I’ve found that writing these posts have been instrumental to my own studying and knowledge retention because I’m looking at the information through frameworks rather than brute force reading as much as I can. Also it is a form of a commitment trap to say that I will write a post every week so I’m forced to go over my notes.
Quantitative Methods is broken down into two study sessions, 2: Basic Concepts and 3: Application. Another way of looking at this is study session 2 is descriptive and study session 3 is inferential. Inferential is what is most important in real life as you try to draw conclusions from a sample size rather than having the whole population of data.
However, for the CFA level 1 exam, I am deliberately choosing to focus on study session 2 and adding study session 3, except for technical analysis, to my ignore list. Frankly, I just find study session 3 boring and difficult. That is my personal choice. Technical analysis may be boring as well, but the types of questions you get from it are so easy, it would be silly to throw those points away. Also, it is only 15 pages to read in the Schweser Notes. What are the basic concepts of study session 2? Time value of money, discounted cash flow applications, statistical concepts and market returns, and probability concepts. With that being said, I want to focus on some of the key points or confusing points, at least for me, in Quantitative Methods.
One of the key components to remember is the conversion and relationship between Holding Period Yield (HPY) < Bank Discount Yield (BDY) < Money Market Yield (MMY) < Effective Annual Yield (EAY). Another relationship is Bond Equivalent Yield (BEY) < EAY. EAY is king as it accounts for full compounding by exponents. BEY is accounting for 6 months of compounding and then multiplied by 2, so you can understand why that is less than EAY. MMY is compounding by multiplication. HPY is usually for a period less than a year. Rate and yield are interchangeable, rate is from the borrower’s perspective and yield is from the lender’s perspective.
HPY = (Ending value – Beginning Value) / Beginning Value (not including dividends for now)
MMY= (HPY)*(360/t)
EAY = (1 + HPY)^{365/t} – 1
The HPY is the common factor between all four so it is useful to determine that first before transforming between.
BDY = (Discount/Face)*(360/t)
BEY = [(Face ValuePrice)/Price]*(365/d); d=days to maturity
P 
opulation (Greek, upper case letters) 
S 
ample (English, lower case letters) 
a 
μ: Population mean 
t 
x̄: Sample mean (x bar on top) 
r 
σ: Population standard deviation 
a 
s: Sample standard deviation 
a 
ρ: Population correlation 
t 
r: Sample correlation 
m 
σ²: Population variance 
i 
s²: Sample variance 
e 
N: Population size 
s 
n: Sample size 
t 

t 

e 

i 

r 

c 

s 

s 

N 
Nominal – Categories, but no order; ex. Colors 
O 
Ordinal – Ordered categories, ex. Bond ratings, AAA, AA+, AA. AAA is better than AA+, but the change from AAA to AA+ vs AA+ to AA isn’t the same, there is no defined value 
I 
Interval – Ordered categories with defined intervals, ex. temperature: 40 degrees is twice the amount of 20 degrees, 0 doesn’t mean absence of temperature, it is only a reference point 
R 
Ratio – Money 
Risk Adjusted Return
Risk = variation; Return = central tendency
Risk adjusted return = CV, Sharpe Ratio, Roy’s Safety First Ratio
A) Coefficient of variation (CV) = s (risk)/ x̄ (return). Low CV = good
B) Sharpe Ratio = (R_{p}R_{f})/ σ_{p} = (Return of portfolio – risk free rate) / portfolio standard deviation. High Sharpe Ratio = good
R_{f }= Nothing is technically risk free, but 3month U.S. Tbill is close enough to serve as a proxy for R_{f}
Sharpe Ratio assumes normal distribution so it is unsuitable for alternative investments
Sharpe Ratio is a specific case of Roy’s Safety First, with the R_{l} being the R_{f}
C) Roy’s Safety First = (R_{p}R_{l})/ σ_{p = }(Return of portfolio – personal lower bound return rate) / portfolio standard deviation. High Roy’s Safety First Ratio = good.
Statistical Concepts
Harmonic mean < Geometric mean (timeweighted rate of return) ≤ Arithmetic mean
The ≤ is due to the fact that if the annual returns are constant, i.e. 50% every year, the arithmetic and geometric returns will be the same
Harmonic mean is used with the average cost per share, i.e. if you purchase 1000 shares at $20 and later, another 1000 shares at $30, the average cost isn’t $25. It is slightly less, $24 in this case. 2/[(1/20)+(1/30)]
Arithmetic mean is upward biased and deceiving
Example, you invest $100 today, gain 50% in year 1, and lose 50% in year 2, what is the arithmetic and geometric mean?
Arithmetic mean: x̄=[1.5+(1.5)]/2=100; However, you have $75 in your pocket, not $100
Geometric mean: (1.5)(.5)^1/2=0.866, which means 13.39% return per year
The units for variance is %² so that is useless for interpretation purposes, which is why standard deviation is more useful, it uses % which matches returns.
Sample uses n1, reason = unbiased, real reason = too mathematically complex for me to understand
Chebyshev’s Inequality – for any distribution, 1[1/(k^2)] = amount of area covered under the distribution, with k = # of standard deviations. Example, for any distribution, two standard deviations will cover at least 75% of the distribution. 1[1/(2^2)]=.75. For a normal distribution, two standard deviations will cover ~95% of all the distribution.
Skew = left and right lean of a distribution, skewed to the right means mean > median > mode, skewed to the left means mode > median > mean
Many hedge funds are built upon skew
Kurtosis = up and down, excess >1 kurtosis = leptokurtic, excess <1 kurtosis = platykurtic
Risk management more concerned with tails
Real life stock market is leptokurtic, does not use normal distribution
Normal distribution: μ=0, σ²=1, skew=0, kurtosis=3
Modern portfolio theory assumes μ=0, σ²=1 and ignores skew (and kurtosis? To be determined)
Covariance is an unscaled measure, ∞ < cov < ∞
Correlation is the scaled version of covariance 1 < ρ < 1
Anything .7 < x > .7 is considered a “strong” correlation
Counting
If order matters – Permutations
If order doesn’t matter – Combination
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My Level 1 Status as of 2/08/2012:
Hours studied: 14.5
Practice questions taken: 30
Ethics: Quiz 1: 23/30;