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[email protected] Investment and Portfolio Management Session 1 Nature and scope of investment What is investment? Investment is the sacrifice of certain present value for (possibly uncertain) future value. It deals with financial markets and security pricing. Investment fields comprise the buy and the sell side. The buy side include management of pension funds, insurance companies, mutual funds as well as providing advice and management of individuals retirement funds and other savings. A manager of a pension fund for example may participate in determining how much of the fund should be invested in stocks and bonds, which stocks and bonds should be purchased, and when specific stocks and bonds should be sold. The sell side includes security analysis which requires analysing economic, market and financial information and brokerage – related tasks which involve selling securities and executing trades for customers. A security analyst for example may specialize in a particular focus on the major firms in that industry. Using economic market and company specific information, the analyst may evaluate the performance of a particular company’s stock and make forecasts with respect to the company’s future earnings. Investment vs. savings Savings represent forgone consumption, with the former restricted to real investment of the sort that increases national output in the future. While this definition may prove useful in other contexts, it is not especially helpful for analysing the specifics of particular investments for even large classes of investment media. A deposit in a savings account at a bank is investment in the eyes of the depositor. Even cash stored in the mattress can be viewed as investment: one yielding a dollar for every dollar invested (or less in the event of fire or theft)
[email protected] Real VS Financial Investment Some investments are simply transactions among people. Others involve nature. The latter are “real” investment; the former are not. Buy a bond/loan
Buy factory
Financial investment
Real investment
In a complex modern economy, much investment is of the financial rather than the real variety. But highly developed institutions for financial investments greatly facilitate real investment. By far and large, the two forms are complimentary, not competitive.
Investment Speculation and Gambling Investmen - to commit (money) in order to earn a financial return. t - To assume a business risk in hope of gain; especially to buy or speculate sell in expectation of profit from market fluctuations - To bet on an uncertain outcome. gamble
All the three definitions fall within the scope of investment as defined. The term speculate is sometimes used to identify the horizon of the investor, e.g. someone who buys a piece of land on which to build a house in which to live might be termed an investor, while a real estate agent also buys the land and builds a house for almost immediate resale might be termed a specular. The former direct benefits the later others evaluate of those benefits (i.e. price at the onset).
[email protected] A speculator trades on the basis of information she believes is not yet known to or properly evaluated by others. An investor makes no such assumption. Some people use the term ‘speculative’ to refer to high risk investments, possibly without commensurately high return. e.g. a new stock issue may be denoted a “speculative investment”. A final use of the term “speculative” is simply to denote activities of which you disapprove. “ones friends are investors, one’s enemies are speculators. Gamble A person might be considered a gambler if he or she takes on a risk that is greater than commensurate with expected return. e.g. playing lotto – the risk is great, yet on average the players return is negative. Investment at he stock market is also risk but the return is positive on average. FOCUS OF THE OUTLINE - Investment Environment and the part if plays in successful decision making. o Characteristic of investment assets and markets in which they are traded. We summarise the general nature of financial investments how the markets for them operate. o Summary measures of the returns the investment have generated in the past. - Capital Market Theory o Develop an integrative theory of optimal decision making and shows how prices are formed in competitive financial markets in a way that reflects the expectations and preferences of all markets participants. o Models how rational investors make decision in an uncertain environment. - Valuation of Securities o Valuation of instruments e.g. stocks, bonds, options o Characteristics that determine value of securities o Forecasting, financial planning and portfolio choice and performance measurement.
[email protected] CAREERS IN INVESTMENT FILED 1. Securities Analyst – analyse and report on companies and industries considering economic and market conditions, making recommendations that assist investors with their investment decisions. 2. Portfolio Review Associate – review the performance o f mutual funds portfolios and present the review to clients. 3. Financial Planner – analyse and advice clients on asset allocation and investment selection to help them achieve their investment goals. 4. Hedging and Arbitrage Manager for fixed investment securities – create a profit centre through trading and hedging in fixed income securities and derivatives. 5. Options Specialist – identify and analyse option hedging strategies that include listed options, commodities options, commodities and stock options. 6. Pension Fund Portfolio Manager manage assets held by the pension fund for the future benefit if employees, determining the assets allocation and selection of securities appropriate for the investment objectives and policies of the pension funds. 7. Stockbroker – advise clients regarding potential investments and execute clients’ trade orders to buy or sell investment. Session 2 FINANCIAL MARKETS A financial market is a market where financial assets are exchanged (i.e. traded). Although the existence of a financial market is not a necessary condition for the creation and exchange of a financial asset, in most economies financial assets are created and subsequently traded in some type of organised financial market structure. Properties of Financial Assets Financial assets have certain properties that determine or influence their attractiveness to different classes of investors. The ten properties of financial assets are 1. moneyness 2. divisibility and denomination 3. reversibility 4. term to maturity
[email protected] 5. liquidity 6. convertibility 7. currency 8. cash flow and return predictability 9. complexity 10. tax status. Each property will be described below. Moneyness some financial assets are used as a medium of exchange or in settlement of transactions. These assets are called money. In Zimbabwe they consist of currency and all forms of deposits that permit cheque writing. Other financial assets, although not money, are very close to money in that they can be transformed into money at little cost, delay, or risk. They are referred to as near money. In Zimbabwe, these include time and savings deposits and a security issued by the government with a maturity of three months called a threemonth Treasury Bills. Moneyness is clearly a desirable property for investors. Divisibility and denomination – divisibility relates to the minimum size at which a financial asset can be liquidated and exchanged for money. The smaller the size, the more the financial asset is divisible. A financial asset such as a deposit at a bank is typically infinitely divisible (down to the penny), but other financial assets have varying degrees of divisibility depending on their denomination, which is the dollar value of the amount that each unit of the asset will pay at maturity. Thus many bonds come in $1 00 denominations. In general, divisibility is desirable for investors. Reversibility – reversibility refers to the cost of investing in a financial asset and then getting out of it and back into cash again. Consequently, reversibility is also referred to as roundtrip cost. A financial asset such as a deposit at a bank is obviously highly reversible because usually there is no charge for adding to or withdrawing from it. Other transaction costs may be unavoidable, but these are small. For financial assets traded in organized markets or with “market makers, the most relevant component of round trip cost is the so called bidask spread, to which might be added commissions and the time and cost, if any, of delivering the asset. The
[email protected] bidask spread is the difference between the price that a market maker is willing to sell a financial asset for (i.e., the price it is asking) and the price that a market maker is willing to buy the financial asset for (i.e., the price it is bidding). For example, if a market maker is willing to sell some financial asset for $70.50 (the ask price) and buy it for $70.00 ( the bid price), the bid ask spread is $0.50. The bid ask spread is also referred to as the bid offer spread. The spread charged by a market maker varies sharply from one financial asset to another, reflecting primarily the amount of risk the market maker is assuming by making a market. This market making risk can be related to two main forces. One is the variability of the price as measured, say, by some measure of dispersion of the relative price over time. The greater the variability, the greater the probability of the market maker incurring a loss in excess of a stated bound between the time of buying and reselling the financial asset. The variability of prices differs widely across financial assets. Threemonth treasury bills, for example have a very stable price, while a common stock generally tends to exhibit much larger short run variations. The second determining factor of the bidask spread charged by a market maker is what is commonly refereed to as the thickness of the market which is essentially the prevailing rate at which buying and selling orders reach the market maker (i.e. the frequency of transactions). A “thin market” is one which has few trades on a regular or continuing basis. Clearly, the greater the frequency of orders coming into the market for the financial asset (referred to as the “order flow”), the shorter the time that the financial asset will have to be held in the market maker’s inventory, and hence the smaller the probability of an unfavourable price movement while held. Thickness also varies from market to market. A three month Treasury bill is easily the thickest market in the world. In contrast, trading in stock of small companies is said to be thin. Because treasury bills dominate other instruments both in price stability and thickness, the bid ask spread tends to be the smallest in the market. A low round trip cost is clearly a desirable property of a financial asset, and as a result, thickness itself is a valuable property. This explains the potential advantage of a large over smaller markets (economies of scale), and a market’s endeavour to standardise the instruments offered to the public.
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Term to maturity – the term to maturity is the length of the interval until the date when the instrument is scheduled to make its final payment, or the owner is entitled to demand liquidation. Often, term to maturity is simply referred to as maturity, and this is the practice that we will follow in this course. Instruments for which the creditor can ask for repayment at any time, such as checking accounts and many savings accounts, are called demand instruments. Maturity is an important characteristic of financial assets such as debt instruments and in the Zimbabwe can range from one day to 100 years. Many other instruments, including equities, have no maturity and are thus a form of perpetual instruments. It should be understood that even a financial asset with a stated maturity may terminate before its stated maturity. This may occur for several reasons including bankruptcy or reorganisation, or because of provisions entitling the debtor to repay in advance, or the investor may have the privilege of asking for early repayment. Liquidity – liquidity is an important and widely used notion, although there is at present no uniformly accepted definition of liquidity. A useful way to think of liquidity and illiquidity proposed by Professor James Tobin, is in terms of how much sellers stand to lose if they wish to sell immediately as against engaging in a costly and time consuming search. An example of a quite illiquid financial asset is the stock of a small corporation or the bond issued by a small school district, for which the market is extremely thin, and one must search for one of a very few suitable buyers. Less suitable buyers including speculators and market makers, may be located more promptly, but will have to be enticed to invest in the illiquid financial asset by an appropriate discount in price. For many other financial assets, liquidity is determined by contractual arrangements. Ordinary deposits at a bank, for example, are perfectly liquid because the bank has a contractual obligation to convert them at par on demand. In contrast, financial contracts representing a claim on a private pension fund
[email protected] may be regarded as totally illiquid, because these can be cashed only at retirement. Liquidity may depend not only on the financial asset but also on the quantity one wishes to sell(or buy). While a small quantity may be quite liquid, a large lot may run into illiquidity problems. Note that liquidity is again closely related to whether a market is thick or thin. Thinness always has the effect of increasing the round trip cost, even of a liquid financial asset. But beyond some point it becomes an obstacle to the formation of a market, and has a direct effect on the illiquidity of the financial asset Convertibility – An important property of some financial assets is that they are convertible into other financial assets. In some cases, the conversion takes place within one class of financial assets, as when a bond is converted into another bond. In other situations, the conversion spans classes. For example, a corporate convertible bond is a bond that the bondholder can change into equity shares. There is preferred stock that may be convertible into common stock. The timing, costs, and conditions for conversion are clearly spelled out in the legal descriptions of the convertible security at the time it is issued. Currency – most financial assets are denominated in one currency, such as dollars or yen or deutsche marks, and investors must choose them with that feature in mind. Some issuers, responding to investors’ wishes to reduce foreign exchange risk, have issued dual currency securities. For example, some pay interest in one currency but principal or redemption value in a second. Further, some bonds carry a currency option which allows the investor to specify that payments of either interest or principal be made in either one of two major currencies. Cash flow and return predictability – as explained earlier, the return that an investor will realise by holding a financial asset depends on the cash flow that is expected to be received. This includes dividend payments on stock and interest payments on debt instruments, as well as the repayment of principal for a debt instrument and the expected sale price of a stock. Therefore, the predictability of the expected return depends on the predictability is a basic property of financial assets, in that it is a major determinant of their value. Assuming investors are risk
[email protected] averse, as we will see in later chapters, the riskness of an asset can be equated with the uncertainty or unpredictability of its return. In a world of nonnegligible inflation, it is further important to distinguish between nominal expected return and the real expected return. The nominal expected return considers the dollars that re expected to be received but does not adjust those dollars to take into consideration changes in their purchasing power. The real expected return is the nominal expected return but after adjustment for the loss of purchasing power of the financial asset as a result of inflation. For example, if the nominal expected return for a one year investment of $1 000 is 6%, then at the end of one year, the investor expects to realize $1060, consisting of interest of $60 and the repayment of the $1 000 investment. However, if the inflation rate over the same period of time is expected to be 4%, then the purchasing power of $1060 is only $1019.23 ($1060 divided by 1.04). Thus the return in terms of purchasing power, or real expected return, is 1.9%. In general, the real expected return can be approximated by subtracting from the nominal expected return the expected inflation rate. In our example, it is approximately 2% (6% minus 4%). Complexity some financial assets are complex in the sense that they are actually combinations of two or more simpler assets. To find the true value of such an asset, one must “decompose” it into its component parts and price each component separately. We will encounter numerous complex financial assets throughout this book. Indeed, there have been many innovations involving debt instruments since the early 1990s that have resulted in complex financial assets. Most complex financial assets involve a choice or option granted to the issuer or investor to do something to alter the cash flow. Because the value of such financial assets depends on the value of the choices or options granted to the issuer or investor, it becomes essential to understand how to determine the value of an option. Tax status – an important feature of any financial asset is its tax status. Governmental codes for taxing the income from the ownership or sale of financial assets vary widely if not wildly. Tax rates differ from year to year, country to
[email protected] country, and even among municipal units within a country (as with state and local taxes in the Zimbabwe). Moreover, tax rates may differ from financial asset to financial asset, depending on the type of issuer, the length of time the asset is held, the nature of the owner, and so on.
The role of financial Assets Financial markets have two principal economic functions. The first is to transfer funds from those who have surplus funds to invest to those who need funds to invest in tangible assets. The second function is transferring funds in such a way as to redistribute the unavoidable risk associated with the cash flow generated by tangible assets among those seeking and those providing the funds. However, the claims held by the final wealth holders are generally different from the liabilities issued by the final demanders of funds because of the activity of entities operating in financial markets, called financial intermediaries, who seek to transform the final liabilities into different financial assets which the public prefers. Financial markets provided three additional economic functions. First the interactions of buyers and sellers in the financial market determine the price of the traded asset; or equivalently, the required return on a financial asset is determined. The inducement for firms to acquire funds depends on the required return that investors demand, and it is this feature of the financial markets that signals how the funds in the economy should be allocated among financial assets. This is called the price discovery process. Second, financial markets provide a mechanism for an investor to sell a financial asset. Because of this feature, it is said that a financial market offers liquidity, an attractive feature when circumstances either force or motivate an investor to sell. In the absence of liquidity, the owner will be forced to hold a debt instrument until it matures and an equity instrument until the company is either voluntarily or involuntarily liquidated. While all financial markets provide some form of liquidity, the degree of liquidity is one of the factors that characterize different markets. The third economic function of a financial market is that it reduces the search and information costs of transacting. Search costs represent explicit costs, such as the money spent to advertise the desire to sell or purchase a financial asset, and
[email protected] implicit costs, such as the value of time spent in locating a counterpart. The presence of some form of organized financial market reduces search costs. Information costs are those entailed with assessing the investment merits of a financial asset, that is, the amount and the likelihood of the cash flow expected to be generated. In an efficient market, prices reflect the aggregate information collected by all market participants. Classification of financial markets There are many ways to classify financial markets. One way is by the type of financial claim. The claims traded in a financial market may be either for a fixed dollar amount or a residual amount. the former financial assets are referred to as debt instruments, and the financial market in which such instruments are traded is referred to as the debt market. The latter financial assets are call equity instruments and the financial market where such instruments are traded is referred to as the equity market. Alternatively, this market is referred to as the stock market. Preferred stock is an equity claim that entitles the investor to receive a fixed dollar amount. Consequently, preferred stock shares characteristic of instruments classified as part of the debt market and the equity market. Generally, debt instruments and preferred stock are classified as part of the fixed income market. The sector of the stock market that does not include preferred stock is called the common stock market. These classifications are summarized in figure 12. Classification of Financial Markets by Type of Claim. Fixed dollar amount claim
Debt instrument
Fixed income market
Residual or equity claim
Preferred stock
Equity (stock) market
Common stock
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Common stock market
Another way to classify financial markets is by the maturity of the claims. For example, there is a financial market for shortterm financial assets, called the money market and one for longer maturity financial assets, called the capital market. The traditional cut off between short term and long term is one year. That is a financial asset with a maturity of one year or less is considered short term and therefore part of the money market. A financial asset with a maturity of more than one year is part of the capital market. Thus, the debt market can be divided into those debt instruments which are part of the money market and part of the capital market depending on the number of years to maturity. Since equity instruments are generally perpetual, they are classified as part of the capital market. This is depicted in figure 13. Figure 1 3 Classification of Financial Markets by Maturity of Claim Debt instruments
Maturity one year or less
Money market
Common stock & preferred stock Maturity greater than one year
Capital market
A third way to classify financial markets is by whether the financial claims are newly issued or not. When an issuer sells a new financial asset to the public, it is said to “issue” the financial asset. The market for a newly issued financial asset is called the primary market. After a certain period of time, the financial asset is bought and sold, (i.e. Exchanged or traded) amongst investors. The market where this takes place is referred to as the secondary market.
[email protected] Finally, a market can be classified by its organisational structure. These organisational structures can be classified as auction markets, over the counter markets and intermediate markets. Globalisation of financial markets. Globalisation means the integration of financial markets throughout the world into an international financial market. Because of the globalisation of financial markets, entities in any country seeking to raise funds need not be limited to their domestic financial market. Nor are investors in a country limited to the financial assets issued in domestic market. The factors that have led to the integration of financial markets are: deregulation or liberalisation of markets and the activities of market participants in key financial centres of the world, technological advances for monitoring world markets, executing orders and analysing financial opportunities, and increased institutionalisation of financial markets. These factors are mutually exclusive. Global competition has forces governments to deregulate or liberalise various aspects of their financial markets so that their financial enterprises can compete effectively around the world. Technological advances have increased the integration and efficiency of the global financial market. Advances in telecommunication systems link market participants throughout the world with the result that orders can be executed within seconds. Advances in computer technology, coupled with advanced telecommunication systems allow the transmission of real time information on security prices and other key information to many participants in many places. Therefore, many investors can monitor global markets and simultaneously assess how this information will impact the risk/reward profile of their portfolios. Significantly, improved computing power allows the instant manipulation of real time market information so that attractive investment opportunities can be identified. Once these opportunities are identified, telecommunication systems permit the rapid execution of orders to capture them. The shifting of the roles of the two types of investors, retail and institutional investors, in financial markets is the third factor that has led to the integration of financial markets. Figure 14: Classification of Global Financial Markets
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Internal market (also called national market)
Domestic market
External market (also called international market, offshore market and Euro market)
Foreign market
The foreign market of a country is where the securities of issuers not domicile in the country are sold and traded. The rules governing the issuance of foreign securities are those imposed by regulatory authorities where the security is issued. The external market, also called the international market includes securities with the following distinguishing features: at issuance they are offered simultaneously to investors in a number of countries, and they are issued out side the jurisdiction of any single country. The external market is commonly referred to as the offshore market or more popularly the Euro market (even though this market is not limited to Europe, it began there). Derivative markets Some contracts give the contract holder either the obligation or the choice to by or sell a financial asset. Such contracts derive the value from the price of the underlying financial asset. Consequently these contracts are called derivative instruments. The array of derivative instruments include options contracts, future contracts, forward contracts, swap agreements, cap and floor agreements. The existence of derivative instruments is the key reason why investors can more effectively implement investment decisions to achieve their financial goals and issuers can more effectively raise funds on more satisfactory terms.
[email protected] As with any financial asset, derivative instruments can be used for speculative purposes as well as for accomplishing a specific financial or investment objective. Unfortunately, there have been several financial fiascos that have involved the use of derivative instruments. As a result, some regulators and lawmakers have been concerned with derivative instruments, viewing them as the product of the devil. The problem with the derivative instruments is not with the instruments per se but the lack of understanding of their risk/return characteristics by some users. Hopefully, the discussion in this book will help dispel the misconceptions associated with derivative instruments. RISK AND RETURN THEORIES 1 Objectives - How to calculate the historical single period investment return for a security or portfolio of securities. - The different methods for calculating the return over several unit periods. - What is meant by an efficient portfolio. - How to calculate the expected return and risk of a single asset and portfolio of assets. - Why the expected return of a portfolio of assets is a weighted average of the expected return of the assets included in the portfolio. - How the portfolio theory assumes that investors make investment decisions. - The difference between systematic risk and unsystematic risk. - The impact of diversification on total risks. - The importance of the correlation of two assets in measuring a portfolio’s risk. - What is meant by a feasible portfolio and a set of feasible portfolios. - What is the Markowitz efficient frontier. - What is meant by an optimal portfolio and how an optimal portfolio is selected from all the portfolios available on the Markowitz efficient frontier. Introduction to the Portfolio theory Valuation is the process of determining the fair value of a financial asset. The fundamental principle of valuation is that the value of any financial asset is the present value of the cash flow expected. The process requires two steps: estimating the cash flow and determining the appropriate interest rate that should be used to calculate the present value. The appropriate interest rate is the minimum interest rate plus a risk premium. The amount of the risk premium depends on the risk associated with realizing the cash flow. Determination of the appropriate risk premium is done by demonstrating the theoretical relationship between risk and expected return that should prevail in capital markets. The development of the theoretical relationship between risk and expected return is built on two economic theories: portfolio theory and capital market theory. Portfolio theory deals with the selection of portfolios that maximise expected returns consistent with
[email protected] individually acceptable levels of risk. Capital market theory deals with the effects of investor decisions on security prices. More specifically, it shows the relationship that should exist between security returns and risk, if investors constructed portfolios as indicated by portfolio theory. Together portfolio and capital market theories provide a framework to specify and measure investment risk and to develop relationships between risk and expected return (and hence between risk and the required return on an investment). These theories have revolutionalized the world of finance, by allowing portfolio managers to quantify the investment risk and expected return of a portfolio and allowing corporate treasures to quantify the cost of capital and risk of a proposed capital investment. In this session, we begin with the basic concepts of portfolio theory and then build upon these concepts in the next session to develop the theoretical relationship between the expected return of an asset and risk. Because the risk and return relationship indicates how much an asset’s expected return should be given its relevant risks, it also tells us how an asset should be priced. Hence the risk and return relationship is also referred to as an asset pricing model. Prior to the development of the theories we present, investors would often speak of risk and return, but the failure to quantify these important measures made the goal of constructing a portfolio of assets highly subjective and provided no insight as to the return investment should expect. Moreover, investors would focus on the risks of individual assets without understanding how combining them into a portfolio can affect the portfolio’s risk. The theories we present here quantify the relationship between risk and expected return. in October 1990, as confirmation of the importance of these theories, the Nobel Prize in Economic Science was awarded to Professor Harry Markowitz, the developer of portfolio theory and to Professor William Sharpe, who is one of the developers of capital market theory. Measuring Investment Return Before proceeding with the theories, we will explain how the actual investment return of a portfolio should be measured. The return on an investor’s portfolio during a given interval is equal to the change in value of the portfolio plus any distributions received from the portfolio, expressed as a fraction of the initial portfolio value. It is important that any capital or income distributions made to the investor be included, or the measure of return will be deficient. Another way to look at return is as the amount (expressed as a fraction of the initial portfolio value) that can be withdrawn at the end of the interval while maintaining the initial portfolio value intact. The return on the investor’s portfolio, designated RP =
v1 − v0 + D V0
Where v1 = portfolio market value at the end of the internal
[email protected] v0 = the portfolio market value at the beginning of the interval D = the cash distributions to the investor during the interval The calculations assumes that any interest or dividend income received on the portfolio of securities and not distributed to the investors is reinvested in the port (and thus reflected in V1 ). Further, the calculation assumes that any distributions occurs at the end of the interval. If the distributions were reinvested prior to the end of the interval, the calculation would have to be modified to consider the gains or losses or the amount reinvested. The formula also assumes no capital inflows during the interval. Otherwise, the calculation would have to be modified to reflect the increased investment base. Capital inflows at the end of the interval (or held in cash until the end), however, can be treated as just the reverse of distributions in the return calculation. Thus, given the beginning and ending portfolio values, plus any contributions from or distributions to the investor (assumed to occur at the end of an interval), For example, if the XYZ pension fund had a market value of $100 million at the end of June, benefit payments of $5 million made at the end of July, and an end of July market value of $103 million the return for the month would be 8%. RP =
103,000,000 − 100,000,000 + 5,00,000 = 0.08 100,000,000
In principle, this sort of calculation of returns could be carried out for any interval of time, say, for one month or ten years. Yet there are several problems with this approach. First, it is apparent that a calculation made over a long period of time, say, more than a few months, would not be very reliable because of the underlying assumption that all cash payments and inflows are made and received at the end of the period. Clearly if two investments have the same return but one investment makes cash payment early and the other late, the one with early payment will be understated. Second, we cannot rely on the formular above to compare return on a one month investment with that on a ten year portfolio. For purpose of comparison, the return must e expressed per unit of time – say per year. In practice, we handle these two problems by first computing the return over a reasonably short unit of time. Perhaps a quarter of a year or less. The return over the relevant horizon, consisting of several unit periods, is computed by averaging the return over the unit intervals. There are three generally used methods of averaging: (1) the arithmetic average rate of return, (2) the time weighted rate of return (also referred to as the geometric rate of return), and (3) the dollar-weighted return. the averaging produces a measure of return per unit of time period. The measure can be converted to an annual or other period return by standard procedures. Arithmetic Average Rate of Return The arithmetic average rate of return is an unweighted average of the returns achieved during a series of such measurement intervals. The general formula is:
[email protected] RA =
RP1 + RP 2 + .... + RPN N
RA = the arithmetic average return R pk = the portfolio return in interval k as measured by Equation (8.1), k = 1,...., N N = the number of intervals in the performance evaluation period. For example, if the portfolio returns were –10%, 20% and 5% in July, August, and September respectively, the arithmetic average monthly return is 5%. The arithmetic average can be thought of as the mean value of the withdrawals (expressed as a fraction of the initial portfolio value) that can be made at the end of each interval while maintaining the initial portfolio value intact. In the example above, he investor must add 10% of the initial portfolio value at the end of the first interval and can withdraw 20% and 5% of the initial portfolio value per period. Time weighted rate of return The time weighted rate of return measures the compounded rate of growth of the initial portfolio during the performance evaluation period, assuming that all cash distributions are reinvested in the portfolio. It is also commonly referred to as the “geometric rate of return”. It is computed by taking the geometric average of the portfolio returns computed from Equation (8.1). The general formula is: RT = [ (1 + R P1 )(1 + R PN )]
1 / N −1
Where RT is the time weighted rate of return and R PK and N are as defined earlier. For example, if the portfolio returns were –10%, 20%, and 5% in July, August and September, as in the example above, then the time weighted rate of return is: RT = [ (1 + (−0.10))(1 + 0.20)(1 + 0.05)]
= [ (0.90)(1.20)(1.05)]
1/ 3
1/ 3
−1
− 1 = 0.043
As the time weighted rate of return is 4.3% per month, one dollar invested in the portfolio at the end of June would have grown at a rate of 4.3% per month during the three month period. In general, the arithmetic and time weighted average returns do not provide the same answers. This is because computation of the arithmetic average assumes the initial amount invested to be maintained (through additions or withdrawals)at its initial portfolio value. The tie weighted return on the other hand, is the return on a portfolio that varies size because of the assumption that all proceeds are reinvested.
[email protected] We can use an example to show how the two averages fail to coincide. Consider a portfolio with a $100 million market value at the end of 19992, a $200 million value at the end of 1993 and a $100 million value at the end of 1994. The annual returns are 100% and –50%. The arithmetic return is 25%, while the time-weighted average return is 0%. The arithmetic average return consists of the average of the $100 million withdrawn at the end of 1993 and however, the 100% in 1993 being exactly offset by the 50% loss in 1994 on the larger investment base. In this example, the arithmetic average exceeds the time weighted average return. this always proves to be true, except in the special situation where the returns in each interval are the same, in which case the averages are identical. Dollar-Weighted Rate of Return The dollar weighted rate of return (also called the internal rate of return) is computed by finding the interest rate that will make the present value of the cash flows from all the interval periods plus the terminal market value of the portfolio. The internal rate of return calculation, as explained in earlier chapters, is calculated exactly the same way the yield to maturity on a bond is. The general formula for the dollar-weighted return is: v0 =
c (1 + R D
+
C + VN C2 + ..... + N (1 + R D ) (1 + RD ) n
where R D = the dollar weighted rate of return V0 = the initial market value of the portfolio V N = the terminal market value of the portfolio C K = the cash flow for the portfolio (cash inflows minus cash outflows) for interval k , k = 1,..., n for example, consider a portfolio with a market value of $100 million at the end of 1990, capital withdrawals of $5 million at the end of 1991, 1992 and 1993 and a market value at the end of 1993 of $110 million. Then V0 = $100,000,000; N = 3; C1 = C 2 = C 3 = $5,000,000;V3 = $110,000,000 and R D is the interest rate that satisfies the equation: $110,000,000 =
$5,000,000 $5,000,000 $5,000,000 + $110,000,000 + + (1 + R D )1 (1 + R D ) 2 (1 + R D ) 3
It can be verified that the interest rate that satisfies this expression is 8.1%. This is the dollar-weighted return. Under the special conditions, both the dollar weighted return and the time weighted return produce the same result. This will occur when no further additions or withdrawals occur when no further additions or withdrawals occur and all dividends are reinvested. Throughout this chapter, the rate of return is generally used to refer to an appropriately standardised measure.
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Different Type Risk With any financing or investment decision, there is some uncertainty about its outcome. Uncertainty is about not knowing exactly what will happen in the future. There is uncertainty in almost everything we do as financial managers because no one knows precisely what changes will occur in such things as tax laws, consumer demand, the economy, or interest rates. Though the terms "risk" and "uncertainty" can be used to mean the same, there is a distinction between them. Uncertainty is about not knowing what's going to happen. Risk is how we characterize how much uncertainty exists: the greater the uncertainty, the greater the risk. Thus risk can be defined as the degree of uncertainty. In financing and investment decisions there are many types of risk we are going consider in this module.
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Types of risk The types of risk a financial manager faces include: cash flow risk; reinvestment risk; interest rate risk; purchasing power risk and currency risk.
Cash Flow Risk Cash flow risk is the risk that the cash flows of an investment will not materialize as expected. For any investment, the risk that cash flows may not be as expected in timing, amount, or both is related to the investment's business risk. Business Risk Business risk is the risk associated with operating cash flows. Operating cash flows may not be certain because neither do the revenues nor the expenditures may comprise the cash flows. Regarding revenues: depending on economic conditions and the actions of competitors, prices or quantity of sales (or both) may be different from what is expected (sales risk). Regarding expenditures: operating costs are comprised of fixed costs and variable costs. The greater the fixed component of operating costs, the less easily a company can adjust its operating costs to changes in sales.
We refer to the risk that comes about from the mixture of fixed and variable costs as operating risk. The greater the fixed operating costs, relative to variable operating costs, the greater the operating risk.
[email protected] Let us take a look at how operating risk affects cash flow risk. Remember back in economics when you learned about elasticity? That is a measure of the sensitivity of changes in one item to changes in another. We can look at how sensitive a firm's operating cash flows are to changes in demand, as measured by unit sales. We will calculate the operating cash flow elasticity, which we call the degree of operating leverage (DOL). The degree of operating leverage is the ratio of the percentage change in operating cash flows to the percentage change in units sold. If a firm sells all its output in one period then,
Suppose the price per unit is $30, the variable cost per unit is $20, and the total fixed costs are $5,000. If we go from selling 1,000 units to selling 1,500 units, an increase of 50% of the units sold, operating cash flows change from: Item
Selling 1,000 units
Selling 1,500 units
Sales
$30,000
$45,000
less variable costs
20,000
30,000
less fixed costs
5,000
5,000
Operating cash flow
$ 5,000
$10,000
Operating cash flows doubled when units sold increased by 50%. What if the number of units decreases by 25%, from 1,000 to 750? Operating cash flows decline by 50%. What is happening is that for a 1% change in units sold, the operating cash flow changes by two times that percentage, in the same direction. So if units sold increased by 10%, operating cash flows would increase by 20%; if units sold decreased by 10%, operating cash flows would decrease by 20%.
[email protected] Both sales risk and operating risk influence a firm's operating cash flow risk. Both sales risk and operating risk are determined in large part by the type of business the firm is in. Financial Risk Financial risk is the risk associated with how a company finances its operations. If a company finances with debt, it is legally obligated to pay the amounts comprising its debts when due. By taking on fixed obligations, such as debt and longterm leases, the firm increases its financial risk. A company that finances its business with equity does not incur fixed obligations. The more fixedcost obligations (debt) incurred by the firm, the greater its financial risk. The sensitivity of the cash flows available to owners when operating cash flows change is referred to as the degree of financial leverage (DFL).
. The cash flows to owners are equal to operating cash flows, less interest and taxes. If operating cash flows change, how do cash flows to owners change? Suppose operating cash flows change from $5,000 to $6,000 and suppose the interest payments are $1,000 and, for simplicity and wishful thinking, the tax rate is 0%:
Operating cash flow
Operating cash flow
of $5,000
of $6,000
Operating cash flow
$ 5,000
$ 6,000
less interest
1,000
1,000
Cash flows to owners
$ 4,000
$ 5,000
A change in operating cash flow from $5,000 to $6,000 a 20% increase increased cash flows to owners by $1,000 a 25% increase.
[email protected] The greater the use of financing sources that require fixed obligations, such as interest, the greater the sensitivity of cash flows to changes in operating cash flows. Default Risk When you invest in a bond, you expect interest to be paid (usually semiannually) and the principal to be paid at the maturity date. But not all interest and principal payments may be made in the amount or on the date expected: interest or principal may be late or the principal may not be paid at all! The more burdened a firm is with debt (required interest and principal payments) the more likely it may be unable to make payments promised to bondholders and the more likely there may be nothing left for the owners. We refer to the cash flow risk of a debt security as default risk or credit risk. Technically, default risk on a debt security depends on the specific obligations comprising the debt. Default may result from: failure to make an interest payment when promised (or within a specified period), failure to make the principal payment as promised, failure to make sinking fund payments (that is, amounts set aside to pay off the obligation), if these payments are required, failure to meet any other condition of the loan, or bankruptcy. Financial managers need to worry about default risk because they invest their firms’ funds in the debt securities of other firms and they want to know what default risk lurks in those investments.. The greater the risk of a firm's securities, the greater the firm's cost of financing. Default risk is affected by both business risk and financial risk. We need to consider the effects operating and financing decisions have on the default risk of the securities a firm issues, since the risk accepted through the financing decisions affects the firm's cost of financing. Reinvestment Rate Risk
[email protected] Another type of risk is the uncertainty associated with reinvesting cash flows, not surprisingly called reinvestment rate risk. If we look at an investment that produces cash flows before maturity or sale, such as a stock (with dividends) or a bond (with interest), we face a more complicated reinvestment problem. In this case we are concerned not only with the reinvestment of the final proceeds (at maturity or sale), but also with the reinvestment of the intermediate dividend or interest cash flows (between purchase and maturity or sale). Two types of risk closely related to reinvestment risk of debt securities are prepayment risk and call risk. In the case of mortgagebacked securities securities that represent a collection of home mortgages a homeowner may pay off her or his mortgage early. If paid off early, investors in mortgages get paid off early so they will have to scramble to reinvest earlier than expected. Therefore, investors in securities that can be paid off earlier than maturity face prepayment risk the risk that the borrower may choose to prepay the loan which causes the investor to have to reinvest the funds. Call risk is the risk that a callable security will be called by the issuer. If you invest in a callable security, there is a possibility that the issuer may call it in (buy it back). While you may receive a call premium (a specified amount above the par value), you have to reinvest the funds you receive. There is reinvestment risk for assets other than stocks and bonds, as well. If you are investing in a new product investing in assets to manufacture and distribute it you expect to generate cash flows in future periods. You face a reinvestment
[email protected] problem with these cash flows: What can you earn by investing these cash flows? What are your future investment opportunities? If we assume that investors do not like risk a safe assumption then they will want to be compensated if they take on more reinvestment rate risk. The greater the reinvestment rate risk, the greater the expected return demanded by investors. Reinvestment rate risk is relevant in our investment decisions no matter the asset and we must consider this risk in assessing the attractiveness of investments. The greater the cash flows during the life of an investment, the greater the reinvestment rate risk of the investment. If an investment has a greater reinvestment rate risk, this must be factored into our decision. Interest rate risk Interest rate risk is the sensitivity of the change in an asset's value to changes in market interest rates. Market interest rates determine the rate we must use to discount a future value to a present value. The value of any investment depends on the rate used to discount its cash flows to the present. If the discount rate changes, the investment's value changes. Interest rate risk is present in debt securities. If you buy a bond and intend to hold it until its maturity, you don't need to worry about its value as interest rates change: your return is the bond's yieldtomaturity. But if you do not intend to hold the bond to maturity, you need to worry about how changes in interest rates affect the value of your investment. As interest rates go up, the value of your bond goes down. As interest rates go down, the value of your bond goes up. Purchasing Power Risk
[email protected] Purchasing power risk is the risk that the pricelevel may increase unexpectedly. If your firm locks in a price on its supply of raw materials through a longterm contract and the pricelevel increases, it benefits from the change in the price level and your supplier loses (the firm pays the supplier in cheaper currency). If a firm borrows funds by issuing a longterm bond with a fixed coupon rate and the pricelevel increases, the firm benefit from an increase in the price level and its creditor is harmed since interest and the principal are repaid in a cheaper currency. Purchasing power risk is the risk that future cash flows may be worth less or more in the future because of inflation or deflation, respectively, and that the return on the investment will not compensate for the unanticipated inflation. If there is risk that the purchasing power of a currency will change, investors who do not like risk will demand a higher return. Currency Risk If we are considering making an investment that generates cash flows in another currency (some other nation's currency), there is some risk that the value of that currency will change relative to the value of our domestic currency. We refer to the risk of the change in the value of the currency as currency risk. Consider a Zimbabwean firm making an investment that produces cash flows in British pounds, £. Suppose we invest 10,000 £ today and expect to get 12,000 £ one year from today. Further suppose that 1£ = $1.48 today, so you are investing $1.48 times 10,000 = $14,800. If the British pound does not change in value, relative to the Zimbabwean dollar, you would have a return of 20%:
[email protected] But what if one year from now the British pound is worth $1.30 instead? Your return would be less than 20% because the value of the pound has dropped vis àvis the Zimbabwean dollar. You are making an investment of 10,000 £, or $14,800, and getting not $17,760, but rather $1.30 times 12,000 = $15,600 in one year. If the pound loses value from $1.48 to $1.30, your return on your investment is ($15,60014,800)/$14,800 = 5.41%. Currency risk is the risk that the relative values of the domestic and foreign currencies will change in the future, changing the value of the future cash flows. As financial managers, we need to consider currency risk in our investment decisions that involve other currencies and make sure that the returns on these investments are sufficient compensation for the risk of changing values of currencies. Expected Return of an asset We refer to both future benefits and future costs as expected returns. The expected return is a measure of the tendency of returns on an investment. This does not mean that these are the only returns possible but just our best measure of what we expect. Suppose we are evaluating the investment in a new product. We do not know and cannot know precisely what the future cash flows will be. But from past experience, we can at least get an idea of possible flows and the likelihood the probability they will occur. After consulting with colleagues in marketing and production management, we figure out that there are two possible cash flow outcomes, success or failure, and the probability of each outcome. Next, consulting with colleagues in production and marketing for sales prices, sales
[email protected] volume, and production costs, we develop the following possible cash flows in the first year:
Scenario
Cash Flow
Probability of cash flow
Product success
$4,000,000
40%
Product flop
2,000,000
60%
But what is the expected cash flow in the first year? The expected cash flow is the average of the possible cash flows, weighted by their probabilities of occurring: Expected cash flow = 0.40 ($4,000,000) + 0.60($2,000,000) = $400,000 The expected value is a guess about the future outcome. It is not necessarily the most likely outcome. The most likely outcome is the one with the highest probability. In the case of our example, the most likely outcome is $2,000,000. The general formula for any expected value is: Expected value = E(x) = p1x1 + p2x2 + p3x3 +...+pnxn. where E(x) is the expected value; n is the number of possible outcomes; pi is the probability of the ith outcome; and xi is the value of the ith outcome. We can abbreviate this formula by using summation notation: E(x)= ∑pixi
.Applying the general formula to our example,
[email protected] n = 2 (there are two possible outcomes) p1 = 0.40 p2 = 0.60 x1 = $4,000,000 x2 = $2,000,000 E(cash flow) = 0.40 ($4,000,000) + 0.60 ($2,000,000) = $400,000. Considering the possible outcomes and their likelihood, we expect a $400,000 cash flow. Risk of an asset The expected return gives us an idea of the tendency of the future outcomes what we expect to happen, considering all the possibilities. But the expected return is a single value and does not tell us anything about the diversity of the possible outcomes. Are the possible outcomes close to the expected value? Are the possible outcomes much different than the expected value? Just how much uncertainty is there about the future? Since we are concerned about the degree of uncertainty (risk), as well as the expected return, we need some way of quantifying the risk associated with decisions. Suppose we are considering two products, Product A and Product B, with estimated returns under different scenarios and their associated probabilities: Product A Scenario
Probability of
Outcome
outcome Success
25%
24%
Moderate success
50
10
[email protected] Failure
25
4
Probability of
Outcome
Product B Scenario
outcome Success
10%
40%
Moderate success
30
30
Failure
60
5
We refer to a product's set of the possible outcomes and their respective probabilities as the probability distribution for those outcomes. We can calculate the expected cash flow for each product as follows:
Product A Scenario
Pi
Xi
Pi xi
Success
0.25
0.24
0.0600
Moderate
0.50
0.10
0.0500
Failure
0.25
0.04
0.0100
Expected return
0.1000 or 10%
Scenario
Pi
Xi
Pixi
Success
0.10
0.40
0.0400
Moderate
0.30
0.30
0.0900
Scenario success
Product B
success
[email protected] Failure
0.60
0.05
0.0300
Expected return
0.1000 or 10%
Both Product A and Product B have the same expected return. However the possible returns for Product A range from 4% to 24%, where the possible returns for Product B range from 5% to 40%. The range is the span of possible outcomes. For Product A the span is 28%; for Product B it is 45%. A wider span indicates more risk, so Product B has more risk than Product A. But the range by itself doesn't tell us much about the possible cash flows at these extremes or within the extremes. The range tells us nothing about the probabilities at or within the extremes. A measure of risk that does tell us something about how much to expect and the probability that it will happen is the standard deviation. The standard deviation is a measure of dispersion that considers the values and probabilities for each possible outcome. The higher the standard deviation, the greater would be the dispersion of possible outcomes from the expected value. The standard deviation considers the deviation, or distance, of each possible outcome from the expected value and the probability associated with it. The standard deviation of possible returns, represented by σ(x), is given by
σ ( x) =
∑ p ( x − E ( x)) i
2
i
Standard deviation is calculated in six steps as follows: Step 1: Calculate the expected value. Step 2: Calculate the deviation of each possible outcome from the expected value
[email protected] The deviation tells us how far each possible outcome is from the expected value. Step 3: Square each deviation. Step 4: Weight each squared deviation, multiplying it by the probability of the outcome Step 5: Sum these weighted squared deviations. This is the variance of the possible outcomes, σ 2(x). Step 6: Take the square root of the sum of the squared deviations. This is the standard deviation of the possible outcomes, σ (x). Example Let's calculate the standard deviation of the expected cash flows for Product A: Step 1: Calculate the expected value E(x) = [0.24 (0.25)] + [0.10 (0.50)] + [0.04 (0.25)] = 0.10 or 10% Step 2: Calculate the deviation of each possible outcome from the expected outcome Success: 0.2400 0.1000 = 0.1400 Moderate success: 0.1000 0.1000 = 0.0000 Failure 0.0400 0.1000 = 0.1400 Step 3: Square each of these deviations Success: 0.14002 = 0.0196 Moderate success: 0.00002 = 0.0000 Failure 0.14002 = 0.0196 Step 4: Weight each of the squared deviations by multiplying the probability of the outcome by the squared deviations Success 0.0196 (0.25) = 0.0049
[email protected] Moderate success: 0.0000 (0.50) = 0.0000 Failure 0.0196 (0.25) = 0.0049 Step 5: Sum these weighted squared deviations. Variance = 0.0049 + 0.0000 + 0.0049
σ 2(x) = 0.0098 Step 6: Take the square root of the sum of the squared deviations Standard deviation = (0.0098)1/2
σ (x) = 0.0990 or 9.90% . The standard deviation Product A's returns is 9.90%.
The standard deviation for Products A and B are:
Expected return
Standard deviation
Product A
10%
9.90%
Product B
10%
18.57%
While the expected value of both products is the same, there is a different distribution of possible outcomes for the two products. When we calculate the standard deviation around the expected value, we see that Product B has a larger standard deviation. The larger standard deviation for Product B tells us that Product B has more risk than Product A since its possible outcomes are more distant from its expected value.
Returns and the Tolerance for Bearing Risk Which product investment do you prefer, A or B? Most people would choose A since it provides the same expected return, with less risk. Most people do not like
[email protected] risk they are risk averse. Risk aversion is the dislike for risk. Does this mean a risk averse person will not take on risk? No they will take on risk if they feel they are compensated for it. A risk neutral person is indifferent towards risk. Risk neutral persons do not need compensation for bearing risk. A risk lover likes risk someone even willing to pay to take on risk. Are there such people? Yes. Consider people who play the state lotteries, where the expected value is always negative: the expected value of the winnings is less than the cost of the lottery ticket. When we consider financing and investment decisions, we assume that most people are risk averse. Managers, as agents for the owners, make decisions that consider risk "bad" and that if risk must be borne, they make sure there is sufficient compensation for bearing it. As agents for the owners, managers cannot have the "fun" of taking on risk for the pleasure of doing so. Risk aversion is the link between return and risk. To evaluate a return you must consider its risk: Is there sufficient compensation (in the form of an expected return) for the investment's risk?
PORTFOLIO THEORY In constructing a portfolio of assets, investors seek to maximise the expected return from their investment given some level of risk they are willing to accept. Portfolios that satisfy this requirement are called efficient portfolios. Portfolio theory tells us how this should be done. Because Markowitz is the developer of portfolio theory, efficient portfolios are sometimes referred to as “Markowitz efficient portfolios”. To construct an efficient portfolio of risky assets, it is necessary to make some assumption about how investors behave in making investment decisions. A reasonable assumption is that investors are risk averse. A risk – averse investor is one who, when faced with two investments with the same expected return but two different risks, will
[email protected] prefer the one with the lower risk. Given a choice of efficient portfolios from which an investor can select, an optimal portfolio is the one that is most preferred. To construct an efficient portfolio, it is necessary to understand what is meant by “expected return and risk”. The latter concept, risk, could mean any one of many types of risk.
Selecting among different investments The two basic approaches that can be used to choose between investments in individual assets are the mean variance criterion and the coefficient of variation (C.V) approach. Mean variance rule According to Harry Markowitz investors can choose between investments based on the expected returns (mean) and risk (variance) if the following assumptions hold: Investors are rational and seek to maximize the utility of wealth Investors are risk averse, that is they aim to minimize risk Returns of securities are normally distributed. To me these assumptions are necessary but insufficient to come up with mean variance efficient securities. To make the rule sufficient and for the purpose of this study I will add other assumptions as follows: Investment sets are complete, that is investors are able to compare between or among assets Investors’ choices are transitive, that is if asset X is preferred to asset Y and asset Y is preferred to asset Z it therefore implies that asset X is better than asset Z. The mean variance rule states that:
[email protected] If there are two assets X and Y asset X is preferred to asset Y if: X has expected returns higher than or equal to that of Y and the risk of X is lower than that of Y i.e. E(x) ≥ E(y) and σ 2(x) E(y) and σ 2(x)≤ σ 2(y) Example A firm is faced with the following investment alternatives that are mutually exclusive.
Expected return
Variance
Product A
10%
9%
Product B
10%
17%
Of the two investments Product A is mean varianceefficient because it has the lower risk though it yields the same expected return as that of B.
Coefficient of Variation (C.V). C.V measures risk per each unit of expected return. When investments are of different scale it is sometimes proper to use C.V which is a relative risk measure. C.V=
s tan dard deviation × 100 exp ected value
Based on C.V we prefer an investment with lower C.V i.e. lower risk per each unit of expected value. Example Of the two products which one is preferred in terms of C.V?
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Expected return
Standard deviation
Product A
20%
40%
Product B
10%
30%
Product A is preferred to B because its C.V is lower than that of B i.e. C.VA=200% and C.VB=300% Activity Consider the following investments: Investment
Expected return
Standard deviation
A
5%
10%
B
7%
11%
C
6%
12%
D
6%
10%
Which investment would you prefer in terms of (a) mean variance rule and (b) coefficient of variation
Diversification and Risk In any portfolio, one investment may do well while another does poorly. The projects' cash flows may be "out of synch" with one another. Let's look at the idea of "outofsynchness" in terms of expected returns, since this is what we face when we make financial decisions. Expected Portfolio Return The expected return is simply the weighted average of a possible outcomes where the weights are the relative chances of occurrence. In general, the expected return on the portfolio, denoted E ( R P )1 is given by: E ( R P ) = P1 R1 + P2 R2 + ..... + Rn Pn n
E ( R P ) = ∑ Pj R j j =1
[email protected] where R j ’s are the possible returns, the Pj ’s the associated probabilities, and n the number of possible outcomes. Consider Investment C and Investment D and their probability distributions:
Scenario
Probability of
Return on
Return on
Scenario
Investment C
Investment D
Boom
30%
20%
10%
Normal
50
0
0
Recession
20
20
45
We see that when Investment C does well, in the boom scenario, Investment D does poorly. Also, when Investment C does poorly, as in the recession scenario, Investment D does well. In other words, these investments are out of synch with one another. Now let's look at how their "outofsynchness" affects the risk of the portfolio of C and D. If we invest an equal amount in C and D, the portfolio's return under each scenario is the weighted average of C & D's returns, where the weights are 50%: Scenario
Probability
Weighted average return
Boom
0.30
[0.5 ( 0.20)] + [0.5 (0.10)] = 0.0500 or 5%
Normal
0.50
[0.5 ( 0.00)] + [0.5 ( 0.00)] = 0.0000 or 0%
Recession
0.20
[0.5 (0.20)] + [0.5 ( 0.45)] = 0.1250 or 12.5%
[email protected] The calculation of the expected return and standard deviation for Investment C, Investment D, and the portfolio consisting of C and D results in the following the statistics,
Scenario
Probability Return on
Return on
Return on a
of
Investment Investment portfolio
Scenario
C
D
comprised of C and D
Boom
30%
20%
10%
5%
Normal
50%
0
0
0
Recession
20%
20
45
12.5
Expected return
2%
6%
4%
Standard
14.00%
19.97%
4.77%
deviation The expected return on Investment C is 2% and the expected return on Investment D is 6%. The return on a portfolio comprised of equal investments of C and D is expected to be 4%. The standard deviation of Investment C's return is 14% and of Investment D's return is 19.97%, but the portfolio's standard deviation, calculated using the weighted average of the returns on Investment C and D in each scenario, is 4.77%. This is less than the standard deviations of each of the individual investments because the returns of the two investments do not move in the same direction at the same time, but rather tend to move in opposite directions. The portfolio comprised of Investments C and D has less risk than the individual investments because each moves in opposite directions with respect to the other. A statistical measure of how two variables in this case, the returns on two
[email protected] different investments move together is the covariance. Covariance is a statistical measure of how one variable changes in relation to changes in another variable. Covariance between X and Y is given by CovXY= ∑ Pi ( xi − E ( x))( yi − E ( y )) Covariance in this example is calculated in four steps: Step 1: For each scenario and investment, subtract the investment's expected value from its possible outcome; Step 2: For each scenario, multiply the deviations for the two investments; Step 3: Weight this product by the scenario's probability; and Step 4: Sum these weighted products to arrive at the covariance. Scenario Probability Deviation of
Deviation of
Product Weight the
Investment
Investment
of the
C's return
D's return
deviations the
from its
from its
expected
expected
return
return
product by
probability
Boom
0.30
0.1800
0.1600
.0288
0.00864
Normal
0.50
0.0200
0.0600
.0012
0.00060
Recessio 0.20
0.2200
0.3900
.0858
0.01716
covariance
n
=0.02520 As you can see in these calculations, in a boom economic environment, when Investment C is above its expected return (deviation is positive), Investment D is below its expected return (deviation is negative). In a recession, Investment C's
[email protected] return is below its expected value and Investment D's return is above its expected value. The tendency is for the returns on these portfolios to covary in opposite directions producing a negative covariance of 0.0252. Let's see the effect of this negative covariance on the risk of the portfolio. The portfolio's variance depends on: (a) the weight of each asset in the portfolio; ( b) the standard deviation of each asset in the portfolio; and (c) the covariance of the assets' returns. Calculating Portfolio based on a probability for the return of the individual asset’s return. in practice, the variance of a portfolio’s return - which we shall simply refer to as the portfolio’s return – which shall simply refer to as the portfolio variance –is calculated from historical data, generally monthly. It can be shown that the variance of a two asset portfolio is: 2 2 Var ( R P ) = Wi var(Ri ) + W j var( R j ) + 2 wi wi std ( Ri ) std ( Ri )cor ( Ri , R j )
where var ( R P ) = portfolio variance Wi = percentage of the portfolio’s funds invested in asset i W j = percentage of the portfolio’s funds invested in asset j var ( Ri ) = variance of asset i var ( R j ) = variance of asset j std ( Ri ) = standard deviation of asset i std ( R j ) = standard deviation of asset j cor ( Ri R j ) = correlation between the return for assets i and j In words, equation (8.4) states that the portfolio variance sum of the weighted variances of the two assets plus the weighted correlation between the two assets plus the weighted correlation between the two assets. Given our earlier discussion, it should not be surprising that the correlation between the two assets affects the portfolio variance. Notice from the equation (8.4) that the lower the correlation between the return on two assets, the lower the portfolio variance. The portfolio variance is the lowest if the two assets have a correlation of –1. The equation for the portfolio variance when there are more than two assets in the portfolio is more complicated. The extension to three assets - i, j and k - is al follows:
[email protected] var( R p ) = wi2 var(Ri ) + wl2 var( R j ) + wk2 var( Rk ) + 2 wi w j std ( Ri ) std ( R j )cor ( Ri R j ) + 2 wi wk std ( Ri ) std ( Rk )cor ( Ri Rk ) + 2 w j wk std ( R j ) std ( Rk )cor ( R j Rk ) where wk = percentage of the portfolio’s funds invested in asset k std ( Rk ) = standard deviation of asset k cor ( Ri , Rk ) =correlation between the return for assets i and k cor ( R j , Rk = correlation between the return for assets j and k In words, Equation states that the portfolio variance is the sum of the weighted variances of the individual assets plus the sum of the weighted correlations of the assets. Hence, the portfolio variance is the weighted sum of the individual variances of the assets in the portfolio plus weighted sum of the degree to which the assets vary together. The formular for the portfolio variance of any size will involve the variances and standard deviations of all the assets and each pair of correlations. The portfolio standard deviation is:
. We can apply this general formula to our example, with Investment C's characteristics indicated with a 1 and Investment D's with a 2, w1 = 0.50 or 50% w2 = 0.50 or 50% σ1 = 0.1400 or 14.00% σ2 = 0.1997 or 19.97% cov1,2 = 0.0252. The portfolio variance is: portfolio variance = 0.502(0.14002) + 0.502(0.19972) + 2 (0.0252) 0.50 (0.50) = 0.002275
[email protected] and the portfolio standard deviation is 0.0477 or 4.77%, which, not coincidentally, is what we got when we calculated the standard deviation directly from the portfolio returns under the three scenarios. The standard deviation of the portfolio is lower than the standard deviations of each of the investments because the returns on Investments C and D are negatively related: when one is doing well the other may be doing poorly, and viceversa. That is, the covariance is
Example: The Portfolio Variance and Standard Deviation Problem Consider a portfolio comprised of two securities, F and G: Statistic
Expected return
Standard deviation Percentage of portfolio invested
Security F
10%
5%
40%
Security G
20%
8%
60%
The covariance between the two securities' returns is 0.002. What is the portfolio's standard deviation? Solution variance =0.16 (0.0025) + 0.36 (0.0064) + [(2) (0.002) (0.40) (0.60)] variance = 0.0004 + 0.0023 + 0.00096 = 0.00366 Portfolio standard deviation = 0.06050The investment in assets whose returns are out of step with one another is the whole idea behind diversification. Diversification is the combination of assets whose returns do not vary with one another in the same direction at the same time.
Common Sense Diversification
[email protected] Whereas the benefits of diversification can be achieved through random selection of a number of stocks, a number of common sense procedures can be usefully employed to construct a diversified portfolio. For example: Diversify across industries: Investing in a number of different stocks within the same industry does not generate a diversified portfolio since the returns of firms within an industry tend to be highly correlated. Diversification benefits can be increased by selecting stocks from different industries. Diversify across industry groups: Some industries themselves can be highly correlated with other industries and hence diversification benefits can be maximized by selecting stocks from those industries that tend to move in opposite directions or have very little correlation with each other. Diversify across geographical regions: Companies whose operations are in the same geographical region are subject to the same risks in terms of natural disasters and state or local tax changes. Investing in companies can diversify these risks whose operations are not in the same geographical region. Diversify across economies: Stocks in the same country tend to be more highly correlated than stocks across different countries. This is because many taxation and regulatory issues apply to all stocks in a particular country. International diversification provides a means for diversifying these risks. Diversify across asset classes: Investing across asset classes such as stocks, bonds, and real property also produces diversification benefits. The returns of two stocks tend to be more highly correlated, on average, than the returns of a stock and a bond or a stock and an investment in real estate.
Correlation Coefficient
[email protected] If the returns on investments move together, we say that they are correlated with one another. Correlation is the tendency for two or more sets of data in our case returns to vary together. The returns on two investments are: positively correlated if one tends to vary in the same direction at the same time as the other. negatively correlated if one tends to vary in the opposite direction with respect to the other. uncorrelated if there is no relation between the changes in one with changes in the other. Statistically, we can measure correlation with a correlation coefficient. The correlation coefficient reflects how the returns of two securities vary together and is measured by the covariance of the two securities' returns, divided by the product of their standard deviations:
By construction, the correlation coefficient is bounded between 1 and +1. We can interpret the correlation coefficient as follows: A correlation coefficient of +1 indicates a perfect, positive correlation between the two assets' returns. A correlation coefficient of 1 indicates a perfect, negative correlation between the two assets returns. A correlation coefficient of 0 indicates no correlation between the two assets returns.
[email protected] A correlation coefficient falling between 0 and +1 indicates positive, but not perfect positive correlation between the two assets returns. A correlation coefficient falling between 1 and 0 indicates negative, but not perfect negative correlation between the two assets returns. In the case of Investments C and D, the covariance of their returns is:
Therefore, the returns on Investment C and Investment D are negatively correlated with one another.
By investing in assets with less than perfectly correlated cash flows, you are getting rid of diversifying away some risk. The less correlated the cash flows, the more risk you can diversify away to a point. Let's see how the correlation and portfolio standard deviation interact. Consider two investments, E and F, whose standard deviations are 5% and 3%, respectively. Suppose our portfolio consists of an equal investment in each; that is, w1=w2=50%.
If the correlation between ... this means that the
and the portfolio's
the assets' returns is ...
standard deviation is
covariance is ...
... +1.0
+0.00150
4.00%
+0.5
+0.00075
3.50
[email protected] 0.0
0.00000
2.92
0.5
0.00075
2.18
1.0
0.00150
0.00
The less perfectly positively correlated are two assets' returns, the lower the risk of the portfolio comprised of these assets.
Portfolio Size and Risk The idea that we can reduce the risk of a portfolio by introducing assets whose returns are not highly correlated with one another is the basis of Modern Portfolio Theory (MPT). MPT tells us that by combining assets whose returns are not correlated with one another, we can determine combinations of assets that provide the least risk for each possible expected portfolio return. Though the mathematics involved in determining the optimal combinations of assets are beyond this module, the basic idea is provided in Exhibit 3. Each red colored point in the graph represents a possible portfolio that can be put together comprising different assets and different weights. The points in this graph represent every possible portfolio. As you can see in this diagram: some portfolios have a higher expected return than other portfolios with the same level of risk; some portfolios have a lower standard deviation than other portfolios with the same expected return. Since investors like more return to less and prefer less risk to more, some portfolios are better than others. The best portfolios are those that are mean variance efficient those that can't be better in terms of either the level of return for the amount of risk or the amount of risk for the level of return. The locus of mean variance efficient investment will make up what is called the efficient
[email protected] frontier. If investors are rational, they will go for the portfolios that fall on this efficient frontier. All the possible portfolios and the efficient frontier (shown in green) are diagrammed in Exhibit 3. Exhibit 3
So what is the relevance of MPT to financial managers? MPT tells us that we can manage risk by judicious combinations of assets in our portfolios; and there are some combinations of assets that are preferred over others. Risky assets versus risk-free assets It is important to distinguish between risk assets and risk-free assets. A risky asset is one for which the return that will be realized in the future is uncertain. For example, suppose an investor purchases the stock of Old Mutual today and plans to hold the stock for one year. At the time she purchased the stock, she does not know what return will be realised. The return will depend on the price of Old Mutual stock one year from now and the dividends that the company pays during the year. Thus, Old Mutual stock, and indeed the stock of all companies, is a risky asset. Even securities issued by the government are risky assets. For example, an investor who purchases a government bond that matures in 30 years do not know the return that will be realised if this bond is to be held for only one year. This is because a change in interest rates will affect the price of the bond one year from now and therefore the return from investing in that bond for one year.
[email protected] There are assets, however, in which the return that will be realized in the future is known with certainty today. Such assets are referred to as risk-free or riskless assets. The riskfree asset is commonly defined as short-term obligations of the government. For example, if an investor buys a government security that matures in one year and plans to hold that security for one year, then there is no uncertainity about the return that will be realized. The investor knows that in one year, on the maturity date of security the government will pay a specific amount to retire the debt. Notice how this situation differs for the government security that matures in 30 year securities are obligations of the government, the former matures in one year so that there is no uncertainty about the return that will pay at the end of 30 years for the 30 year bond he does not know what the price of the bond will be one year from now.
Modern Portfolio Theory and Asset Pricing Two Nobel Laureates in Economics, Harry Markowitz and William Sharpe recognized the relation between portfolio returns and portfolio risk. Harry Markowitz tuned us into the idea that investors hold portfolios of assets and therefore their concern is focused upon the portfolio return and the portfolio risk, not on the return and risk of individual assets. Is this reasonable? Probably. Not many businesses consist of a single asset. Nor do investors invest in only one asset. The relevant risk to an investor is the portfolio's risk, not the risk of an individual asset. If an investor holds assets in a portfolio and is considering buying an additional asset or selling an asset from the portfolio, what must be considered is how this change will affect the risk of the portfolio
The Capital Asset Pricing Model (CAPM) William Sharpe took the idea that portfolio return and risk are the only elements to consider and developed a model that deals with how assets are priced. This model is referred to as the Capital Asset Pricing Model (CAPM).
[email protected] The CAPM is a ceteris paribus model. It is only valid within a special set of assumptions. These are: Investors are risk averse individuals who maximize the expected utility of their end of period wealth. Implication: The model is a one period model. Investors have homogenous expectations (beliefs) about asset returns. Implication: all investors perceive identical opportunity sets. This is, everyone has the same information at the same time. Asset returns are distributed by the normal distribution. There exists a risk free asset and investors may borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate (rf). There is a definite number of assets and their quantities are fixed within the one period world. All assets are perfectly divisible and priced in a perfectly competitive market. Implication: e.g. human capital is nonexisting (it is not divisible and it can’t be owned as an asset). Asset markets are frictionless and information is costless and simultaneously available to all investors. Implication: the borrowing rate equals the lending rate. There are no market imperfections such as taxes, regulations, or restrictions. All the assets in each portfolio, even on the frontier, have some risk. Now let's see what happens when we add an asset with no risk referred to as the risk free asset. Suppose we have a portfolio along the efficient frontier that has a return of 4% and a standard deviation of 3%. Suppose we introduce into this portfolio the riskfree asset, which has an expected return of 2% and, by definition, a standard deviation of zero. If the riskfree asset's expected return is
[email protected] certain, there is no covariance between the risky portfolio's returns and the returns of the riskfree asset. A portfolio comprised of 50% of the risky portfolio and 50% of the riskfree asset has an expected return of (0.50) 4% + (0.50) 2% = 3% and a portfolio standard deviation calculated as follows: Portfolio variance = 0.502(0.03) + 0.502(0.00) + 2 (0.00) 0.50 (0.50) = 0.0075. Portfolio standard deviation = √0.0075 = 0.0866. If we look at all possible combinations of portfolios along the efficient frontier and the riskfree asset, we see that the best portfolios are no longer along the entire length of the efficient frontier, but rather are the combinations of the riskfree asset and one and only one portfolio of risky assets on the frontier. The combinations of the riskfree asset and this one portfolio is shown in Exhibit 4. These combinations differ from one another by the proportion invested in the risk free asset; as less is invested the riskfree asset, both the portfolio's expected return and standard deviation increase. Exhibit 4
[email protected]
William Sharpe demonstrates that this one and only one portfolio of risky assets is the market portfolio a portfolio that consists of all assets, with the weights of these assets being the ratio of their market value to the total market value of all assets. If investors are all risk averse they only take on risk if there is adequate compensation and if they are free to invest in the risky assets as well as the riskfree asset, the best deal lie along the line that is tangent to the efficient frontier. This line is referred to as the capital market line (CML), shown in Exhibit 4. If the portfolios along the capital market line are the best deals and are available to all investors, it follows that the returns of these risky assets will be priced to compensate investors for the risk they bear relative to that of the market portfolio. Since the portfolios along the capital market line are the best deals, they are as diversified as they can get no other combination of risky assets or riskfree asset provides a better expected return for the level of risk or provides a lower risk for the level of expected return.
The equation to this line which represents the possible sets of portfolios of the riskless asset and portfolio is
[email protected]
where rf is the intercept and
is the slope of the line and represent
price of risk. We could also write
where E[rm]rf is the risk premium. The capital market line tells us about the returns an investor can expect for a given level of risk. The CAPM uses this relationship between expected return and risk to describe how assets are priced. The CAPM specifies that the return on any asset is a function of the return on a riskfree asset plus a risk premium. The return on the riskfree asset is compensation for the time value of money. The risk premium is the compensation for bearing risk. Putting these components of return together, the CAPM says: Expected return on an asset = expected return on a risk free asset + risk premium or, return of any asset is given by a formula of the form: E[ri] = rf + [Number of Units of Risk][Risk Premium per Unit] We have established above that the appropriate measure of risk is cov(Ri, Rm) and hence, the equation can be rewritten as:
We already know the details surrounding two points on this line. The riskless asset has expected return of rf and covariance with the market portfolio of zero
[email protected] (since rf is constant). The market portfolio has expected return E[rm] and covariance with the market of
(the covariance of a variable with itself is its
variance). Since these two points must lie on the line, the equation to the line must be:
Note that for the riskless asset this becomes:
and for the market portfolio it is
It is common to standardize the units of this equation by defining
and
rewriting the equation as
It is this equation that is known as the SharpeLintner CAPM. The beta of the market portfolio is one:
This provides a reference point against which the risk of other assets can be measured. The average risk (or beta) of all assets is the beta of the market, which is one. Assets or portfolios that have a beta greater than one have above average risk, tending to move more than the market. For example, if the riskless rate of interest (Tbill rate) is 5% p.a. and the market rises by 10%, assets with a
[email protected] beta of 2 will tend to increase by 15%. If however, the market falls by 10%, assets with a beta of 2 will tend to fall by 25% on average. Conversely, assets with betas less than one are of below average risk and tend to move less than the market portfolio. Assets that have betas less than zero tend to move in the opposite direction to the market. These assets are known as hedge assets.
If we assume that investors hold welldiversified portfolios (approximating the market portfolio), the only risk they have is nondiversifiable risk. If assets are priced to compensate for the risk of assets and if the only risk in your portfolio is nondiversifiable risk, then it follows that the compensation for risk is only for non diversifiable risk. Let's refer to this nondiversifiable risk as market risk. Since the market portfolio is made up of all assets, each asset possesses some degree of market risk. Since market risk is systematic across assets, it is often referred to as systematic risk and diversifiable risk is referred to as unsystematic risk. Further, the risk that is not associated with the market as a whole is often referred to as companyspecific risk when referring to stocks, since it is risk that is specific to the company's own situation such as the risk of lawsuits and labor strikes and is not part of the risk that pervades all securities.
The risk that reflects an asset's
The risk that reflects an asset's
returns moving with asset returns in returns not moving along with asset general is referred to as ...
returns in general is referred to as ...
nondiversifiable risk
diversifiable risk
market risk
companyspecific risk
systematic risk
Unsystematic risk
[email protected] The measure of an asset's return sensitivity to the market's return, its market risk, is the asset's beta. The expected return on an individual asset is the sum of the expected return on the riskfree asset and the premium for bearing market risk. Let ri represent the expected return on asset i, rf represented the expected return on the riskfree asset, and β i represent the degree of market risk for asset i. Then:
The term (rm rf), is the market risk premium if you owned all the assets in the market portfolio, you would expect to be compensated (rm rf) for bearing the risk of these assets. β is measure of market risk, which serves to finetune the risk premium for the individual asset. For example, if the market risk premium were 2% and the β for an individual asset were 1.5, you would expect to receive a risk premium of 3% since you are taking on 50% more risk than the market. Security market line (SML) For each asset there is a beta. If we represent the expected return on each asset and its beta as a point on a graph, and we do the same for every asset in the market, and connect all the points, the result is the security market line, SML, as shown in Exhibit 5. Exhibit 5
[email protected] Assets that plot on the security market line are correctly priced (equilibrium). Along the SML expected returns will be equal to required returns. In this case it pays the individual to hold on to the security. Assets that plot off the SML are mispriced (disequilibria). Above the SML securities will be underpriced because expected returns are greater than required returns. In this case it pays the investor to buy the security. On the other hand asset that plot below the SML are overpriced because expected returns will be lower than the required return and hence the investor will prefer to sell the security. As you can see in this graph: The greater the β, the greater the expected return. If there were no market risk (beta = 0.0) on an asset its expected return would be the expected return on the riskfree asset. If the asset's risk is similar to the risk of the market as a whole (beta = 1.0), that asset's expected return is the return on the market portfolio. For an individual asset, beta is a measure of sensitivity of its returns to changes in return on the market portfolio. If beta is one, we expect that for a given change of 1% in the market portfolio return, the asset's return is expected to change by 1%. If beta is less than one, then for a 1% change in the expected market return, the asset's return is expected to change by less than 1%. If the beta is greater than one, then for a 1% change in the expected market return, the asset's return is expected to change by more than 1%.
6.10 Portfolio Beta The beta of an individual asset is:
[email protected]
Now consider a portfolio with weights wp. The portfolio beta is:
β p = w1β 1 + w2β2 + w3β3 + ... + wSβS,
The beta of the portfolio is the weighted average of the individual asset betas where the weights are the portfolio weights. So we can think of constructing a portfolio with whatever beta we want. All the information that we need is the betas of the underlying asset. For example, if I wanted to construct a portfolio with zero market (or systematic) risk, then I should choose an appropriate combination of securities and weights that delivers a portfolio beta of zero. Suppose we have three securities in our portfolio, with the amount invested in each and their security beta as follows:
Security
Security beta
Amount invested
AAA
1.00
$10,000
BBB
1.50
$20,000
CCC
0.75
$20,000
[email protected] The portfolio's beta is:
Criticism and reality of CAPM The above is a world version of the Sharpe (1964) CAPM. There are, however, potential problems with these tests. The beta may not be constant through time. The alpha may not be constant through time. The error variance may not be constant through time (this is known as heteroskedasticity). The errors may be correlated through time (this is known as autocorrelation or serial correlation). Returns may be nonlinearly related to market returns rather than the linear relation that is suggested in the statistical model. The returns on the market portfolio and the risk free rate may be measured incorrectly. There may be other sources of risk. The world CAPM may not hold in all countries. A beta is an estimate. For stocks, the beta is typically estimated using historical returns. But the proxy for market risk depends on the method and period in which is it is measured. For assets other than stocks, beta estimation is more difficult. The CAPM includes some unrealistic assumptions. For example, it assumes that all investors can borrow and lend at the same rate.
[email protected] The CAPM is really not testable. The market portfolio is a theoretical and not really observable, so we cannot test the relation between the expected return on an asset and the expected return of the market to see if the relation specified in the CAPM holds. In studies of the CAPM applied to common stocks, the CAPM does not explain the differences in returns for securities that differ over time, differ on the basis of dividend yield, and differ on the basis of the market value of equity (the so called "size effect"). Though it lacks realism and is difficult to apply, the CAPM makes some sense regarding the role of diversification and the types of risk we need to consider in investment decisions.
The multifactor CAPM The CAPM described above assumes that the only risk that an investor is concerned with is uncertainty about the future price of a security. Investors, however, usually are concerned with other risks that will affect their ability to consume goods and services in the future. Three examples would be the risks associated with future labour income, the future relative price of consumer goods, and future investment opportunities. Recognizing these other risks that investors face, Robert Merton has extended the CAPM based on consumers deriving their optimal lifetime consumption when they face these “extramarket” sources of risk. These extramarket sources of risk are also referred to as “factors” hence the model derived by Merton is called a multifactor CAPM and is given below: E ( R P ) = R F + β P , M [ E ( R M ) − R F ] + β P , F 1 [ E ( R F 1 ) − R F ] + β P , F 2 [ E ( R F 2 ) − R F ] + ...
+ β P , Fk [ E ( R Fk ) − R F ]
[email protected] where R F = The risk free return F1 , F2 , Fk = factors of extramarket sources of risk, 1 to k K = number of factors or extra market sources of risk B P 'FK = the sensitivity of the portfolio to the k E ( R FK ) = the expected return of factor k The total extramarket sources of risk is equal to:
β P , F 1 [ E ( RF 1 ) − RF ] + β P , F 2 [ E ( RF 2 ) − R ] + ... + β P , Fk [ E ( RFk ) − RF k ] this expression says that investors want to be compensated for the risk associated with each source of extramarket risk, in addition to market risk. Note that if there are no extra market sources of risk, then equation reduces to the expected return for the portfolio as predicted by the CAPM: E ( R P ) = R F + β P [ E ( RM ) − R F ] In the case of the CAPM, investors hedge the uncertainty associated with future security prices by diversification. This is done by holding the market portfolio which can be thought of as a mutual fund that invests in all securities based on their relative capitalizations. In the multifactor CAPM, in addition to investing in the market portfolio, investors will also allocate funds to something equivalent to a mutual fund that hedges a particular extra market risk. While not all investors are concerned with the same sources of extra market risk, those that are concerned with a specific extra market will basically hedge them in the same way. We have just described the multifactor model for a portfolio. How can this model be used to obtain the expected return for an individual security? Since individual securities are nothing more than portfolios consisting of only one security, equation below must hold each security, i . That is: E ( Ri ) = R F + β i , M [ E ( RM ) − R F )] + β i , F 1 [ E ( R F 1 ) − R F )] + β i , F 2 [ E ( R F 2 ) − RF )] + ...
+ β i , FK [ E ( R FK ) − R FK )]
[email protected] The multifactor CAPM is an attractive model because it recognizes nonmarket risks. The pricing of an asset by the market place, then must reflect risk premiums to compensate for these extra market risks. Unfortunately, it may be difficult to identify all the extra market risk and to value each of these risks empirically. Furthermore, when these risks are taken together , the multifactor CAPM begins to resemble the arbitrage pricing theory model described next.
The Arbitrage Pricing Model An alternative to CAPM in relating risk and return is the Arbitrage Pricing Model. The Arbitrage Pricing Model developed by Stephen Ross, is an asset pricing model that is based on the idea that identical assets in different markets should be priced identically. While the CAPM is based on a market portfolio of assets, the Arbitrage Pricing Model doesn't mention a market portfolio at all. Instead, the Arbitrage Pricing Model states that an asset's returns should compensate the investor for the risk of the asset, where the risk is due to a number of economic influences, or economic factors. Therefore, the expected return on the asset i, ri, is: ri = rf + δ1f1 + δ2f2 + δ3 f3+ ... where each of the δ's reflect the asset's return's sensitivity to the corresponding economic factor, f. The Arbitrage Pricing Model looks much like the CAPM, but the CAPM has one factor the market portfolio. There are many factors in the Arbitrage Pricing Model. What if an asset's price is such that it is out of line with what is expected? That's where arbitrage comes in. Any time an asset's price is out of line with how market participants feel it should be priced based on the basic economic influences
[email protected] investors will enter the market and buy or sell the asset until its price is in line with what investors think it should be. What are these economic factors? They are not specified in the original Arbitrage Pricing Model, though evidence suggests that these factors include: unanticipated changes in inflation; unanticipated changes in industrial production; unanticipated changes in risk premiums; and unanticipated changes in the difference between interest rates for short and long term securities. Anticipated factors are already reflected in an asset's price. It is the unanticipated factors that cause an asset's price to change. For example, consider a bond with a fixed coupon interest. The bond's current price is the present value of expected interest and principal payments, discounted at some rate that reflects the time value of money, the uncertainty of these future cash flows, and the expected rate of inflation. If there is an unanticipated increase in inflation, what will happen to the price of the bond? It will go down since the discount rate increases as inflation increases. If the price of the bond goes down, the return on the bond will decrease. Therefore, the sensitivity of a bond's price to changes in unanticipated inflation is negative.
