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6-1

Risk and Risk Risk and Risk AversionAversion

Chapter 6

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6-2 CHAPTER OVERVIEW

This chapter introduces the concepts of risk, risk measurement and the basic elements or concepts on which portfolio theory is built.

The chapter begins by introducing the concepts of expected return an expected future value for uncertain investments. The concept of risk premiums and certainty equivalents are developed.

The chapter concludes with the introduction of the basic rules for portfolio theory.

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6-3CHAPTER OVERVIEW

This chapter introduces three themes in portfolio theory, all centering on risk.

The first is the basic tenet that investors avoid risk and demand a reward for engaging in risky investments. The reward is taken as a risk premium, the difference between the expected rate of return and that available on alternative risk-free investments.

The second theme allows us to quantify investors ’ personal tradeoff between portfolio risk and expected return. To do this ,we introduce the utility function, which assumes that investors can assign a welfare or “utility ” score to any investment portfolio depending on its risk and return.

The third fundamental principle is that we cannot evaluate the risk of an asset separate from the portfolio of which it is a part; the proper way to measure the risk of an individual asset is to assess its impact on the volatility of the entire portfolio of investments. Taking this approach , we find that seemingly risky securities may be portfolio stabilizers and actually low-risk assets.

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6-4 LEARNING OBJECTIVES

After covering the chapter, the students should understand how to measure

expected return and expected future value for uncertain investments.

They should understand the concept of risk aversion and utility and know why we need utility function .

They should be able to apply the concept of risk aversion in measuring a utility function.

The students should be able to describe the basic statistical measurements and properties that are used to develop portfolio theory.

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6-5 Definitions of risk

There are many definitions of risk that vary by specific application and situational context. One is that risk is an issue, which can be avoided or mitigated (wherein an issue is a potential problem that has to be fixed now.) Risk is described both qualitatively and quantitatively. In some texts risk is described as a situation which would lead to negative consequences.

Qualitatively, risk is proportional to both the expected losses which may be caused by an event and to the probability of this event. Greater loss and greater event likelihood result in a greater overall risk.

Frequently in the subject matter literature, risk is defined in pseudo-formal forms where the components of the definition are vague and ill-defined, for example, risk is considered as an indicator of threat, or depends on threats, vulnerability, impact and uncertainty.

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6-6 Definitions of risk-In finance

In finance, risk is the probability that an investment's actual return will be different than expected. This includes the possibility of losing some or all of the original investment. It is usually measured by calculating the standard deviation of the historical returns or average returns of a specific investment

In finance, risk has no one definition, but some theorists, notably Ron Dembo, have defined quite general methods to assess risk as an expected after-the-fact level of regret. Such methods have been uniquely successful in limiting interest rate risk in financial markets. Financial markets are considered to be a proving ground for general methods of risk assessment.

However, these methods are also hard to understand. The mathematical difficulties interfere with other social goods such as disclosure, valuation and transparency. In particular, it is often difficult to tell if such financial instruments are “hedging" (purchasing/selling a financial instrument specifically to reduce or cancel out the risk in another investment) or “gambling" (increasing measurable risk and exposing the investor to catastrophic loss in pursuit of very high windfalls that increase expected value).

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6-7 Definitions of risk-In finance

As regret measures rarely reflect actual human risk-aversion, it is difficult to determine if the outcomes of such transactions will be satisfactory. Risk seeking describes an individual whose utility function's second derivative is positive. Such an individual would willingly (actually pay a premium to) assume all risk in the economy and is hence not likely to exist.

In financial markets, one may need to measure credit risk, information timing and source risk, probability model risk, and legal risk if there are regulatory or civil actions taken as a result of some “investor’s regret".

"A fundamental idea in finance is the relationship between risk and return. The greater the amount of risk that an investor is willing to take on, the greater the potential return. The reason for this is that investors need to be compensated for taking on additional risk."

"For example, a US Treasury bond is considered to be one of the safest investments and, when compared to a corporate bond, provides a lower rate of return. The reason for this is that a corporation is much more likely to go bankrupt than the U.S. government. Because the risk of investing in a corporate bond is higher, investors are offered a higher rate of return."

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6-8

W = 100

W1 = 150 Profit = 50

W2 = 80 Profit = -20

p = .6

1-p = .4

E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122

2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 =

.6 (150-122)2 + .4(80=122)2 = 1,176,000

Risk - Uncertain Outcomes

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6-9

W1 = 150 Profit = 50

W2 = 80 Profit = -20

p = .6

1-p = .4100

Risky Inv.

Risk Free T-bills Profit = 5

Risk Premium = 17

Risky Investments with Risk-Free Investment

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6-10 Risk aversion

Risk aversion is a concept in economics, finance, and psychology related to the behavior of consumers and investors under uncertainty. Risk aversion is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff.

The inverse of a person's risk aversion is sometimes called their risk tolerance .

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6-11 Risk aversion -Example

A person is given the choice between two scenarios, one certain and one not. In the uncertain scenario, the person is to make a gamble with an equal probability between receiving $100 or nothing. The alternative scenario is to receive a specific dollar amount with certainty (probability of 1).

Investors have different risk attitudes. A person is risk-averse if he or she would accept a certain payoff of less than $50 (for

example, $40) rather than the gamble. risk neutral if he or she is indifferent between the bet and a certain $50

payment. risk-seeking (or risk-loving) if the certain payment must be more than $50

(for example, $60) to induce him or her to take the certain option over the gamble.

The average payoff of the gamble, known as its expected value, is $50. The dollar amount accepted instead of the bet is called the certainty equivalent, and the difference between it and the expected value is called the risk premium.

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6-12

Investor’s view of risk- Risk Averse:A risk averse investor will demand

compensation for uncertainty or risk.

- Risk Neutral:A risk neutral investor will be willing to accept a fair bet or would be willing to analyze investments in terms of expected value.

- Risk Seeking:A risk seeking investor will take an unfair bet, that is, would be willing to take on an uncertain investment that has a lower expected value for the chance of securing a large profit.

Risk Aversion & Utility

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6-13 Utility

In economics, utility is a measure of the relative satisfaction from or desirability of consumption of various goods and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility. For illustrative purposes, changes in utility are sometimes expressed in units called utils.

In neoclassical economics, rationality is precisely defined in terms of imputed utility-maximizing behavior under economic constraints. As a hypothetical behavioral measure, utility does not require attribution of mental states suggested by "happiness", "satisfaction", etc.

Utility is applied by economists in such constructs as the indifference curve, which plots the combination of commodities that an individual or a society requires to maintain a given level of satisfaction. Individual utility and social utility can be construed as the dependent variable of a utility function (such as an indifference curve map) and a social welfare function respectively. When coupled with production or commodity constraints, these functions can represent Pareto efficiency, such as illustrated by Edgeworth boxes and contract curves. Such efficiency is a central concept of welfare economics.

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6-14 Cardinal and ordinal utility

Economists distinguish between cardinal utility and ordinal utility. When cardinal utility is used, the magnitude of utility differences is

treated as an ethically or behaviorally significant quantity. On the other hand, ordinal utility captures only ranking and not strength of preferences. An important example of a cardinal utility is the probability of achieving some target.

When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility values assigned encode a full behavioral ordering between members of a choice set, but nothing about strength of preferences. In the above example, it would only be possible to say that coffee is preferred to tea to water, but no more.

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6-15 Utility functions

While preferences are the conventional foundation of microeconomics, it is often convenient to represent preferences with a utility function and reason indirectly about preferences with utility functions. Let X be the consumption set, the set of all mutually-exclusive packages the consumer could conceivably consume (such as an indifference curve map without the indifference curves). The consumer's utility function ranks each package in the consumption set. If u(x) ≥ u(y), then the consumer strictly prefers x to y or is indifferent between them.

RXu :

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6-16 Risk Aversion & Utility

Utility:Utility is a measure of an investor’s welfare.

Utility Function:A function that is used to assign utility for risk and return .

U = E ( r ) - .005 A 2

A is measures the degree of risk aversion.

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6-17 What’s Your Risk Tolerance?

How to get the A’ value?

Questionary:投资问卷.doc

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6-18 Risk Aversion and Value: Using the Sample Investment

U = E ( r ) - .005 A 2

= .22 - .005 A (34%) 2

Risk Aversion A Value

High 5 -6.90

3 4.66

Low 1 16.22

T-bill = 5%

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6-19 Risk Aversion and Value: Using the Sample Investment

Examples to calculate utility using the basic investment in the text are presented. Discussion of this example is a good method of describing how the utility equation can be used to rank investments. In the sample case, a highly risk averse investor with an A value of 5, would assign a low value to the investment. If given a choice, such an investor would prefer a riskless T-bill. A less risk investor, with an A value of one would prefer the risky investment. A moderate investor, with an A value of three would be close to being indifferent between the riskless investment and the uncertain investment.

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6-20 Dominance Principle

1

2 3

4

Expected Return

Variance or Standard Deviation

The concept of dominance is also good to reinforce the types of

decisions that risk averse investors would make. • 2 dominates 1; has a higher return• 2 dominates 3; has a lower risk• 4 dominates 3; has a higher return

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6-21 Utility and Indifference Curves

The indifference curve is a curve on which all points are of the same utility value. The example shows how an indifference curve is developed using the utility function. The graph shows indifference curves with higher levels of utility.

Represent an investor’s willingness to trade-off return and risk. Example

Exp Ret St Deviation U=E ( r ) - .005A2

10 20.0 2

15 25.5 2

20 30.0 2

25 33.9 2

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6-22 Indifference Curves

Expected Return

Standard Deviation

Increasing Utility

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6-23 Expected Return

Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

s

srsPrE )()()(

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6-24 Variance of Return

Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

])()()[( 22

srEsrsP

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6-25 Return on a Portfolio

Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights.

rp = W1r1 + W2r2

W1 = Proportion of funds in Security 1

W2 = Proportion of funds in Security 2

r1 = Expected return on Security 1

r2 = Expected return on Security 2

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6-26 Portfolio Risk with Risk-Free Asset

Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.

riskyassetriskyassetp w

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6-27

Rule 5: When two risky assets with variances 12 and

22, respectively, are combined into a portfolio with

portfolio weights w1 and w2, respectively, the portfolio variance is given by:

p2

= w121

2 + w222

2 + 2W1W2 Cov(r1r2)

Cov(r1r2) = Covariance of returns for

Security 1 and Security 2

Portfolio Risk