Wiley CFA Level I Formula Sheets

99 Pages·2016·6.27 MB·English

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Quantitative Methods

The Time Value of money

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2

The Time Value of Money

The Future Value of a Single Cash Flow

FV
PV (1
r)
N
N
=
+

The Present Value of a Single Cash Flow

PV
FV
(1
r)N
=
+

=
×
+

=
×
+
PV
PV
(1
r)
FV
FV
(1
r)

Annuity Due
Ordinary Annuity

Annuity Due
Ordinary Annuity

Present Value of a Perpetuity

PV(perpetuity)
PMT
I/Y
=

Continuous Compounding and Future Values

FV
PVe
N
r N
s
=
⋅

Effective Annual Rates

=
+
−
EAR
(1
Periodic interest rate)
1
N

DiscounTeDcash floWapplicaTions

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3

Discounted Cash Flow Applications

Net Present Value

∑(
)

=
+
NPV
CF
1
r

t
t
t=0

N

where:
CFt = the expected net cash flow at time t
N = the investment’s projected life
r = the discount rate or appropriate cost of capital

Bank Discount Yield

=
×
r
D
F
360
t
BD

where:
rBD = the annualized yield on a bank discount basis
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

=
−
+
=
+
−
HPY
P
P
D
P
P
D
P
1
1
0
1

0

1
1

0

where:
P0 = initial price of the investment.
P1 = price received from the instrument at maturity/sale.
D1 = interest or dividend received from the investment.

Effective Annual Yield

=
+
−
EAY
(1
HPY)
1
365/t

where:
HPY = holding period yield
t = numbers of days remaining till maturity

=
+
−
HPY
(1
EAY)
1
t/365

DiscounTeDcash floWapplicaTions

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4

Money Market Yield

=
×
−
×
R
360
r
360
(t
r
)
MM
BD

BD

=
×
R
HPY
(360/t)
MM

Bond Equivalent Yield

=
+
−
×
BEY
[(1
EAY)
1]
2
0.5

sTaTisTical concepTs

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5

Statistical Concepts

Population Mean

∑
µ =
=
x

N

i
i 1

N

where:
xi = is the ith observation.

Sample Mean

∑
=
=
X
x

n

i
i 1

n

Geometric Mean

1
R
(1
R )
(1
R )
(1
R )
OR
G
X X X
X

with X
0 for i
1, 2,
, n.

R
(1
R )
1

G
1
2
T
T
1
2
3
n
n

i

G
t
t 1

T
1
T
∏

+
=
+
×
+
×…×
+
=
…

>
=
…

=
+
�

�
�
�

�
�
−

=

Harmonic Mean

∑

=
>
=
…

=

Harmonic mean: X
N
1
x

with X
0 for i
1,2,
,N.
H

i
i 1

N
i

Percentiles

(
)
=
+
L
n
1 y
100
y

where:
y = percentage point at which we are dividing the distribution
Ly = location (L) of the percentile (Py) in the data set sorted in ascending order

Range

=
−
Range
Maximum value
Minimum value

sTaTisTical concepTs

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6

Mean Absolute Deviation

∑
=
−

=
MAD

X
X

n

i
i 1

n

where:
n = number of items in the data set
X = the arithmetic mean of the sample

Population Variance

(X
)

N

2
i
2

i 1

N
∑
σ =
− µ

=

where:
Xi = observation i
μ = population mean
N = size of the population

Population Standard Deviation

∑
σ =
− µ

=
(X
)

N

i
2

i 1

N

Sample Variance

∑
=
=
−

−

=
Sample variance
s
(X
X)

n
1

2
i
2

i 1

n

where:
n = sample size.

Sample Standard Deviation

∑
=
−

−

=
s
(X
X)

n
1

i
2

i 1

n

sTaTisTical concepTs

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7

Coefficient of Variation

Coefficient of variation
s
X
=

where:
s = sample standard deviation
X = the sample mean.

Sharpe Ratio

Sharpe ratio
r
r

s

p
f

p
=
−

where:
rp = mean portfolio return
rf = risk‐free return
sp = standard deviation of portfolio returns

Sample skewness, also known as sample relative skewness, is calculated as:

∑

(
)(
)
=
−
−
�

��
�

��

−

=
S
n
n
1 n
2

(X
X)

s
K

i
3

i 1

n

3

As n becomes large, the expression reduces to the mean cubed deviation.

∑
≈
−

=
S
1
n

(X
X)

s
K

i
3

i 1

n

3

where:
s = sample standard deviation

Sample Kurtosis uses standard deviations to the fourth power. Sample excess kurtosis is
calculated as:

∑
=
+
−
−
−

−
�

�

�
�
�
�

�

�

�
�
�
�

−
−

−
−

=
K
n(n
1)
(n
1)(n
2)(n
3)

(X
X)

s
3(n
1)
(n
2)(n
3)
E

i
4

i 1

n

4

2

sTaTisTical concepTs

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8

As n becomes large the equation simplifies to:

∑
≈
−
−
K
1
n

(X
X)

s
3
E

i
4

i=1

n

4

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.

pRobabiliTy concepTs

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9

Probability Concepts

Odds for an Event

P E
a

a
b
(
)
( ) =
+

Where the odds for are given as “a to b”, then:

Odds for an Event

P E
b
(a
b)
( ) =
+

Where the odds against are given as “a to b”, then:

Conditional Probabilities

P(A B)
P(AB)
P(B) given that P(B)
0
=
≠

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 or B)
P(A)
P(B)
P(AB)

P(A and B)
P(A)
P(B)

The Total Probability Rule

P(A)
P(AS)
P(AS )

P(A)
P(A S)
P(S)
P(A S )
P(S )

c

c
c

=
+

=
×
+
×

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 )

where the set of events {S , S ,
, S } is mutually exclusive and exhaustive.

1
1
2
2
n
n

1
2
n

�
=
×
+
×
+
+
×

…

pRobabiliTy concepTs

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10

Expected Value

=
+
+ …
E(X)
P(X )X
P(X )X
P(X )X
1
1
2
2
n
n

E(X)
P(X )X
i
i

i 1

n
∑
=

=

where:
Xi = one of n possible outcomes.

Variance and Standard Deviation

σ
=
−
(X)
E{[X
E(X)] }
2
2

∑
σ
=
−

=
(X)
P(X ) [X
E(X)]
2
i
i
2

i 1

n

The Total Probability Rule for Expected Value

1.
E(X) = E(X| S)P(S) + E(X| Sc)P(Sc)
2.
E(X) = E(X| S1) × P(S1) + E(X| S2) × P(S2) + . . . + E(X| Sn) × P(Sn)

where:
E(X) = the unconditional expected value of X
E(X| S1) = the expected value of X given Scenario 1
P(S1) = the probability of Scenario 1 occurring
The set of events {S1, S2, . . . , Sn} is mutually exclusive and exhaustive.

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

= ρ
=
σ
σ
Corr(R ,R )
(R ,R )
Cov(R ,R )

(
)(
)
A
B
A
B

A
B

A
B

pRobabiliTy concepTs

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11

Expected Return on a Portfolio

E(R )
w E(R )
w E(R )
w E(R )
w E(R )
p
i
i
1
1
2
2
N
N
i 1

N
�
∑
=
=
+
+
+

=

where:

Weight of asset i
Market value of investment i

Market value of portfolio
=

Portfolio Variance

∑
∑
=

=
=
Var(R )
w w Cov(R ,R )
p
i
j
i
j
j 1

N

i 1

N

Variance of a 2 Asset Portfolio

Var(R )
w
(R )
w
(R )
2w w Cov(R ,R )
p
A
2
2
A
B
2
2
B
A
B
A
B
=
σ
+
σ
+

Var(R )
w
(R )
w
(R )
2w w
(R ,R ) (R ) (R )
p
A
2
2
A
B
2
2
B
A
B
A
B
A
B
=
σ
+
σ
+
ρ
σ
σ

Variance of a 3 Asset Portfolio

=
σ
+
σ
+
σ

+
+
+

Var(R )
w
(R )
w
(R )
w
(R )

2w w Cov(R ,R )
2w w Cov(R ,R )
2w w Cov(R ,R )

p
A
2
2
A
B
2
2
B
C
2
2
C

A
B
A
B
B
C
B
C
C
A
C
A

Bayes’ Formula

=
×
P(Event Information)
P (Information Event)
P (Event)
P (Information)

Counting Rules

The number of different ways that the k tasks can be done equals
.
1
2
3
n
n
n
nk
×
×
×…

Combinations

C
n
r
n!
n
r ! r!
n
r
( )
(
)
= �
�
�
� =
−

Remember: The combination formula is used when the order in which the items are
assigned the labels is NOT important.

Permutations

P
n!
n
r !
n
r
(
)
=
−

common pRobabiliTy DisTRibuTions

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12

Common Probability Distributions

Discrete Uniform Distribution

F(x)
n
p(x) for the th observation.
n
=
×

Binomial Distribution

=
−
P(X=x)
C (p) (1
p)
n
x

x
n-x

where:
p = probability of success
1 − p = probability of failure

nCx = number of possible combinations of having x successes in n trials. Stated differently,
it is the number of ways to choose x from n when the order does not matter.

Variance of a Binomial Random Variable

σ =
×
×
−
n
p
(1
p)
x
2

Tracking Error

Tracking error
Gross return on portfolio
Total return on benchmark index
=
−

The Continuous Uniform Distribution

P(X
a), P (X
b)
0
<
>
=

≤
≤
=
−
−
P (x
X
x )
x
x
b
a
1
2

2
1

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 ± 1σ

•
Approximately 95% of all observations lie in the interval
μ ± 2σ

•
Approximately 99% of all observations lie in the interval
μ ± 3σ

common pRobabiliTy DisTRibuTions

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13

z‐Score

=
−
=
− µ σ
z
(observed value
population mean)/standard deviation
(x
)/

Roy’s Safety‐First Criterion

Minimize P(RP< RT)

where:
RP = portfolio return
RT = target return

Shortfall Ratio

(
)
=
−
σ
Shortfall ratio (SF Ratio) or z-score
E R
R
P
T

P

Continuously Compounded Returns

EAR
e
1
r
continuously compounded annual rate
r
cc
cc
=
−
=

=
−
×
HPR
e
l
t
r
t
cc