CFA Level 1 (2009) - 1

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

7.The probability chat the DJIA will increase tomorrow is 2/3. The probability of an increase in the DJIA stated as odds is:

A.two-to-one.

B.one-to-three.

C.two-to-three.

8.A discrete uniform distribution (each event has an equal probability of occurrence) has the following possible outcomes for X [1, 2, 3, 4]. The variance of this distribution is closest to:

A.1.00.

B.1.25.

C.2.00.

9.If events A and B are mutually exclusive, then:

A.P(A IB) = P(A).

B.P(AB) = P(A) x P(B).

C.peA or B) = peA) + P(B).

I 1. Among 900 taxpayers with incomes below $100,000, 35 were audited by the IRS. The probability that a randomly chosen individual wirh an income below $100.000 was audited is closest ro:

A.0.039.

B.0.125.

C.0.350.

12.Which of the following values cannot be the probability of an event?

A.0.00.

B.1.00.

e.1.25.

13.Two mutually exclusive events:

A.always occur together.

B.cannot occur together.

C.can sometimes occur together.

14. 1,-vo events are said to be independent if the occurrence of one event:

A.means that the second event cannor occur.

B.means rhat the second event is certain to occur.

C. does not affect the probabili ty of the occurrence of the other event.

Srudy Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

Use the following conditional probabilities to answer Questions 15 through 18.

15.What is the conditional probability of having good srock performance in a poor economic environlllcnt?

A.0.02.

B.0.10.

e.0.30.

16.What is the joint prohability of having a good economy and a neutral srock performance?

A.0.09.

B.0.20.

e.0.30.

17. What is the total probability of having a good performance in the stock?

A.0.35. B. 0.65.

e.1.00.

18.Given that the stock had good performance, the probability the state of the economy was good is closest to:

A. 0.35.

B. 0.46.

e.0.51.

19.Consider a universe of ten bonds from which an investor will ultimately purchase six bonds for his portfolio. If the order in which he buys these bonds is not important, how many potential 6-bond combinations arc there?

A. 7. B. 210.

e.5,040.

Study Session 2

Cros~-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

20.The correlation of returns between Stocks A and B is 0.50. The covariance between these two securities is 0.0043, and the standard deviation of the return of Stock B is 26%. The variance of returns for Stock A is:

A.0.0011.

B.0.0331.

C.0.2656.

21.There are ten sprinters in the finals of a race. How many different ways can the gold, silver, and bronze medals be awarded?

A.120.

B.720.

C.1,440.

22.Which of the following is least likely a probability distribution?

COMPREHENSIVE PROBLEMS

1.Given the following probability data for the return on the market and the return on Best Oil, calculate the covariance of returns between Best Oil and the marker.

Joint Probability Table

2.The correlation of returns between the returns on Cape Products and Dogger Industries is 0.6. The standard deviation of returns for Cape is 15% and the standard deviation of returns for Dogger is 20%. The expected return for Dogger is 18% and the expected return for Cape is 12%. Calculate the expected returns and standard deviation of returns on a portfolio that has $300,000 invested in Dogger and $200,000 invested in Cape.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

3.M. Atwood, an analyst, has developed a scoring system for bonds and found that if the score from a bond is less than 20, there is a probability of 85% that it will default within five years. If a bond's score is greater than or equal co 20, there is

only a 40% chance that it will defaulr within five years. Given that a randomly chosen bond currendy has a 25% probability of a score less than 20, what is the probability that a bond that defaults within the next five years had a score of 20 or higher?

4.A bond that matures in one year is priced at $950 coday. You estimate that it has

a 10% probability of defaulr. If the bond defaults, you expect co recover $600. If it does not default, it will pay $1,080 at maturity. The nominal I-year risk-free rate is 7.5%.

A. What are the odds against this bond defaulring?

B. \X'hat is the expecred paYoff on the bond in one year?

C.What is the expected rerurn on the bond?

D.What would be the price of the bond if its expected return were equal co the risk-free rate?

Srudy Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

5. You are considering a portfolio of three stocks:

•Stock A (55% of the portfolio) has an expected return of 8% with a standard deviation of 24%.

•Stock B (25% of the portfolio) has an expected return of 4% with a standard deviation of 18%.

•Stock C (20% of the portfolio) has an expected return of 3% with a standard deviation of 15%.

The correlations between these stocks' returns are:

•Stock A with Stock B: 0.85.

•Stock A with Stock C: 0.30.

•Stock B with Stock C: -0.15.

A.Based on these data, construct a covariance matrix for the returns on the three stocks.

B. Calculate the expected return and standard deviation of the portfolio.

C. Provide a set of three mutually exclusive and exhaustive events with respect to the relation between this portfolio's realized return and its expected return.

D.If you add three more stocks to the portfolio, how many variances and how many unique covariances will you need to calculate the portfolio variance?

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

6.You are forecasting the sales of a building materials supplier by assessing the expansion plans of its largest customer, a homebuilder. You estimate the

probability that the customer will increase its orders for building materials to 25%. If the customer does increase its orders, you estimate the probability that the homebuilder will start a new development at 70%. If the customer does not increase its orders from this supplier, you estimate only a 20% chance that it will start the new development. Later you find out that the homebuilder will start the new development. In light of this new information, what is your new

(updated) probability that the builder will increase its orders from this supplier?

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

ANSWERS - CONCEPT CHECKERS

1.B E(X!Y = I) = (0.2)(0) + (0.4)(5) + (0.4)(10) = 6 and E(XIY = 2) = (0.1 )(0) + (0.8)(5) + (0.1)(10) = 5

E(X) = (0.3)(6) + (0.7)(5) = 5.30

2. A Expecred rerum of Srock A = (0.4)(-0.1) + (0.3)(0.1) + (0.3)(0.3) = 0.08

3.C Expecred return of Srock B = (0.4)(0.5) + (0.3)(0.2) + (0.3)(-0.3) = 0.17

Var(RB) = 0.4(0.5 - 0.17)2 + 0.3(0.2 - 0.17)2 + 0.3(-0.3 - 0.17)2 = 0.1101

Srandard deviarion = .JQ.l101 = 0.3318

4.A E(R.•) = 0.08 (from #2 above)

VariR) = 0.4(-0.1 - 0.08)2 + 0.3(0.1 - 0.08)2 + 0..'(0.3 - n.oS)2 = 0.0276

). B C:ov(RA.IZ n) = O.LJ(-O.l - 0.08)(0. ') - 0.17) + 0.3(0.1 - 0.(8)(0.2 - 0.17) + 0.3(0.3 -

0.(8)( -0.3 - 0.17) = -0.0546

Cov(RA,RB) = -0.0546 (from #5 above)

Var(R,,) = 0.0276 (from #4 above); a(RA) = JO.0276 = 0.1661

a(R B) = 0.3318 (from #3 above)

Corr(RA,RB) = -0.0546 / (0.1661)(0.3318) = -0.9907

1/3

8.B Expecred value = (1/4)(1 + 2 + 3 + 4) = 2.5

Variance = (1/4)[(1 - 2.5)2 + (2 - 2.5)2 + (3 - 2.5)2 + (4 - 2.5)2]= 1.25

Note that since each observation is equally likely, each has 25% (l/4) chance of occurrence.

9.C There is no intersecrion of events when events are mutually exclusive. P(AIB) = P(A) x P(B) is only true for independent events. Note rhat since A and B are mutually exclusive (cannot borh happen), P(AIB) and P(AB) musr both be equal to zero.

10.B P(name 1 or name 2 or name 3 or name 4) = 1/800 + 11800 + 1/800 + 11800 = 4/800 = 0.005

11.A 35 /900 = 0.0389

12.C Probabilities may range from 0 (meaning no chance of occurrence) rhrough 1 (which means a sure thing).

13.B One or the other may occur, not borh.

. Study Session 2

,Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

14.C Two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other event.

15.B Go to the poor state and read off the probabiliry of good performance

[i.e., P(poor performance Igood economy) = 0.10].

16.A P(good economy and neutral performance) = P(good economy)P(neutral performance I good economy) = (0.3)(0.3) = 0.09.

17. A (0.3)(0.6) = 0.35. This is the sum of all the joint probabilities for good performance over all states [i.e., L:P(economic statej) P(good performance I economic statej)].

18.C This is an application of Bayes' formula. P(good economy Igood performance) = P(good stock performance Igood economy) x P(good economy)/P(good stock

performance).

19.B

20.A

21.B Since the order of the top three finishers malters, we need to lise the pcrmulJ1'ion formula.

for a probability distribution. The others have P[Xj] be1'ween zero and one and LP[X;J = 1, and thus sa1'isfy both requirements for a probabili1'Y dis1'ribu1'ion.

ANSWERS - COMPREHENSIVE PROBLEMS

1.E(RBesr) = 0.4(20%) + 0.2(10%) + 0.4(5%) = 12%

E(RMkr) = 0.4(15%) + 0.2(10%) + 0.4(0%) = 8%

Cov(RBesr' RMkr)= 0.4(20% - 12%)(15% - 8%)

+0.2(10% - 12%)(10% - 8%)

+0.4(5% - 12%)(0% - 8%)

0.4(8)(7) + 0.2(-2)(2) + 0.4(-7)(-8) = 44

30% ( c 20, default)

3.75% ( < 20, no default)

21.25% ( < 20, default)

Total probability of default = 30% + 21.25% = 51.25%. Percenr of defaulting bonds

1-0.10 9

bond defaulting are the reciprocal, or 9 to 1.

B.The expected payoff on the bond at maturity is:

P(default) x bond value if it defaults + P(no default) x bond value if it docs not default = 0.1 (600) + 0.9(1,080) = 60 + 972 = $1,032.

C. The expected return is 1.032 I 950 - 1 = 0.0863, or 8.63%.

D.1,0.32 -1 = 0.075 so price would need to be 1,032 = $960.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #8 - Probability Concepts

5.A. First calculate the variances on each of the three stocks: Var(A) = (0.24)2 = 0.0576

Var(B) = (0.18) 2 = 0.0324

Var(C) = (0.15) 2 = 0.0225

These will be the diagonal entries in the covariance matrix:

Figure 1: Covariance Matrix of Returns for Stocks A, B, and C

Next calculate the covariance for each pair of stocks. The correlation (Pxyl =

Cov(x,y)lc\ay" Rearranging that, we get Cov(x,y) = r\y (J"/\,. So:

Cov(A,B) = 0.85 x 0.24 x 0.18 = 0.0367

Cov(A,C) = 0.30 x 0.24 x 0.15 = 0.Ql 08

Cov(B.C) = -0.15 x 0.18 x 0.15 = -0.0041

These results complete the covariance matrix:

Figure 2: Covariance Matrix of Returns for Stocks A, B, and C

B.The expected rerum on the portfolio is a weighted average of the individual stock returns:

E[Rp] = 0.55(0.08) + 0.25(0.04) + 0.20(0.03) = 0.06 or 6% For a 3-asset portfolio, the portfolio variance is calculated as:

Var(Rp) = WA 2a 2(RA) + W B2a 2 (RB) + W e2a2 (Rc) + 2WAW BCov(RA ,RB) + 2WAWcCov(RN Rc) + 2WBW eCov(RB,Rc)