## Quadratic Utility

The quadratic utility function can be defined as U(x) = ax — \bx2, where a > 0 and b > 0. This function is shown in Figure 9.6.

This utility function is really meaningful only in the range ,v < a/b, for it is in this range that the function is increasing. Note also that for b > 0 the function is strictly concave everywhere and thus exhibits risk aversion

We assume that all random variables of interest lie in the feasible range ..v < a/b; that is, within the meaningful range of the quadratic utility function.

Tilts Risk Quiz" is intended as a starting point at sessions between a client and a financial planner to help evaluate your tolerance for risk li shouldn't be used to make specific investment decisions Even if you haven't experienced a specific situation addressed here, answer based on what you think your decision would be if you faced the issue today

1 My salary ami overall earnings from my job arc Mkely to grow significantly in ¡he coming years a) Disagree siroii|',ty to) Dis.TfjreL'

2 If I were deciding how to invest contributions in my retirement plan, I would choose investments that offered fixed yields and stability a) Agree strongly h) Agiee cl Neither agree nor disagree d> Disagree c>) Disnniee strongly

3 t believe investing in today's volatile stock market is like spinning a roulette wheel in Las Vegas—the odds are against you

,i) Agree strongly b) Agree

C) Neither agree nor disagree d) Disagree e) Disagree strongly

4. If I were picking a stock lo invest in, t would look for companies that are involved in developing the hot products of the future, such as the next penicillin a) Disagree strongly b) Disagree c) Neither agree nor disagree (I) Agree e) Agree strongly

5 If I were selecting a» investment for my child's college education fund, t would chooser nl Certificate of (i»i|H>sit b) Government-hacked mortgage securities or municipal bonds c) Corporate bonds fl) Slocks equity mutual funds el Commodities it times contrasts

6 The following number of dependents rely on me for (heir financial welfare a I four or more b)Three <:) Two (I) One e) Only myself

7 The number of years remaining until I expect to retire is approximately: at Currently reiired b) less than 5 years e) 3-1-f years d) 15-24 years e) 25 or more fl My total nel worth (the value of my assets less my debts) is:

at UnderS15.000

c)550.00l-S150,000

'J. Tlte amount I have saved !o handle emergencies, such ns a job loss or unexpected medical expenses, equates to:

a) One month's salary Or less b) Two lo sty months' salary c) Seven months' to one year's salary d) One to two years salary o) Mote than two years' salary

10 I would rather invest in a stock mutual fund than buy individual slocks because a mutual fund provides professional management and diversification.

a) Agree shortly b) Agree e| Neither agree nor disagree d> Disagree e) Disagree strongly

11 I want and need lo reduce the overall level of debt in my personal finances nl Agree stron;;ly b| A ¡¡ret*

c| Neither agr;;e nor disagree (¡1 Disagree o) Disagree strongly

12 When making investments, I am willing to settle for a lower yield if it is guaranteed, as opposed lo higher yields that arc less certain ai Strongly agree b} Agree c) Neither agree nor disagree d) Disagree e) Strongly disagree

### Scoring System

SCORING: Give yourself one point for every a" answer, two points for every I),' three points for every "c." four points far every "d" and five points for every "e "

4G AND HIGHER: You probably have the money and the inclination to take risks High-risk investments include growth slocks, start-up companies, commodities, junk bonds and fimited partnerships, as well as stock options and investment real estate Uul be sure lo diversify at teasl some of your portfolio into safer investments Even you could lose everything and regret your higlwtsk tolerance

41—15: You have an above-average tolerance for risk and probably enough time and income lo cover your losses Investors in Ihis category are wise to mix high-risk and low-risk options 36-40; You have an average tolerance for risk, bui don't like to gamble Consider a mix of long-term investments thai have a history of strong and steady performance Blue chip stocks, high-grade corporate bonds, mutual funds and real estate are all possible options

31-35: You have below-average tolerance tor risk, either because of your age or your income and family circumstances Comfortabfe investments for you would probably include your home, high-quality bonds, government-backed securities and federally insured savings accounts 30 and below: You have virtually no tolerance for risk Look for investments that have government backing, such as bank and thrift certificates of deposit Treasury bills, bonds and notes

FIGURE 9.5 Risk quiz. An investor's attitude toward risk and toward type of investment might be inferred from responses to a questionnaire such as this one Source: Fidelity Investments, 1991 Developed in association with Andrew Comrey, Ph D , Professor of Psychology. University of California at Los Angeles

9.5 UTILITY FUNCTIONS AND THE MEAN-VARIANCE CRITERION* 2,39

U f FIGURE 9.6 Quadratic utility function. This function is meaningful as a utility function only for < a/b

Suppose that a portfolio has a random wealth value of y, Using the expected utility criterion we evaluate the portfolio using the value

The optimal portfolio is the one that maximizes this value with respect to all feasible choices of the random wealth variable y,.

This can be seen to be equivalent to a mean-variance approach First, for convenience, suppose that the initial wealth is I Then y corresponds exactly to the return R. Suppose also that the solution has an expected value E(y) = M Then clearly, y must have minimum variance with respect to all feasible y's with E(y) = M ~ 1 +m (where m is the mean rate of return) Since y — R, it follows that the solution must correspond to a mean-variance efficient point,

Different mean-variance efficient points are obtained by selecting different values for the parameters a and b. Likewise, if the initial wealth is not I, a different factor is introduced, (See Exercise 5 )

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