CFA Level 1 (2009) - 1

Study Session 2

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

Answer:

This is the simplest type of example because the most tedious calculations have already been performed. Simply extract the appropriate values from the covariance matrix and insert them into the variance formula.

Recall that the covariance of an asset with itself is its variance. Thus, the terms along the diagonal in the covariance matrix are return variances.

The portfolio return variance can be computed as:

Var(Rp) = (0.90)2(0.0011) + (0.10)2(0.016) + 2(0.90)(0.10)(-0.0036)

= 0.000403

LOS R.m: Calculate and interpret an updated probability, using Bayes' formula.

Bayes' formula is used to update a given set of prior probabilities for a given event in response to the arrival of new information. The rule for updating prior probability of an event IS:

x prior probability of event

Note in the following example of the application of Bayes' formula that we can essentially reverse a given set of conditional probabilities. This means that given P(B), P(A I B), and P(A I BC), it is possible to use Bayes' formula to compute P(B I A).

Example: Bayes' formula

Electcomp Corporation manufactures electronic components for computers and other devices, There is speculation that Electcomp is about to announce a major expansion into overseas markets. The expansion will occur, however, only if Electcomp's managers estimate overseas demand to be sufficient to support the necessary sales. Furthermore, if demand is sufficient and overseas expansion occurs, Electcomp is likely to raise its prices.

Using 0 to represent the event of overseas expansion, I to represent a price increase, and ;C to represent no price increase, an industry analyst has estimated the unconditional and conditional probabilities shown as follows:

Study Session 2

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

~'~Wh~::;n#y~~'sestimates for P(I) and P(Ic) are called the priors. because they reflect .

:'·~RP1.r~~tr6n,6fBayes',forinulaallowsustocompute-·PJI·I:O);the:probability that. ,

1"p49~s;~iM~i~crease'giventhat Electcornp::anllou~c-~sthat itwill expa.-nd overseas (the ...• i·-i£~W~~~fbrihatiOri).Using:the·.multiplkation.nue;.we-canexpress thejoirit··probabiliry···.

(:,fiorfi~ihisexairiple~asindicatedbelow [I.e., substitute me second equation into the'

Inorde! to solve this equation, P(O) must be determined. This can be done using the total probability rule:

P(O) = P(O I1) x P(I) + P(O I IC) x P(Ic)

P(O) = (0.6 x 0.3) + (0.4 x 0.7)

P(O) = 0.46

. Now the updated probability of the increase in prices given that Electcomp expands overseas can be computed:

. This\:Jneansthatifthe new information of "expand overseas" is announced, the prior:

.. prdbabillty estimate ofP(I) = 0.30 must be increased to 0.3913.

Study Session 2

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

Example: Bayes' formula

":,,",

. Anodler.illustration,()f.the use of Bayes' formula n1afIT1akeit~easierto'-re~emb~r.e~Tii;

and apply. 'Considerthe following possibilities: The~ds a 600/0:prob:iBilitythe ';:}:\:,':'

·~conomrwill outperform,. and if it does, there is a}OO(ochance a stocKwiH'betip"~>:,;:,::

. ~nda~Q%chance the·stock will go down. rhereis_a40%chancetheeconomy'wi4':::\~<:

und~~perform, and ifit;does, there is a 20% 'chance the.,stock)n,q\lestion witLincr~~.li;§~.;

'w ·vaIliejhave gains)~:d an 80% chanceit will not. :L~tis.diagr~ thls:situation,.,.t;fCb~.:

::e?~:~\y'

42% (outperform + gains)

" "- .-~

18% (outperform + no gains)

8% (underperform + gains)

32% (underperform + no gains)

In the figure above, we have multiplied the probabilities to calculate the probabilities of each of the four outcome pairs. Note thanhesesum to 1. Given thatthe stock has gains, what is our updated probability of an outperforming economy? We sum the probability of stock gains in both states (outperformandunderperform) tbge'i:

42% +8% = 50%. Given that the stock hasgains,Lheprobability that the economy.: .

economic outperformance =P(O) = 60%, the probability of stock gains given economic outperformance is P(G I0) = 70%, and the (unconditional) probability of a gain in stock price is 50%.

We are seeking P(O I G), the probability of outperformance given gains. Bayes' form ula says:

Study Session 2

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

LOS B.n: Identify the most appropriate method to solve a particular counting problem, and solve counting problems using the factorial, combination, and permutation notations.

Labeling refers to the situation where there are n items that can each receive one of k different labels. The number of items that receives label 1 is n j and the number that receive label 2 is n2 , and so on such that n\ + nz + n3 + ... + nk = n. The total number of ways that the labels can be assigned is:

n!

(nt !)x(nz !)x ... x(nk!)

where:

the symbol "!" stands for factorial. For example, 4! = 4 x 3 x 2 x 1 = 24, and 2! = 2 x

The general expression for n factorial is:

n! = 11 X (n - I) x (11 - 2) x (n - 3) x .. , xl, where by ddll1ition, ()! = 1

Calculator help: On the TI, factorial is [2nd] [x!] (above the multiplication sign). On the HP, factorial is [g] [n!]. To compute 4! on [he Tl. enter [4J[2ndJ[x!] = 24. On the HE pres.; [4HENTER][gHn!].

Example: Labeling

Consider a portfolio consisting of eight stocks, Your goal is ro designate four of the stocks as "long-term holds," three of the stocks as "short-term holds," and one stock as "sell." How many ways can these eight stocks be labeled?

Answer:

There are 8! = 40,320 total possible sequences that can be followed to assign the three labels to the eight stocks. However, the order that each srock is assigned a label does not matter. For example, it does not matter which of the first three stocks labeled "long-term" is the first to be labeled. Thus, there are 4! ways to assign the long-term label. Continuing this reasoning to the other categories, there are 4! x 3! x 1! equivalent sequences for assigning the labels. To eliminate the counting of these

redundant sequences, the total number of possible sequences (8!) must be divided by the number of redundant sequences (4! x 3! x I!).

Thus, the number of different ways to label the eight stocks is:

Study Session 2

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

n J and n2 = n - r. Since there are only twO categories, we usually talk about choosing r items. Then (n - r) are not chosen. The general formula for labeling when k = 2 is cal1ed the combination formula (or binomial ftrmula) and is expressed as:

where nCr is the number of possible ways (combinations) of selecting r items from a set

of 12 items when the order of selection is not important. This is also written (n) and read

Another useful formula is the permutation formula. A permutation is a specific ordering of a group of objects. The question of how many different groups of size r in specific order can be chosen from n objects is answered by the permutation formula. The

current policy permits you to bring both calculators to the exam ifyou choose.

Example: Number of choices in any order

How many ways can 3 stocks be sold from an 8-srock portfolio?

Answer:

This is similar to the preceding labeling example. Since order does not matter, we take the total number of possible ways to select three of the eight stocks and divide by the number of possible redundant selections. Thus, the answer is:

_8_!_= 56

5! x 3!

In the preceding two examples, ordering did not matter. The order of selection could, however, be important. For example, suppose we want to liquidate only one srock position per week over the next three weeks. Once we choose three particular stocks to

Study Session 2

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

sell, the order in which they are sold must be determined. In this case, the concept of permutation comes into play. The permutation formula is:

where nPr is the number of possible ways (permutations) to select r items from a set of 11 items when the order of selection is important. The permutation formula implies that

there are r! more ways to choose r items if the order of selection is important than if order is not important.

Example: Permutation

How many ways are there to sell three stocks out of eight if the order of the sales is important?

This is 3! times the 56 possible combinations computed in the preceding example for selecting the three stocks when the order was not important.

There are five guidelines that may be used to determine which counting method to employ when dealing with counting problems:

•The multiplication rule of counting is used when there are tlUO or more groups. The

key is that only one item may be selected from each group. If there are k steps required to complete a task and each step can be done in n ways, the number of different ways to complete the task is n J! x n2 l x ... x nk!'

• Factorial is used by itself when there are no groups-we are only arranging a given set of n items. Given n items, there are n! ways of arranging them.

•The labelingformula applies to three or more sub-groups of predetermined size. Each element of the entire group must be assigned a place, or labeL in one of the three or more sub-groups.

•The combination formula applies to only tlUO groups of predetermined size. Look for the word "choose" or "combination."

•The permutation formula applies to only two groups of predetermined size. Look for a specific reference to "order" being important.

Study Session 2

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

KEy CONCEPTS

LOS 8.a

A random variable is an uncertain value determined by chance.

An outcome is the realization of a random variable.

An event is a set of one or more outcomes. Two events that cannot both occur are termed "mutually exclusive" and a set of events that includes all possible outcomes is an "exhaustive" set of events.

LOS 8.b

The two properrics of probability arc:

The sum of the probabilities of all possible mutually exclusive events is ].

• The probability of any evenr cannot be greater than 1 or less than O.

A priori probability mcasures probabilities based on well-defined inpurs; empirical probability measures probability from observations or experimenrs: and subjective probahility is an informed guess.

LOS S.c

If the probability of an event is A out of B rrials (A/B). the "odds for" are A to

(R - A) and the "odds against" arc (R - A) to A.

LOS B.d

Unconditional probability (marginal probability) is the probability of an event occurring; conditional probability. P(AIB), is the probability of an event (A) occurring given that another event (B) has occurred.

LOS B.e

The joint probability of two events, P(AB), is the probability that they will both occur. P(AB) = P(AIB) x P(B). For independent events, P(A I B) = P(A) so that P(AR) =

P(A) x P(B).

The probability that at least one of twO events will occur is P(A or B) = P(A) + P(B) -

P(AB). For mutually exclusive events, P(A or B) = P(A) + P(B), since P(AB) = O.

The joint probability of any number of independent events is the product of their individual probabilities.

LOS 8.[

The probability of an independent event is unaffected by the occurrence of other events, bu t the probability of a dependent event is changed by the occurrence of another event.

Study Session 2

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

LOS 8.g

Using the total probability rule, the unconditional probability of A is the probability weighted sum of the conditional probabilities:

n

P(A) = I:[Pj (B j )] x P(A IBj) , where Bj is a set of mutually exclusive and

j=1

exhaustive events.

LOS 8.h

Conditional expectations are used in investments to update expectations when a conditioning event has occurred.

LOS 8.i

A tree diagram shows the probabilities of tWO events and the conditional probabilities of twO subsequent events.

A

Be wirh Prob (BC)

B

BD with Prob (BD)

LOS 8.j

Covariance measures the extent to which two random variables tend to be above and below their respective means for each joint realization. It can be calculated as:

N

Cov(A,B) = I:Pj (Ai - A)(Bi - B)

i=1

Correlation is a standardized measure of association between two random variables; it

i=1

Standard deviation, ax = g .

The expected returns and variance of a 2-asset portfolio are given by:

E(Rp )

Var(R p )

Study Session 2

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

LOS 8.1

Given the J'oint probabilities for X. and Y, i.e., P(XY), the covariance is calculated as:

n

L:P(XjYj)(X j -X)(Yj - V)

i=\

LOS 8.m

Bayes' formula for updating probabilities based on the occurrence of an event 0 is:

P(I I0) = p(O II) xP(I).

P(O)

P(AC)

Note that based on the tree diagram below, P(A IC) =

P(AC) + P(BC)

AC with Prob (AC)

A

AD with Prob (AD)

/~c with Proh (BCl

BD with I'roh (HO)

L02, 6.n

The number of ways to order 71 ohjects is 71 factoriaL n! = n x (n -1) x (n - 2) x .. ' xl.

N!

Study Session 2

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

CONCEPT CHECKERS

1.Given the conditional probabilities in the cable below and the unconditional

probabilities P(Y = 1) = 0.3 and P(Y = 2) = 0.7, what is the expeered value of X?

Use the following data to answer Questions 2 through 6.

Joint Probabilities

4. Given the joint probability table, the variance of Stock A is closest to:

A.0.03.

B.0.12.

e. 0.17.

5.Given the joint probability table, the covariance between A and B is closest to: A. -0.160.

B.-0.055. e. 0.004.

6.Given the joint probability table, the correlation between RA and RB is closest to:

A.-0.99.

B.0.02.

e. 0.86.