Quantitative Methods - Probability (Schweser Version)

Start - 11:00 AM

Based on yesterday's session and the pain that was CFAI probability, I think that a recap of probability using Schweser is in order.  Scanning over the Schweser notes it seems like they do a much better job explaining the concepts so I am going to try an quickly do the Schweser for the same Reading #8 that we did yesterday.  This will also be useful for getting an idea of how Schweser compares to CFAI in terms of depth, content, and difficulty.

Intro

• Random variable - uncertain quantity/number (number that comes up on a dice)
• Outcome - observed value of a random variable (if you roll a 4)
• Event - single outcome or set of outcomes (rolling a 4)
• Mutually exclusive - events that can't both happen at same time (rolling a 4 and rolling a 6)
• Exhaustive events - those that include all possibilities (rolling an even and rolling an odd)

Probability P(Ei) for any event Ei must be 0<=P(Ei)<=1
If events are mutually exclusive and exhaustive, sum of all probabilities = 1

Types of Probability

• Empirical - established on past data (DJIA closes higher than previous close 2/3 days => future probability is 2/3)
• A priori - using a formal reasoning process (yesterday 24/30 stocks rose; if you pick on at random there is a 24/30 chance its value increased yesterday)
• Subjective - based on personal feeling

Odds For and Against

• "Odds" that an event will occur are P / (1-P) = the p it will occur to the p it won't occur
• Given a set of odds a/b, to calculate the P, it is a/(a+b)
• E.g. if odds of winning are 1 to 7, P(winning) = 1 / (1+7) = 1/8 or 12.5%

Unconditional/Conditional

• Unconditional  Probability - probability of an event regardless of past or future occurrences of other event
• Conditional - probability of an event given that another event has happened - look for words "given" or "conditional upon"

Rules

• Multiplication - determine joint probability: P(AB) = P(A|B)*P(B)
• Addition - determine that at least 1 of 2 events occurs
• P(A or B) = P(A) + P(B) - P(AB)
• Total Probability Rule - determines the unconditional prob of an event, given conditionals
• P(A) = P(A|B1)(B1) + P(A|B2)(B2) + ... + P(A|Bn)(Bn)
• B1 to Bn is a set of MECE outcomes

Interpretation

• Joint probability of two events - probability they will both occur - multiplication rule
• Prob that at least one of two events will occur - addition rule
• Joint probability of more than two events - keep multiplying
• Dependent/Independent
• For independent events, P(A|B)=P(A).  Likewise P(B|A)=P(B)

Calculating an unconditional prob using the total probability rule

• Sum of the joint probabilities is the unconditional probability of an event
• Example
• P(I) = 0.4, the prob that the authorities raise interest rates
• P(R|I) = 0.7, prob of a recession given that rates rise
• P(R|Ic) = 0.1, prob of a recession if rates do NOT rise
• What is the unconditional probability of a recession?
• P(R|I)*P(I) + P(R|Ic)*P(Ic)
• We know P(Ic) = 1-P(I) = 1 - 0.4 = 0.6
• So 0.7*0.4 + 0.1*0.6 = 0.28 + 0.06 = 0.34

Expected value and variance/sdev 

• EV is simply a weighted average of the outcomes, weighted by probability
• To calculate variance:
• Subtract EV from each outcome and square the difference
• Multiply by probability
• Add all these up
• Standard deviation is the square root of this
• Expected values can be conditional - i.e. 'conditional expected values'
• Analyst would use a conditional expected value to revise estimate when new information arrives
• Using a tree to find expected value:

Covariance and Correlation

• Covariance - measure of how two assets move together - Cov(X,Y).  We are usually concerned with the covariance of returns.
• Covariance is the expected value of the product of deviations of two random variables from their respective expected values
• Cov (Ri, Rj) = E{[Ri - E(Ri)]*[Rj - E(Rj)]} 
• The covariance of a variable with itself is equal to its variance
• Covariance can range from negative infinity to positive infinity

To calculate covariance for two assets in a portfolio:

Assume probabilities and returns as follows:

• First find expected returns of each stock A and B
• E(Ra) = 0.3*0.2 + 0.5*0.12 + 0.2*0.05 = 0.13
• E(Rb) = 0.14
• To get covariance, now do Return minus Expected for each row/security, and multiply together and then by probability, and then sum
• Ex. in row Boom, do 0.3 * (0.20 - 0.13) * (0.30 - 0.14) = 0.00336
• Same approach in rows Normal and Slow give 0.00020 and 0.00224
• Covariance is the sum of these three numbers: 0.00580
• Covariance though is difficult to interpret - so we use correlation
• Correlation = Covariance / (SdevA * SdevB)
• Ex. given the covariance of 0.00580 from previous, and Var(Ra) = 0.0028 and Var(Rb) = 0.0124, what is correlation?
• Remember to convert Variances to sdevs (take sqrts)
• 0.0058 / (0.0028^0.5*0.0124^0.5) = 0.9824

Calculating EV and variance for a portfolio

• EV and variance for a portfolio can be determined using the properties of the individual assets in the portfolio
• First, establish portfolio weight for each asset: w = mkt value of investment / mkt value portfolio
• Expected value and expected return are just a weighted average
• (This is a very difficult formula and I don't expect to study it frankly

There is more information and examples on how to calculate variance, covariance, etc for portfolios that I don't want to go into at this point.

Bayes' formula

• This is used to update a given set of prior probabilities for a given event in response to the arrival of new information
• updated prob = (probability of new information / unconditional prob of new information) * prior prob of the event
• Tree model made sense to me but the Electrocomp example made no sense - perhaps ask a friend for help on Bayes

Counting Problems

• Labeling - n items can each receive one of k different labels
• Ex - you have a portfolio of 8 stocks.  You want to label 4 as long holds, 3 as short term holds, and 1 as sell.
• Number of ways = n! / (n1!*n2!*...*nk!)
•  = 8! / (4! * 3! * 1!) = 40,320 / (24*6*1) = 280 combinations
• Note - there is a special case when n = 2
• Number of ways = n! / ((n-r)!r!)
• Permutation - a specific ordering of a group of objects
• 'How many different groups of size r can be chosen from n objects'
• Permutations = n! / (n-r)!
• Ex how many ways can 3 stocks be sold from an 8 stock portfolio if order is important?
• Permutations = 8! / (8-3)! = 40,320 / 120 = 336
• if order not important, 8! / ((8-3)!*3!) = 40,320 / 720 = 56
• Rules and hints
• Multiplication rule - if there are k steps to do something and each step can be done n ways, there are n1! * n2! * ... *nk! ways to do it
• Factorial - given n items, there are n! ways to arrange them
• Labeling is for when you have 3 or more subgroups of predetermined size and each element must be assigned a place (label) in one of the subgroups
• Combination - only applies to when there are only 2 groups of predetermined size - look for 'choose' or 'combination'

End of reading.

1:30 pm

About 2.5 hours