Reilly Chapter 6

Why standard deviation is used

(1)this measure is somewhat intuitive, (2)it is widely recognized risk measure, and (3)it has been used in most of the theoretical asset pricing models.

Because capital market theory builds on the Markowitz portfolio model, it requires the same assumptions, along with some additional ones:

-All investors seek to invest in portfolios representing tangent points on the Markowitz efficient frontier. The exact location of this tangent point and, therefore, the specific portfolio selected will depend on the individual investor's risk-return utility function. -Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). (Clearly, it is always possible to lend money at the nominal risk-free rate by buying risk-free securities such as government T-bills. It is not possible in practice for everyone to borrow at this level.) -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 or one year. The model will be developed for a single hypothetical period, and its results could be affected by a different assumption since it requires investors to derive risk measures and risk-free assets that are consistent with their investment horizons. -All investments are infinitely divisible, so it is possible to buy or sell fractional shares of any asset or portfolio. This assumption allows us to discuss investment alternatives as continuous curves. -There are no taxes or transaction costs involved in buying or selling assets. This is a reasonable assumption in many instances. Neither pension funds nor charitable foundations have to pay taxes, and the transaction costs for most financial institutions are negligible on most investment instruments. -Either there is no inflation or any change in interest rates, or inflation is fully anticipated. This is a reasonable initial assumption, and it can be modified. -Capital markets are in equilibrium. This means that we begin with all investments properly priced in line with their risk levels.

Covariance of returns

-Covariance is a measure of the degree to which two variables move together relative to their individual mean values over time. -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 opposite directions relative to their means during specified time intervals over time. -The magnitude of the covariance depends on the variances of the individual return series, as well as on the relationship between the series. -Although the rates of return for the two assets moved together during some months, in other months they moved in opposite directions. The covariance statistic provides an absolute measure of how they moved together over time.

covariance and correlation

-Covariance is affected by the variability of the two individual return indexes. -Therefore, a number such as −0.68 in our example might indicate a weak negative relationship if the two individual indexes were volatile, but would reflect a strong negative relationship if the two indexes were relatively stable. Obviously, we want to standardize this covariance measure by taking into consideration the variability of the two individual return indexes, as follows: R = cov / standard dev rit x standard dev rjt -Standardizing the covariance by the product of the individual standard deviations yields the correlation coefficient, , which can vary only in the range −1 to +1. A value of +1 indicates a perfect positive linear relationship between ri and rj, meaning the returns for the two assets move together in a completely linear manner. A value of −1 indicates a perfect negative relationship between the two return indexes, so that when one asset's rate of return is above its mean, the other asset's rate of return will be below its mean by a proportional amount. -As noted, a correlation of +1.0 indicates perfect positive correlation, and a value of −1.0 means that the returns moved in completely opposite directions. A value of zero means that the returns had no linear relationship, that is, they were uncorrelated statistically. That does not mean that they are independent.

Estimation issues

-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, we must estimate its expected returns and standard deviation. We must also estimate the correlation coefficient among the entire set of assets. The number of correlation estimates can be significant—for example, for a portfolio of 100 securities, the number correlation estimates is 4,950 (that is, 99 + 98 + 97 + ...). The potential source of error that arises from these approximations is referred to as estimation risk. -We can reduce the number of correlation coefficients that must be estimated by assuming that stock returns can be described by the relationship of each stock to the same market index—that is, a single index market model

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 dispersed the potential returns, the greater the uncertainty of the potential outcomes. -Another measure of risk is the range of returns. It is assumed that a larger range of possible returns, from the lowest to the highest, means greater uncertainty regarding future expected returns. -some observers believe that investors should be concerned only with returns below some threshold level. These are sometimes called downside risk measures because they only consider potential returns that fall beneath that target rate. A measure that only considers deviations below the expected return is the semi-variance. Extensions of the semi-variance measure only computes return deviations below zero (that is, negative returns), or returns below the returns of 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 (regret) from returns less than some target rate. Assuming that investors would welcome returns above some target rate, the returns above such a target rate are not considered when measuring risk.

The CML and the separation theorem

-The CML leads all investors to invest in the same risky asset Portfolio M. Individual investors should only differ regarding their position on the CML, which depends on their risk preferences. In turn, how they get to a point on the CML is based on their financing decisions. If you are relatively risk averse, you will lend some part of your portfolio at the RFR by buying some risk-free securities and investing the remainder in the market portfolio of risky assets -This financing decision provides more risk but greater expected returns than the market portfolio. Because portfolios on the CML dominate other portfolio possibilities, the CML becomes the new efficient frontier of portfolios, and investors decide where they want to be along this efficient frontier. -Tobin (1958) called this division of the investment decision from the financing decision the separation theorem. Specifically, to be somewhere on the CML efficient frontier, you initially decide to invest in the market Portfolio M. This is your investment decision. Subsequently, based on your risk preferences, you make a separate financing decision either to borrow or to lend to attain your preferred risk position on the CML.

The Markowitz portfolio theory

-The basic portfolio model was developed by Harry Markowitz (1952, 1959), who derived the expected rate of return for a portfolio of assets as well as a 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. More important, he derived the formula for computing the variance of a portfolio. -This portfolio variance formula not only indicated the importance of diversifying investments to reduce the total risk of a portfolio but also showed how to effectively diversify. Based on several assumptions -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. -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.

Impact of a new security in a portfolio

-The first is the asset's own variance of returns, and the second is 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. 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 new security's own variance but the average covariance of this asset with all other investments in the portfolio.

The efficient frontier and investor utility

-as an investor, you will target a point along the efficient frontier based on your utility function, which reflects 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. -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 best portfolio is the mean-variance efficient portfolio 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

Diversification and the elimination of unsystematic risk

-the purpose of diversification is to reduce the standard deviation of the total portfolio. This assumes less-than-perfect correlations among securities. Ideally, as you add securities, the average covariance for the portfolio declines. How many securities must be included to arrive at a completely diversified portfolio? For the answer, you must observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation. -Subsequent studies have modified this finding. Statman (1987) considered the trade-off between the diversification benefits and additional transaction costs involved with increasing the size of a portfolio. He concluded that a well-diversified portfolio must contain at least 30-40 stocks. Campbell, Lettau, Malkiel, and Xu (2001) demonstrated that because the unique portion of an individual stock's total risk has been increasing in recent years, it now takes more stocks to diversify a portfolio. -The important point is that, by adding to a portfolio new stocks that are not perfectly correlated with stocks already held, you can reduce the overall standard deviation of the portfolio, which will eventually reach the level in the market portfolio. At that point, you will have diversified away all unsystematic risk, but you still have market or systematic risk. You cannot eliminate the variability and uncertainty of macroeconomic factors that affect all risky assets. Further, you can attain a lower level of systematic risk by diversifying globally versus only diversifying within the United States because some of the systematic risk factors in the U.S. market (such as U.S. monetary policy) are not perfectly correlated with systematic risk variables in other countries, like Germany and Japan.

A risk measure for the CML

-the relevant risk to consider when adding a security to a portfolio is its average covariance with all other assets in the portfolio -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.

completely diversified portfolio-

A portfolio in which all unsystematic risk has been eliminated by diversification.

risky asset

An asset with uncertain future 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 rate of return (RFR), which, as we discussed in Chapter 1, should equal the expected long-run growth rate of the economy with an adjustment for short-run liquidity.

Risk return possibilities with leverage

An investor may want to attain a higher expected return than is available at Point M—which represents a 100 percent allocation to Portfolio M—in exchange for accepting higher risk. One alternative would be to invest in one of the risky asset portfolios on the Markowitz frontier beyond Point M such as the portfolio at Point D. A second alternative is to add leverage to the portfolio by borrowing money at the risk-free rate and investing the proceeds in the risky asset portfolio at Point M; this is depicted as Point E. What effect would this have on the return and risk for your portfolio? -Because the CML is a straight line, it implies that all the portfolios on the CML are perfectly positively correlated. This occurs because all portfolios on the CML combine the risky asset Portfolio M and the risk-free asset. You either invest part of your money in the risk-free asset (that is, you lend at the RFR) and the rest in the risky asset Portfolio M, or you borrow at the risk-free rate and invest these funds in the risky asset portfolio. In either case, all the variability comes from the risky asset M portfolio.

one basic assumption of portfolio theory is that investors want to maximize the returns from the total set of investments for a given level of risk.

First, your portfolio should include all of your assets and liabilities, not only your marketable securities but also less marketable investments such as real estate, art, and antiques. The full spectrum of investments must be considered because the returns from all these investments interact, and this relationship among the returns for assets in the portfolio is important

unsystematic risk

Risk that is unique to an asset, derived from its particular characteristics. It can be eliminated in a diversified portfolio.

mean-variance optimization

The approach to forming portfolios to help the investor to minimize portfolio risk for a given expected (mean) return goal.

capital market line (CML)-

The line from the intercept point that represents the risk-free rate tangent to the original efficient frontier; it becomes the new efficient frontier since investments on this line dominate all the portfolios on the original Markowitz efficient frontier. -You can continue to draw lines from RFR to portfolios on the efficient frontier with increasingly higher slopes until you reach the point of tangency at Portfolio M. The set of portfolio possibilities along line RFR-M—which is the CML—dominates all other feasible combinations that investors could form.

Separation theorem

The proposition that the investment decision, which involves investing in the market portfolio on the capital market line, is separate from the financing decision, which targets a specific point on the CML based on the investor's risk preference.

efficient frontier

The set of portfolios that has the maximum rate of return for every given level of risk, or the minimum risk for every potential rate of return. 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. -Because of the benefits of diversification among less-than-perfectly 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 the asset with the lowest risk. -The general method for solving the formula in Equation 6.11 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: (i)the portfolio must produce an expected return at least as large as the return goal, ; and (ii)all of the investment weights must sum to 1.0.

Diversification-

The situation when the risk of a portfolio is lower than the risk of either of the assets held in the portfolio.

Markowitz (1959) derived the general formula for the standard deviation of a portfolio as follows:

This 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 very important point is that the standard deviation for a portfolio of assets encompasses not only the variances of the individual assets but also includes the covariances between all the 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

risk averse

given a choice between two assets with equal rates of return, they will select the asset with the lower level of risk. -Further evidence of risk aversion is the difference in promised yield (the required rate of return) for different grades of bonds with different degrees of credit risk.

risk

means the uncertainty of future outcomes. An alternative definition might be the probability of an adverse outcome. -Risk and uncertainty are used interchangeably

The expected rate of return for a portfolio of investments

s 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 individual investment.

Variance (Standard deviation) of returns for an individual investment

sum of [(RI - ER)^2] x p

market portfolio

the portfolio that includes all risky assets with relative weights equal to their proportional market values.

true or false. A notable result is that with low, zero, or negative correlations, it is possible to derive portfolios that have lower risk than either single asset. In our set of examples where , this occurs in Cases h, i, j, and k. As we saw earlier, this ability to reduce risk is the essence of diversification.

true

true or false. 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. As we will see, capital market theory also has important implications for how portfolios are managed in practice

true

true or false. Investors should only invest their funds in two types of assets—the risk-free security and the risky collections of assets representing Portfolio M—with the weights of these two holdings determined by the investors' tolerance for risk. Because of the special place that the market Portfolio M holds for all investors, it must contain all risky assets that exist in the marketplace. This includes not just U.S. common stocks, but also non-U.S. stocks, U.S. and non-U.S. bonds, real estate, private equity, options and futures contracts, art, antiques, and so on. Further, these assets should be represented in Portfolio M in proportion to their relative market values.

true

true or false. This combination of risk preference and risk aversion can be explained by an attitude toward risk that depends on the amount of money involved. Researchers such as Friedman and Savage (1948) have speculated 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 losses such as fire or accidents.

true

true or false. This implies that only systematic risk, defined as the variability in all risky assets caused by marketwide variables, remains in Portfolio M. Systematic risk can be measured by the standard deviation of returns to the market portfolio, and it changes over time whenever there are changes in the underlying economic forces that affect the valuation of all risky assets, such as variability of money supply growth, interest rate volatility, and variability in industrial production or corporate earnings

true

true or false. f the rates of return for one asset are above (below) its mean rate of return during a given period and the returns for the other asset are likewise above (below) its mean rate of return during this same period, then the product of these deviations from the mean is positive.

true