile. Greater heights ‘cancels’ smaller heights

Average Bounce

Bouncing Ball Standard Deviation

Diversification: An Analogy
‘Cancellation’ effect = diversification Hold one stock and record daily return The return is very volatile. As we add more stock… Average return less volatile Larger returns ‘cancels’ smaller returns

Diversification: The Dis-Analogy
Stocks are not identical to balls. Drop more balls, volatility will Eventually go to zero. Add more stocks, volatility will Decrease, but Level out at a point well above zero.

Stock Diversification
Key Idea: No matter how many stocks in my portfolio, the volatility will not get to zero!

What Different about Stocks?
As I start adding stocks The non-market risks of some stocks cancel the non-market risks of other stocks. The volatility begins to go down.

What Different about Stocks?
At some point, all non-market risks cancel each other. But there is still market risk! But volatility can never reach zero. Diversification cannot reduce market risk.

Diversification and Market Risk
Impact on all firms in the market No cancellation effect Example: Government doubles the corporate tax All firms worse off Holding many different stocks would not help. Diversification can eliminate my portfolio’s exposure to non-markets risks, but not the exposure to market risk.

What Happens in Stock Diversification?▪
Non-Market Risk Volatility of Portfolio Market Risk Number of Stocks

Empirical Diversification Example
Five Companies Ford (F) Walt Disney (DIS) IBM Marriott International (MAR) Wal-Mart (WMT)

Diversification Example (cont’d)
Five Equally Weighted Portfolios Portfolio Equal Value in… F Ford F,D Ford, Disney F,D,I, Ford, Disney, IBM F,D,I,M Ford, Disney, IBM, Marriott F,D,I,M,W Ford, Disney, IBM, Marriott, Wal-Mart Minimum Variance Portfolio (MVP)

Individual Returns

F Portfolio

F,D Portfolio

F,D,I Portfolio

F,D,I,M Portfolio

F,D,I,M,W Portfolio

F,D,I,M,W versus F Portfolio

Equally Weighted versus MVP

MVP versus F Portfolio

Decreasing Risk

A Well-Diversified Portfolio
Non-market risks eliminated by diversification Assumption: All investors hold well-diversified portfolios. Index funds S&P 500 Russell 2000 Wilshire 5000

Implications If investors hold well-diversified portfolios…
Ignore non-market risk No compensation for non-market risk Only concern is market risk Risk Identification. If you hold a well diversified portfolio, then your only exposure is to market risk. Risk Analysis, Step 1 Revised

6.2 Markowitz Portfolio Theory

The Markowitz model is based on several assumptions regarding investor behavior: Investors consider each investment alternative as being represented by a probability distribution of potential 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 potential returns Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the variance (or standard deviation) of returns only For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk

Using 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

6.2.1 Alternative Measures of Risk
Variance or standard deviation of expected return Range of returns Returns below expectations Semi-variance – a measure that only considers deviations below the mean These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate

6.2.1 Alternative Measures of Risk
Advantages of using variance or standard deviation of returns: Measure is somewhat intuitive Widely recognized risk measure Has been used in most of the theoretical asset pricing models

6.2.2 Expected Rates of Return
Expected rate of return For an individual investment Equal to the sum of the potential returns multiplied with the corresponding probability of the returns For a portfolio of investments Equal to the weighted average of the expected rates of return for the individual investments in the portfolio

6.2.2 Expected Rates of Return

6.2.3 Variance of Returns for an Individual Investment
The variance is a measure of the variation of possible rates of return:

6.2.3 Variance of Returns for an Individual Investment
The standard deviation is also a measure of the variation of possible rates of return:

6.2.3 Variance (Standard Deviation) of Returns for an Individual Investment

6.2.4 Variance of Returns for a Portfolio
Covariance of Returns A measure of the degree to which two variables “move together” relative to their individual mean values over time For two assets, x and y, the covariance of rates of return is defined as:

Covariance and Correlation The correlation coefficient (r) is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations

The coefficient can vary only in the range +1 to -1 A value of +1 would indicate perfect positive correlation. This means that returns for the two assets move together in a positively and completely linear manner A value of –1 would indicate perfect negative correlation. This means that the returns for two assets move together in a completely linear manner, but in opposite directions

Portfolio Variance Formula

Two-Asset Portfolio Variance Formula

6.2.5 Standard Deviation of Returns for a Portfolio
Portfolio Standard Deviation Formula

6.2.5 Standard Deviation of Returns for a Portfolio
Two-Asset Standard Deviation Variance Formula

6.2.5 Standard Deviation of a Portfolio
Impact of a New Security in a Portfolio Two effects to the portfolio’s standard deviation when we add a new security to such a portfolio: The asset’s own variance of returns The covariance between the returns of this new asset and the returns of every other asset that is already in the portfolio The relative weight of these numerous covariances is substantially greater than the asset’s unique variance; the more assets in the portfolio, the more this is true The important factor to consider when adding an investment to a portfolio that contains a number of other investments is not the new security’s own variance but the average covariance of this asset with all other investments in the portfolio

Portfolio Standard Deviation Calculation Any asset or portfolio of assets can be described by two characteristics: The expected rate of return The standard deviation of returns

Equal Risk and Return—Changing Correlations The expected return of the portfolio does not change because it is simply the weighted average of the individual expected returns Demonstrates the concept of diversification, whereby the risk of the portfolio is lower than the risk of either of the assets held in the portfolio Risk reduction benefit occurs to some degree any time the assets combined in a portfolio are not perfectly positively correlated (that is, whenever ri,j < +1) Diversification works because there will be investment periods when a negative return to one asset will be offset by a positive return to the other, thereby reducing the variability of the overall portfolio return

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, meaning that the average combined return for the two securities over time would be a constant value (that is, have no variability) Thus, a pair of completely negatively correlated assets provides the maximum benefits of diversification by completely eliminating variability from the portfolio

Combining Stocks with Different Returns and Risk Consider two assets (or portfolios) with different expected rates of return and individual standard deviations With perfect negative correlation, the portfolio standard deviation is not zero This is because the different examples have equal weights, but the asset standard deviations are not equal

Constant Correlation with Changing Weights If the weights of the two assets are changed while holding the correlation coefficient constant, a set of combinations is derived that trace an ellipse The benefits of diversification are critically dependent on the correlation between assets

Two Asset Portfolio: Example
Return s Weight r A 7% 19% 80% 0.8 B 11% 22% 20% NOTE: sp < sA and sp < sB

Example If I hold a stock with a standard deviation of 20%, would I get more diversification by adding a stock with a standard deviation of 10% or 30%? If I added two stocks each with a standard deviation of 25%, the standard deviation of the portfolio could be anywhere from 25% to 0%–depending on the correlation. If r = 1, s = 25% If r = -1, s = 0% (with the optimal weights)

Standard Deviation and Stock Risk
Standard deviation tells nothing about… Stock’s diversification effect on a portfolio; or Whether including that stock will increase or decrease the exposure to market risk. Thus, standard deviation (and variance) Not a correct measure of market risk, and Cannot be used as our measure of risk in the analysis of stocks.

6.2.6 A Three-Asset Portfolio
The results presented earlier for the two-asset portfolio can be extended to a portfolio of n assets As more assets are added to the portfolio, more risk will be reduced (everything else being the same) The general computing procedure is still the same, but the amount of computation has increase rapidly For the three-asset portfolio, the computation has doubled in comparison with the two-asset portfolio

6.2.7 Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Estimates of Expected returns Standard deviation Correlation coefficient Among entire set of assets With 100 assets, 4,950 correlation estimates Estimation risk refers to potential errors

6.2.7 Estimation Issues With the assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets Single index market model: Ri = ai + biRm + ei

6.2.7 Estimation Issues If all the securities are similarly related to the market and a bi derived for each one, it can be shown that the correlation coefficient between two securities i and j is given as:

6.3 The Efficient Frontier

The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk or the minimum risk for every level of return Every portfolio that lies on the efficient frontier has either a higher rate of return for the same risk level or lower risk for an equal rate of return than some portfolio falling below the frontier See Exhibit 6.13 Portfolio A in Exhibit 6.13 dominates Portfolio C because it has an equal rate of return but substantially less risk Portfolio B dominates Portfolio C because it has equal risk but a higher expected rate of return

Markowitz defined the basic problem that the investor needs to solve as: Select {wi} so as to: subject to the following conditions: i) E(Rport) = Swi E(Ri) = R* ii) Swi = 1.0

The general method for solving the formula is called a constrained optimization procedure because the task the investor faces is to select the investment weights that will “optimize” the objective (minimize portfolio risk) while also satisfying two restrictions (constraints) on the investment process: The portfolio must produce an expected return at least as large as the return goal, R; and All of the investment weights must sum to 1.0 The approach to forming portfolios according to this equation is often referred to as mean-variance optimization because it requires the investor to minimize portfolio risk for a given expected (mean) return goal

6.3.1 The Efficient Frontier: An Example
What would be the optimal asset allocation strategy using these five asset classes?

6.3.1 The Efficient Frontier: An Example

6.3.2 The Efficient Frontier and Investor Utility
An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upward The interactions of these two curves will determine the particular portfolio selected by an individual investor The optimal portfolio has the highest utility for a given investor

The optimal lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility See Exhibit 6.16 Investor X with the set of utility curves will achieve the highest utility by investing the portfolio at X With a different set of utility curves, Investor Y will achieve the highest utility by investing the portfolio at Y Which investor is more risk averse?

6.4 Capital Market Theory

6.4 Capital Market Theory: An Overview
Capital market theory builds directly on the portfolio theory we have just developed by extending the Markowitz efficient frontier into a model for valuing all risky assets Capital market theory also has important implications for how portfolios are managed in practice The development of this approach depends on the existence of a risk-free asset, which in turn will lead to the designation of the market portfolio

6.4.1 Background for Capital Market Theory
Assumptions: All investors are Markowitz efficient investors who want to target points on the efficient frontier Investors can borrow or lend any amount of money at the risk-free rate of return (RFR) All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return All investors have the same one-period time horizon such as one month, six months, or one year

6.4.1 Background for Capital Market Theory
Assumptions (Continued) All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio There are no taxes or transaction costs involved in buying or selling assets There is no inflation or any change in interest rates, or inflation is fully anticipated Capital markets are in equilibrium, which implies that all investments are properly priced in line with their risk levels

6.4.2 Developing the Capital Market Line
A risky asset is one for which future returns are uncertain Uncertainty is measured by the standard deviation of expected returns Because the expected return on a risk-free asset is entirely certain, the standard deviation of its expected return is zero The rate of return earned on such an asset should be the risk-free

Covariance with a Risk-Free Asset Covariance between two sets of returns is Because the returns for the risk free asset are certain, the covariance between the risk-free asset and any risky asset or portfolio will always be zero Similarly, the correlation between any risky asset and the risk-free asset would be zero

Combining Risk-Free Asset with a Risky Portfolio Expected Return Is the weighted average of the two returns: E(Rport) = wrf rf + (1 - wrf)E(Rm) Standard Deviation Is the linear proportion of the standard deviation of the risky asset portfolio

The Risk–Return Combination Investors who allocate their money between a riskless security and the risky Portfolio M can expect a return equal to the risk-free rate plus compensation for the number of risk units (σport) they accept

The risk–return relationship holds for every combination of the risk-free asset with any collection of risky assets This relationship holds for every combination of the risk-free asset with any collection of risky assets However, when the risky portfolio, M, is the market portfolio containing all risky assets held anywhere in the marketplace, this linear relationship is called the Capital Market Line (CML)

Risk–Return Possibilities with Leverage One can attain a higher expected return than is available at point M One can invest along the efficient frontier beyond point M, such as point D With the risk-free asset, one can add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M to achieve a point like E Clearly, point E dominates point D Similarly, one can reduce the investment risk by lending money at the risk-free asset to reach points like C

6.4.3 Risk, Diversification, and the Market Portfolio
Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML It must include all risky assets Because the market is in equilibrium, all assets in this portfolio are in proportion to their market values Because it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away

Systematic risk Only systematic risk remains in the market portfolio Systematic risk can be measured by the standard deviation of returns of the market portfolio and can change over time Systematic risk is the variability in all risky assets caused by macroeconomic variables: Variability in growth of money supply Interest rate volatility Variability in factors like industrial production, corporate earnings, cash flow

Diversification and the Elimination of Unsystematic Risk The purpose of diversification is to reduce the standard deviation of the total portfolio This assumes that imperfect correlations exist among securities As you add securities, you expect the average covariance for the portfolio to decline How many securities must you add to obtain a completely diversified portfolio?

The CML and the Separation Theorem The CML leads all investors to invest in the M portfolio Individual investors should differ in position on the CML depending on risk preferences How an investor gets to a point on the CML is based on financing decisions Risk averse investors will lend at the risk-free rate, while investors preferring more risk might borrow funds at the RFR and invest in the market portfolio The investment decision of choosing the point on CML is separate from the financing decision of reaching there through either lending or borrowing

A Risk Measure for the CML Capital market theory now shows that the only relevant portfolio is the market Portfolio M. Together, this means that the only important consideration for any individual risky asset is its average covariance with all the risky assets in Portfolio M or the asset’s covariance with the market portfolio This covariance, then, is the relevant risk measure for an individual risky asset

Because all individual risky assets are a part of the market portfolio, one can describe their rates of return in relation to the returns to Portfolio M using a linear model: Rit = ai +biRMt + e The variance of returns for a risky asset can similarly be described as: Var(Rit) = Var(ai +biRMt + e) = Var(ai) + Var(biRMt) + Var(e) = 0+ Var(biRMt) + Var(e)

Note that Var(biRMt) is the variance of return for an asset related to the variance of the market return, or the asset’s systematic variance or risk Also, Var(ε) is the residual variance of return for the individual asset that is not related to the market portfolio This residual variance is what we have referred to as the unsystematic or unique risk because it arises from the unique features of the asset Therefore: Var(Rit) = Systematic Variance + Unsystematic Variance

6.4.4 Investing with the CML: An Example
Suppose you have a riskless security at 4% and a market portfolio with a return of 9% and a standard deviation of 10%. How should you go about investing your money so that your investment will have a risk level of 15%? Portfolio Return E(Rport) = rf+σport[(E(RM) - rf)/σM) = 4%+15%[(9%-4%)/10%] = 11.5% Money invested in riskless security, wRF 11.5% = wRF (4%) + (1-wRF )(9%) ----> wRF = -0.5 The investment strategy is to borrow 50% and invest 150% of equity in the market portfolio

6.4.4 Investing with the CML: An Example

FIN 377: Investments Topic 4: An Introduction to Portfolio Management

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