CFA level i formula sheet

2015

CFA® EXAM REVIEW

COVERS
ALL TOPICS
IN LEVEL I

LEVEL I CFA­
®

FORMULA SHEETS


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Quantitative Methods 

Quantitative Methods 
The Future Value of a Single Cash Flow
FVN = PV (1 + r) N

The Present Value of a Single Cash Flow
PV =

FV
(1 + r) N

PVAnnuity Due = PVOrdinary Annuity × (1 + r)
FVAnnuity Due = FVOrdinary Annuity × (1 + r)

Present Value of a Perpetuity

PV(perpetuity) =

PMT
I/Y

Continuous Compounding and Future Values
FVN = PVe r ⋅N
s

Effective Annual Rates
EAR = (1 + Periodic interest rate) N - 1

Net Present Value
N

CFt
t
t=0 (1 + r )

NPV = ∑

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
D 360
rBD = ×
F
t

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 =

P1 - P0 + D1 P1 + D1
=
-1
P0
P0

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

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3


Quantitative Methods 

Effective Annual Yield
EAY = (1 + HPY)365/ t - 1

where:

HPY = holding period yield
t = numbers of days remaining till maturity
HPY = (1 + EAY) t /365 - 1

Money Market Yield
R MM =

360 × rBD
360 - (t × rBD )

R MM = HPY × (360/t)

Bond Equivalent Yield
BEY = [(1 + EAY)0.5 - 1] × 2

Population Mean
N

µ=

∑ xi
i =1

N

where:
xi = is the ith observation.
Sample Mean
n

X=

∑ xi
i =1

n

Geometric Mean
1 + R G = T (1 + R1 ) × (1 + R 2 ) ×…× (1 + R T )

OR

G = n X1X 2 X 3 … X n
with X i > 0 for i = 1, 2,…, n.

1

T
 T
R G =  ∏ (1 + R t )  − 1
 t =1


Harmonic Mean
Harmonic mean: X H =

4

N
with X i > 0 for i = 1,2,…,N.

1
∑x
i =1 i
N

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Quantitative Methods 

Percentiles
Ly =

( n + 1) y
100

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

Mean Absolute Deviation
n

MAD =

∑ Xi − X
i =1

n

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

σ2 =

∑ (X i − µ)2
i =1

N

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

σ=

∑ (X i −

µ)2

i =1

N

Sample Variance
n

Sample variance = s2 =

∑ (X i −
i =1

X)2

n −1

where:
n = sample size.

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5


Quantitative Methods 

Sample Standard Deviation
n

s=

∑ (X i − X)2

i =1

n −1

Coefficient of Variation
Coefficient of variation =

s
X

where:
s = sample standard deviation
X = the sample mean.
Sharpe Ratio
Sharpe ratio =

rp − rf
sp

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:
n

(X i - X)3
∑



n
i =1
SK = 

 ( n - 1)( n - 2 ) 

s3

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

SK ≈

(X i - X)3
∑
1
i =1

n

s3

where:
s = sample standard deviation

6

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Quantitative Methods 

Sample Kurtosis uses standard deviations to the fourth power. Sample excess kurtosis is
calculated as:
n


(X i - X)4 
∑

n(n + 1)
3(n - 1)2
i =1

KE = 
−
s4
 (n - 1)(n - 2)(n - 3)
 (n - 2)(n - 3)





As n becomes large the equation simplifies to:
n

KE ≈

(X i - X)4
∑
1
i=1

n

s4

−3

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
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:

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7


Quantitative Methods 

Conditional Probabilities
P(A B) =

P(AB)
given that P(B) ≠ 0
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 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(ASc )
P(A) = P(A S) × P(S) + P(A Sc ) × P(Sc )

The Total Probability Rule for n Possible Scenarios

P(A) = P(A S1 ) × P(S1 ) + P(A S2 ) × P(S2 ) +

+ P(A Sn ) × P(Sn )

where the set of events {S1 , S2 ,…, Sn } is mutually exclusive and exhaustive.

Expected Value
E(X) = P(X1 )X1 + P(X 2 )X 2 + … P(X n )X n
n

E(X) = ∑ P(X i )X i
i =1

where:
Xi = one of n possible outcomes.

8

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Quantitative Methods 

Variance and Standard Deviation
σ 2 (X) = E{[X - E(X)]2}
n

σ 2 (X) = ∑ P(X i ) [X i - E(X)]2
i =1

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 A ,R B ) = E{[R A - E(R A )][R B - E(R B )]}

Correlation Coefficient
Corr(R A ,R B ) = ρ(R A ,R B ) =

Cov(R A ,R B )
(σ A )(σ B )

Expected Return on a Portfolio
N

E(R p ) = ∑ wi E(R i ) = w1E(R1 ) + w2 E(R 2 ) +
i =1

+ w N E(R N )

where:
Weight of asset i =

Market value of investment i

Market value of portfolio

Portfolio Variance
N N

Var(R p ) = ∑ ∑ wi w jCov(R i ,R j )
i =1 j=1

Variance of a 2 Asset Portfolio
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w B Cov(R A ,R B )
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w Bρ(R A ,R B )σ (R A )σ (R B )

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9


Quantitative Methods 

Variance of a 3 Asset Portfolio
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + w2C σ 2 (R C )
+ 2w A w B Cov(R A ,R B ) + 2w B wC Cov(R B ,R C ) + 2wC w A Cov(R C ,R 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 n1 × n2 × n3 × … nk .
Combinations
n Cr

n
n!
=  =
 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 Pr

=

n!
( n − r )!

Discrete Uniform Distribution
F(x) = n × p(x) for the nth observation.

Binomial Distribution
P(X=x) = n Cx (p)x (1-p)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 2x = n × p × (1- p)

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

10

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Quantitative Methods 

The Continuous Uniform Distribution
P(X < a), P (X > b) = 0
P (x1 ≤ X ≤ x 2 ) =

x 2 - x1
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 
Approximately 68% of all observations lie in the interval 
Approximately 95% of all observations lie in the interval 
Approximately 99% of all observations lie in the interval 

μ ± (2/3)σ
μ ± 1σ
μ ± 2σ
μ ± 3σ

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

Roy’s Safety‐First Criterion
Minimize P(RP< RT)
where:
RP = portfolio return
RT = target return
Shortfall Ratio
Shortfall ratio (SF Ratio) or z-score =

E (RP ) - RT
σP

Continuously Compounded Returns
EAR = e r − 1
cc

HPR t = e r

cc

×t

rcc = continuously compounded annual rate

-l

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11


Quantitative Methods 

Sampling Error
Sampling error of the mean = Sample mean - Population mean = x - µ

Standard Error of Sample Mean when Population Variance is known
σx = σ

n

where:
σ x = the standard error of the sample mean
σ = the population standard deviation
n = the sample size
Standard Error of Sample Mean when Population Variance is not known

s
n

sx =

where:
s x = standard error of sample mean
s = sample standard deviation.
Confidence Intervals
Point estimate ± (reliability factor × standard error)

where:
Point estimate = value of the sample statistic that is used to estimate the population
parameter
Reliability factor = a number based on the assumed distribution of the point estimate and
the level of confidence for the interval (1 - α).
Standard error = the standard error of the sample statistic (point estimate)
x ± z α /2

σ
n

where:
x = The sample mean (point estimate of population mean)
zα/2 = The standard normal random variable for which the probability of an observation
lying in either tail is σ / 2 (reliability factor).
σ
= The standard error of the sample mean.
n
x ± tα

2

s
n

where:
x = sample mean (the point estimate of the population mean)
tα
= the t‐reliability factor
2
s
= standard error of the sample mean
n
s = sample standard deviation

12

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Quantitative Methods 

Test Statistic
Test statistic =

Sample statistic - Hypothesized value
Standard error of sample statistic

Power of a Test
Power of a test = 1 - P(Type II error)

Decision Rules for Hypothesis Tests
Decision
Do not reject H0

H0 is True
Correct decision

Reject H0

Incorrect decision
Type I error
Significance level =
P(Type I error)

H0 is False
Incorrect decision
Type II error
Correct decision
Power of the test
= 1 - P(Type II error)

Confidence Interval
 sample 
 statistic

x

 critical   standard    population   sample   critical   standard  
-

≤
≤
+
 value   error    parameter   statistic  value   error  
- (z α /2 )
(s n)
x
(s n)
≤
µ0
≤
+ (z α /2 )

Summary

H0 : μ ≤ μ0

Alternate
hypothesis
Ha : μ > μ0

One tailed
(lower tail)
test

H0 : μ ≥ μ0

Ha : μ < μ0

Test statistic <

critical value

Test statistic ≥
critical value

Probability that lies
below the computed test
statistic.

Two‐tailed

H0 : μ = μ0

Ha : μ ≠ μ0

Test statistic <
lower critical
value
Test statistic >
upper critical
value

Lower critical
value ≤ test
statistic ≤
upper critical
value

Probability that lies
above the positive

value of the computed
test statistic plus the
probability that lies
below the negative
value of the computed
test statistic.

Type of test
One tailed
(upper tail)
test

Null
hypothesis

Fail to reject
null if

Reject null if
Test statistic >
critical value

Test statistic ≤
critical value

P‐value represents
Probability that lies
above the computed test
statistic.

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13


Quantitative Methods 

t‐Statistic
t-stat =

x - µ0
s n

where:
x = sample mean
μ0 = hypothesized population mean
s = standard deviation of the sample
n = sample size
z‐Statistic
z-stat =

x - µ0
σ n

z-stat =

where:
x = sample mean
μ = hypothesized population mean
σ = standard deviation of the population

n = sample size

x - µ0
s n

where:
x = sample mean
μ = hypothesized population mean
s = standard deviation of the sample
n = sample size

Tests for Means when Population Variances are Assumed Equal
t=

(x1 - x2 ) − (µ1 - µ 2 )
 s2p s2p 
n +n 
 1
2

1/2

where:
s2p =

(n1 - 1)s12 + (n 2 - 1)s22
n1 + n 2 - 2

s12 = variance of the first sample
s22 = variance of the second sample

n1 = number of observations in first sample
n2 = number of observations in second sample
degrees of freedom = n1 + n2 -2

14

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Quantitative Methods 

Tests for Means when Population Variances are Assumed Unequal
t-stat =

df =

(x1 - x2 ) − (µ1 - µ 2 )
 s12 s22 
 n + n 
1
2
 s12 s22 
 n + n 
1
2

1/2

where:

s12 = variance of the first sample

2

s22 = variance of the second sample

(s12 n1 )2 + (s22 n2 )2
n1

n1 = number of observations in first sample

n2

n2 = number of observations in second sample

Paired Comparisons Test
t=

d - µ dz
sd

where:
d = sample mean difference
sd
s d = standard error of the mean difference =
n
sd = sample standard deviation
n = the number of paired observations
Hypothesis Tests Concerning the Mean of Two Populations ‐ Appropriate Tests
Population

distribution
Normal

Relationship
between
samples
Independent

Assumption
regarding
variance
Equal

Normal

Independent

Unequal

t‐test with
variance not
pooled

Normal

Dependent

N/A

t‐test with

paired
comparisons

Type of test
t‐test pooled
variance

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15


Quantitative Methods 

Chi Squared Test‐Statistic
χ2 =

( n-1) s2
σ 20

where:
n = sample size
s2 = sample variance
σ 20 = hypothesized value for population variance
Test‐Statistic for the F‐Test
F=

s12
s22

where:
s12 = Variance of sample drawn from Population 1
s22 = Variance of sample drawn from Population 2
Hypothesis tests concerning the variance
Hypothesis Test Concerning
Variance of a single, normally distributed
population

Appropriate Test Statistic
Chi‐square stat

Equality of variance of two independent,
normally distributed populations

F‐stat

Setting Price Targets with Head and Shoulders Patterns
Price target = Neckline - (Head - Neckline)

Setting Price Targets for Inverse Head and Shoulders Patterns
Price target = Neckline + (Neckline - Head)

Momentum or Rate of Change Oscillator
M = (V - Vx ) × 100

where:
M = momentum oscillator value
V = last closing price
Vx = closing price x days ago, typically 10 days

16

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Quantitative Methods 

Relative Strength Index
RSI = 100 −

100
1 + RS

where:
RS =

Σ (Up changes for the period under consideration)
Σ(| Down changes for the period under consideration|)

Stochastic Oscillator
C − L14 
%K = 100 
 H14 − L14 

where:
C = last closing price
L14 = lowest price in last 14 days
H14 = highest price in last 14 days
%D (signal line) = Average of the last three %K values calculated daily.
Short Interest ratio

Short interest ratio =

Short interest
Average daily trading volume

Arms Index
Arms index =

Number of advancing issues / Number of declining issues
Volume of advancing issues / Volume of declining issues

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17


Economics

Economics
Demand and Supply Analysis: Introduction
The demand function captures the effect of all these factors on demand for a good.

Demand function: QDx = f(Px, I, Py , …) … (Equation 1)
Equation 1 is read as “the quantity demanded of Good X (QDX) depends on the price of
Good X (PX), consumers’ incomes (I) and the price of Good Y (PY), etc.”
The supply function can be expressed as:

Supply function: QSx = f(Px,W, …) … (Equation 5)
The own‐price elasticity of demand is calculated as:
EDPx =

%∆QDx
… (Equation 16)
%∆Px

If we express the percentage change in X as the change in X divided by the value of X,
Equation 16 can be expanded to the following form:
Slope of demand
function.
Coefficient on
own‐price in
market demand
function

EDPx

%∆QDx
=
=
%∆Px

∆QDx
∆Px

QDx
Px

 ∆QDx   Px 
=
… (Equation 17)

 ∆Px   QDx 

Arc elasticity is calculated as:
(Q 0 - Q1 )
× 100
% change in quantity demanded % ∆ Q d (Q 0 + Q1 )/2
=
=
EP =
(P0 - P1 )
% change in price
%∆P
× 100
(P0 + P1 )/2

18

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Economics

Income Elasticity of Demand
Income elasticity of demand measures the responsiveness of demand for a particular good
to a change in income, holding all other things constant.

%∆QDx
ED I =
=
%∆I

EI =

∆QDx
∆I

QDx
I

= 


∆QDx   I 
… (Equation 18)

∆I   QDx 

Same as coefficient
on I in market
demand function
(Equation 11)

% change in quantity demanded
% change in income

Cross‐Price Elasticity of Demand
Cross elasticity of demand measures the responsiveness of demand for a particular good to
a change in price of another good, holding all other things constant.

EDPy

EC =

%∆QDx
=
=
%∆Py

∆QDx
∆Py

QDx
Py

 ∆QDx   Py 
=

 … (Equation 19)
 ∆Py   QDx 

Same as coefficient
on PY in market
demand function
(Equation 11)

% change in quantity demanded
% change in price of substitute or complement

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19


Economics

Demand and Supply Analysis: Consumer Demand
The Utility Function
In general a utility function can be represented as:

U = f(Q x ,Q x , … ,Q x )
1

2

n

Demand and Supply Analysis: The Firm
Accounting Profit
Accounting profit (loss) = Total revenue − Total accounting costs.
Economic Profit
Economic profit (also known as abnormal profit or supernormal profit) is calculated as:
Economic profit = Total revenue − Total economic costs
Economic profit = Total revenue − (Explicit costs + Implicit costs)
Economic profit = Accounting profit − Total implicit opportunity costs
Normal Profit
Normal profit = Accounting profit - Economic profit
Total, Average and Marginal Revenue
Table: Summary of Revenue Terms2

20

Revenue

Calculation

Total revenue (TR)

Price times quantity (P × Q), or the sum of individual
units sold times their respective prices; Σ(Pi × Qi)

Average revenue (AR)

Total revenue divided by quantity; (TR / Q)

Marginal revenue (MR)

Change in total revenue divided by change in quantity;
(ΔTR / ΔQ)

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Economics

Total, Average, Marginal, Fixed and Variable Costs
Table: Summary of Cost Terms3
Costs
Calculation
Total fixed cost (TFC)

Sum of all fixed expenses; here defined to include all
opportunity costs

Total variable cost (TVC)

Sum of all variable expenses, or per unit variable cost
times quantity; (per unit VC × Q)

Total costs (TC)

Total fixed cost plus total variable cost; (TFC + TVC)

Average fixed cost (AFC)

Total fixed cost divided by quantity; (TFC / Q)

Average variable cost (AVC)

Total variable cost divided by quantity; (TVC / Q)

Average total cost (ATC)

Total cost divided by quantity; (TC / Q) or (AFC + AVC)

Marginal cost (MC)

Change in total cost divided by change in quantity;
(ΔTC / ΔQ)

Marginal revenue product (MRP) of labor is calculated as:

MRP of labor = Change in total revenue / Change in quantity of labor
For a firm in perfect competition, MRP of labor equals the MP of the last unit of labor
times the price of the output unit.
MRP = Marginal product * Product price
A profit‐maximizing firm will hire more labor until:
MRPLabor = PriceLabor
Profits are maximized when:
MRP1
MRPn
=…=
Price of input 1
Price of input n

2 Exhibit

3, Volume 2, CFA Program Curriculum 2012

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21


Economics

The Firm And Market Structures
The relationship between MR and price elasticity can be expressed as:

MR = P[1 − (1/E p )]
In a monopoly, MC = MR so:
P[1 − (1/E p )] = MC

N‐firm concentration ratio: Simply computes the aggregate market share of the N
largest firms in the industry. The ratio will equal 0 for perfect competition and 100 for a
monopoly.
Herfindahl‐Hirschman Index (HHI): Adds up the squares of the market shares of each of
the largest N companies in the market. The HHI equals 1 for a monopoly. If there are M
firms in the industry with equal market shares, the HHI will equal 1/M.

Aggregate Output, Price, And Economic Growth
Nominal GDP refers to the value of goods and services included in GDP measured at
current prices.
Nominal GDP = Quantity produced in Year t × Prices in Year t

Real GDP refers to the value of goods and services included in GDP measured at
base‐year prices.
Real GDP = Quantity produced in Year t × Base-year prices

GDP Deflator

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GDP deflator =

Value of current year output at current year prices
× 100
Value of current year output at base year prices

GDP deflator =

Nominal GDP

× 100
Real GDP

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Economics

The Components of GDP
Based on the expenditure approach, GDP may be calculated as:
GDP = C + I + G + (X − M)

C = Consumer spending on final goods and services
I = Gross private domestic investment, which includes business investment in capital goods
(e.g. plant and equipment) and changes in inventory (inventory investment)
G = Government spending on final goods and services
X = Exports
M = Imports
Expenditure Approach
Under the expenditure approach, GDP at market prices may be calculated as:
GDP = Consumer spending on goods and services
+ Business gross fixed investment
+ Change in inventories
+ Government spending on goods and services
+ Government gross fixed investment
+ Exports − Imports
+ Statistical discrepancy

This equation is just
a breakdown of the

expression for GDP
we stated in the
previous LOS, i.e.
GDP = C + I + G +
(X − M).

Income Approach
Under the income approach, GDP at market prices may be calculated as:
GDP = National income + Capital consumption allowance
+ Statistical discrepancy
… (Equation 1)

National income equals the sum of incomes received by all factors of production used to
generate final output. It includes:

• Employee compensation
• Corporate and government enterprise profits before taxes, which includes:
○○ Dividends paid to households
○○ Corporate profits retained by businesses
○○ Corporate taxes paid to the government
• Interest income
• Rent and unincorporated business net income (proprietor’s income): Amounts
earned by unincorporated proprietors and farm operators, who run their own
businesses.
• Indirect business taxes less subsidies: This amount reflects taxes and subsidies that
are included in the final price of a good or service, and therefore represents the
portion of national income that is directly paid to the government.

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23


Economics

The capital consumption allowance (CCA) accounts for the wear and tear or depreciation
that occurs in capital stock during the production process. It represents the amount that
must be reinvested by the company in the business to maintain current productivity levels.
You should think of profits + CCA as the amount earned by capital.
Personal income = National income
− Indirect business taxes
− Corporate income taxes
− Undistributed corporate profits
+ Transfer payments
… (Equation 2)

Personal disposable income = Personal income − Personal taxes … (Equation 3)
Personal disposable income = Household consumption + Household saving



… (Equation 4)
Household saving = Personal disposable income
− Consumption expenditures
− Interest paid by consumers to businesses
− Personal transfer payments to foreigners … (Equation 5)
Business sector saving = Undistributed corporate profits
+ Capital consumption allowance … (Equation 6)

GDP = Household consumption + Total private sector saving + Net taxes

The equality of expenditure and income

S = I + (G − T) + ( X − M) … (Equation 7)
The IS Curve (Relationship between Income and the Real Interest Rate)
Disposable income = GDP − Business saving − Net taxes
S − I = (G − T) + ( X − M) … (Equation 7)

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Economics

The LM Curve
Quantity theory of money: MV = PY
The quantity theory equation can also be written as:
M/P and MD/P = kY
where:
k = I/V
M = Nominal money supply
MD = Nominal money demand
MD/P is referred to as real money demand and M/P is real money supply.
Equilibrium in the money market requires that money supply and money demand be equal.
Money market equilibrium: M/P = RMD
Solow (neoclassical) growth model
Y = AF(L,K)

where:

Y = Aggregate output
L = Quantity of labor
K = Quantity of capital
A = Technological knowledge or total factor productivity (TFP)
Growth accounting equation

Growth in potential GDP = Growth in technology + WL (Growth in labor)
+ WK (Growth in capital)
Growth in per capital potential GDP = Growth in technology
+ WK (Growth in capital-labor ratio)

Measures of Sustainable Growth
Labor productivity = Real GDP/Aggregate hours
Potential GDP = Aggregate hours × Labor productivity
This equation can be expressed in terms of growth rates as:
Potential GDP growth rate = Long‐term growth rate of labor force + Long‐term labor
productivity growth rate

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