Bosto’s expected return and standard deviation of returns for the coming year are closest to: Expected return Standard deviation B.. The probability that a continuously distributed ran
Trang 1
SELF-TEST: QUANTITATIVE METHODS
15 Questions: 22.5 Minutes
1, Allan Jabber invested $400 at the beginning of each of the last 12 months in the
shares of a mutual fund that paid no dividends Which method will he correctly
choose in order to calculate his average price per share from the monthly share
Central limit theorem Chebyshev’s inequality
C Any distribution Any distribution
Colonia has only two political parties, the Wigs and the Wags If the Wags are
elected, there is a 32% probability of a tax increase over the next four years If
the Wigs are elected, there is a 60% probability of a tax increase Based on the
current polls, there is a 20% probability that the Wags will be elected The sum
of the (unconditional) probability of a tax increase and the joint probability that
the Wigs will be elected and there will be no tax increase are closest to:
A 55%
B 70%
C 85%
Analysts at Wellborn Advisors are considering two well-diversified portfolios
based on firm forecasts of their expected returns and variance of returns James
argues that Portfolio 1 will be preferred by the client because it has a lower
coefficient of variation Samantha argues that Portfolio 2 would be preferred by
the client because it has a higher Sharpe ratio The client states that he wishes
to minimize the probability that his portfolio will produce returns less than the
risk-free rate Based on this information, the client would most likely prefer: ———
A 100% in Portfolio 1
B 100% in Portfolio 2
C some combination of Portfolios 1 and 2
Ralph will retire 15 years from today and has saved $121,000 in his investment
account for retirement He believes he will need $37,000 at the beginning of
each year for 25 years of retirement, with the first withdrawal on the day he
retires Ralph assumes that his investment account will return 8% The amount
he needs to deposit at the beginning of this year and each of the following 14
years (15 deposits in all) is closest to:
Trang 2Self-Test: Quantitative Methods
6
10
The current price of Bosto shares is €50 Over the coming year, there is a 40% probability that share returns will be 10%, a 40% probability that share returns will be 12.5%, and a 20% probability that share returns will be 30% Bosto’s expected return and standard deviation of returns for the coming year are closest to:
Expected return Standard deviation
B The probability that a continuously distributed random variable will take on
a specific value is always zero
C A normally distributed random variable divided by its standard deviation will follow a standard normal probability distribution
Market technician Christine Collies uses the Barron’s confidence index as a
“smart money” indicator and uses the CBOE put-call ratio as a contrarian indicator Given that both of these indicators have recently risen sharply, her market outlook based on each indicator is most likely:
Confidence index Put-call ratio
Given the following data:
* There is a 40% probability that the economy will be good next year and a 60% probability that it will be bad
* Ifthe economy is good, there is a 50% probability of a bull market, a 30% probability of an average market, and a 20% probability of a bear market
* Ifthe economy is bad, there is a 20% probability of a bull market, a 30% probability of an average market, and a 50% probability of a bear market The unconditional probability of a bull market is closest to:
A 20%
B 32%
C 50%
Trang 3Self-Test: Quantitative Methods
X, Y, and Z are independently distributed The probability of X is 30%,
the probability of Y is 40%, and the probability of Z is 20% Which of the
following is closest to the probability that either X or Y will occur?
A 70%
B 58%
C 12%
Which will be egual for a 1-year T-bill with 360 days to maturity?
A Bank discount yield and money market yield
B Money market yield and holding period yield
C Effective annual yield and bond equivalent yield
The percentage changes in annual earnings for a company are approximately
normally distributed with a mean of 5% and a standard deviation of 12% The
probability that the average change in earnings over the next five years will be
greater than 15.5% is closest to:
A It has a symmetric distribution
B The natural logarithms of the random variable are normally distributed
C It is a univariate distribution
A discrete random variable x can take on the values 1, 2, 3, 4, or 5 The
probability function is Prob(x) = x/15, so the cumulative distribution function is
Trang 4Self-Test: Quantitative Methods
The harmonic mean of the 12 purchase prices will be his average price paid per share
Both the central limit theorem and Chebyshev’s inequality apply to any distribution
The unconditional probability of a tax increase is: 0.2(0.32) + 0.8(0.6) = 54.4%
The joint probability that the Wigs will be elected and there will be no tax increase is:
0.8(0.4) = 32% The sum is: 54.4 + 32 = 86.4%
A portfolio that has a higher Sharpe ratio will have a lower probability of generating returns less than the risk-free rate With a target return equal to the risk-free rate, the safety-first ratio for a portfolio is (E{R,} — Ra / Op» which is also the Sharpe ratio
Portfolio 2 will have a lower probability of returns less than the risk-free rate Since both portfolios are well diversified and Portfolio 1 has a lower Sharpe ratio than Portfolio 2, any allocation to Portfolio 1 would decrease the overall portfolio’s Sharpe and safety-first ratios, increasing the probability of returns less than the risk-free rate
Step 1: Calculate the amount needed at retirement at t = 15, with calculator in BGN mode
N = 25, FV =0, UY = 8, PMT = 37,000, CPT PV = —426,564 Step 2: Reduce this by the t = 15 value of current savings
©2009 Kaplan, Inc
Trang 5Self-Test: Quantitative Methods
A An increase in the confidence index typically indicates that high-grade bond yields and
average bond yields are moving closer together, which is bullish when used as a smart
money indicator An increase in the put-call ratio indicates that options traders are
buying more puts than calls, which would be bullish when used as a contrary indicator
Using the total probability rule, rhe unconditional probability of a bull market is
0.50(0.40) + 0.20(0.60) = 32%
Probability of X or Y is P(X) + P(Y) - PQXY)
0.3 + 0.4 — (0.3)(0.4) = 58%
Since the money market yield is the holding period yield times #days / 360,
HPY x 360 / 360 = HPY = MMY
The standard error of a 5-year average of earnings changes is 2 Fg = 5.366%
15.5% is 15.575 = 1.96 standard errors above the mean, and the probability of a
5-year average more than 1.96 standard errors above the mean is 2.5% for a normal
distribution
A lognormal distribution is skewed to the right (positively skewed)
F(4) is the probability that x < 4, which is (1 +2 +344) /15 = 0.667, or 1-5 /15=
Trang 6effective annual rate = (1 + periodic rate)™ — 1
continuous compounding: e'—-1=EAR
general formula for the IRR: 0 = CKy + + pee
14+IRR (1+IRR)? (1+IRR)N
360 bank discount yield = = x——
t
PR -Pp +D, _ P,+D, _
effective annual yield = (1 + HPY)265 Ít— ]
60 money market yield = nrv | °°)
1
i=l *i
Trang 7Book 1 — Ethical and Professional Standards and Quantitative Methods
range = maximum value — minimum value
excess kurtosis = sample kurtosis — 3
joint probability: P(AB) = P(A | B) x P(B)
addition rule: P(A or B) = P(A) + P(B) — P(AB)
multiplication rule: P(A and B) = P(A) x P(B)
total probability rule:
P(R) = P(R|S,) x P(S,) + P(RI S,) x P(S,) + + P(R| Sy) x P(SQ)
expected value: E(X) = UP (x,)x, = P(x,)x, + P(x,)x,+ + Px,)x,
Cov(R,,R,) = EILR;— E(R,)]LR, — E(R))lÌ
Trang 8Book 1 — Ethical and Professional Standards and Quantitative Methods
binomial probability: p(x) = (aowixi? (1—p)” *
for a binomial random variable: E(X) = np
for a normal variable:
90% confidence interval for X is X — 1.65s to X + 1.65s 95% confidence interval for X is X — 1.96s to X + 1.96s 99% confidence interval for X is X — 2.58s to X + 2.58s
observation — population mean x—|t
So
(x2 = x1)
for a uniform distribution: P(x <X< x2) = (b )
—a sampling error ofthe mean = sample mean — population mean = x— [A
x prior probability of event
standard error of the sample mean, known population variance: Øy =—
Trang 9Book 1 — Ethical and Professional Standards and Quantitative Methods
standard error of the sample mean, unknown population variance: s- =
confidence interval: point estimate + (reliability factor x standard error)
test of mean differences = 0: t-statistic =
test for equality of means:
( —%z)-|U, -p,)
t-Statistic = (sample variances assumed unequal)
2 s2 12 3L 4 2
Trang 10APPENDIX A:
AREAS UNDER THE NORMAL CURVE
Most of the examples in this book have used one version of the z-table to find the area
under the normal curve This table provides the cumulative probabilities (or the area
under the entire curve to left of the z-value)
Probability Example Assume that the annual earnings per share (EPS) for a large sample of firms is normally distributed with a mean of $5.00 and a standard deviation of $1.50 What is the approximate probability of an observed EPS value falling between $3.00 and $7.25?
If EPS = x = $7.25, then z = (x—)/o = ($7.25 — $5.00)/$1.50 = +1.50
If EPS = x = $3.00, then z = (x— H)/ơ = ($3.00 — $5.00)/$1.50 = —1.33 Solving Using The Cumulative Z-Table
For z-value of 1.50: Use the row headed 1.5 and the column headed 0 to find the value 0.9332 This represents the area under the curve to the left of the critical value 1.50 For z-value of -1.33: Use the row headed 1.3 and the column headed 3 to find the value
0.9082 This represents the area under the curve to the left of the critical value +1.33
The area to the left of -1.33 is 1 — 0.9082 = 0.0918
The area between these critical values is 0.9332 — 0.0918 = 0.8414, or 84.14%
Hypothesis Testing - One-Tailed Test Example
A sample of a stock’s returns on 36 non-consecutive days results in a mean return of 2.0 percent Assume the population standard deviation is 20.0 percent Can we say with 95 percent confidence that the mean return is greater than zero percent?
Ho: b < 0.0%, H,: p> 0.0% The test statistic = z-statistic = X= Ho (2.0—0.0) /
The significance level = 1.0 — 0.95 = 0.05, or 5% Since we are Interested in a return
greater than 0.0 percent, this is a one-tailed test
Using The Cumulative Z-Table Since this is a one-tailed test with an alpha of 0.05, we need to find the value 0.95 in the cumulative z-table The closest value is 0.9505, with a corresponding critical z-value of 1.65 Since the test statistic is less than the critical value, we fail to reject Ho
Trang 11Hypothesis Testing — Two-Tailed Test Example
Using the same assumptions as before, suppose that the analyst now wants to determine
if he can say with 99% confidence that the stock’s return is not equal to 0.0 percent
Ho: p = 0.0%, H,: u # 0.0% The test statistic (z-value) = (2.0 — 0.0) / (20.0 / 6) = 0.60
The significance level = 1.0-0.99 = 0.01, or 1% Since we are interested in whether or
not the stock return is nonzero, this is a two-tailed test
Using The Cumulative Z-Table
Since this is a two-tailed test with an alpha of 0.01, there is a 0.005 rejection region in
both tails Thus, we need to find the value 0.995 (1.0 — 0.005) in the table The closest
value is 0.9951, which corresponds to a critical z-value of 2.58 Since the test statistic is
less than the critical value, we fail to reject Hy and conclude that the stock’s return
equals 0.0 percent
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0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 | 0.7157 | 0.7190 0.7224 | 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 | 0.7454 | 0.7486 | 0.7517 0.7549 | 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 | 0.7764 | 0.7794 | 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 | 0.8315 | 0.8340 0.8365 0.8389
1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 | 0.8554 | 0.8577 0.8599 | 0.8621 1.1 | 0.8643 | 0.8665 | 0.8686 | 0.8708 | 0.8729 | 0.8749 | 0.8770 | 0.8790 | 0.8810 | 0.8830 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015 | 1.3 | 09032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 | 0.9131 | 0.9147 | 0.9162 [| 0.9177 _| [1⁄4 | 09192 | 0.9207 | 0.9222 | 0.9236 | 0.9251 | 0.9265 | 0.9279 | 0.9292 | 0.9306 | 0.9319 |
1.5 | 0.9332 | 0.9345 | 0.9357 | 0.9370 | 0.9382 | 0.9394 | 0.9406 | 0.9418 | 0.9429 | 0.9441 | 1.6 | 0.9452 | 0.9463 | 0.9474 | 0.9484 | 0.9495 | 0.9505 | 0.9515 | 0.9525 | 0.9535 | 0.9545 | 1.7 | 0.9554 | 0.9564 | 0.9573 | 0.9582 | 0.9591 | 0.9599 | 0.9608 | 0.9616 | 0.9625 | 0.9633 | 1.8 | 0.9641 | 0.9649 | 0.9656 | 0.9664 | 0.9671 | 0.9678 | 0.9686 | 0.9693 | 0.9699 | 0.9706 | 1.9 | 09713 | 0.9719 | 0.9726 | 0.9732 | 0.9738 | 0.9744 | 0.9750 | 0.9756 | 0.9761 | 0.9767 |
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 | 0.9803 | 0.9808 | 0.9812 0.9817 | 2.1 0.9821 0.9826 0.9830 0.9834 | 0.9838 0.9842 | 0.9846 | 0.9850 0.9854 0.9857 _| 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 | 0.9887 0.9890 | 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 | 0.9909 | 0.9911 0.9913 0.9916
| 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 | 0.9934 0.9936
| 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 | 0.9948 | 0.9949 | 0.9951 0.9952 | 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 | 0.9961 0.9962 | 0.9963 0.9964 | 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 | 0.9971 0.9972 | 0.9973 0.9974 | 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 | 0.9979 | 0.9979 | 0.9980 0.9981 | 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 | 0.9985 | 0.9985 0.9986 | 0.9986 | 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 | 0.9989 | 0.9989 | 0.9990 0.9990 |
Trang 13
CUMULATIVE Z-ÏABLE (CONT.) :
STANDARD NORMAL DISTRIBUTION
Trang 14APPENDIX B:
STUDENT'S f-DISTRIBUTION
Trang 15
APPENDIX C:
F-TABLE aT 5 PERCENT (Upper TAIL)
F-TABLE, CRITICAL VALUES, 5 PERCENT IN UPPER TAIL
Degrees of freedom for the numerator along top row Degrees of freedom for the denominator along side row
Trang 16APPENDIX D:
F-TABLE AT 2.5 PERCENT (UPPER TAIL)
F-TsBLE, CriTICAL VALUES, 2.5 PERCENT IN PPER TAILS
Degrees of freedom for the numerator along top row
Degrees of freedom for the denominator along side row
6 | 8.81 | 7.26 | 6.60 | 6.23 | 5.99 | 5.82 | 5.70 | 5.60 | 5.52 | 5.46 | 5.37 | 5.27 | 5.17 | 5.12 | 5.07 | 5.01 7_| 8.07 | 6.54 | 5.89 | 5.52 | 5.29 | 5.12 | 4.99 | 4.90 | 4.82 | 4.76 | 4.67 | 4.57 | 4.47 | 4.41 | 4.36 | 4.31
14 | 6.30 | 4.86 | 4.24 | 3.89 | 3.66 | 3.50 | 3.38 | 3.29 | 3.21 | 3.15 | 3.05 | 2.95 | 2.84 | 2.79 | 2.73 | 2.67
15 | 6.20 | 4.77 | 4.15 | 3.80 | 3.58 | 3.41 | 3.29 | 3.20 | 3.12 | 3.06 | 2.96 | 2.86 | 2.76 | 2.70 | 2.64 | 2.59
_
16 | 6.12 | 4.69 | 4.08 | 3.73 | 3.50 | 3.34 | 3.22 | 3.12 | 3.05 | 2.99 | 2.89 | 2.79 | 2.68 | 2.63 | 2.57 | 2.51 17_| 6.04 | 4.62 | 4.01 | 3.66 | 3.44 | 3.28 | 3.16 | 3.06 | 2.98 | 2.92 | 2.82 | 2.72 | 2.62 | 2.56 | 2.50 | 2.44
Trang 17APPENDIX E:
CHI-SQUARED TABLE
Values of x7 (Degrees of Freedom, Level of Significance)
Probability in Right Tail
Trang 18
INDEX
A absolute frequency 159 additional compensation arrangements 47
addition rule of probability 196, 198
additivity principle 123 advance-decline line 341 alternative hypothesis 297 amortization 114
CD equivalent yield 145 central limit theorem 275 Chebyshev’s inequality 173 chi-square distribution 319 Code of Ethics 13
coefficient of variation 174 combination formula 218 communication with clients 53 composites 70, 74
compounding frequency 112 compound interest 95, 96 conditional probability 196 conduct as members and candidates 65 confidence index 341
confidence interval 249, 278 confidence interval for the population mean 281
consistent estimator 279 continuous compounding 257
continuous distribution 239 continuous random variable 238
continuous uniform distribution 246 contrarian view 339
cumulative relative frequency 161
D
data mining 285 debit balances in brokerage accounts 341 decile 168
decision rule 298, 303 default risk premium 98 degrees of freedom 279 descriptive statistics 157 desirable properties of an estimator 278 diligence and reasonable basis 50 disclosure of conflicts 57 discount factor 101 discounting 96, 101 discount rate 97, 101, 119 discrete distribution 239 discretely compounded returns 257 discrete random variable 238 discrete uniform random variable 241 dispersion 170
distribution function 240 down transition probability 244
E
effective annual rate (EAR) 98
effective annual yield (EAY) 144 efficient estimator 279
efficient market hypothesis (EMH) 337 empirical probability 195
equality of population means 312 equality of variances 321
event 194
excess kurtosis 178 excess return 175 exhaustive events 194 expected value 202
expected value and variance for a portfolio of assets 210
of a binomial random variable 243