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COVERS 2015 ®

ALL TOPICS CFA EXAM REVIEW

IN LEVEL I ®

LEVEL I CFA FORMULA SHEETS

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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