wiley cfa level i formula sheets

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Quantitative Methods

The Time Value of money

© Wiley 2017 all Rights Reserved. any unauthorized copying or distribution will constitute an infringement of copyright.2

The Time Value of Money

The Future Value of a Single Cash Flow

FV PV (1 r)NN= +

The Present Value of a Single Cash Flow

PVFV

(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 PVeNr Ns= ⋅

Effective Annual Rates

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

DiscounTeD cash floW applicaTions

© Wiley 2017 all Rights Reserved. any unauthorized copying or distribution will constitute an infringement of copyright. 3

Discounted Cash Flow Applications

Net Present Value

∑( )=

+NPV

CF

1 rt

tt=0

N

where:CFt = the expected net cash flow at time tN = the investment’s projected lifer = the discount rate or appropriate cost of capital

Bank Discount Yield

= ×rD

F

360

tBD

where:rBD = the annualized yield on a bank discount basisD = the dollar discount (face value – purchase price)F = the face value of the billt = number of days remaining until maturity

Holding Period Yield

=− +

= + −HPYP P D

P

P D

P11 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) 1365/ t

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

= + −HPY (1 EAY) 1t/365

DiscounTeD cash floW applicaTions


Money Market Yield

= ×− ×

R360 r

360 (t r )MMBD

BD

= ×R HPY (360/t)MM

Bond Equivalent Yield

= + − ×BEY [(1 EAY) 1] 20.5

sTaTisTical concepTs


Statistical Concepts

Population Mean

∑µ = =

x

N

ii 1

N

where:xi = is the ith observation.

Sample Mean

∑= =X

x

n

ii 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 TT

1 2 3 nn

i

G tt 1

T1T

∏

+ = + × + ×…× + = …

> = …

= +

−

=

Harmonic Mean

∑= > = …

=

Harmonic mean: XN

1x

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

ii 1

N i

Percentiles

( )= +L

n 1 y

100y

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

Range

= −Range Maximum value Minimum value



Mean Absolute Deviation

∑=

−=MAD

X X

n

ii 1

n

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

Population Variance

(X )

N2

i2

i 1

N

∑σ =

− µ=

where:Xi = observation iμ = population meanN = size of the population

Population Standard Deviation

∑σ =

− µ=

(X )

N

i2

i 1

N

Sample Variance

∑= =

−

−=Sample variance s

(X X)

n 12

i2

i 1

n

where:n = sample size.

Sample Standard Deviation

∑=

−

−=s

(X X)

n 1

i2

i 1

n



Coefficient of Variation

Coefficient of variations

X=

where:s = sample standard deviationX = the sample mean.

Sharpe Ratio

Sharpe ratior r

sp f

p

=−

where:rp = mean portfolio returnrf = risk‐free returnsp = standard deviation of portfolio returns

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

∑( )( )

=− −

−=S

n

n 1 n 2

(X X)

sK

i3

i 1

n

3

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

∑≈

−=S

1

n

(X X)

sK

i3

i 1

n

3

where:s = sample standard deviation

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

∑= +

− − −

−

− −− −

=Kn(n 1)

(n 1)(n 2)(n 3)

(X X)

s

3(n 1)

(n 2)(n 3)E

i4

i 1

n

4

2



As n becomes large the equation simplifies to:

∑≈

−−K

1

n

(X X)

s3E

i4

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


Probability Concepts

Odds for an Event

P Ea

a b( )( ) =

+

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

Odds for an Event

P Eb

(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

�= × + × + + ×…



Expected Value

= + + …E(X) P(X )X P(X )X P(X )X1 1 2 2 n n

E(X) P(X )Xi ii 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)]2i 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 XE(X | S1) = the expected value of X given Scenario 1P(S1) = the probability of Scenario 1 occurringThe 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 BA B

A B



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 Ni 1

N

�∑= = + + +=

where:

Weight of asset iMarket value of investment i

Market value of portfolio=

Portfolio Variance

∑∑===

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

N

i 1

N

Variance of a 2 Asset Portfolio

Var(R ) w (R ) w (R ) 2w w Cov(R ,R )p A2 2

A B2 2

B A B A B= σ + σ +

Var(R ) w (R ) w (R ) 2w w (R ,R ) (R ) (R )p A2 2

A B2 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 2A B

2 2B C

2 2C

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 3n n n nk× × × …

Combinations

Cn

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

Pn!

n r !n r ( )=

−

common pRobabiliTy DisTRibuTions


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 xx n-x

where:p = probability of success1 − 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)x2

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 a1 22 1

Confidence Intervals

For a random variable X that follows the normal distribution:The 90% confidence interval is x − 1.65s to x + 1.65sThe 95% confidence interval is x − 1.96s to x + 1.96sThe 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


z‐Score

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

Roy’s Safety‐First Criterion

Minimize P(RP< RT)

where:RP = portfolio returnRT = target return

Shortfall Ratio

( )=

−σ

Shortfall ratio (SF Ratio) or z-scoreE R RP T

P

Continuously Compounded Returns

EAR e 1 r continuously compounded annual ratercc

cc= − =

= −×HPR e ltr tcc

sampling anD esTimaTion


Sampling and Estimation

Sampling Error

= − = − µSampling error of the mean Sample mean Population mean x

Standard Error of Sample Mean when Population Variance is known

σ = σnx

where:σx = the standard error of the sample meanσ = the population standard deviationn = the sample size

Standard Error of Sample Mean when Population Variance is not known

=ss

nx

where:sx = standard error of sample means = 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 parameterReliability 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)

sampling anD esTimaTion


± σαx z

n/2

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).σn

= The standard error of the sample mean.

± αx ts

n2

where:x = sample mean (the point estimate of the population mean)αt2 = the t‐reliability factor

s

n = standard error of the sample mean

s = sample standard deviation

hypoThesis TesTing


Hypothesis Testing

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 H0 is True H0 is FalseDo not reject H0 Correct decision Incorrect decision

Type II errorReject H0 Incorrect decision

Type I errorSignificance level =

P(Type I error)

Correct decisionPower of the test

= 1 − P(Type II error)

Confidence Interval

−

≤

≤

+

− ≤ µ ≤ +α α

samplestatistic

criticalvalue

standarderror

populationparameter

samplestatistic

criticalvalue

standarderror

x (z ) (s n) x (z ) (s n)/2 0 /2

Summary

Type of testNull

hypothesisAlternate hypothesis Reject null if

Fail to reject null if P‐value represents

One tailed (upper tail) test

H0 : μ ≤ μ0 Ha : μ > μ0 Test statistic > critical value

Test statistic ≤ critical value

Probability that lies above the computed test statistic.

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 valueTest 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.

hypoThesis TesTing


t‐Statistic

=− µ

t-statx

s n0

where:x = sample meanμ0 = hypothesized population means = standard deviation of the samplen = sample size

z‐Statistic

=− µ

σz-stat

x

n0 =

− µz-stat

x

s n0

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

Tests for Means when Population Variances are Assumed Equal

= − − µ − µ

+

t(x x ) ( )

s

n

s

n

1 2 1 2

p2

1

p2

2

1/2

where:

= − + −+ −

s(n 1)s (n 1)s

n n 2p2 1 1

22 2

2

1 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

hypoThesis TesTing


Tests for Means when Population Variances are Assumed Unequal

= − − µ − µ

+

t-stat(x x ) ( )

sn

sn

1 2 1 2

12

1

22

2

1/2

df

sn

sn

s n

n

s n

n

12

1

22

2

2

12

12

1

22

22

2

( ) ( )=

+

+

where:

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

Paired Comparisons Test

=− µ

td

sdz

d

where:d = sample mean differencesd = standard error of the mean difference =

s

nd

sd = sample standard deviationn = the number of paired observations

Hypothesis Tests Concerning the Mean of Two Populations ‐ Appropriate Tests

Population distribution

Relationship between samples

Assumption regarding variance Type of test

Normal Independent Equal t‐test pooled variance

Normal Independent Unequal t‐test with variance not pooled

Normal Dependent N/A t‐test withpairedcomparisons

hypoThesis TesTing


Chi Squared Test‐Statistic

( )χ = −σ

n 1 s22

02

where:n = sample sizes2 = sample variance

02σ = hypothesized value for population variance

Test‐Statistic for the F‐Test

Fs

s12

22=

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 Appropriate Test Statistic

Variance of a single, normally distributed population

Chi‐square stat

Equality of variance of two independent, normally distributed populations

F‐stat

Technical analysis


Technical Analysis

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 V ) 100x

where:M = momentum oscillator valueV = last closing priceVx = closing price x days ago, typically 10 days

Relative Strength Index

RSI 100100

1 RS= −

+

where:

RS(Up changes for the period under consideration)

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

Σ

Stochastic Oscillator

%K 100C L14

H14 L14= −

−

where:C = last closing priceL14 = lowest price in last 14 daysH14 = highest price in last 14 days

%D (signal line) = Average of the last three %K values calculated daily.

Short Interest ratio

Short interest ratioShort interest

Average daily trading volume=

Arms Index

Arms indexNumber of advancing issues / Number of declining issues

Volume of advancing issues / Volume of declining issues=

Economics

Topics in DemanD anD supply analysis


Topics in Demand and Supply Analysis

The demand function captures the effect of all these factors on demand for a good.

= …Demand function: QD f(P , I, P , )x x y … (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 own‐price elasticity of demand is calculated as:

ED% QD

% PPxx

x

= ∆∆

… (Equation 6)

If we express the percentage change in X as the change in X divided by the value of X, Equation 6 can be expanded to the following form:

ED% QD

% P

QDQD

PP

QD

P

P

QDPxx

x

xx

xx

x

x

x

x

= ∆∆

=∆

∆ = ∆∆

… (Equation 7)

Arc elasticity is calculated as:

= =∆∆

= +×

+×

E% change in quantity demanded

% change in price

% Q

% P

(Q - Q )

(Q Q )/2100

(P - P )

(P P )/2100

Pd

0 1

0 1

0 1

0 1

Slope of demand function.

Coefficient on own‐price in market demand function



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.

ED% QD

% I

QDQD

II

QD

I

I

QDIx

xx x

x

= ∆∆

=∆

∆ = ∆∆

… (Equation 8)

=E% change in quantity demanded

% change in incomeI

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.

ED% QD

% P

QDQD

PP

QD

P

P

QDPyx

y

xx

y

y

x

y

y

x

= ∆∆

=∆

∆ = ∆∆

… (Equation 9)

E% change in quantity demanded

% change in price of substitute or complementC =

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

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



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

Normal profit = Accounting profit − Economic profit

Economic profit = Accounting profit − Total implicit opportunity costs

Economic profit = Total revenue − (Explicit costs + Implicit costs)

Normal Profit

Total, Average and Marginal Revenue

Table: Summary of Revenue Terms2

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)



MRPLabor = PriceLabor

2 Exhibit 3, Volume 2, CFA Program Curriculum 2012

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

MRP = Marginal product * Product price

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.

A profit‐maximizing firm will hire more labor until:

Profits are maximized when:

MRP

Price of input 1

MRP

Price of input n1 n= … =

The FiRm anD maRkeT sTRucTuRes


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 )] MCp

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


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

GDP deflatorValue of current year output at current year prices

Value of current year output at base year prices100= ×

GDP deflatorNominal GDP

Real GDP100= ×

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 servicesI = 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 servicesX = ExportsM = Imports

Expenditure Approach

Under the expenditure approach, GDP at market prices may be calculated as:

=+++++ −+

GDP Consumer spending on goods and servicesBusiness gross fixed investmentChange in inventoriesGovernment spending on goods and servicesGovernment gross fixed investmentExports ImportsStatistical 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 allowanceStatistical 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.

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 incomeIndirect business taxesCorporate income taxesUndistributed corporate profitsTransfer 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 incomeConsumption expendituresInterest paid by consumers to businessesPersonal transfer payments to foreigners … (Equation 5)

=+

Business sector saving Undistributed corporate profitsCapital 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)

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/VM = Nominal money supplyMD = Nominal money demandMD/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 outputL = Quantity of laborK = Quantity of capitalA = Technological knowledge or total factor productivity (TFP)



Growth accounting equation

= ++

Growth in potential GDP Growth in technology W (Growth in labor)W (Growth in capital)

L

K

=+

Growth in per capital potential GDP Growth in technologyW (Growth in capital-labor ratio)K

Measures of Sustainable Growth

Potential GDP = Aggregate hours × Labor productivity

Labor productivity = Real GDP/Aggregate hours

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

unDeRsTanDing Business cycles


Understanding Business Cycles

Unit labor cost (ULC) is calculated as:

ULC W/O=

where:O = Output per hour per workerW = Total labor compensation per hour per worker

moneTaRy anD Fiscal policy


Monetary And Fiscal Policy

Required reserve ratio = Required reserves / Total deposits

Money multiplier = 1/ (Reserve requirement)

The Fischer effect states that the nominal interest rate (RN) reflects the real interest rate (RR) and the expected rate of inflation (IIe).

R RN Re= + Π

The Fiscal Multiplier

Ignoring taxes, the multiplier can also be calculated as:

○1

(1 MPC)

1

(1 0.9)10

−=

−=

Assuming taxes, the multiplier can also be calculated as:

− −1

[1 MPC(1 t)]

inTeRnaTional TRaDe anD capiTal FloWs


International Trade And Capital Flows

Balance of Payment Components

A country’s balance of payments is composed of three main accounts:• The current account balance largely reflects trade in goods and services.• The capital account balance mainly consists of capital transfers and net sales of

non‐produced, non‐financial assets.• The financial account measures net capital flows based on sales and purchases of

domestic and foreign financial assets.

cuRRency exchange RaTes


Currency Exchange Rates

The real exchange rate may be calculated as:

Real exchange rate S (P /P )DC/FC DC/FC FC DC= ×

where:SDC/FC = Nominal spot exchange ratePFC = Foreign price level quoted in terms of the foreign currencyPDC = Domestic price level quoted in terms of the domestic currency

The forward rate may be calculated as:

F1

S

(1 r )

(1 r )or F S

(1 r )

(1 r )DC/FCFC/DC

DC

FCDC/FC DC/FC

DC

FC

= ×++

= ×++

Forward rates are sometimes interpreted as expected future spot rates.

F St t 1= +

(S )

S1 %S(DC/FC)

(r r )

(1 r )t 1

t 1DC FC

FC

− = ∆ =−

++

+

Exchange Rates and the Trade Balance

The Elasticities Approach

Marshall-Lerner condition: ( 1) 0x x M Mω ε + ω ε − >

where:ωx = Share of exports in total tradeωM = Share of imports in total tradeεx = Price elasticity of demand for exportsεM = Price elasticity of demand for imports

This version of the formula is perhaps easiest to remember because it contains the DC term in numerator for all three components: FDC/FC, SDC/FC and (1 + rDC)

Financial Reporting and Analysis

Financial RepoRting Mechanics


Financial Reporting Mechanics

2 Exhibit 10, Vol 3, CFA Program Curriculum 2012

UndeRstanding the incoMe stateMent


Understanding the Income Statement

Basic EPS

=−

Basic EPSNet income Preferred dividends

Weighted average number of shares outstanding

Diluted EPS

Diluted EPS

Net incomePreferreddividends

Convertiblepreferred

dividends

Convertibledebt

interest1 t

Weightedaverageshares

Shares fromconversion of

convertiblepreferred shares

Shares fromconversion of

convertibledebt

Sharesissuable fromstock options

( )

=

−

+ + × −

+ + +

Comprehensive Income

+ =Net income Other comprehensive income Comprehensive income

Ending Shareholders’ Equity

Ending shareholders’ equity Beginning shareholders’ equity Net incomeOther comprehensive income Dividends declared

= + +−

UndeRstanding the Balance sheet


Understanding the Balance Sheet

Gains and Losses on Marketable Securities

Held‐to‐Maturity Securities Available‐for‐Sale Securities Trading Securities

Balance Sheet Reported at cost or amortized cost.

Reported at fair value. Reported at fair value.

Unrealized gains or losses due to changes in market values are reported in other comprehensive income within owners’ equity.

Items recognized on the income statement

Interest income.

Realized gains and losses.

Dividend income.

Interest income.


Dividend income.

Interest income.


Unrealized gains and losses due to changes in market values.

UndeRstanding cash FloW stateMents


Understanding Cash Flow Statements

Cash Flow Classification under U.S. GAAP

IFRS U.S. GAAP

Classification of Cash Flows

Interest and dividends receivedInterest paid

CFO or CFICFO or CFF

CFOCFO

Dividend paidDividends receivedTaxes paid

CFO or CFFCFO or CFICFO, but part of the tax can be categorized as CFI or CFF if it is clear that the tax arose from investing or financing activities.

CFFCFOCFO

Bank overdraft Included as a part of cash equivalents. Not considered a part of cash equivalents and included in CFF.

Presentation Format

CFO(No difference in CFI and CFF presentation)

Direct or indirect method. The former is preferred.

Direct or indirect method. The former is preferred. However, if the direct method is used, a reconciliation of net income and CFO must be included.

Disclosures

Taxes paid should be presented separately on the cash flow statement.

If taxes and interest paid are not explicitly stated on the cash flow statement, details can be provided in footnotes.

CFOInflows OutflowsCash collected from customers.Interest and dividends received.Proceeds from sale of securities held for trading.

Cash paid to employees.Cash paid to suppliers.Cash paid for other expenses.Cash used to purchase trading securities.Interest paid.Taxes paid.

CFIInflows OutflowsSale proceeds from fixed assets.Sale proceeds from long‐term investments.

Purchase of fixed assets.Cash used to acquire LT investment securities.

CFFInflows OutflowsProceeds from debt issuance.Proceeds from issuance of equity instruments.

Repayment of LT debt.Payments made to repurchase stock.Dividends payments.

Cash Flow Statements under IFRS and U.S. GAAP

UndeRstanding cash FloW stateMents


Free Cash Flow to the Firm

FCFF NI NCC [Int * (1 tax rate)] FCInv WCInv= + + − − −

FCFF CFO [Int * (1 tax rate)] FCInv= + − −

Free Cash Flow to Equity

= − +FCFE CFO FCInv Net borrowing

Financial analysis techniqUes


Financial Analysis Techniques

Inventory Turnover

Cost of goods sold

Average inventoryInventory turnover =

Days of Inventory on Hand

= 365

Inventory turnoverDays of inventory on hand (DOH)

Receivables Turnover

= Revenue

Average receivablesReceivables turnover

Days of Sales Outstanding

=( )365

Receivables turnoverDays of sales outstanding DSO

Payables Turnover

= Purchases

Average trade payablesPayables turnover

Number of Days of Payables

= 365

Payables turnoverNumber of days of payables

Working Capital Turnover

= Revenue

Average working capitalWorking capital turnover

Fixed Asset Turnover

= Revenue

Average fixed assetsFixed asset turnover

Total Asset Turnover

Revenue

Average total assetsTotal asset turnover =



Current Ratio

=Current assets

Current liabilitiesCurrent ratio

Quick Ratio

=+ +Cash Short-term marketable investments Receivables

Current liabilitiesQuick ratio

Cash Ratio

=+Cash Short-term marketable investments

Current liabilitiesCash ratio

Defensive Interval Ratio

=+ +Cash Short-term marketable investments Receivables

Daily cash expendituresDefensive interval ratio

Cash Conversion Cycle

= + −DSO DOH Number of days of payablesCash conversion cycle

Debt‐to‐Assets Ratio

=- -Total debt

Total assetsDebt to assets ratio

Debt‐to‐Capital Ratio

- -Total debt

Total debt Shareholders’ equityDebt to capital ratio =

+

Debt‐to‐Equity Ratio

- -Total debt

Shareholders’ equityDebt to equity ratio =

Financial Leverage Ratio

=Average total assets

Average total equityFinancial leverage ratio

Interest Coverage Ratio

= EBIT

Interest paymentsInterest coverage ratio



Fixed Charge Coverage Ratio

=+

+EBIT Lease payments

Interest payments Lease paymentsFixed charge coverage ratio

Gross Profit Margin

=Gross profit

RevenueGross profit margin

Operating Profit Margin

=Operating profit

RevenueOperating profit margin

Pretax Margin

=EBT (earnings before tax, but after interest)

RevenuePretax margin

Net Profit Margin

=Net profit

RevenueNet profit margin

Return on Assets

=Net income

Average total assetsROA

=+ −Net income Interest expense (1 Tax rate)

Average total assetsAdjusted ROA

Operating income or EBIT

Average total assetsOperating ROA =

Return on Total Capital

=+ +

EBIT

Short-term debt Long-term debt EquityReturn on total capital

Return on Equity

=Net income

Average total equityReturn on equity

Return on Common Equity

=−Net income Preferred dividends

Average common equityReturn on common equity



DuPont Decomposition of ROE

Net income

Average shareholders’ equityROE =

2‐Way Dupont Decomposition

Net income

Average total assets


Average shareholders’ equity

ROA Leverage

ROE = ×

↓ ↓


Net income

Revenue

Revenue



Average shareholders’ equity

Net profit margin Asset turnover Leverage

ROE = × ×

↓ ↓ ↓


Interest burden Asset turnover

Net income

EBT

EBT

EBIT

EBIT

Revenue

Revenue



Avg. shareholders’ equity

Tax burden EBIT margin Leverage

↓ ↓

= × × × ×

↓ ↓ ↓

ROE

Price‐to‐Earnings Ratio

=/Price per share

Earnings per shareP E

Price to Cash Flow

=/Price per share

Cash flow per shareP CE

Price to Sales

/Price per share

Sales per shareP S =

Price to Book Value

/Price per share

Book value per shareP BV =



Per Share Ratios

Cash flow from operations

Average number of shares outstandingCash flow per share =

EBITDA

Average number of shares outstandingEBITDA per share =

=Common dividends declared

Weighted average number of ordinary sharesDividends per share

Dividend Payout Ratio

=Common share dividends

Net income attributable to common sharesDividend payout ratio

Retention Rate

Net income attributable to common shares Common share dividends

Net income attributable to common sharesRetention Rate =

−

Growth Rate

= ×Retention rate ROESustainable growth rate

inventoRies


Inventories

LIFO versus FIFO (with rising prices and stable inventory levels.)

LIFO versus FIFO when Prices are Rising

LIFO FIFOCOGS Higher Lower

Income before taxes Lower Higher

Income taxes Lower Higher

Net income Lower Higher

Cash flow Higher Lower

EI Lower Higher

Working capital Lower Higher

Type of RatioEffect on Numerator

Effect on Denominator Effect on Ratio

Profitability ratios NP and GP margins

Income is lower under LIFO because COGS is higher

Sales are the same under both

Lower under LIFO

Debt-to-equity Same debt levels Lower equity under LIFO

Higher under LIFO

Current ratio Current assets are lower under LIFO because EI is lower

Current liabilities are the same

Lower under LIFO

Quick ratio Assets are higher as a result of lower taxes paid

Current liabilities are the same

Higher under LIFO

Inventory turnover COGS is higher under LIFO

Average inventory is lower under LIFO

Higher under LIFO

Total asset turnover Sales are the same Lower total assets under LIFO

Higher under LIFO

The LIFO Method and the LIFO Reserve:

= +EI EI LRFIFO LIFO

whereLR = LIFO Reserve

COGSFIFO = COGSLIFO − (Change in LR during the year)

Net income after tax under FIFO will be greater than LIFO net income after tax by:

× −Change in LIFO Reserve (1 Tax rate)

inventoRies


When converting from LIFO to FIFO assuming rising prices:

Equity (retained earnings) increase by:

LIFO Reserve × (1 − Tax rate)

LIFO Reserve × (Tax rate)

LIFO Reserve

Liabilities (deferred taxes) increase by:

Current assets (inventory) increase by:

long-lived assets


Long-Lived Assets

Financial Statement Effects of Capitalizing versus Expensing

Effect on Financial StatementsInitially when the cost is capitalized

• Noncurrent assets increase.• Cash flow from investing activities decreases.

In future periods when the asset is depreciated or amortized

• Noncurrent assets decrease.• Net income decreases.• Retained earnings decrease.• Equity decreases.

When the cost is expensed • Net income decreases by the entire after‐tax amount of the cost.

• No related asset is recorded on the balance sheet and therefore, no depreciation or amortization expense is charged in future periods.

• Operating cash flow decreases.• Expensed costs have no financial statement

impact in future years.

Capitalizing Expensing

Net income (first year) Higher Lower

Net income (future years) Lower Higher

Total assets Higher Lower

Shareholders’ equity Higher Lower

Cash flow from operations Higher Lower

Cash flow from investing Lower Higher

Income variability Lower Higher

Debt-to-equity Lower Higher

long-lived assets


Straight Line Depreciation

=−

Depreciation expenseOriginal cost Salvage value

Depreciable life

Accelerated Depreciation

DDB depreciation in Year X2

Depreciable lifeBook value at the beginning of Year X= ×

Estimated Useful Life

=Estimated useful lifeGross investment in fixed assets

Annual depreciation expense

Average Cost of Asset

=Average age of assetAccumulated depreciation


Remaining Useful Life

=Remaining useful lifeNet investment in fixed assets


Gross investment in �xed assets = Accumulated depreciation + Net investment in �xed assetsAnnual depreciation expense Annual depreciation expense Annual depreciation expense

Estimated useful or depreciablelife

The historical cost of an assetdivided by its useful life equals

annual depreciation expense underthe straight line method. Therefore,

the historical cost divided byannual depreciation expense

equals the estimated useful life.

Average age of asset

Annual depreciation expensetimes the number of years thatthe asset has been in use equals

accumulated depreciation.Therefore, accumulated

depreciation divided by annualdepreciation equals the average

age of the asset.

Remaining useful life

The book value of the asset dividedby annual depreciation expense

equals the number of years the assethas remaining in its useful life.

incoMe taxes


Income Taxes

Effective Tax rate

=Effective tax rateIncome tax expense

Pretax income

Income Tax Expense

= + −Income tax expense Taxes Payable Change in DTL Change in DTA

Treatment of Temporary Differences

Balance Sheet Item Carrying Value versus Tax Base Results in…

Asset Carrying amount is greater. DTLAsset Tax base is greater. DTA

Liability Carrying amount is greater. DTALiability Tax base is greater. DTL

Income Tax Accounting under IFRS versus U.S. GAAP

IFRS U.S. GAAPISSUE SPECIFIC TREATMENTSRevaluation of fixed assets and intangible assets.

Recognized in equity as deferred taxes.

Revaluation is prohibited.

Treatment of undistributed profit from investment in subsidiaries.

Recognized as deferred taxes except when the parent company is able to control the distribution of profits and it is probable that temporary differences will not reverse in future.

No recognition of deferred taxes for foreign subsidiaries that fulfill indefinite reversal criteria.No recognition of deferred taxes for domestic subsidiaries when amounts are tax‐free.

Treatment of undistributed profit from investments in joint ventures.

Recognized as deferred taxes except when the investor controls the sharing of profits and it is probable that there will be no reversal of temporary differences in future.

No recognition of deferred taxes for foreign corporate joint ventures that fulfill indefinite reversal criteria.

Treatment of undistributed profit from investments in associates.

Recognized as deferred taxes except when the investor controls the sharing of profits and it is probable that there will be no reversal of temporary differences in future.

Deferred taxes are recognized from temporary differences.

incoMe taxes


IFRS U.S. GAAPDEFERRED TAX MEASUREMENTTax rates. Tax rates and tax laws

enacted or substantively enacted.

Only enacted tax rates and tax laws are used.

Deferred tax asset recognition.

Recognized if it is probable that sufficient taxable profit will be available in the future.

Deferred tax assets are recognized in full and then reduced by a valuation allowance if it is likely that they will not be realized.

DEFERRED TAX PRESENTATIONOffsetting of deferred tax assets and liabilities.

Offsetting allowed only if the entity has right to legally enforce it and the balance is related to a tax levied by the same authority.

Same as in IFRS.

Balance sheet classification.

Classified on balance sheet as net noncurrent with supplementary disclosures.

Classified as either current or noncurrent based on classification of underlying asset and liability.

non-cURRent (long-teRM) liaBilities


Non-Current (Long-Term) Liabilities

Income Statement Effects of Lease Classification

Income Statement Item Finance Lease Operating Lease

Operating expenses Lower Higher

Nonoperating expenses Higher Lower

EBIT (operating income) Higher Lower

Total expenses‐ early years Higher Lower

Total expenses‐ later years Lower Higher

Net income‐ early years Lower Higher

Net income‐ later years Higher Lower

Balance Sheet Effects of Lease Classification

Balance Sheet Item Capital Lease Operating Lease

Assets Higher Lower

Current liabilities Higher Lower

Long term liabilities Higher Lower

Total cash Same Same

Cash Flow Effects of Lease Classification

CF Item Capital Lease Operating Lease

CFO Higher Lower

CFF Lower Higher

Total cash flow Same Same

Impact of Lease Classification on Financial Ratios

Ratio

Numerator under Finance

Lease

Denominator under Finance

Lease Effect on Ratio

Ratio Better or Worse under

Finance Lease

Asset turnover Sales‐ same Assets‐ higher Lower Worse

Return on assets*

Net income lower in early

years

Assets‐ higher Lower Worse

Current ratio Current assets-same

Current liabilities-

higher

Lower Worse

Leverage ratios (D/E and D/A)

Debt‐ higher Equity same Assets higher

Higher Worse

Return on equity*

Net income lower in early

years

Equity same Lower Worse

* In early years of the lease agreement.

non-cURRent (long-teRM) liaBilities


Financial Statement Effects of Lease Classification from Lessor’s Perspective

Financing Lease Operating Lease

Total net income Same SameNet income (early years) Higher LowerTaxes (early years) Higher LowerTotal CFO Lower HigherTotal CFI Higher LowerTotal cash flow Same Same

Definitions of Commonly Used Solvency Ratios

Solvency Ratios Description Numerator Denominator

Leverage Ratios

Debt‐to‐assets ratio Expresses the percentage of total assets financed by debt

Total debt Total assets

Debt‐to‐capital ratio Measures the percentage of a company’s total capital (debt + equity) financed by debt.

Total debt Total debt + Total shareholders’ equity

Debt‐to‐equity ratio Measures the amount of debt financing relative to equity financing

Total debt Total shareholders’ equity

Financial leverage ratio Measures the amount of total assets supported by one money unit of equity

Average total assets Average shareholders’ equity

Coverage Ratios

Interest coverage ratio Measures the number of times a company’s EBIT could cover its interest payments.

EBIT Interest payments

Fixed charge coverage ratio Measures the number of times a company’s earnings (before interest, taxes and lease payments) can cover the company’s interest and lease payments.

EBIT + Lease payments

Interest payments + Lease payments

Financial RepoRting qUality


Financial Reporting Quality

Relationship between Financial Reporting Quality and Earnings Quality

Financial Reporting Quality

Low High

Earnings (Results) Quality

High

LOW financial reporting quality impedes assessment of earnings quality and impedes valuation.

HIGH financial reporting quality enables assessment. HIGH earnings quality increases company value.

Low HIGH financial reporting quality enables assessment. LOW earnings quality decreases company value.

Corporate Finance

Capital Budgeting


Capital Budgeting

Net Present Value (NPV)

NPVCF

(1 r)Outlayt

tt 1

n

∑=+

−=

where:CFt = after‐tax cash flow at time, t.r = required rate of return for the investment. This is the firm’s cost of capital adjusted for

the risk inherent in the project.Outlay = investment cash outflow at t = 0.

Internal Rate of Return (IRR)

CF

(1 IRR)Outlay

CF

(1 IRR)Outlay 0t

tt 1

nt

tt 1

n

∑ ∑+=

+− =

= =

Average Accounting Rate of Return (AAR)

AARAverage net income

Average book value=

Profitability Index

PIPV of future cash flows

Initial investment1

NPV

Initial investment= = +

Cost of Capital


Cost of Capital

Weighted Average Cost of Capital

WACC (w )(r )(1 t) (w )(r ) (w )(r )d d p p e e= − + +

where:wd = Proportion of debt that the company uses when it raises new fundsrd = Before‐tax marginal cost of debtt = Company’s marginal tax ratewp = Proportion of preferred stock that the company uses when it raises new fundsrp = Marginal cost of preferred stockwe = Proportion of equity that the company uses when it raises new fundsre = Marginal cost of equity

To Transform Debt‐to‐equity Ratio into a Component’s Weight

DE

1 DE

D

D Ew

w w 1

d

d e

+=

+=

+ =

Valuation of Bonds

PPMT

1r2

FV

1r2

0d

tt 1

n

dn∑=

+

++

=

where:P0 = current market price of the bond.PMTt = interest payment in period t.rd = yield to maturity on BEY basis.n = number of periods remaining to maturity.FV = Par or maturity value of the bond.

Valuation of Preferred Stock

VD

rpp

p=

where:Vp = current value (price) of preferred stock.Dp = preferred stock dividend per share.rp = cost of preferred stock.

Cost of Capital


Required Return on a Stock

Capital Asset Pricing Model

= + β −r R [E(R ) R ]e F i M F

where:[E(RM) − Rf] = Equity risk premium.RM = Expected return on the market.βi = Beta of stock. Beta measures the sensitivity of the stock’s returns to changes in market

returns.RF = Risk‐free rate.re = Expected return on stock (cost of equity).

Dividend Discount Model

PD

r g01

e

=−

where:P0 = current market value of the security.D1 = next year’s dividend.re = required rate of return on common equity.g = the firm’s expected constant growth rate of dividends.

Rearranging the above equation gives us a formula to calculate the required return on equity:

rD

Pge

1

0

= +

Sustainable Growth Rate

g 1D

EPSROE( )= −

×

Where (1 − (D/EPS)) = Earnings retention rate

Bond Yield plus Risk Premium Approach

re = rd + risk premium

Cost of Capital


To Unlever the beta

1

1 1 tDE

ASSET EQUITY

( )β = β

+ −

To Lever the beta

1 1 tD

EPROJECT ASSET ( )β = β + −

Country Risk Premium

r R [E(R ) R CRP]e F M F= + β − +

Country riskpremium

Sovereign yieldspread

Annualized standard deviation of equity index

Annualized standard deviation of sovereignbond market in terms of the developed market

currency

= ×

Break pointAmount of capital at which a component’s cost of capital changes

Proportion of new capital raised from the component=

MeasuRes of leveRage


Measures of Leverage

Degree of Operating Leverage

DOLPercentage change in operating income

Percentage change in units sold=

DOLQ P V

Q P V F

( )( )

= × −× − −

where:Q = Number of units soldP = Price per unitV = Variable operating cost per unitF = Fixed operating costQ × (P − V) = Contribution margin (the amount that units sold contribute to covering fixed

costs)(P − V) = Contribution margin per unit

Degree of Financial Leverage

DFLPercentage change in net income

Percentage change in operating income=

DFL[Q(P V) F](1 t)

[Q(P V) F C](1 t)

[Q(P V) F]

[Q(P V) F C]= − − −

− − − −= − −

− − −

where:Q = Number of units soldP = Price per unitV = Variable operating cost per unitF = Fixed operating costC = Fixed financial costt = Tax rate

MeasuRes of leveRage


Degree of Total Leverage

DTLPercentage change in net income

Percentage change in the number of units sold=

DTL DOL DFL= ×

DTLQ (P V)

[Q(P V) F C]= × −

− − −

where:Q = Number of units produced and soldP = Price per unitV = Variable operating cost per unitF = Fixed operating costC = Fixed financial cost

Break point

PQ VQ F C= + +

where:P = Price per unitQ = Number of units produced and soldV = Variable cost per unitF = Fixed operating costsC = Fixed financial cost

The breakeven number of units can be calculated as:

QF C

P VBE = +−

Operating breakeven point

PQ VQ FOBE OBE= +

QF

P VOBE =−

WoRking Capital ManageMent


Working Capital Management

Current RatioCurrent assets

Current liabilities=

Quick RatioCash Short term marketable investments Receivables

Current liabilities= + +

Accounts receivable turnoverCredit sales

Average receivables=

Number of days of receivablesAccounts receivable

Average days sales on credit

Accounts receivable

Sales on credit / 365

=

=

Inventory turnoverCost of goods sold

Average inventory=

Number of days of inventoryInventory

Average day’s cost of goods sold

Inventory

Cost of goods sold / 365

=

=

Payables turnoverPurchases

Average trade payables=

Number of days of payablesAccounts payables

Average day’s purchases

Accounts payables

Purchases / 365

=

=

Operating cycle = Number of days of inventory + Number of days of receivables

Net operating cycle = Number of days of inventory + Number of days of receivables − Number of days of payables

Purchases = Ending inventory + COGS − Beginning inventory

WoRking Capital ManageMent


Money market yieldFace value price

Price

360

DaysHolding period yield

360

Days= −

×

= ×

Bond equivalent yieldFace value price

Price

365

DaysHolding period yield

365

Days= −

×

= ×

Discount basis yieldFace value price

Face value

360

Days% discount

360

Days= −

×

= ×

% DiscountFace value Price

Price= −

Inventory turnoverCost of goods sold

Average inventory=

Number of days of inventoryInventory

Average days cost of goods sold

Inventory

Cost of goods sold / 365

365

Inventory turnover

=

=

=

Implicit rate Cost of trade credit 1Discount

1 Discount1

365Number of days

beyond discount period= = +−

−

Number of days of payablesAccounts payable

Average day’s purchases

Accounts payable

Purchases / 365

365

Payables turnover

=

=

Line of credit costInterest Commitment fee

Loan amount= +

Banker’s acceptance costInterest

Net proceeds

Interest

Loan amount Interest= =

−

Interest Dealer’s commission Backup costs

Loan amount Interest

+ +−

Portfolio Management

Risk ManageMent: an intRoduction


Risk Management: An Introduction

Figure 1-1: The Risk Management Framework in an Enterprise

Risk Drivers

Board

Goals

RiskTolerance

(Allocate to)Risky Activities

Risk Mitigation& Management

Establish RiskManagementInfrastructure

Identify Risks

Measure Risks

Montitor RisksRisks

inLine?

Reports(Communications)

StrategicAnalysis

YES

NO

Risk ExposuresRisk

Budgeting

Ris

k G

over

nanc

e

Policies & Processes

MODIFY

Strategies

Management

PoRtfolio Risk and RetuRn: PaRt i


Portfolio Risk and Return: Part I

Holding Period Return

RP P D

P

P P

P

D

PCapital gain Dividend yield

P D

P1

t t 1 t

t 1

t t 1

t 1

t

t 1

T T

0

=− +

=−

+ = +

= + −

−

−

−

− −

where:Pt = Price at the end of the periodPt-1 = Price at the beginning of the periodDt = Dividend for the period

Holding Period Returns for more than One Period

R [(1 R ) (1 R ) . . . (1 R )] 11 2 n= + × + × × + −

where:R1, R2, . . . , Rn are sub‐period returns

Geometric Mean Return

R {[(1 R ) (1 R ) . . . (1 R )] } 11 2 n1/n= + × + × × + −

Annualized Return

= + −r (1 r ) 1annual periodn

where:r = Return on investmentn = Number of periods in a year



Portfolio Return

R w R w Rp 1 1 2 2= +

where:Rp = Portfolio returnw1 = Weight of Asset 1w2 = Weight of Asset 2R1 = Return of Asset 1R2 = Return of Asset 2

Variance of a Single Asset

∑σ =

− µ=

(R )

T2

t2

t 1

T

where:Rt = Return for the period tT = Total number of periodsμ = Mean of T returns

Variance of a Representative Sample of the Population

∑=

−

−=s

(R R)

T 12

t2

t 1

T

where:R = mean return of the sample observationss2 = sample variance

Standard Deviation of an Asset

∑ ∑σ =

− µ=

−

−= =

(R )

Ts

(R R)

T 1

t2

t 1

T

t2

t 1

T

Variance of a Portfolio of Assets

w w Cov(R ,R )

w Var(R ) w w Cov(R ,R )

P2

i j i ji, j 1

N

P2

i2

i i j i ji, j 1, i j

N

i 1

N

∑

∑∑

σ =

σ = +

=

= ≠=



Standard Deviation of a Portfolio of Two Risky Assets

w w 2w w or w w 2w w Covp 12

12

22

22

1 2 1 2 1,2 12

12

22

22

1 2 1,2σ = σ + σ + σ σ ρ σ + σ +

Utility Function

U E(R)1

2A 2= − σ

where:U = Utility of an investmentE(R) = Expected returnσ2 = Variance of returnsA = Additional return required by the investor to accept an additional unit of risk.

PoRtfolio Risk and RetuRn: PaRt ii


Portfolio Risk and Return: Part II

Capital Allocation Line

The CAL has an intercept of RFR and a constant slope that equals:

[E(R ) RFR]i

i

−σ

Expected Return on portfolios that lie on CML

= + −E(R ) w R (1 w ) E(R )p 1 f 1 m

Variance of portfolios that lie on CML

σ = σ + − σ + −w (1 w ) 2w (1 w )Cov(R R )212

f2

12

m2

1 1 f , m

Equation of CML

= + −σ

× σE(R ) RE(R ) R

p fm f

mp

where:y‐intercept = Rf = risk‐free rate

= −σ

=slopeE(R ) R

market price of risk.m f

m

Systematic and Nonsystematic Risk

Total Risk = Systematic risk + Unsystematic risk

Return‐Generating Models

∑∑− = β = β − + β==

E(R ) R E(F ) [E(R ) R ] E(F )i f ij j i1 m f ij jj 2

k

j 1

k

The Market Model

R R ei i i m i= α + β +

Calculation of Beta

Cov(R ,R )i

i m

m2

i,m i m

m2

i,m i

m

β =σ

=ρ σ σ

σ=

ρ σσ

PoRtfolio Risk and RetuRn: PaRt ii


The Capital Asset Pricing Model

E(R ) R [E(R ) R ]i f i m f= + β −

Sharpe ratio

Sharpe ratioR Rp f

p

=−

σ

Treynor ratio

Treynor ratioR Rp f

p

=−

β

M‐squared (M2)

M (R R ) (R R )2p f

m

pm f= − σ

σ− −

Jensen’s alpha

R [R (R R )]p p f p m fα = − + β −

Security Characteristic Line

R R (R R )i f i i m f− = α + β −

Equity

Market OrganizatiOn and Structure

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Market Organization and Structure

The price at which an investor who goes long on a stock receives a margin call is calculated as:

P(1 Initial margin)

(1 Maintenance margin)0 × −−

Security Market indiceS

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Security Market Indices

The value of a price return index is calculated as follows:

V

n P

DPRI

i ii 1

N

∑= =

where:VPRI = Value of the price return indexni = Number of units of constituent security i held in the index portfolioN = Number of constituent securities in the indexPi = Unit price of constituent security iD = Value of the divisor

Price Return

The price return of an index can be calculated as:

PRV V

VIPRI1 PRI0

PRI0

=−

where:PRI = Price return of the index portfolio (as a decimal number)VPRI1 = Value of the price return index at the end of the periodVPRI0 = Value of the price return index at the beginning of the period

The price return of each constituent security is calculated as:

PRP P

Pii1 i0

i0

=−

where:PRi = Price return of constituent security i (as a decimal number)Pi1 = Price of the constituent security i at the end of the periodPi0 = Price of the constituent security i at the beginning of the period

The price return of the index equals the weighted average price return of the constituent securities. It is calculated as:

PRI = w1PR1 + w2PR2 + . . . + wNPRN

where:PRI = Price return of the index portfolio (as a decimal number)PRi = Price return of constituent security i (as a decimal number)wi = Weight of security i in the index portfolioN = Number of securities in the index



Total Return

The total return of an index can be calculated as:

TRV V Inc

VIPRI1 PRI0 I

PRI0

=− +

where:TRI = Total return of the index portfolio (as a decimal number)VPRI1 = Value of the total return index at the end of the periodVPRI0 = Value of the total return index at the beginning of the periodIncI = Total income from all securities in the index held over the period

The total return of each constituent security is calculated as:

TRP P Inc

Pi1i 0i i

0i

=− +

where:TRi = Total return of constituent security i (as a decimal number)P1i = Price of constituent security i at the end of the periodP0i = Price of constituent security i at the beginning of the periodInci = Total income from security i over the period

The total return of the index equals the weighted average total return of the constituent securities. It is calculated as:

TRI = w1TR1 + w2TR2 + . . . + wNTRN

where:TRI = Total return of the index portfolio (as a decimal number)TRi = Total return of constituent security i (as a decimal number)wi = Weight of security i in the index portfolioN = Number of securities in the index

Calculation of Index Returns over Multiple Time Periods

Given a series of price returns for an index, the value of a price return index can be calculated as:

V V (1 PR ) (1 PR ) . . . (1 PR )PRIT PRI0 I1 I2 IT= + + +

where:VPRI0 = Value of the price return index at inceptionVPRIT = Value of the price return index at time tPRIT = Price return (as a decimal number) on the index over the period



Similarly, the value of a total return index may be calculated as:

V V (1 TR ) (1 TR ) . . . (1 TR )TRIT TRI0 I1 I2 IT= + + +

where:VTRI0 = Value of the index at inceptionVTRIT = Value of the index at time tTRIT = Total return (as a decimal number) on the index over the period

Price Weighting

wP

PiP i

ii 1

N

∑=

=

Equal Weighting

w1

NiE =

where:wi = Fraction of the portfolio that is allocated to security i or weight of security iN = Number of securities in the index

Market‐Capitalization Weighting

wQ P

Q PiM i i

j jj 1

N

∑=

=

where:wi = Fraction of the portfolio that is allocated to security i or weight of security iQi = Number of shares outstanding of security iPi = Share price of security iN = Number of securities in the index

The float‐adjusted market‐capitalization weight of each constituent security is calculated as:

wf Q P

f Q PiM i i i

j j jj 1

N

∑=

=

where:fi = Fraction of shares outstanding in the market floatwi = Fraction of the portfolio that is allocated to security i or weight of security iQi = Number of shares outstanding of security iPi = Share price of security iN = Number of securities in the index



Fundamental Weighting

wF

FiF i

jj 1

N

∑=

=

where:Fi = A given fundamental size measure of company i

OvervieW Of equity SecuritieS


Overview of Equity Securities

Return Characteristics of Equity Securities

Total Return, Rt = (Pt – Pt−1 + Dt) / Pt−1

where:Pt‐1 = Purchase price at time t − 1Pt = Selling price at time tDt = Dividends paid by the company during the period

Accounting Return on Equity

= =+ −

ROENI

Average BVE

NI

(BVE BVE )/2tt

t

t

t t 1

equity valuatiOn cOnceptS and BaSic tOOlS


Equity Valuation Concepts and Basic Tools

Dividend Discount Model (DDM)

ValueD

(1 k )

D

(1 k )

D

(1 k )1

e1

2

e2

e

�=+

++

+ ++

∞∞

ValueD

(1 k )t

et

t 1

n

∑=+=

One year holding period:

Valuedividend to be received

(1 k )

year-end price

(1 k )e1

e1=

++

+

Multiple‐Year Holding Period DDM

VD

(1 K )

D

(1 k )

P

(1 k )1

e1

2

e2

n

en�=

++

++ +

+

where:Pn = Price at the end of n years.

Infinite Period DDM (Gordon Growth Model)

PVD (1 g )

(1 k )

D (1 g )

(1 k )

D (1 g )

(1 k )

D (1 g )

(1 k )00 c

1

e1

0 c2

e2

0 c3

e3

0 c

e

�=+

++

++

++

++ +

++

∞

∞

This equation simplifies to:

=+

−=

−PV

D (1 g )

(k g )

D

k g0 c

1

e c1

1

e c

The long‐term (constant) growth rate is usually calculated as:

gc = RR × ROE



Multi‐Stage Dividend Discount Model

ValueD

(1 k )

D

(1 k )

D

(1 k )

P

(1 k )1

e1

2

e2

n

en

n

en�=

++

++ +

++

+

where:

PD

k gnn 1

e c

=−

+

Dn = Last dividend of the supernormal growth periodDn+1 = First dividend of the constant growth period

The Free‐Cash‐Flow‐to‐Equity (FCFE) Model

FCFE = CFO − FC Inv + Net borrowing

Analysts may calculate the intrinsic value of the company’s stock by discounting their projections of future FCFE at the required rate of return on equity.

VFCFE

(1 k )0t

et

t 1∑=

+=

∞

Value of a Preferred Stock

When preferred stock is non‐callable, non‐convertible, has no maturity date and pays dividends at a fixed rate, the value of the preferred stock can be calculated using the perpetuity formula:

VD

r00=

For a non‐callable, non‐convertible preferred stock with maturity at time, n, the value of the stock can be calculated using the following formula:

VD

(1 r)

F

(1 r)0t

t nt 1

n

∑=+

++=

where:V0 = value of preferred stock today (t = 0)Dt = expected dividend in year t, assumed to be paid at the end of the yearr = required rate of return on the stockF = par value of preferred stock



Price Multiples

P

E

D /E

r g0

1

1 1=−

Price to cash flow ratioMarket price of share

Cash flow per share=

Price to sales ratioMarket price per share

Net sales per share=

Price to sales ratioMarket value of equity

Total net sales=

P/BVCurrent market price of share

Book value per share=

P/BVMarket value of common shareholders’ equity

Book value of common shareholders’ equity=

where:Book value of common shareholders’ equity =(Total assets – Total liabilities) – Preferred stock

Enterprise Value Multiples

EV/EBITDA

where:EV = Enterprise value and is calculated as the market value of the company’s common stock plus the market value of outstanding preferred stock if any, plus the market value of debt, less cash and short term investments (cash equivalents).

Fixed Income

Fixed-income SecuritieS: deFining elementS

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Fixed-Income Securities: Defining Elements

Bond Coupon

Coupon = Coupon rate × Par value

Coupon Rate (Floating)

Coupon Rate = Reference rate + Quoted margin

Value of embedded call option = Value of non‐callable bond − Value of callable bond

Value of callable bond = Value of non‐callable bond − Value of embedded call option

Coupon Rate (Inverse Floaters)

Callable Bonds

Putable Bonds

Value of putable bond = Value of non‐putable bond + Value of embedded put option

Value of embedded put option = Value of putable bond − Value of non‐putable bond

Traditional Analysis of Convertible Securities

Conversion value = Market price of common stock × Conversion ratio

Market conversion priceMarket price of convertible security

Conversion ratio=

Coupon rate = K − L × (Reference rate)

Market conversion premium per share = Market conversion price − Current market price

Market conversion premium ratioMarket conversion premium per share

Market price of common stock=

Premium payback periodMarket conversion premium per share

Favorable income differential per share=

Favorable income differential per shareCoupon interest (Conversion ratio Common stock dividend per share)

Conversion ratio= − ×

Premium over straight valueMarket price of convertible bond

Straight value1= −

introduction to Fixed-income VAluAtion

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Introduction to Fixed-Income Valuation

Pricing Bonds with Spot Rates

PVPMT

(1 Z )

PMT

(1 Z )

PMT FV

(1 Z )11

22

NN=

++

++…+ +

+

z1 = Spot rate for Period 1z2 = Spot rate for Period 2zN = Spot rate for Period N

Flat Price, Accrued Interest and the Full Price

Figure: Valuing a Bond between Coupon‐Payment Dates

PV PV AIFull Flat= +

AI t/T PMT= ×

PV PV (1 r)Full t/T= × +

Semiannual bond basis yield or semiannual bond equivalent yield

1SAR

M1

SAR

NM

MN

N

+

= +

Important: What we refer to as stated annual rate (SAR) is referred to in the curriculum as APR or annual percentage rate. We stick to SAR to keep your focus on a stated annual rate versus the effective annual rate. Just remember that if you see an annual percentage rate on the exam, it refers to the stated annual rate.

Current yield

Current yieldAnnual cash coupon payment

Bond price=

introduction to Fixed-income VAluAtion


Option‐adjusted price

Value of non‐callable bond (option‐adjusted price) = Flat price of callable bond + Value of embedded call option

Pricing formula for money market instruments quoted on a discount rate basis:

PV FV 1Days

yearDR= × − ×

DRYear

Days

FV PV

FV=

× −

Pricing formula for money market instruments quoted on an add‐on rate basis:

PV=FV

1DaysYear

AOR+ ×

AORYear

Days

FV PV

PV=

× −

Yield Spreads over the Benchmark Yield Curve

PVPMT

(1 z Z)

PMT

(1 z Z). . .

PMT FV

(1 z Z)11

22

NN=

+ ++

+ ++ + +

+ +

• The benchmark spot rates z1, z2, zN are derived from the government yield curve (or from fixed rates on interest rate swaps).

• Z refers to the z‐spread per period. It is constant for all time periods.

Option‐adjusted Spread (OAS)

OAS = z‐spread − Option value (bps per year)

introduction to ASSet-BAcked SecuritieS


Introduction to Asset-Backed Securities

Parties to the Securitization

Party DescriptionParty in Illustration

Seller Originates the loans and sells loans to the SPV ABC Company

Issuer/Trust The SPV that buys the loans from the seller and issues the asset-backed securities

SPV

Servicer Services the loans Servicer

SMMPrepayment in month

Beginning mortgage balance for month Scheduled principal payment in month

t

t tt =

−

underStAnding Fixed-income riSk And return


Understanding Fixed-Income Risk and Return

Macaulay Duration

MacDur1 r

r

1 r [N (c r)]

c [(1 r) 1] r(t/T)N= + − + + × −

× + − +

−

c = Coupon rate per period (PMT/FV)

Modified Duration

ModDurMacDur

1 r=

+

Modified duration has a very important application in risk management. It can be used to estimate the percentage price change for a bond in response to a change in its yield‐to‐maturity.

% PV AnnModDur YieldFull∆ ≈ − × ∆

If Macaulay duration is not already known, annual modified duration can be estimated using the following formula:

ApproxModDur(PV ) (PV )

2 ( Yield) (PV )0

= −× ∆ ×

− +

We can also use the approximate modified duration (ApproxModDur) to estimate Macaulay duration (ApproxMacDur) by applying the following formula:

ApproxMacDur ApproxModDur (1 r)= × +

Effective Duration

EffDur(PV ) (PV )

2 ( Curve) (PV )+

0

= −× ∆ ×

−



Duration of a Bond Portfolio

Portfolio duration w D w D w D1 1 2 2 N N= + +…+

Annual ModDurAnnual MacDur

1 r=

+

Money Duration

MoneyDur = AnnModDur × PVFull

The estimated (dollar) change in the price of the bond is calculated as:

ΔPVFull = – MoneyDur × ΔYield

Price Value of a Basis Point

PVBP(PV ) (PV )

2+= −−

A related statistic is basis point value (BPV), which is calculated as:

BPV = MoneyDur × 0.0001 (1 bps expressed as a decimal)

Annual Convexity

ApproxCon(PV ) (PV ) [2 (PV )]

( Yield) (PV )+ 0

20

=+ − ×

∆ ×−

Once we have an estimate for convexity, we can estimate the percentage change in a bond’s full price as:

% PV ( AnnModDur Yield)1

2AnnConvexity ( Yield)Full 2∆ ≈ − × ∆ + × × ∆



Money convexity

PV ( MoneyDur Yield)1

2MoneyCon ( Yield)Full 2∆ ≈ − × ∆ + × × ∆

Effective convexity

EffCon[(PV ) (PV )] [2 (PV )]

( Curve) (PV )+ 0

20

=+ − ×

∆ ×−

Yield Volatility

% PV ( AnnModDur Yield)1

2AnnConvexity ( Yield)Full 2∆ ≈ − × ∆ + × × ∆

Duration Gap

Duration gap Macaulay duration Investment horizon= −

FundAmentAlS oF credit AnAlySiS


Fundamentals of Credit Analysis

Expected Loss

Expected loss Default probability Loss severity given default= ×

Yield on a corporate bond:

Yield on a corporate bond Real risk-free interest rate Expected inflation rateMaturity premium Liquidity premium Credit spread

= ++ + +

Yield Spread:

Yield spread Liquidity premium Credit spread= +

For small, instantaneous changes in the yield spread, the return impact (i.e. the percentage change in price, including accrued interest) can be estimated using the following formula:

Return impact Modified duration Spread≈ − × ∆

For larger changes in the yield spread, we must also incorporate the (positive) impact of convexity into our estimate of the return impact:

Return impact (MDur Spread) (1/2 Convexity Spread )2≈ − × ∆ + × × ∆

Derivatives

Basics of Derivative Pricing anD valuation


Basics of Derivative Pricing and Valuation

Fundamental value of an asset (S0) that incurs costs (θ) and generates benefits (γ):

=+ + λ

− θ + γSE(S )

(1 r )0T

T

Arbitrage and Replication:

Asset Derivative Risk-free asset+ =

− = −Asset Risk-free asset Derivative

− = −Derivative Risk-free asset Asset

Forward Contract Payoffs:

ST > F(0,T) ST < F(0,T)

Long position ST – F(0,T)(Positive payoff)

ST – F(0,T)(Negative payoff)

Short position –[ST – F(0,T)](Negative payoff)

–[ST – F(0,T)](Positive payoff)

Forward price:

= +F(0,T) S (1 r)0T

F(0,T) (S )(1 r) or F(0,T) S (1 r) ( )(1 r)0T

0T T= − γ + θ + = + − γ − θ +

*Note that benefits (γ) and costs (θ) are expressed in terms of present value.

Value of a forward contract:

= − + −V (0,T) S [F(0,T) / (1 r) ]t tT t

= − γ − θ + − + −V (0,T) S ( )(1 r) [F(0,T) / (1 r) ]t tt T t

Time Long Position Value Short Position Value

At initiation Zero, as the contract is priced to prevent arbitrage

Zero, as the contract is priced to prevent arbitrage

During life of the contract ( )

−

−S

F(0,T)

1+rt T t ( )

−−

F(0,T)

1+rST t t

At expiration ST – F(0,T) F(0,T) – ST



Net payment made (received) by the fixed‐rate payer on a swap:

Net fixed rate payment [Swap fixed rate (LIBOR spread)] (No. of days/360) NPt t 1* *= − +−

Call Option Payoffs

Option Position Descriptions Payoff

ST > X ST < X

Option holder exercises the option

Option holder does not exercise the option

Call option holder Choice to buy the underlying asset for X

ST – X 0

Call option writer Obligation to sell the underlying asset for X if the option holder chooses to exercise the option

– (ST – X) 0

Moneyness and Exercise Value of a Call Option

Moneyness

Current Market Price (St) versus Exercise Price (X)

Intrinsic Value Max [0, (St – X)]

In‐the‐money St is greater than X St – X

At‐the‐money St equals X 0

Out‐of‐the‐money St is less than X 0

Put Option Payoffs

Option Position Descriptions Payoff

ST < X ST > XOption holder exercises the option

Option holder does not exercise the option

Put option holder Choice to sell the underlying asset for X

X – ST 0

Put option writer Obligation to buy the underlying asset for X if the option holder chooses to exercise the option

– (X – ST) 0



Moneyness and Exercise Value of a Put Option

Moneyness

Current Market Price (St) versus Exercise Price (X)

Intrinsic Value Max [0, (X – St)]

In‐the‐money St is less than X X – St

At‐the‐money St equals X 0

Out‐of‐the‐money St is greater than X 0

Fiduciary Call and Protective Put Payoffs

Security Value if ST > X Value if ST < X

Call option ST – X Zero

Zero coupon bond X X

Fiduciary call payoff ST X

Put option Zero X – ST

Stock ST ST

Protective put payoff ST X

Put‐Call Parity

++

= +cX

(1 R )p S0

FT 0 0

Combining Portfolios to Make Synthetic Securities

StrategyConsisting of Value Equals Strategy Consisting of Value

fiduciary call

long call + long bond +

+c

X

(1 R )0F

T

= Protective put long put + long underlying asset

p0 + S0

long call long call c0 = Synthetic call long put + long underlying asset + short bond

+ −+

p SX

(1 R )0 0F

T

long put long put p0 = Synthetic put long call + short underlying asset + long bond

− ++

c SX

(1 R )0 0F

T

long underlying asset

long underlying asset

S0 = Synthetic underlying asset

long call + long bond + short put +

+−c

X

(1 R )p0

FT 0

long bond long bond

+X

(1 R )FT

= Synthetic bond

long put + long underlying asset + short call

p0 + S0 – c0



Lowest Prices of European Calls and Puts

≥ −+

c Max[0,SX

(1 R )]0 0

FT

≥+

−p Max[0,X

(1 R )S ]0

FT 0

Put‐Call Forward Parity

− = −+

p c[X F(0,T)]

(1 R )0 0F

T

Binomial Option Pricing

= + −+

+ −c

πc (1 π)c

(1 r)

= + −−

π(1 r d)

(u d)

Hedge ratio

= −−

−

−nc c

S S

+

+

Lowest Prices of American Calls and Puts

C Max[0, S X/(1 + RFR) ]0 0T≥ −

P Max[0, (X S )]0 0≥ −



Summary of Options Strategies

Call Put

Holder

CT = max(0,ST − X)Value at expiration = CT

Profit: Π = CT − C0

Maximum profit = ∞Maximum loss = C0

Breakeven: ST* = X + C0

PT = max(0,X − ST)Value at expiration = PT

Profit: Π = PT − P0

Maximum profit = X − P0 Maximum loss = P0

Breakeven: ST* = X − P0

Writer

CT = max(0,ST − X)Value at expiration = –CT

Profit: Π = –CT − C0

Maximum profit = C0

Maximum loss = ∞Breakeven: ST* = X + C0

PT = max(0,X − ST)Value at expiration = –PT

Profit: Π = –PT − P0

Maximum profit = P0

Maximum loss = X − P0 Breakeven: ST* = X − P0

Where: C0, CT = price of the call option at time 0 and time TP0, PT = price of the put option at time 0 and time TX = exercise priceS0, ST = price of the underlying at time 0 and time TV0, VT = value of the position at time 0 and time TΠ = profit from the transaction: VT − V0

r = risk‐free rate

Covered Call

Value at expiration: VT = ST − max(0,ST − X)Profit: Π = VT − S0 + C0

Maximum profit = X − S0 + C0

Maximum loss = S0 − C0

Breakeven: ST* = S0 − C0

Protective Put

Value at expiration: VT = ST + max(0,X − ST)Profit: Π = VT − S0 − P0

Maximum profit = ∞Maximum loss = S0 + P0 − XBreakeven: ST* = S0 + P0

Alternative Investments

IntroductIon to AlternAtIve Investments


Introduction to Alternative Investments

Pricing of Commodity Futures Contracts

Futures price = Spot price (1 + r) + Storage costs − Convenience yield

r = Short‐term risk‐free rate

wiley cfa level i formula sheets

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