CFA Level I Formula Sheet

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  COVERS 2015 ®
  ALL TOPICS CFA EXAM REVIEW
  IN LEVEL I ®
  
LEVEL I CFA FORMULA SHEETS
  Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. The material was previously published by Elan Guides. Published simultaneously in Canada.
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  Quantitative Methods  The Future Value of a Single Cash Flow
  FV PV (1 r) =
N
  N The Present Value of a Single Cash Flow
  FV PV =
  N
(1 r)
  PV PV (1 r) = × +
  Annuity Due Ordinary Annuity
  FV FV (1 r) = × +
  Annuity Due Ordinary Annuity Present Value of a Perpetuity
  PMT PV(perpetuity) =
  I/Y
  Continuous Compounding and Future Values r N ⋅ s
  FV = PVe
  N Effective Annual Rates
  N
  EAR = + (1 Periodic interest rate) - 1
  Net Present Value N
  CF
  t
  NPV =
  t ∑
  1 r
t=0
  ( )
  where: CF = the expected net cash flow at time t
  t
  N = the investment’s projected life r = the discount rate or appropriate cost of capital
  Bank Discount Yield
  D 360 r = ×
  BD
  F t where: r = the annualized yield on a bank discount basis
  BD
  D = the dollar discount (face value – purchase price) F = the face value of the bill t = number of days remaining until maturity
  Holding Period Yield
  P - P D
P D
  1
  1
  11 HPY = = - 1
  P P where: P = initial price of the investment. P = price received from the instrument at maturity/sale.1 D = interest or dividend received from the investment.
  1
  Effective Annual Yield 365/ t
  EAY (1 HPY) - 1 = + where:
  HPY = holding period yield t = numbers of days remaining till maturity
  t /365
  HPY (1 EAY) - 1 = +
  Money Market Yield
  360 × r
  BD
  R =
  MM
  360 - (t × r )
  BD
  R = HPY (360/t) ×
  MM Bond Equivalent Yield
0.5 BEY = + [(1 EAY) - 1] 2 ×
  Population Mean N
  x
  i ∑i 1 =
  µ = N where: x = is the ith observation.
  i Sample Mean n
  x
  i ∑i 1 =
  X = n Geometric Mean
  T n
  X X X …
  X G
  12 T
  1
  1 R = (1 R ) (1 R ) × + ×…× + (1 R ) OR G = + +
  2 3 n with X > 0 for i = 1, 2, … , n. i
  1 T T
    = + R (1 R )
  1 G  t  −
  ∏ t 1 =
   
  Harmonic Mean
  N Harmonic mean: X H = with X > 0 for i = 1,2, … , N.
  N i
  1
  ∑
  x
  i i 1 =
  Percentiles
n 1 y
  ( )
  L =
  y
  100 where: y = percentage point at which we are dividing the distribution L = location (L) of the percentile (P ) in the data set sorted in ascending order
  y y Range
  Range = Maximum value - Minimum value
  Mean Absolute Deviation n
  X −
  X
  i ∑i 1 =
  MAD = n where: n = number of items in the data set X = the arithmetic mean of the sample
  Population Variance N
  2
  (X ) − µ
  i ∑
  2 i 1 =
  σ = N where:
  X = observation i
  i
  μ = population mean N = size of the population
  Population Standard Deviation N
  2
  (X − µ )
  i ∑i 1 =
  σ = N
  Sample Variance n
  2
  (X X) −
  i ∑
  2 i 1 =
  Sample variance = s = n 1 − where: n = sample size.
  Sample Standard Deviation n
  2
  (X − X)
  i ∑i 1 =
  s = n 1 −
  Coefficient of Variation
  s Coefficient of variation =
  X where: s = sample standard deviation X = the sample mean.
  Sharpe Ratio
  r r −
  p f
  Sharpe ratio = s
  p
  where: r = mean portfolio return
  p
  r = risk‐free return
  f
  s = standard deviation of portfolio returns
  p Sample skewness, also known as sample relative skewness, is calculated as: n
  3
  (X - X)
  
i ∑
   n i 1 =
  S =
  K
  
3
    n - 1 n - 2 s
  ( )( )
    As n becomes large, the expression reduces to the mean cubed deviation.
  n
  3
  (X - X)
  i ∑
  1i 1 =
  S ≈
  K
  3
  n s where: s = sample standard deviation
  Sample Kurtosis uses standard deviations to the fourth power. Sample excess kurtosis is
  calculated as:
  n
   
  4
  (X - X)
  i
  2
   ∑ 
n(n 1) 3(n - 1)i 1 =
    K = −
  E
  4
  (n - 1)(n - 2)(n - 3) s (n - 2)(n - 3)      
  As n becomes large the equation simplifies to:
  n
  4
  (X - X)
  i ∑
  1
  i=1
  K
  3 ≈ −
  E
  4
  n s where: s = sample standard deviation For a sample size greater than 100, a sample excess kurtosis of greater than 1.0 would be considered unusually high. Most equity return series have been found to be leptokurtic.
  Odds for an Event
  a P E =
  ( )
a b
  ( )
  Where the odds for are given as “a to b”, then:
  Odds for an Event
  b P E =
  ( )
(a
  b) Where the odds against are given as “a to b”, then:
  Conditional Probabilities
  P(AB) P(A B) given that P(B)
  = ≠ P(B)
  Multiplication Rule for Probabilities
  P(AB) = P(A B) P(B) ×
  Addition Rule for Probabilities
P(A or B) = P(A) P(B) P(AB) −
  For Independant Events
  P(A B) = P(A), or equivalently, P(B A) = P(B)
  = P(A and B) P(A) P(B)
P(A or B) P(A) P(B) - P(AB)
  = ×
  The Total Probability Rule c
  =
P(A) P(AS) P(AS )
  c c
  = × ×
P(A) P(A S) P(S) P(A S ) P(S )
  The Total Probability Rule for n Possible Scenarios
  = × + P(A) P(A S ) P(S ) P(A S ) P(S ) × × + + P(A S ) P(S )
  ⋯
  1
  1
  2 2 n n where the set of events {S , S , , S } is mutually exclusive and exhaustive.
  …
  1 2 n Expected Value
  = + …
E(X) P(X )X P(X )X P(X )X
  1
  1
  2
2 n n n
  E(X) P(X )X =
  i i ∑i 1 =
  where: X = one of n possible outcomes.
  i
  Variance and Standard Deviation
  2
  2
  σ (X) = E{[X - E(X)] }
  n
  2
  2
  σ (X) = P(X ) [X - E(X)]
  i i ∑i 1 =
  The Total Probability Rule for Expected Value c c
  1. E(X) = E(X | S)P(S) + E(X | S )P(S )
  2. E(X) = E(X | S ) × P(S ) + E(X | S ) × P(S ) + . . . + E(X | S ) × P(S )
  1
  1
  2 2 n n
  where: E(X) = the unconditional expected value of X E(X | S ) = the expected value of X given Scenario 1
  1 P(S ) = the probability of Scenario 1 occurring
  1 The set of events {S , S , . . . , S } is mutually exclusive and exhaustive.
  1 2 n Covariance
  Cov(XY) E{[X - E(X)][Y - E(Y)]} =
  Cov(R ,R ) = E{[R - E(R )][R - E(R )]}
  A B A A B B Correlation Coefficient
  Cov(R ,R )
  A B
  Corr(R ,R ) = ρ (R ,R ) =
  A B A B
  ( )( ) σ σ
  A B Expected Return on a Portfolio
  N
E(R ) = w E(R ) = w E(R ) w E(R ) + + w E(R )
  ⋯
  p i i
  1
  1
  22 N N
  ∑i 1 =
  where: Market value of investment i
  Weight of asset i = Market value of portfolio
  Portfolio Variance N N
  Var(R ) w w Cov(R ,R ) =
  p i j i j ∑ ∑i 1 = j 1 =
  Variance of a 2 Asset Portfolio
  2
  2
  2
  2
p A A B B A B A BVar(R ) = w σ (R ) w σ (R ) 2w w Cov(R ,R )
  2
  2
  2
  2 Var(R ) = w σ (R ) w σ (R ) 2w w ρ (R ,R ) (R ) (R ) σ σ + + p A A B B A B A B A B
  Variance of a 3 Asset Portfolio
  2
  2
  2
  2
  2
  2 Var(R ) w (R ) w (R ) w (R )
  = σ σ σ + +
  p A A B B C C
  2w w Cov(R ,R ) 2w w Cov(R ,R ) 2w w Cov(R ,R )

  A B A B B C B C C A C A Bayes’ Formula
  P (Information Event) P (Event) × P(Event Information) =
  P (Information)
  Counting Rules The number of different ways that the k tasks can be done equals n n n n .
  × × × …
  1
  2 3 k Combinations
  n n! 
  C
  n r
  =   = r n − r ! r!
  ( ) ( ) Remember: The combination formula is used when the order in which the items are assigned the labels is NOT important. Permutations
  n! P =
  n r
  n r ! −
  ( ) Discrete Uniform Distribution
  F(x) = × n p(x) for the th observation. n
  Binomial Distribution x n-x
  P(X=x) = C (p) (1-p)
  n x
  where: p = probability of success 1 - p = probability of failure
  C = number of possible combinations of having x successes in n trials. Stated differently,
  n x it is the number of ways to choose x from n when the order does not matter.
  Variance of a Binomial Random Variable
  2
  n p (1- p) σ = × ×
  x Tracking Error
  Tracking error = Gross return on portfolio Total return on benchmark index −
  The Continuous Uniform Distribution
  P(X
  a), P (X
  b) < > = x - x
  21 P (x ≤
  X ≤ x ) =
  1
  2
  b - a
  Confidence Intervals
  For a random variable X that follows the normal distribution: The 90% confidence interval is x - 1.65s to x + 1.65s The 95% confidence interval is x - 1.96s to x + 1.96s The 99% confidence interval is x - 2.58s to x + 2.58s The following probability statements can be made about normal distributions
Approximately 50% of all observations lie in the interval μ ± (2/3)
  σ
Approximately 68% of all observations lie in the interval μ ± 1
  σ
Approximately 95% of all observations lie in the interval μ ± 2
  σ
Approximately 99% of all observations lie in the interval μ ± 3
  σ
  z‐Score
  z = (observed value - population mean)/standard deviation = (x − µ σ )/
  Roy’s Safety‐First Criterion
  Minimize P(R < R )
  P T
  where: R = portfolio return
  P
  R = target return
  T Shortfall Ratio
  E R - R
  ( ) P T
  Shortfall ratio (SF Ratio) or z-score = σ
  P Continuously Compounded Returns r cc
  EAR = e − 1 r = continuously compounded annual rat

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