sapm mod-4

Krupanidhi school of management Prof. V.N.V.SASTRY

SAPM-4ANALYSIS OF RISK AND RETURN

Investment decisions are influenced by various decisions. Some people invest in a business to acquire control and enjoy the prestige associated with it. Some people invest in expensive yatches and famous villas to display their wealth. Most investors are guided by the pecuniary motive of earning a return on their investment.For earning returns investors have to almost invariably bear some risk. In general risk and return go hand in hand. Investment decisions involve a trade off between risk and return.Return: Return is the primary motivating force that drives investment. It represents the reward for undertaking investment. Since the game of investing is about returns (after allowing for risk), measurement of realized (historical) returns is necessary to asses how well the investment manager has done. In addition historical returns are often used as an important an input in estimating future (prospective returns) Components of return: it has two components:Current return: the first component that often comes to mind when one is thinking about return is periodic cash flow ( income), such as dividend or interest, generated by the investment. Current returns is measured as the periodic income in relation to the beginning prices of the investment.Capital return: the second component of returns is reflected in the price change called the capital return- it is simply the price appreciation ( or depreciation) divided by the beginning price of the asset. For assets like equity stocks, the capital return predominates.Thus the total return for any security ( or for that matter any asset) is defined as : Total return =current return + capital returnThe current return can be zero, or +ve , where as capital return can be –ve , zero or +ve.Risk: Risk refers to the actual outcome of an investment will differ from its expected outcome. Forces that contribute to variations in return price or dividend (interest)- constitute elements of risk. Some influences are external to the firm cannot be controlled, and affect large number of securities. Other influences are internal to the firm and are controllable to a large degree. In investments, forces that are uncontrollable, external, and broad in their effect are called ‘sources of systematic risk”. Conversely, controllable, internal factors some what peculiar to industries and/ or firms are referred t as ‘ sources of unsystematic risk’.Systematic risk refers to that portion of total variability in return caused by factors affecting the prices of all securities. Economic, political and sociological changes are sources of systematic risk. Their effect is to cause prices of nearly all


individual common stocks and or all individual bonds to move together in the same manner. Unsystematic risk is the portion of total risk that ,is unique to a firm or industry. Factors such as management capability, consumer preferences, and labour strikes cause systematic variability of returns in a firm. Unsystematic factors are largely independent of factors that are affecting securities markets. Because these factors affect one firm, they must be examined for each firm. Three major sources of risk= business risk , interest rate risk, and market risk

Unsystematic risk: business risk, financial risk

Systematic risk: market risk, interest rate risk , purchasing power risk

Interest rate risk: the changes in interest rate have a bearing on the welfare of investors. As the interest rates go up, the market price of existing fixed income securities falls, and vice versa. This happens because the buyer of a fixed income security would not buy it at its par value or face value if its fixed interest rate is lower than the prevailing interest rate on a similar security. For ex. A debenture that has a face value of Rs.100 and a fixed rate of 12%will sell at a discount if the interest rate moves up from , say 12% to 14%. While the changes in interest rate have direct bearing on the prices of fixed income securities, they affect equity prices too, albeit some what indirectly. The changes in the relative yields of debenture and equity shares influence equity prices.

Market rate risk: even if the earning power of the corporate sector and the interest rate structure remain more or less unchanged, prices of securities, equity shares in particular, tend to fluctuate. While there can be several reasons for this fluctuations, a major cause appears to be the changing psychology of the investors. There are periods when investors become bullish and their investment horizons lengthen. Investors optimism, which may border on euphoria, during such periods drives share prices to great heights. The buoyancy created in the wake of this development is pervasive, affecting all most all the shares. On the other hand when a wave of pessimism ( which often is an exaggerated response to some unfavorable political or economic development) sweeps the market, investors turn bearish and myopic. Prices of almost all equity shares register decline as fear and uncertainty pervade the market.The market tends to move in cycles. As John Train explains “the ebb and flow of mass emotion is quite regular. panic is followed by relief, and relief by optimism, then comes enthusiasm ; then euphoria and rapture, then the bubble bursts, and public felling slides off again into concern, desperation, and finally a new panic”.One would expect large scale participation of institutions to dampen the price fluctuations in the market. After all institutional investors have core professional


expertise to do fundamental analysis and greater financial resources to act on fundamental analysis. However , nothing of this kind has happened. On the contrary, price fluctuations seem to have become wider after the arrival of institutional investors in large numbers. Why? , perhaps the institutions and their analysts have not displayed more prudence and rationality than the general investing public and have succumbed in equal measure to the temptation to speculate. As John Marynard Keynes had argued, factor that contribute to the volatility f the market are not likely to diminish when expert professionals possessing better judgement and knowledge compete in the market place. Why? According to Keynes, even these people are concerned with “speculation” (the activity of forecasting the psychology of the market) and not “enterprise” ( the activity of forecasting the prospective yield of assets over their whole life).

(Fischer and Jordan)Purchasing power risk: market risk and interest risk can be defined in terms of uncerainities as to the amount of current dollars to received by an investor. Purchasing power risk is the uncertainit;y of the purchasing power of the amounts to be received. In more everyday terms, purchasing power risk refers to the impact of inflation or deflation on an investment.If we think investment is the postponement of consumption, we can see that when a person purchases a stock, he has foregone the opportunity to buy some good or service for as long as he owns the stock. If during the holding period prices on desired goods and services rise, the investor actually losses purchasing power. Rising prices on goods and services are associated with what is referred to as inflation. Falling prices on good and services are termed as deflation. Both inflation and deflation are covered in the all-encompassing term purchasing power risk. Generally purchasing power risk has come to be identified with inflation(rising prices); the incidence of declining prices in most countries has been slight.

Financial risk: risk is associated with the way in which a company finances it activities. We usually gauge financial risk by looking at the capital structure of a firm. The presence of borrowed money or debt in the capital structure creates fixed payments in the form of interest hat must be sustained by the firm. The presence of these interest commitments – fixed interest payments due to debt or fixed dividend payments on preferred stock – causes the amount of residual earnings available for common stock dividends to be more variable than if no interest payments were required. Financial risk is avoidable risk t the extent that managements have the freedom to decide to borrow or not to borrow funds. A firm with no debt financing has no financial risk.By engaging in debt financing , the firm changes the characteristic of the earning streams available to the common stock holders. Specifically, the reliance on debt


financing called ‘financial leverage” has at least three important effects on common stock holder. 1. increases the variability of their returns 2. affects their expectations concerning their returns. 3. increases their risk of being ruined.

Business risk: as a holder of corporate securities (equity shares or debentures), you are exposed to the risk of poor business performance. This many be caused by a variety of factors like heightened competition , emergence of new technologies, development of substitute products, shifts in consumer preference , in adequate supply of essential inputs , changes in governmental polices an so on. Often of course, the principal factor maybe inept and incompetent management. The poor business performance definitely affects the interest of equity share holders, who have a residual claim on the income and wealth of the firm. It can also affect the interest of a debenture holders if he ability of the firm to meet its interest and principal payments obligation is impaired. In such a case, debenture holders face the prospect of default risk.

Types of risk: modern portfolio theory looks at risk from a different perspective. It divides total risk as follows: total risk = unique risk + market risk

The ‘unique risk’ of security represents that portion of its total risk which stems from firm specific factors like the development of a new product, labour strike , or the emergence of a new competitor. Evens of this nature primarily affects the specific firm and not all firms in general. Hence the unique risk of a stock can be washed away by combining it with other stocks. In a diversifies portfolio , unique risks of different stocks tend to cancel each other- a favourable development in one firm may offset an adverse happening in another and vice versa. Hence unique risk is also referred to as a diversifiable risk or unsystematic risk.The ‘market risk’ of a stock represents that portion of its risk which is attributable to economy-wide factors like the growth rate of GDP, the level of government spending, money supply, interest rate structure, and inflation rate. Since these factors affect all firms to a greater of lesser degree, investors cannot avoid the risk arising from them, however diversified their portfolio maybe. Hence it is also referred to as ‘systematic risk’ (as it affects all securities) or non-diversiable risk.

Bodie et.al.

Risk and risk aversion: The presence of risk means that more than one outcome is possible. A simple prospect is an investment opportunity in which a certain wealth is placed at risk, and there are only two possible outcomes. For the sake of simplicity, it is useful to elucidate some basic concepts using simple prospect.


Take as an example initial wealth , W of $100,000 and assume two possible results, with a probability p=0.6, the favourable outcome will occur leading to final wealth W1= $150,000, other wise, with probability 1-p = 0.4, a less favourable outcome, W2 = $80,000, will occur. We can represent the simple prospect using an event tree.

Fig -1

Suppose an investor is offered an investment portfolio with a payoff of 1 year described by such a simple prospect. How can you evaluate this portfolio? First try to summarise it using descriptive statistics. For instance, the mean or expected end-of-year wealth denoted by E(W), is

E(W) =pW1 +(1-p)W2= (0.6 x 150,000) + (0.4 x 80,000) = $122,000

The expected profit on the $100,000 investment portfolio is $22,000 (122000-100000).The variance , σ2 , of the portfolio’s pay off is calculated as the expected value of the squared deviation of each possible outcome from the mean:

σ2 = p[W1-E(W)]2 + (1-p) [W2 – E(W)]2

=0.6 [150,000-122000]2 + 0.4[80,00-122000]2 = 1,176,000,000

The standard deviation σ = $34292.86

Clearly this is a risky business. The standard deviation of the pay off is large much larger than the expected profit of $22,000. whether the expected profit is large enough to justify such risk depends on alternatives to this portfolio.Let us suppose, treasury bills are one alternative to the risky portfolio. Suppose that at the time of the decision, a 1 year T-bill offers a rate of return of 5% ; $100,000 can be invested to yield a sure profit of $5000. we can now draw decision tree.

Fig -2

Earlier we showed that the expected profit on the prospect to be $22000. therefore the expected marginal, or incremental, profit of the risky portfolio over investing in safe T-bill is 22000-5000=$17000, meaning that one can earn a ”risk premium” of $17000 as a compensation of the investment. The question of whether a given risk premium provides adequate compensation for an investment ‘s risk isage old. Indeed one of the central concerns of finance theory ( and much of this text) is the


measurement of risk and the determination of the premium that investor can expect of risky assets in well functioning capital markets.

Risk, speculation and gambling:One definition of ‘ speculation’ is the “assumption of considerable business risk in obtaining commensurate gain”. Although this definition is fine linguistically, it is useless with out first specifying what is meant by “commensurate gain” and “commensurate risk”.By commensurate gain” we mean a positive premium, that is an expected profit greater than the risk free alternative. In our example, the dollar risk premium is $17000, the incremental expected gain from taking on the risk. By “considerable risk” we mean that the risk is sufficient to affect the decision. An individual might reject a prospect that has a positive risk premium because the added gain is insufficient to make up for risk involved.To gamble is “to bet or wager on an uncertain outcome”. If your compare this definition to that of speculation, you will see that the central difference is the lack of “commensurate gain”. Economically speaking, a gamble is the assumption of risk for no purpose, but enjoyment of the risk itself, here as speculation is undertaken ‘inspite’ of the risk involved because one perceives a favourable risk-return tradeoff. To turn a gamble into a speculative prospect requires an adequate risk premium to compensate risk –averse investors for the risk they bear. Hence “ risk aversion and speculation are not inconsistent” In some cases a gamble may appear to the participants as speculation. Suppose two investors disagree sharply about the future exchange rate of US dollar against British pound. They may choose to bet on the outcome. Suppose that Paul will pay Mary $100 if the value of a pound exceeds $1.70 one year from now, where as Mary will pay Paul if the pound worth is less than $ 1.70 . There are only two relevant outcomes: 1. the pound will exceed $1.70 or 2. it will fall below $1.70. if both Paul and Mary agree on the probabilities of two possible outcomes. And if neither party anticipates a loss, it must be that they assign p=0.5 to each outcome. In that case the expected profit to both is zero and each has entered one side of a gambling prospect.What is more likely, however is the bet results from differences in the probabilities that Paul and Mary assign to the outcome. Mary assigns it p> 0.5 where as Paul assessment is p<0.5. they perceive subjectively, two different prospects. Economists call this case of differing beliefs “heterogeneous expectations”. In such cases investors on each side of a financial position see themselves as speculating rather than gambling.Both Paul and Mary should be asking, why is the other willing to invest in the side of a risky prospect that I believe offers a negative expected profit?. The ideal way to resolve heterogeneous belief is for Paul and Mary to “merge heir


information”, that is for each party to verify that he or she posses all relevant information and processes that information properly. Of course , the acquisition of information and the extensive communication that is required to eliminate all hetrogeneity in expectations is costly, and thus up to a point of heterogeneous expectation cannot be taken as irrational. If however , Paul and Mary enter such contracts frequently, they would recognize the information problem in one of two ways: either they will realize that they are creating gambles when each wins half of the bets, or the consistent losses will admit that he or she has been betting on the basis of inferior forecasts.

Risk aversion and utility values: we have discussed risk with simple prospect and how risk premium bear on speculation. A prospect that has zero risk premium is called a ‘ fair game’. Investors who are “ risk averse” reject investment portfolios that are fir games or worse. Risk-averse investors are willing to consider only risk- free speculative prospects with positive risk premia. Loosely speaking, a risk-averse investor “penalizes” the expected rate of return of a risky portfolio by a certain percentage (or penalizes the expected profit by a dollar amount) to account for the risk involved. The greater the risk, the larger the penalty. One might wonder why we assume risk aversion as fundamental. We believe that most investors would accept this view from simple introspection. We can formalize the notion of a risk penalty system. To do so we will assume that each investor can assign a welfare or ‘utility ‘ score to competing investment portfolios based on the expected return and risk of those portfolios. The utility score may be viewed as a means of ranking portfolios. Higher utility values are assigned to portfolios with more attractive risk-return profiles. Portfolios receive higher utility scores for higher expected returns and lower scores for higher volatility. Many particular ‘scoring’ system are legitimate. One reasonable function that is commonly employed by financial theorists and the AIMR (Association of Investment Management and Research) assign a portfolios with expected return E(r) and variance of return σ2 the following utility score.

U = E® - 0.005 A σ2 where U is the utility value and A is an index of the investors risk aversion. The factor of 0.005 a scaling conversion that allows us to express the expected return and standard deviations in the above equation as percentage rather than decimal.The equation is consistent with the notion that utility is enhanced by high expected returns and diminished by high risk. The extent to which variance lowers utility depends on A, the investors degree of risk aversion. More risk-aversion investors (who have larger As) penalize risky investments more severely. Investors choosing among competing investment portfolios will select the one providing the high utility. Risk aversion obviously will have a major impact on the investor’s appropriate risk-return trade-off.


Notice in the above equation that the utility provided by risk-free portfolio is simply the rate of return on the portfolio, because there is no penalization for risk. This provides us with a convenient bench mark for evaluating portfolios.

Evaluating investments by using utility scores: in the earlier example of choosing between portfolio with an expected return of 22% and standard deviation σ =34% and T bills providing risk-free return of 5%. Although the risk premium on the risky portfolio was large ,17% the risk of project is so great that an investment would not need to be very risk averse to choose the safe all bills strategy . Even A=3, a moderate risk aversion parameter as per the equation show that the risky portfolio utility value as, 22 – (0.005 x 3 x 342) = 4.66% , which is slightly lower than the risk free rate. In this case one would reject the portfolio in faovour of T bills.The downward adjustment of the expected return as a penalty for risk is 0.005 x 3x 342= 17.34%. if the investors were less risk averse (more risk tolerant), for ex. With A=2, she would adjust the expected rate of return downward by only 11.56%. in that case the utility level of the portfolio would be 10.44%, higher than the risk free rate, leading her to accept the profit.

Ex. A portfolio has an expected rate of return of 20% and std. deviation 20%. Bills offer a sure rate of 7%. Which investment alternative will be choosen by an investor whose A=4? What if A=8?U= 20 – (0.005x 4 x 202) = 12% and U= 20 – (0.005 x 8 x 202) = 4%Choose the investment where A = 4.

Because we can compare utility values to the rate offered on risk free investments when choosing between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its ‘certainty equivalent’ rate of return to an investor. That is the ‘certainty equivalent rate’ of a portfolio is the rate that risk free investments would need to offer with certainty to be considered equally attractive as the risky portfolio.Now we can say that a portfolio is desirable only if its certainty equivalent return exceeds that of the risk free alternative. A sufficiently risk averse investor may assign any risky portfolio, even one with a positive risk premium, a certainty equivalent rate of return that is below the risk free rate, which will cause the investor to reject the portfolio. At the same time a less risk averse (more risk tolerant) investor may assign the same portfolio a certainty equivalent rate that exceeds the risk free rate and thus will prefer the portfolio to the risk free alternative. If the risk premium is zero or negative to begin with , any downward adjustment to utility makes the portfolio look worse. Its certainty equivalent rate will be below that of risk free alternative for all risk averse investors.


In contrast to risk averse investors, “risk –neutral” investors judge risky prospects solely by their expected rates of return. The level of risk is irrevalent to the risk – neutral investor, meaning that there is no penalization for risk. For this investor a portfolio’s certainty equivalent rate is simply its expected rate of return.A “risk lover” is willing to engage in fair games and gamble ; this investor adjusts the expected return upward to take into account the ’fun’ of confronting the prospects is. Risk lovers will always take a fair game because their upward adjustment of utility for risk gives the fair game a certainty equivalent that exceeds the alternative of the risk free investment.We can depict the individual’s trade off between risk and return by plotting the characteristics of potential investment portfolios that the individual would view as equally attractive on a graph ( pl. see graph below).

Fig -3

With axes measuring expected values and standard deviation of portfolio returns.

Portfolio ‘p’, which has expected return E(rp) and standard deviation σp is preferred by risk averse investors to any portfolio in quadrant IV. Because it has an expected return equal to or greater than any portfolio in that quadrant and a standard deviation equal to smaller than any portfolio in that quadrant. Conversely any portfolio in quadrant I is preferable to portfolio ‘p’ because its expected return is equal to greater than ‘p’s and its standard deviation is equal to or smaller than ‘p’s.This is the mean-standard deviation , or equivalently, mean variance (M-V) criterion. It can be stated as : A dominates B if E(rA) ≥ E(rb) and σA ≤ σB . and at least one inequality is strict (rules out the equality)In the expected return-standard deviation plane in the above graph the preferred direction is north west, because in this direction we can simultaneously increase expected return and decrease he variance of the rate of return. This means that any portfolio that lies north west of ‘p’ is superior to ‘p’. What can be said about portfolios in quadrant II and III? their desirability, compared with ‘p’ depends on the exact nature of the investor’s risk aversion. Suppose an investor identifies all portfolios that are equally attractive as portfolio ‘p’, starting at ‘p’, an increase in standard deviation lower utility ; it must be compensated for by an increase in expected return. Thus point Q in the graph below is equally desirable to this investor as ‘p’. investors will be equally attracted to portfolios with high risk and high expected returns compared with other portfolios with lower risk but lower expected return. These equally preferred portfolios will lie in mean-standard deviation plane on a curve that connects all portfolio points with the same utility value called the ‘indifference curve’.


To determine some of the points that appear on the indifference curve , examine the utility values of several possible portfolios for an investor with A=4, presented in the table below . note that each portfolio offers identical utility , because the high-return portfolios also have high risk.

Utility value of possible portfolios for investor with risk aversion, A=4Expected return E(r) Standard deviation σ Utility= E(r )- 0.005A σ2

10% 20% 10-(.005x4x400)=215% 25.5% 15-(.005x4x650)=220% 30% 20-(.005x4x900)=225% 33.9% 25-(.005x4x1150)=2 Prasanna chandraRisk aversion and required return:You are lucky to be invited by the host of a television show. After the usual introduction , the host shows two boxes to you. He tells you that one box contains Rs. 10,000 and the other box is empty. He does not tell you which one is which. The host asks you t open any one of the boxes and keep what ever you find in it. You are not sure which box you should open. Sensing your vacillation, he says he will offer you a certain sum of Rs. 3000 if you forfeit the option to open a box. You do not accept his offer. He raises his offer to Rs.3500. now you feel indifferent between a certain return of Rs. 3500 and a risky(uncertain) expected return of Rs.5000. This means that a certain return of Rs. 3500 provides you with the same satisfaction as a risky return of Rs.5000. Thus your certainty equivalent (Rs.3500) is less than the risky expected value (Rs 5000).Empirical evidence suggests that most individuals, if placed in a similar situation, would have a certainty equivalent which is less than the risky expected value. The relationship of a person’s certainty equivalent to the expected monetary value of a risky investment defines his attitude towards risk. If the certainty equivalent is equal to the expected value, the person is “risk-averse”; if the certainty equivalent is equal to the expected value, the person is “risk-neutral”; finally if the certainty equivalent is equal to more than the expected value, the person is “risk loving”.In general, investors are risk averse. This means that risky investments must offer higher expected returns than less risky investments to induce people to invest in them. Remember, however that we are talking about “expected” return and


“actual” return on a risky investment may well turn out to be less than “actual” return on a less risky investment.Put differently risk and return go hand in hand. This indeed is a well established empirical fact, particularly over long periods of time. For example, the average annual rates and annual standard deviation for Treasury bills, bonds, and common stocks in the U.S. over 75 years period (1926-2000) as calculated by Ibbotson Associates have been shown below:Port folio Average annual rate of

return%Standard deviation %

Treasury bills 3.9 3.2Govt bonds 5.7 9.4Corporate bonds 6 8.7Common stocks (S & P 500)

13.0 20.2

Small firm common stocks

17.3 33.4

From the above it is clear that a. treasury bills, the least risky of financial assets earned the lowest average annual rate of return b. common stocks, the most risky of financial assets, earned the highest average annual rate of return c. bonds which occupy a middling position on the risk dimension, earned a middling average annual return.

Risk premium:Investors assume risk so that they are rewarded in the form of higher return. Hence risk premium may be defined as the additional return investors expect to get, or investors earned in the past, for assuming additional risk. Risk premium may be calculated between two classes of securities that differ in their risk level. There are three well known risk premiums: Equity risk premium: this is the difference between the return on equity stocks as a class and the risk free represented commonly by the return on treasury bill.Bond horizon premium: this is the difference between the long term government bonds and return on treasury bills.Bond default premium: this is the difference between the return on long – term corporate bonds (which have probability of default) and return on long-term government bonds (which are free from default risk).

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