## R

14In this section we have drawn the efficient frontier as it would look if short sales were not allowed. However, the analysis is general and applies equally well to the case where short sales are allowed. Figure 5.14 Combinations of the riskless asset and various risky portfolios. segment RF G and place some of their money in a riskless asset and some in risky portfolio G. Others who were much more tolerant of risk would hold portfolios along the segment G H, borrowing funds and placing their...

## Info

As discussed earlier, C* is the cut-off rate. All securities whose excess-return-to-risk ratio is above the cut-off rate are selected and all whose ratios are below are rejected. The value of C* is computed from the characteristics of all of the securities that belong in the optimum portfolio. To determine C* it is necessary to calculate its value as if there were different numbers of securities in the optimum portfolio. Designate C. as a candidate for C*. I he value of Ci is calculated when i...

## Stochastic Liability Stream

Stochastic liability streams arise in a number of realistic investment situations. Retired employees often have a cost-of-living adjustment (COLA) in their retirement plans. Their base pension is fixed, but is adjusted upward because of changes in some index such as the consumer price index. Casualty insurance (such as automobile insurance) companies are another example. A casualty insurance company receives premium income. The size of the liabilities (e.g., auto accident claims) takes a number...

## Fixed Liability Stream

A manager is often called on to manage a portfolio of bonds so as to meet a fixed set of liability payments over time. Although liabilities often are not truly fixed, there are circumstances in which acting as if they were fixed is a close approximation to reality. Probably the clearest case was the sale of Guaranteed Insurance Contracts (GIC) by insurance companies, which were very popular in the early 1980s and still are. These contracts required that the insurance company pay a fixed sum at...

## Bibliography

Another Look at Mutual Fund Performance, Journal of Financial and Quantitative Analysis, VI, No. 3 June 1971 , pp. 909-912. 2. Ball, Ray, Kothari, S.P., and Shanken, Jay. Problems in Measuring Portfolio Performance An Application to Contrarian Investment Strategies, The Journal of Financial Economics, 38, No. 1 May 1995 , pp. 79-107. 3. Blake, Christopher, Elton, Edwin, and Gruber, Martin. The Performance of Bond Mutual Funds, Journal of Business, 66, No. 3 July 1993 . 4....

## The Evidence Bond Funds

Bond mutual fund performance has not received nearly as much attention as equity fund performance. The principal study in this area is that of Blake, Elton, and Gruber 3 . Table 24.17 is reproduced from their article. They measure the performance of the bond funds for the five-year period ending in 1991 by comparing the return earned by the fund with Table 24. 7 Performance of Bond Funds Table 24. 7 Performance of Bond Funds No. of Significant Alphas Positive Negative

## The Evidence Stock and Balanced Mutual Funds

A review of the evidence presented earlier in this chapter shows that, in general, mutual funds performed worse than a naive strategy of random selection or mixing passive portfolios with the riskless asset. This conclusion is broadly consistent with the other mutual fund studies that we did not examine of McDonald 61 , Williamson 89 , Crenshaw 15 , Lehmann and Modest 56 , and Elton, Gruber, Das, and Hlavka 25 . There are exceptions. The Securities and Exchange Commission SEC study showed one...

## Mutual Fund Performance

In the earlier section we discussed mutual fund studies to illustrate some of the performance measurement techniques. In this section we intend to summarize the results. When examining mutual fund performance figures, it is important to be careful to state the purpose of the evaluation. One purpose is to evaluate them as an alternative to individual direct investment, and this is the question we initially address. There are three types of transaction costs incurred directly or indirectly by...

## The Use of APT Models to Evaluate and Diagnose Performance

The use of an APT model in combination with a multi-index model allows for better diagnoses of what a portfolio manager is doing, a better development of appropriate benchmarks, and a better measurement and attribution of performance.13 The overall performance and reasons for the performance can be measured using an APT model such as the one discussed in Chapter 16. As an example, consider the model we described in the latter section of Chapter 16. Let's examine the return-generating process...

## The Use of Multiple Benchmark and Multiple Index Models for Performance Evaluation

Almost all of the early studies of portfolio performance compared performance of managed portfolios to a single-market index, usually the S amp P index. In the last decade, this has changed. Researchers have recognized that managers hold a wide range of assets besides the large stocks that are contained in the S amp P index. For example, certain funds hold only small stocks, whereas other funds e.g., balanced funds hold a mixture of bonds and stocks and still others, e.g., international funds...

## Problems in Portfolio Measurement

For most individuals and institutions, riskless or near riskless- investments are available. The purchase of the government bond of the appropriate maturity is riskless except for some potential risk on the rate to be earned on reinvested interest. Similarly, for individuals, savings accounts are essentially riskless. Note, however, that the single-parameter portfolio evaluation techniques, in general, assume lending and borrowing at the same riskless rate. As an example, consider the Sharpe...

## Decomposition Of Overall Evaluation

In the previous section we examined a number of overall measures of performance. A number of attempts have been made to examine various aspects of performance that might affect overall performance. In this section we discuss these techniques. One of the most widely referenced decompositions is one proposed by Fama 27 . Figure 24.9 shows Fama's decomposition. The line plotted in Figure 24.9 is the line representing all combinations of the riskless asset and portfolio M. As discussed earlier, one...

## Oneparameter Performance Measures

There are four different one-parameter performance measures that have been proposed in the literature. We will discuss each measure in turn. These measures differ in their definition of risk and their treatment of the ability of the investor to adjust the risk level of any fund in which he or she might invest. The Excess Return to Variability Measure Consider the original portfolio problem. Figure 24.1 plots the return risk opportunities with riskless lending and borrowing. As shown in Chapter...

## Direct Comparisons

As discussed before, one way to compare portfolios is to examine the return earned by alternative portfolios of the same risk. This is the procedure used by Friend, Blume, and Crockett 30 in their examination of mutual funds. Mutual funds have been evaluated by academics more than any other group of investment vehicles. This attention, which may well be unwelcome, is due, primarily, to the fact that data on mutual funds' portfolios are publicly available. Throughout this chapter we illustrate...

## Measures of Risk

There are two possible measures of risk that can be used total risk or nondiversifiable risk. Consider a college endowment fund. Clearly, the appropriate risk is the risk on the total assets. The college will find very little comfort in the fact that part of the risk could be diversified away if they held other assets when the portfolio under consideration contains their total assets. As an alternative, consider the pension fund of a large corporation. For example, at one time AT amp T...

## Measures of Return

In earlier chapters when we computed return, we calculated the capital gains plus dividends from an initial investment. Thus, if a security paid dividends of 3.00 and had a capital gain of 7.00 on an investment of 100, the return was The 10 return was the return over the period in which the capital gain occurred. When evaluating a portfoEo, generalizing our simple idea of return requires care. A problem occurs because there are many inflows and outflows of funds to the portfolio and very...

## Evaluation of Portfolio Performance

An integral part of any decision-making process should be the evaluation of the decision. This is equally true whether investors make their own investment decisions or employ a manager to make them. A large percentage of investments are made by professional managers. Professionally managed funds include mutual funds, pension funds, college endowments, and discretionary accounts, among others. It is important for an investor utilizing one of these managers not only to evaluate how well the fund...

## Questions And Problems

Given the following data, what is the arbitrage with no transaction costs What is the size of the transaction costs necessary to negate the arbitrage A. S amp P 6-month futures contract 200 B. S amp P current value 190 C. 6-month interest rate 6 D. Present value of dividends on stocks in S amp P index over 6 months 4 2. Assume that General Mills, a user of wheat, and wheat farmers, have the same distributional assumptions about future wheat prices. Does a futures contract make economic sense...

## Nonfinancial Futures And Commodity Funds

This chapter is primarily concerned with a discussion of financial futures. Before closing, though, we should mention that there are a tremendous number of nonfinancial commodity futures. Futures exist on a range of additional assets, from those that are thought of as being close to financial assets like silver and gold to those that are almost never thought of as financial futures like hog bellies. In the late 1960s and 1970s, with the tremendous increase in inflation in the American economy,...

## Creating New Products

Futures have been used to create products that could not exist or were inordinately expensive before futures existed. One such product is an Alpha fund. The idea behind such a fund is to capture the stock selection ability of a set of analysts without being subject to market risks. The implementation of the concept simply involves selling enough futures on the S amp P index so that the sum of the Betas on the futures and the fund's stock portfolio equals zero. Thus the fund has a Beta of zero....

## Changing Investment Policy

Financial futures have transaction costs that are dramatically less than those on stocks and bonds. This implies that they are likely to be the preferred way to change the risk exposure of individual assets or categories of assets. In addition, the use of financial futures allows a direct measure of the value added or subtracted by the policy change. Finally, using financial futures allows a wider choice of assets because of an ability to change risk exposure without having to buy and sell the...

## Valuation Of Financial Futures

The valuation of financial futures is greatly simplified by understanding the relationship between futures prices and the current or spot price of the underlying financial instrument. As we will show, a particular relationship must exist, for if it fails to hold, then an immediate riskless profit could be made. Since there are many individuals continuously looking for opportunities to profit from just such a failure, these basic relationships are reasonably descriptive of real markets. We will...

## Description Of Financial Futures

A financial futures contract calls for the delivery of either a specific financial instrument or a member of a set of financial instruments at a specific date or during a specific period of time for an agreed-upon price. Financial futures are traded on organized exchanges and have standardized contract terms. The exact terms differ from financial future to financial future. Table 23.1 lists some of the financial instruments on which financial futures are Table 23.1 The Underlying Instruments...

## P2c2pipciptc

In exactly the same way we can derive the formula for the three-period case. The possible movements of the share price are shown in Figure 22.10. Notice in this case that two periods before the expiration date the stock price is either uS or dS instead of S, as it was in the two-period example. If it is uS, then from Equation A.2 the value at time 1 is simply If the price of the stock were dS in period 1, then where CndR the value expiration if there are n up movements and R down movements....

## Uses Of Options

In earlier sections of this chapter, we discussed the nature of options and their valuation. In this section, we will examine the major uses of options by individual investors and institutional investors. In Chapter 5 we discussed the efficient frontier with riskless lending and borrowing. The efficient frontier was a straight line such as that shown in Figure 22.7. Note that as we increase the number of Treasury bills in the portfolio, we lower expected return and the standard deviation of...

## Artificial Or Homemade Options

One of the existing insights in modem option theory is that an appropriate mixture of Treasury bills and a security creates a payoff pattern identical to the pattern of an option on the underlying security. This is exciting because options are written only on a limited number of securities and artificially created options can produce the payoff pattern of an option on securities or portfolios where actual options don't exist. Consider, for example, an arbitrary portfolio. Assume further that...

## Types Of Options

An option is a contract entitling the holder to buy or sell a designated security at or within a certain period of time at a particular price. There are a large number of types of option contracts, but they all have one element in common the value of an option is directly dependent on the value of some underlying security. Options represent a claim against the underlying security and thus are often called contingent claim contracts. The two least complex options are called puts and calls. These...

## Exact Matching Programs

One of the ways to reduce sensitivity to changes in interest rates is exact matching. Exact matching is an attempt to find the minimum cost portfolio such that the cash flows in each period are sufficient to cover all liabilities. Define the following elements 1. L t as the liabilities in time t. 2. C t, i as the cash flows in period t from a bond of type i. 3. P i as the price of bond i. 4. N c as the number of bond of type i purchased. The cost of the portfolio of bonds is the number of bonds...

## Djl Su 1 So

D2 measures the sensitivity of bond price to a change in the yield curve where the shift in the yield curve is such that the proportional change in all spot rates is the same. 2. Nonproportlonal Shift in Spot Rates D2 resulted from an assumption that the proportional change in all spot rates is identical. Empirical evidence suggests that long rates change less than short rates. Let K t be the proportional change in the fth-period rate compared to the one-period rate. Then

## Bond Portfolio Management Of Yearly Returns

In the prior sections we have discussed designing portfolios of bonds that are reasonably insensitive to changes in the yield curve. The return on these portfolios can fluctuate dramatically from period to period because the concern is meeting some future liability rather than period-by-period returns. Many managers are interested not in meeting some future liability but in the year-by-year return on the portfolio. Managers of bond funds and many managers of pension funds are concerned with...

## Protecting Against Term Structure Shifts

Shifts in the term structure are viewed by most managers as the major sources of risk to bond portfolios. Just as shifts in the market systematically affect all equity prices, shifts in the term structure affect all bond prices. Two techniques have been devised to try to insulate a portfolio from shifts in the term structure. These techniques are known as exact matching and immunization. Exact matching involves finding the lowest cost portfolio that produces cash flows exactly matching the...

## Estimating Spot Rates

s discussed in the text, spot rates are extremely important in bond valuation and invest-nent decisions, and it is necessary to estimate them. Three techniques have been discussed in the literature. We discuss two of them in this appendix. These two differ in that one of them estimates discrete rates and the other continuous rates. Consider the following equation relating the price of a bond to the cash flows accruing to the bondholder 1 2 1 SJ2 l Sm 2 l S0T 2

## The Determinants Of Bond Prices

Bonds can differ in a number of respects. These differences affect bond prices, spot rate, yields to maturity, the expected return in the next period, and the risk associated with next period's return. Standard bond theory deals with the determination of the yield to maturity or price. The yields to maturity on bonds differ for a number of reasons. Among the more important are the following 1. The length of time before the bond matures. 2. The risk of not receiving coupon and principal...

## Determining Spot Rates

More details on the techniques for determining spot rates or equivalent discount functions are discussed in Appendix B and the associated references. However, because spot rates and discount functions play such an important role in bond pricing, some understanding of how they are obtained is useful. To illustrate how spot rates are estimated, assume we observe the following two bond prices and cash flows Bond A is a one-period pure discount instrument. Thus, the one-period spot rate can be...

## J V 2 A

The return equivalency is an application of the law of one price. Similarly, an investor with a three-period horizon could hold a three-period spot or buy a two-period spot and simultaneously enter into a forward commitment from two to three. For there to be no arbitrage the return must be the same or As a further example, consider the spot rates shown in Table 20.3 for period 1 and period 2. These can be used to determine the forward rate from period 1 to 2. Thus, A number of additional...

## An Introduction To Debt Securities

Bonds are primarily traded over the counter rather than on organized exchanges. For some bond issues the bond markets are highly liquid. Other bonds rarely trade. There are four major categories of long-term fixed income securities Although these were discussed in Chapter 2, we will review their major characteristics here. Government bonds represent the borrowing of the federal government. They represent the largest percentage of the total debt market and are by far the most liquid. Since they...

## Past Earnings and Future Earnings

Two separate issues have been examined with respect to the time-series behavior of earnings. One is whether past growth is an indication of future growth. The second is whether the concept of normal earnings is meaningful. We discuss each of these in turn. One of the popular terms used in the financial literature is the term growth stock. This term often refers to a stock that has had substantial growth in the past and is expected to in the future. Names like IBM and Xerox come to mind. From...

## The Influence of the Economy and industry

In earlier chapters we showed that a stock's returns are strongly affected by market movements and by industry or sector returns. A similar phenomenon exists with respect to earnings. Earnings of a firm are strongly influenced by changes in aggregate earnings for the economy and there is some evidence that they are influenced by changes in the earnings of the industry to which the firm belongs. Table 19.4 illustrates the strength of these influences. The sample used in calculating this table...

## The Elusive Number Called Earnings

The value of any asset is determined by its future earning power and not by what it cost at some time in the past. An economist would define earnings as cash flow plus the change in market value of an asset. Consider a bond originally purchased for 100 that carries a 10 interest rate. Assume the bond is worth 95 after one period. What has the earnings been on this investment over the period The economist would say the earnings were 10 m interest plus the 5 decrease in value or a net of 5. An...

## Portfolios Customized for User Characteristics

Even if the institution employing the type of multi-index model under discussion does not believe that it possess superior forecasting ability, it can take advantage of certain attributes of these models. The simplest use of this type of model is in the construction of index funds from a small number of stocks. Empirical evidence suggests that portfolios can be constructed that more closely mimic a target portfolio of securities e.g., an index when the target portfolio is matched with respect...

## An Evolving System of Security Selection

Just as the capital asset pricing model CAPM can be used as a tool in the stock selection process, the new models evolving from the arbitrage pricing theory APT and multifactor model literature can be used to enrich the stock selection process. Because the models employ more information about the process driving security returns, they allow for a more detailed structure for selecting stocks or designing stock selection systems. To illustrate some of the ways in which a multi-index model can be...

## An Ongoing System

In this chapter we have considered several techniques for valuing common stock. The one with the strongest theoretical base involves the discounting of future dividends where the discount rate is appropriately formulated in terms of risk. In recent years several firms have attempted to implement stock valuation and selection systems that incorporate the DCF approach to stock selection and modern capital market theory. Perhaps the best known is the Wells Fargo stock evaluation system. The first...

## Crosssectional Regression Analysis

While DCF models are enjoying a rapidly increased popularity in the investment community, they have been adopted by only a small fraction of the practicing security analysts.18 The majority of security analysts still values common stocks by applying some sort of earnings multiple price earnings ratio to either present earnings, normalized earnings, or forecasted earnings. Approaches to the establishment of the P E ratio cover a vast range. Some firms use the historical P E ratios for companies...

## Finite Horizon Models

We have just seen that a model based on discounting a finite stream of dividends and a terminal price can be consistent with discounting an infinite stream of dividends. In this case, the finite nature of the model arose from consideration of future growth. Let us now look at a finite horizon mode that arises from the way many organizations work, rather than from discounting an infinite stream of dividends. Many organizations make short-run earnings forecasts for stocks one- and two-year...

## The Three Period Model

The usual two-period model assumes that during the initial period, earnings would continue to grow at some constant rate. At year N the second period started and growth was assumed to drop instantly to some steady-state value. Normally, the change to a new long-term growth rate would not occur instantly rather, it would occur over a period of time. Thus, a logical extension is to assume a third period. The resultant model would assume that in period one growth is expected to be constant at some...

## The Two Period Growth Model

The simplest extension of the one-period model is to assume that a period of extraordinary growth good or bad will continue for a certain number of years, after which growth will change to a level at which it is expected to continue indefinitely. The assumption that growth is constant after some point in time follows from the following line of reasoning. After some point in time 5 years, 10 years, 15 years the analyst has no ability to differentiate between firms on the basis of growth. Many...

## Constant Growth Model

One of the best known and certainly the simplest DCF model assumes that dividends will grow at the same rate g into the indefinite future. Defining PQ as today's price, and D1 as next period's dividend the value of a share of stock is P _ Di , 4 1 , Aft g ', , 4 1 , 0 i l kf l kf l 'See Williams 80 or Gordon 45 for discussion of models of this type. 'See Malkiel 67 for the presentation of a model of this type. 'See Molodovsky, May, and Chottinger 72 for the presentation of a model of this type....

## Discounted Cash Flow Models

Discounted cash flow models are based on the concept that the value of a share of stock is equal to the present value of the cash flow that the stockholder expects to receive from it.3 We will argue that this is equivalent to the present value of all future dividends. To facilitate this argument, let us assume that a stockholder intends to hold a share of stock for one period. In this one period the stockholder will receive a dividend and the value of the stock when he or she sells it. If the...

## The Valuation Process

The search for the correct way to value common stocks, or even one that works, has occupied a huge amount of effort over a long period of time. Attempts have ranged from simple mechanical techniques for picking winners to hypotheses about the broad influences affecting stock prices. At one extreme, the attempt to find a simple rule for selecting stocks that will have above-average performance can be likened to the search for a perpetual motion machine. Just as the laws of thermodynamics tell us...

## Information in Analysts Forecasts

Many authors have analyzed whether or not security analysts have information not incorporated in security prices. The majority of these studies suffer from selection bias and survivorship bias. Selection bias occurs because most studies analyze a set of historical analysts' forecasts, and access to these forecasts is controlled. Security analysts generally work for an investment organization that controls whether outsiders have access to prior analysts' forecasts. Furthermore, the investment...

## Strong Form Efficiency

In this section we discuss two issues. The first issue is whether insiders in their trading earn an excess return. Working at Atlantic Richfield and learning that your geologist had discovered massive oil fields off Alaska then trading on that information clearly leads to excess returns. It also likely leads to jail, since trading on inside information in the United States is illegal. Thus, examining the profitability of insider trading is both an examination of the usefulness of insider...

## Results of Some Event Studies

We will not review all types of event studies in this chapter. Rather, we will concentrate on issues that are especially important for investment strategy. In particular, in this section we ill examine the pattern of abnormal returns around the announcement day and whether here is a long-term abnormal return after the announcement post-announcement drift , ihese questions are concerned with whether an investor can make short-term profits by buying on the announcements or make long-term abnormal...

## Methodology Of Event Studies

The methodology of event studies is fairly standard and proceeds as follows 1. Collect a sample of firms that had a surprise announcement the event . What causes prices to change is an announcement that is a surprise to investors. For many studies, such as an announcement of a merger, any announcement can be treated as a surprise. For other studies such as the impact of earnings announcements, it is more complicated. For these studies it is necessary to define a surprise. This is normally done...

## Interest Rate Variables

Term premium yield on long-term bonds minus yield on short-term bonds 2. Risk premium yield on low-rated debt minus yield on high-rated debt The proportion of long-term return that can be explained by these variables is quite high, Fama and French 80 report that 25 of the returns on a value and equally weighted market index over two to four years can be explained by past dividends price. Furthermore, the sign is positive a high dividend over price low level of price implies high returns....

## Tests Of Return Predictability

In this section we review the studies examining the predictability of return from past data. In the first section we examine seasonal patterns in returns. A number of studies find that returns are different depending on the day of the week or time of the year. In the second section we discuss the predictability of return using past return. We analyze both short-term predictability and long-term predictability. In the third section we examine return and firm characteristics. In particular, we...

## Efficient Markets

One of the dominant themes in the academic literature since the 1960s has been the concept of an efficient capital market.1 Although the reader may well be able to visualize several meanings of the term efficient market and although it has, in fact, been used to denote different phenomena at different times, it has come to have a very specific meaning in finance. When someone refers to efficient capital markets, they mean that security prices fully reflect all available information. This is a...

## Apt And Capm

Hsfore continuing our examination of APT models, we should discuss the fact that the PT model and, in fact, the existence of a multifactor model, including one where more than one factor is priced, is not necessarily inconsistent with the Sharpe-Lintner-Mossin iorm or one of the other forms of the CAPM. The simplest case in which an APT model is consistent with the simple form of the 1 APM is the case where the return-generating function is of the form If returns are generated by a single-index...

## The Arbitrage Pricing Model APTA New Approach to Explaining Asset Prices

All of the equilibrium models discussed in Chapters 13, 14, and 15 have their basis in mean-variance analysis. All require that it is optimal for the investor to choose investments on the basis of expected return and variance. However, definitions of returns for which means and variances are calculated differ between models. For example, in the version of the capital asset pricing model CAPM involving taxes, investors examine means and variances of after-tax returns. As a second example, Elton...

## Some Reservations About Traditional Tests Of General Equilibrium Relationships And Some New Research

In this chapter we have reviewed some of the classic tests of general equilibrium relationships. These tests were intended to validate the theories we have described in the previous two chapters. Roll has argued 65 that general equilibrium models of the form of the CAPM are not amenable to testing or, at least, that the tests performed so far provide little evidence in support of, or against, CAPM. Roll raised some legitimate questions, and his arguments are well worth reviewing. Perhaps the...

## Rf Yo YP h 8 Rf e

Where hit is the dividend divided by price for stock i in month t. This model appears like a test of the two-factor model with the addition of a new term involving dividend yields. The form of this new term is consistent with the post-tax model presented in Chapter 14 with y2 interpreted as t.14 When Litzenberger and Ramaswamy tested this model using maximum likelihood estimates on monthly data, they found the following results for the period 1936-1977.15 Rh - Rf, 0.0063 0.0421 3,, 0.236 5,, -...

## Testing The Posttax Form Of The Capm Model

Iiile a great deal of attention has been paid to tests of the zero Beta two-factor CAPM odel, almost no testing has been done on the other forms of general equilibrium models described in the previous chapter. The one exception to this is tax-adjusted versions of the general equilibrium model. Black and Scholes 5 have tested a form of the CAPM that includes a dividend term and concluded that dividends do not affect the equilibrium relationship. Since a dividend term is present in the post-tax...

## Fmpirical Tests Of The Capm

' here has been a huge amount of empirical testing of the standard form and the two-factor u rn of the CAPM model. A discussion of all empirical work would require a volume by lultaneous test of all thrae of these 1 two-factor model instep of-theCM itself. The approach we have adopted is to review the hypotheses that should be tested, to review some of the early work on testing the CAPM, then to discuss briefly a few of the problems inherent in any test of the CAPM. Finally, we review, in more...

## Empirical Tests of Equilibrium Models

La the two previous chapters we stressed the fact that the construction of a theory necessitates a simplification of the phenomena under study. To understand and model any process, elements in the real world are simplified or assumed away. While a model based on simple assumptions can always be called into question because of these assumptions, the relevant test of how much damage has been done by the simplification is to examine the relationship between the predictions of the model and...

## Derivation Of The General Equilibrium With Taxes

Earlier in this chapter we saw that any security or portfolio has an equilibrium return given by We derived this expression by maximizing for the investor's portfolio P equal to the market portfolio M and the riskless rate defined as the intercept of a line tangent to point M. Rz in the foregoing solution is the return on the minimum variance portfolio that is uncorrelated with the portfolio M. We could have repeated this analysis for any portfolio P different from M, and for assets included in...

## Inflation Risk And Equilibrium

One specific case of a multiperiod genera equilibrium model that has received special attention is the case where all of Fama's assumptions are met except that there is uncertain inflation. Friend, Landskroner, and Losq 39 derive a general equilibrium relationship for the expected return on any asset under uncertain inflation, assuming that all utility functions exhibit constant proportional risk aversion. Their equilibrium appears similar to the simple form of the CAPM, but both the definition...

## The Consumptionoriented Capm

A number of authors, starting with Breeden 5 and Rubinstein 104 , have taken a different approach to defining equilibrium in the capital markets. They start with a set of assumptions investors maximize a multiperiod utility function for lifetime consumption havejljomogeneous eliefs concerning return characteristics of assets there is an infinitely livec fixed population there is a single consumption good and there exists a capital market tnat allows investors to reach a consumption pattern such...

## Personal Taxes

The simple form of the capital asset pricing model ignores the presence of taxes in arriving at an equilibrium solution. The implication of this assumption is that investors are indifferent between receiving income in the form of capital gains or dividends and that all investors hold the same portfolio of risky assets. If we recognize the existence of taxes and, in particular, the fact that capital gains are taxed, in general, at a lower rate than dividends, the equilibrium prices should...

## Modifications Of Riskless Lending And Borrowing

A second assumption of the CAPM is that investors can lend and borrow unlimited sums of money at the riskless rate of interest. Such an assumption is clearly not descriptive of the real world. It seems much more realistic to assume that investors can lend unlimited sums of money at the riskless rate but cannot borrow at a riskless rate. The lending assumption is equivalent to investors being able to buy government securities equal in maturity to their single-period horizon. Such securities...

## Conclusion

In this chapter we have discussed the Sharpe-Lintner-Mossin form of a general equilibrium relationship in the capital markets. This model, usually referred to as the capital asset pricing model or standard CAPM, is a fundamental contribution to understanding the manner in which capital markets function. It is worthwhile highlighting some of the implications of this model. First, we have shown that, under the assumptions of the CAPM, the only portfolio of risky assets that any investor will own...

## Prices And The Capm

Up to now we have discussed equilibrium in terms of rate of return. In the introduction to this chapter we mentioned that the CAPM could be used to describe equilibrium in terms of either return or prices. The latter is of importance in certain situations, for example, the pricing of new assets. It is very easy to move from the equilibrium relationship in terms of rates of return to one expressed in terms of prices. All that is involved is a little algebra. PM as the present price of the market...

## The Standard Capital Asset Pricing Model

All of the preceding chapters have been concerned with how an individual or institution, acting upon a set of estimates, could select an optimum portfolio, or set of portfolios. If investors act as we have prescribed, then we should be able to draw on the analysis to determine how the aggregate of investors will behave, and how prices and returns at which markets will clear are set. The construction of general equilibrium models will allow us to determine the relevant measure of risk for any...

## Other Evidence On Internationally Diversified Portfolios

In prior sections we have presented the considerations that are important in deciding on the reasonableness of international diversification. Obviously, we feel that the type of analysis we have presented is the relevant way to analyze the problem. However, several studies analyze the reasonableness of international diversification by examining the characteristics of international portfolios selected using historical data. The most common approach attempts to show the advantages of...

## The Effect Of Exchange Risk

F.driier we showed how the return on a foreign investment could be split into the return in the security's home market and the return from changes in exchange rates. In each of the prior tables we separated out the effect of changes in the exchange rate on return and risk. In Table 12.8b, the column entitled Exchange Return or Exchange Risk calculated the 1 particular, exchange rates between European currencies are fixed. Although European currencies will continue to fluctuate with the U.S....

## The World Portfolio

In discussing the size of capital markets it is interesting to employ the concept of a world portfolio. The world portfolio represents the total market value of all stocks or bonds that an investor would own if he or she bought the total of all marketable stocks on all the major stock exchanges in the world. Table 12.1 shows the percentage that each nation's equity securities represented of the world portfolio in 2000. Table 12.2 shows similar percentages for the various publicly traded bond...

## Safety First With Riskless Lending And Borrowing

In the text we discuss the choice of portfolios that satisfies each of the three standard formulations of the safety first criteria, assuming choices are to be made from among risky assets. In this appendix we extend the analysis to include riskless lending and borrowing. If a riskless lending and borrowing rate exists and returns are normally distributed, then the Roy criteria lead to infinite borrowing or only investment in the riskless asset depending on the relationship between RL and RF.19...

## Maximizing The Geometric Mean Return

One alternative to mean-variance analysis is simply to select that portfolio that has the highest expected geometric mean return. Many researchers have put this forth as a universal criterion. That is, they advocate its use without qualifications as to the form of utility function or the characteristics of the probability distribution of security returns. The proponents of the geometric mean usually proceed with one of the following arguments. Consider an investor saving for some purpose in the...

## Other Portfolio Selection Models

In all previous chapters and, in fact, in most of those that follow, we have assumed that investors are attempting to maximize the expected utility of the returns from an investment portfolio. Usually we have assumed that the expected utility of any opportunity could be meaningfully measured in terms of means and variances. This is the traditional and widely accepted mean-variance approach to portfolio management. While this is the central approach of this book, we would be remiss if we did not...

## Absolute And Relative Risk Aversion

Assume an investor has wealth W and a security with outcomes represented by the random variable Z. Let Z be a fair gamble so that E Z 0. Let a2_ equal the variance of Z and U the investor's utility function. Let W be the level of wealth such that the investor is indifferent between having Wc and having wealth W plus the gamble Z. Thus, the two choices are By assumption, the investor is indifferent between these positions, thus e u w z eu wc u wc B.l The last equality holds because wc is...

## An Axiomatic Derivation Of The Expected Utility Theorem

The expected utility theorem can be developed from a set of axioms or postulates concerning investor behavior. If an investor acts in accordance with these postulates, then the investor's behavior is indistinguishable from one who makes decisions on the basis of the expected utility theorem. The first two axioms concern the preference ordering of certain outcomes. The remaining two axioms concern rationality when ordering random prospects. The axioms are 1. Comparability. An investor can state...

## Empirical Evidence On The Suitability Of Alternative Preference Functions

In the earlier sections we continuously discussed the consistency of assumptions about investor behavior with observations of actual behavior. In this section, we elaborate on these statements. Empirical evidence as to the form of a utility function that might reasonably represent behavior is of two types 1. Experimental evidence from simple choice situations. 2. Survey data on investor's asset choices. The assumption that investors prefer more to less is consistent with most evidence. Few...

## The Economic Properties Of Utility Functions

The first restriction placed on a utility function is that it be consistent with more being preferred to less. This attribute, known in the economic literature as nonsatiation, simply says that the utility of more X 1 dollars is always higher than the utility of less X dollars. Thus, if we want to choose between two certain investments, we always take the one with the largest outcome. In this section we will formulate utility functions in terms of end of period wealth. This property then states...

## Techniques for Calculating the Efficient Frontier

In Chapters 4 and 5 we discussed the properties of the efficient frontier under alternative assumptions about lending and borrowing and alternative assumptions about short sales. In this chapter we describe and illustrate methods that can be used to calculate efficient portfolios. By necessity this chapter is more mathematically complex than those that preceded it and most of those that follow. The reader who is only concerned with a conceptual approach to portfolio management can skip this...

## The Incorporation Of Additional Constraints

The imposition of short sales constraints has complicated the solution technique, forcing us to use quadratic programming. Once we resort to this technique, however, it is a simple matter to impose other requirements on the solution. Literally any set of requirements that can be formulated as linear functions of the investment weights can be imposed on the solution. For example, some managers wish to select optimum portfolios given that the dividend yield on the optimum portfolios is at least...

## Quadratic Programming And Kuhntucker Conditions

These quadratic programming algorithms are based on a technique from advanced calculus called Kuhn-Tucker conditions. For small-scale problems these conditions may be able to be used directly. Furthermore, an understanding of the nature of the solution to this type of portfolio problem can be gained by understanding the Kuhn-Tucker conditions. Earlier we simply took the derivative of 9 with respect to each and set it equal to zero to find a maximum value of 0. This maximum is indicated by point...

## The Inputs To Portfolio Analysis

Ut us return to a consideration of the portfolio problem. From earlier chapters we know ibat to define the efficient frontier we must be able to determine the expected return and tndard deviation of return on a portfolio. We can write the expected return on any port-iolio as while the standard deviation of return on any portfolio can be written as hese equations define the input data necessary to perform portfolio analysis. From I quation 7.1 we see that we need estimates of the expected return...

## Estimating Beta

The use of the single-index model calls for estimates of the Beta of each stock that is a potential candidate for inclusion in a portfolio. Analysts could be asked to provide subjective estimates of Beta for a security or a portfolio. On the other hand, estimates of future Beta could be arrived at by estimating Beta from past data and using this historical Beta as an estimate of the future Beta. There is evidence that historical Betas provide useful information about future Betas. Furthermore,...

## The Market Model

Although the single-index model was developed to aid in portfolio management, a less restrictive form of it known as the market model has found increased usage in finance. The market model is identical to the single-index model except that the assumption that cov e ep 0 is not made.24 The model starts with the simpler linear relationship of returns and the market and produces an expected value for any stock of Since it does not make the assumption that all covariances between stocks are due to...

## Multiindex Models And Grouping Techniques

a Chapter 7 we argued that because of both the huge number of forecasts required and the necessary restrictions on the organizational structure of security analysts, it was not feasible for analysts to directly estimate correlation coefficients. Instead, some structural or behavioral model of how stocks move together should be developed. The parameters of this model can be estimated either from historical data or by attempting to get subjective stimates from security analysts. We have already...

## Multiindex Models

The assumption underlying the single-index model is that stock prices move together only because of common movement with the market. Many researchers have found that there are influences beyond the market that cause stocks to move together. For example, as early as 1966, King 43 presented evidence on the existence of industry influences. Two different types of schemes have been put forth for handling additional influences. We have called them the general multi-index model and the industry index...

## Average Correlation Models

I lie idea of averaging smoothing some of the data in the historical correlation matrix as forecast of the future has been tested by Elton and Gruber 20 and Elton, Gruber, and .'rich 22 , The most aggregate type of averaging that can be done is to use the average of all pair- a ise correlation coefficients over some past period as a forecast of each pairwise correlation coefficient for the future. This is equivalent to the assumption that the past correlation matrix contains information about...

## Mixed Models

Another model that has received attention is a combination of the models discussed in Chapter 7 and those introduced in this chapter. We call them mixed models. In a mixed model, the single-index model is used as the basic starting point. However, rather than assume that the extramarket covariance is zero, a second model is constructed to explain extramarket covariance. This concept should not be new to the reader. If we consider a general multi-index model, where the first index is the market,...

## Fundamental Multiindex Models

As mentioned earlier, a number of multi-index models have recently been developed relating security return to macroeconomic variables. The first of the recent group of multi-index models of stock returns was published by Chen, Roll, and Ross 13 . Although the purpose of their article was to explain equilibrium returns a subject we will discuss at great length in Chapter 16 , their analysis laid the groundwork for many of the models that were to follow. Chen, Roll, and Ross hypothesized a broad...

## Singleindex Model With Short Sales Not Allowed

In this appendix we derive simple ranking rules when the investor wishes to act as if the .mgle-index model is a reasonable method of describing the structure of security returns, in Chapter 6 we showed that if we could find a solution that met the Kuhn-Tucker conditions, then we could be certain we had the optimum portfolio. In this appendix we show that our simple ranking procedure does, in fact, lead to a solution that meets the Kuhn-Tucker conditions. The Kuhn-Tucker conditions were 1. R,...

## Other Return Structures

We have presented two simple ranking devices based on different correlation structures. As discussed in the last two chapters, there are a number of other models for estimating the covariance structure. For each of these other structures a simple ranking device exists the references listed at the end of the chapter show where. However, a few comments are in order. There are two types of models for estimating correlation structure index models and group models. The single- and multi-index models...

## Zq

Figure 11.8 Sum of cumulative probabilities for two investments. dominance are well known and more easily obtainable in other ways. However, stochastic dominance is a set of tools that is likely to lead to substantial additional breakthroughs. Knowledge of these tools should help the reader understand future research in this area. A. number of authors have proposed selecting portfolios on the basis of the first three moments of return distributions, rather than the first two mean and variance ....

## Jbi

Figure 14.4 The location of portfolios with return R'F. efficient segment of the minimum variance frontier, and the slope at this point must be positive. Thus, as we move along the line tangent to RM toward the vertical axis, we lower return. Since R7 is the intercept of the tangency line and the vertical axis, it has a return less than RM. Second, as we prove below, the minimum variance zero Beta portfolio cannot be efficient. Proof Denote by s the portfolio that has the smallest possible...

## Delineating Efficient Portfolios

In Chapter 4 we examined the return and risk characteristics of individual securities and began to study the attributes of combinations or portfolios of securities. In this chapter we look at the risk and return characteristics of combinations of securities in more detail. We start off with a reexamination of the attributes of combinations of two risky assets. In doing so we emphasize a geometric interpretation of asset combinations. It is a short step from the analysis of the combination of...

## The Shape Of The Portfolio Possibilities Curve

Reexamine the earlier figures in this chapter and note that the portion of the portfolio possibility curve that lies above the minimum variance portfolio is concave while that which lies below the minimum variance portfolio is convex.8 This is not due to the peculiarities of the examples we have chosen but rather is a general characteristic of all portfolio problems. This can easily be demonstrated. Remember that the equations and diagrams we have developed are appropriate for all combinations...

## Nonpricetaking Behavior

Up to now we have assumed that individuals act as price takers in that they ignore the impact of their buying or selling behavior on the equilibrium price of securities and, hence, on their See Lintner 79 , Sharpe 118 , Fama 32 , and Gonedes 41 . 18Lintner assumes the negative exponential utility function given by u w e a'wi. The measure of risk aversion is given by ar optimal portfolio holdings. The obvious question to ask is what happens if there are one or more investors, such as mutual...

## Short Sales Disallowed

One of the assumptions made in deriving the capital asset pricing model is that the investor can engage in unlimited short sales. Furthermore, short sales were defined in the broadest sense of the term in that the investor was allowed to sell any security whether owned or not and to use the proceeds to buy any other security.' This was a convenient assumption and it simplified the mathematics of the derivation, but it was not a necessary assumption. Exactly the same result would have been...