Elan Guides Formula Sheet CFA 2013 Level 2

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Sample covariance = Cov (X,Y) =



n i = 1 (Xi  X)(Yi  Y)/(n  1) where: n = sample size

X_i = ith observation of Variable X X = mean observation of Variable X Y_i = ith observation of Variable Y Y = mean observation of Variable Y

Sample correlation coefficient = r =Cov (X,Y) sXsY

Sample variance = s_X2=



n i = 1

(X_i  X)2_{/(n  1)}

Sample standard deviation = s_X = s_X2

Test-stat = t = r n  2 1  r2 Where: n = Number of observations r = Sample correlation Test statistic

Regression model equation = Y_i = b0 + b1Xi + i, i = 1,...., n

 b1 and b0 are the regression coefficients.

 b₁ is the slope coefficient.

Linear Regression with One Independent Variable

C

ORRELATION AND

R

EGRESSION

Regression line equation = Yˆ_i = bˆ0 + bˆ1Xi , i = 1,...., n



n i = 1 [Y_i  (bˆ₀  bˆ₁X_i)]2 Regression Residuals where:

Y_i = Actual value of the dependent variable bˆ0 + bˆ1Xi = Predicted value of dependent variable

SEE = 1/2

(



_{n  2}

)

n i = 1 (Y_i  bˆ₀  bˆ₁X_i)2 1/2

(



)

n i = 1 (_ˆi)2 n  2 = =

(

SSE

)

n  2 1/2

The Standard Error of Estimate

Hypothesis Tests on Regression Coefficients CAPM: R_ABC = R_F + _ABC(R_M – R_F)

R_ABC – R_F =  + _ABC(R_M – R_F) + 

 The intercept term for the regression, b₀, is .  The slope coefficient for the regression, b₁, is _ABC The Coefficient of Determination

R2 = Explained variation = Total variation

Total variation  Unexplained variation Total variation

= 1 

Total variation Unexplained variation

Total variation = Unexplained variation + Explained variation



n i = 1

(Y^_i  Y )2

RSS =  Explained variation

The regression sum of squares (RSS) QUANTITATIVE METHODS

s_f2 =s2 1  1  n (X  X)2 (n  1) s_x2

[

]

Y  t^ _c s_f Prediction Intervals Source of Variation Regression (explained) Error (unexplained) Total Degrees of Freedom k n k + 1) n 1 Sum of Squares RSS SSE SST

Mean Sum of Squares

MSR = RSS k RSS 1 = RSS = MSE = SSE n 2

k = the number of slope coefficients in the regression. ANOVA Table

Multiple regression equation = Y_i = b₀ + b₁X_1i + b₂X_2i + . . .+ b_k X_ki + _i, i = 1,2, . . . , n Y_i X_ji b₀ b₁, . . . , b_k _i n

= the ith observation of the dependent variable Y

= the ith observation of the independent variable X_j , j = 1,2, . . . , k = the intercept of the equation

= the slope coefficients for each of the independent variables = the error term for the ith observation

= the number of observations Multiple regression equation

ˆ_i = Y_i  Yˆ_i = Y_i  (bˆ₀ + bˆ₁X_1i + bˆ₂X_2i + . . .+ bˆ_k X_ki) Residual Term

M

ULTIPLE

R

EGRESSIONAND

I

SSUESIN

R

EGRESSION

A

NALYSIS

b_j ± (t_c  s_b j)

ˆ _ˆ

estimated regression coefficient ± (critical t-value)(coefficient standard error) Confidence Intervals F-stat = RSS/k SSE/[n k + 1)] MSR MSE =

F-statistic

(1 R2) n 1 n k1

(

)

Adjusted R2 _{= R}2 _{= 1 }

R2 ₌ Total variation  Unexplained variation

Total variation SST  SSE SST = RSS SST = R2 _{and Adjusted R}2

2 _{= nR}2 _{with k degrees of freedom} n = Number of observations

Testing for Heteroskedasticity- The Breusch-Pagan (BP) Test MULTIPLE REGRESSION AND ISSUES IN REGRESSION

Y_i = b₀ + b₁lnX_1i + b₂X_2i +  Model Specification Errors

y_t = b0 + b1t + t, t = 1, 2, . . . , T

where:

y_t = the value of the time series at time t (value of the dependent variable) b₀ = the y-intercept term

b1 = the slope coefficient/ trend coefficient

t = time, the independent or explanatory variable Linear Trend Models

Testing for Serial Correlation- The Durban-Watson (DW) Test

DW  2(1 – r); where r is the sample correlation between squared residuals from one period and those from the previous period.

Value of Durbin-Watson Statistic

Inconclusive d_l d_u (H₀: No serial correlation) Reject H₀, conclude Positive Serial Correlation Inconclusive Do not Reject H₀ Reject H₀, conclude Negative Serial Correlation 0 4  d_u 4  d_l 4

Problems in Linear Regression and Solutions Problem

Heteroskedasticity

Serial correlation

Multicollinearity

Effect

Incorrect standard errors

Incorrect standard errors (additional problems if a lagged value of the dependent variable is used as an independent variable)

High R2and low t-statistics

Solution

Use robust standard errors (corrected for conditional heteroskedasticity) Use robust standard errors (corrected for serial correlation)

Remove one or more independent variables; often no solution based in theory

y_t = eb0+ b1t

ln y_t = b₀ + b₁t + _t, t = 1,2, . . . , T where:

y_t = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term

b₁ = the slope coefficient t = time = 1, 2, 3 ... T

We take the natural logarithm of both sides of the equation to arrive at the equation for the log-linear model:

Log-Linear Trend Models

x_t = b0 + b1xt  1 + t

x_t = b₀ + b₁x_{t  1} + b₂x_{t  2}+ . . . + b_px_{t  p} + _t AUTOREGRESSIVE (AR) TIME-SERIES MODELS

Detecting Serially Correlated Errors in an AR Model

t-stat = Residual autocorrelation for lag Standard error of residual autocorrelation

T

IME

-S

ERIES

A

NALYSIS

Linear Trend Models

y_t = b0 + b1t + t, t = 1, 2, . . . , T where:

y_t = the value of the time series at time t (value of the dependent variable) b₀ = the y-intercept term

b₁ = the slope coefficient/ trend coefficient t = time, the independent or explanatory variable _t = a random-error term

A series that grows exponentially can be described using the following equation:

A pth order autoregressive model is represented as:

where:

Standard error of residual autocorrelation = 1/ T T = Number of observations in the time series TIME SERIES ANALYSIS

x_t = b0 1  b₁ Mean Reversion x_t+1 = b^₀ + b^₁x_t ^

Multiperiod Forecasts and the Chain Rule of Forecasting

x_t = x_{t  1} + _t , E(_t) = 0, E(_t2) = 2,E(_t_s) = 0 if t s

y_t = x_t  x_{t  1} = x_{t  1} + _t  x_{t  1}= _t , E(_t) = 0, E(_t2) = 2,E(_t_s) = 0 for t s Random Walks

The first difference of the random walk equation is given as:

x_t = b₀ + b₁x_{t  1} + _t b₁ = 1, b₀ 0, or

x_t = b0 + xt  1 + t , E(t) = 0

y_t = x_t  x_{t  1} , y_t = b₀ + _t , b₀ 0 Random Walk with a Drift

The first-difference of the random walk with a drift equation is given as:

The Unit Root Test of Nonstationarity

x_t  b₀ + b₁x_{t  1} + _t

x_t  x_{t  1}  b₀ + b₁x_{t  1}  x_{t  1} + _t x_t  x_{t  1}  b₀ + (b₁  1)x_{t  1} + _t x_t  x_{t  1}  b₀ + g₁x_{t  1} + _t

_t2 = a₀ + a₁_t2 _{ 1} + u_t

^ ^

_{t + 1} = a₀ + a₁_t2

^2 _^ ^_^

Autoregressive Conditional Heteroskedasticity Models (ARCH Models) x_t = b₀ + b₁x_{t  1} + . . . + b_px_{t  p} + _t + ₁_{t  1} +. . . + _q_{t  q} E(_t) = 0, E(_t2) = 2,E(_t_s) = 0 for t s

Autoregressive Moving Average (ARMA) Models

The error in period t+1 can then be predicted using the following formula: TIME SERIES ANALYSIS

C

URRENCY

E

XCHANGE

R

ATES:

D

ETERMINATION

A

ND

F

ORECASTING

For example, given the USD/EUR and JPY/USD exchange rates, we can calculate the cross rate between the JPY and the EUR, JPY/EUR as follows:

 = USD EUR JPY EUR USD JPY Currency Cross Rates

Cross Rate Calculations with Bid-Ask Spreads

USD/EURbid = 1.3802

 Represents the price of EUR (base currency).

 An investor can sell EUR for USD at this price (as it is the bid price quoted by the dealer).

USD/EURask = 1.3806

 Represents the price of EUR  An investor can buy EUR with

USD at this price.

Determining the EUR/USDbid cross rate:

EUR/USD_bid = 1/(USD/EUR_ask)

Determining the EUR/USD_ask cross rate:

EUR/USDask = 1 / (USD/EURbid)

Forward exchange rates (F) - One year Horizom

FFC/DC = SFC/DC  (1 + iFC) (1 + iDC) FPC/BC = SPC/BC  (1 + i_PC) (1 + iBC) FPC/BC = SPC/BC  1 + (iBC Actual 360) 1 + (iPC Actual 360) FFC/DC = SFC/DC  1 + (iDC Actual 360) 1 + (iFC Actual 360)

Forward exchange rates (F) - Any Investment Horizom

Currencies Trading at a Forward Premium/Discount

(

)

The expected percentage change in the spot exchange rate can be calculated as: Covered Interest Rate Parity

FPC/BC 

1 + (i_{BC Actual 360}) 1 + (iPC Actual 360)

SPC/BC

=

Forward premium (discount) as a % = FPC/BC  SPC/BC SPC/BC

Forward premium (discount) as a %  FPC/BC  SPC/BC iPC  iBC

Uncovered Interest Rate Parity Expected future spot exchange rate:

(1 + i_FC) (1 + iDC)

Se

FC/DC= SFC/DC 

Expected % change in spot exchange rate = Se_PC/BC =

The expected percentage change in the spot exchange rate can be estimated as: Expected % change in spot exchange rate  Se_PC/BC i_PCi_BC

S_PC/BC Se_PC/BC – S_PC/BC

The forward premium (discount) on the base currency can be expressed as a percentage as:

The forward premium (discount) on the base currency can be estimated as:

Purchasing Power Parity (PPP) Law of one price: PX

FC= P

X

DC  SFC/DC

Law of one price: PX

PC= P

X

BC  SPC/BC

Absolute Purchasing Power Parity (Absolute PPP) S_FC/DC = GPL_FC / GPL_DC

S_PC/BC = GPL_PC / GPL_BC

Relative Purchasing Power Parity (Relative PPP)

Relative PPP: E(STFC/DC) = S 0 FC/DC 1_FC 1 + _DC

(

)

T Ex Ante Version of PPP Ex ante PPP: %Se_FC/DC  e_FC e_DC Ex ante PPP: %Se_PC/BC e_PCe_BC CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING

Real Exchange Rates

The real exchange rate (q_FC/DC) equals the ratio of the domestic price level expressed in the foreign currency to the foreign price level.

q_FC/DC S_FC/DC PFC P_DC in terms of FC P_DC  S_FC/DC PFC = = = PDC PFC

( )

The Fisher Effect

Fischer Effect: i = r + e

International Fisher effect: (iFC – iDC) = ( e

FC –  e

DC)

Figure 1: Spot Exchange Rates, Forward Exchange Rates, and Interest Rates

Ex Ante PPP Forward Rate as an Unbiased Predictor International Fisher Effect Covered Interest Rate Parity Uncovered Interest Rate Parity Expected change in

Spot Exchange Rate %Se_FC/DC Foreign-Domestic Expected Inflation Differential e_FC  e_DC Foreign-Domestic Interest rate Differential i_FC  i_DC Forward Discount F_FC/DC S_FC/DC S_FC/DC Balance of Payment

Current account + Capital account + Financial account = 0

Real Interest Rate Differentials, Capital Flows and the Exchange Rate q_L/H – q_L/H = (i_H – i_L) – (e_H – e_L) – (_H – _L)

The Taylor rule

i = r_n +  + y y*) where

i = the Taylor rule prescribed central bank policy rate r_n = the neutral real policy rate

 = the current inflation rate

* = the central bank’s target inflation rate y = the log of the current level of output

y* = the log of the economy’s potential/sustainable level of output CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING

q_PC/BC = q_PC/BC + ( r_nBCr_nPC)+ _BC_BC_PC_PC yBC y*BC) yPC y*PC)] BC PC)

E

CONOMIC

G

ROWTH

A

ND

T

HE

I

NVESTMENT

D

ECISION P = GDP E GDP

( )

P E

( )

P = Aggregate price or value of earnings. E = Aggregate earnings

This equation can also be expressed in terms of growth rates: P = (GDP) + (E/GDP) + (P/E)

Production Function

Y = AKL1-

Y = Level of aggregate output in the economy L = Quantity of labor

K = Quantity of capital

A = Total factor productivity. Total factor productivity (TFP) reflects the general level

of productivity or technology in the economy. TFP is a scale factor i.e., an increase in TFP implies a proportionate increase in output for any combination of inputs.

 = Share of GDP paid out to capital 1   = Share of GDP paid out to labor

y = Y/L = A(K/L)(L/L)1- = Ak y = Y/L = Output per worker or labor productivity. k = K/L = Capital per worker or capital-labor ratio

Cobb-Douglas production function

Y/Y =A/A + K/K + (1  )L/L

Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity

Labor Supply

Total number of hours available for work = Labor force Average hours worked per worker Relationship between economic growth and stock prices

Potential GDP

Neoclassical Model (Solow’s Model) = +



+ n Y K  1 s

( )

 (1-)

( )

[

]

s = Fraction of income that is saved  = Growth rate of TFP

 = Elasticity of output with respect to capital y = Y/L or income per worker

k = K/L or capital-labor ratio

 = Constant rate of depreciation on physical stock n = Labor supply growth rate.

=  +



+ n (1 )

(

)

[

]

sy k Savings/Investment Equation:

Growth rates of Output Per Capita and the Capital-Labor Ratio

y_e = f(k_e) = ck_e = y y  (1) Y K   + s = k k  (1)

( )

Y K  

(

)

+ s

Production Function in the Endogenous Growth Model ECONOMIC GROWTH AND THE INVESTMENT DECISION

I

NVENTORIES

: I

MPLICATIONS

F

OR

F

INANCIAL

S

TATEMENTS

A

ND

R

ATIOS

Ending Inventory = Opening Inventory + Purchases - Cost of goods sold

LIFO and FIFO Comparison with Rising Prices and Stable Inventory Levels

COGS

Income before taxes Income taxes Net income Total cash flow EI Working capital LIFO Higher Lower Lower Lower Higher Lower Lower FIFO Lower Higher Higher Higher Lower Higher Higher

LIFO versus FIFO with Rising Prices and Stable Inventory Levels

Income is lower under LIFO because COGS is higher

Same debt levels Profitability ratios

NP and GP margins

Solvency ratios Debt-to-equity and debt ratio

Sales are the same under both

Lower equity and assets under LIFO

Lower under LIFO

Higher under LIFO Type of Ratio

Effect on Numerator

Effect on

Denominator Effect on Ratio

Liquidity ratios Current ratio

Quick ratio

Activity ratios Inventory turnover

Total asset turnover

Current assets are lower under LIFO because EI is lower Quick assets are higher under LIFO as a result of lower taxes paid

COGS is higher under LIFO Sales are the same

Current liabilities are the same.

Current liabilities are the same

Average inventory is lower under LIFO Lower total assets under LIFO

Lower under LIFO

Higher under LIFO

Higher under LIFO

Higher under LIFO

LIFO reserve (LR)

EIFIFO = EILIFO + LR

COGS_FIFO = COGS_LIFO  (Change in LR during the year)

Change in LIFO Reserve  (1  Tax rate)

When converting from LIFO to FIFO assuming rising prices: Equity (retained earnings) increases by:

Liabilities (deferred taxes) increase by:

LIFO Reserve  (Tax rate) LIFO Reserve  (1 Tax rate)

Current assets (inventory) increase by:

LIFO Reserve where LR = LIFO Reserve

Net Income after tax under FIFO will be greater than LIFO net income after tax by: COGS_FIFO is lower than COGS_LIFO during periods of rising prices:

Impact of an Inventory Write-Down on Various Financial Ratios

COGS increases so profits fall

Debt levels remain the same

Current assets decrease (due to lower inventory)

COGS increases

Sales remain the same Profitability ratios NP and GP margins Solvency ratios Debt-to-equity and debt ratio Liquidity ratios Current ratio Activity ratios Inventory turnover

Total asset turnover

Sales remain the same

Equity decreases (due to lower profits) and current assets decrease (due to lower inventory)

Current liabilities remain the same.

Average inventory decreases

Total assets decrease

Lower (worsens) Higher (worsens) Lower (worsens) Higher (improves) Higher (improves) Type of Ratio Effect on Numerator Effect on

Denominator Effect on Ratio

L

ONG

-

LIVED

A

SSETS

: I

MPLICATIONS

F

OR

F

INANCIAL

S

TATEMENTS

A

ND

R

ATIOS

Straight Line Depreciation

Depreciation expense =

Depreciable life Original cost  Salvage value

Accelerated Depreciation

DDB depreciation in Year X = 2

Depreciable life × Book value at the beginning of Year X Effects of Expensing

When the item is expensed

Effects on Financial Statements

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.

Initially when the cost is capitalized

In future periods when the asset is depreciated or amortized

Effects on Financial Statements Noncurrent assets increase.

Cash flow from investing activities decreases. Noncurrent assets decrease.

Net income decreases. Retained earnings decrease. Equity decreases.

Effects of Capitalization

Financial Statement Effects of Capitalizing versus Expensing Net income (first year)

Net income (future years) Total assets

Shareholders’ equity

Cash flow from operations activities Cash flow from investing activities Income variability

Debt to equity ratio

Capitalizing Higher Lower Higher Higher Higher Lower Lower Lower Expensing Lower Higher Lower Lower Lower Higher Higher Higher LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS

Remaining useful life

The book value of the asset divided by annual depreciation expense equals the number of years the asset

has remaining in its useful life.

= +

Gross investment in fixed assets Annual depreciation expense

Accumulated depreciation Annual depreciation expense

Net investment in fixed assets Annual depreciation expense

Average age of asset

Annual depreciation expense times the number of years that the

asset has been in use equals accumulated depreciation.

Therefore, accumulated depreciation divided by annual depreciation equals the average

age of the asset.

Estimated useful or depreciable life

The historical cost of an asset divided by its useful life equals annual depreciation expense under the straight line method. Therefore, the historical cost divided by annual depreciation expense equals the

estimated useful life.

Income Statement Item Operating expenses

Nonoperating expenses EBIT (operating income) Total expenses- early years Total expenses- later years Net income- early years Net income- later years

Finance Lease Lower (Depreciation) Higher (Interest expense)

Higher Higher Lower Lower Higher Operating Lease Higher (Lease payment)

Lower (None) Lower Lower Higher Higher Lower Income Statement Effects of Lease Classification

CF Item CFO

CFF

Total cash flow

Finance Lease Higher Lower Same Operating Lease Lower Higher Same Cash Flow Effects of Lease Classification

Effect on Ratio Lower Lower Lower Higher Lower Denominator under Finance Lease Assets- higher Assets- higher Current liabilities-higher Equity- same Assets- higher Equity- same Numerator under Finance Lease Sales- same Net income- lower

Current assets-same

Debt- higher

Net income- lower

Ratio Better or Worse under Finance Lease Worse Worse Worse Worse Worse Ratio Asset turnover Return on assets* Current ratio Leverage ratios (D/E and D/A**)

Return on equity*

Table 9: Impact of Lease Classification on Financial Ratios

**Notice that both the numerator and the denominator for the D/A ratio are higher when classifying the lease as a finance lease. Beware of such exam questions. When the numerator and the denominator of any ratio are heading in the same direction (either increasing or decreasing), determine which of the two is changing more in percentage terms. If the percentage change in the numerator is greater than the percentage change in the denominator, the numerator effect will dominate.

Firms usually have lower levels of total debt compared to total assets. The increase in both debt and assets by classifying the lease as a finance lease will lead to an increase in the debt to asset ratio because the percentage increase in the numerator is greater.

Operating Lease Same Lower Lower Higher Lower Same Financing Lease Same Higher Higher Lower Higher Same Total net income

Net income (early years) Taxes (early years) Total CFO

Total CFI Total cash flow

Financial Statement Effects of Lease Classification from Lessor’s Perspective LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS

Summary of Accounting Treatment for Investments Influence Typical percentage interest Accounting Treatment In Financial Assets Not significant Usually < 20%

Classified into one of four categories based on management intent and type of security. Debt only:  Held-to-maturity (amortized cost, changes in value ignored unless deemed as impaired) Debt and Equity:

 Held for trading (fair value, changes in value recognized in profit or loss)  Available-for-sale

(fair value, changes in value recognized in equity)

 Designated at fair value (fair value, changes in value recognized in profit or loss) In Associates Significant Usually 20%  50% Equity method Business Combinations Controlling Usually > 50% Consolidation In Joint Ventures Shared Control Varies

IFRS: Equity method or proportionate consolidation U.S. GAAP: Equity method (except for unincorporated ventures in specialized industries)

I

NTERCORPORATE

I

NVESTMENTS Combination Merger Acquisition Consolidation Description

Company A + Company B = Company A

Company A + Company B = (Company A + Company B) Company A + Company B = Company C

CFO CFF

Total cash flow Current assets Current liabilities Current ratio

Adjusted Values Upon Reclassification of Sale of Receivables: Lower Higher Same Higher Higher Lower (Assuming it was greater than 1)

Difference between QSPE and SPE

Securitized Transaction: Qualified Special Purpose Entity

 Originator of receivables sells financial assets to an SPE.

 The originator does not own or hold or expect to receive beneficial interest.  SFAS 140 (before 2008 revision)

allowed seller to derecognize the sold assets if transferred assets have been isolated from the transferor and are beyond the reach of bankruptcy, and are financial assets.

Securitized Transaction: Special Purpose Entity

 Originator of receivables sells financial assets to an SPE.

 Seller is primary beneficiary; absorbs risks and rewards.

 Seller maintains some level of control.  Seller is required to consolidate.  Seller’s balance sheet would still

show receivables as an asset.

 Debt of SPE would appear on seller’s balance sheet.

Impact of Different Accounting Methods on Financial Ratios

Leverage Net Profit Margin ROE ROA

Better (lower) as liabilities are lower and equity is the same Better (higher) as sales are lower

and net income is the same Better (higher) as equity is lower

and net income is the same Better (higher) as net income is

the same and assets are lower Equity Method

In-between

In-between

Same as under the equity method

In-between Proportionate Consolidation

Worse (higher) as liabilities are higher and equity is the same Worse (lower) as sales are higher

and net income is the same Worse (lower) as equity is higher

and net income is the same Worse (lower) as net income is

the same and assets are higher Acquisition Method INTERCORPORATE INVESTMENTS

E

MPLOYEE

C

OMPENSATION

: P

OST

-

EMPLOYMENT

A

ND

S

HARE

-

BASED

Types of Post-Employment Benefits

Type of Benefit Defined contribution pension plan Defined benefit pension plan Other post-employment benefits (e.g., retirees’ health care)

Amount of Post-Employment Benefit to Employee

Amount of future benefit is not defined. Actual future benefit will depend on investment performance of plan assets.

Investment risk is borne by employee.

Amount of future benefit is defined, based on the plan’s formula (often a function of length of service and final year’s compensation). Investment risk is borne by company.

Amount of future benefit depends on plan

specifications and type of benefit.

Obligation of Sponsoring Company

Amount of the company’s obligation (contribution) is defined in each period. The contribution, if any, is typically made on a periodic basis with no additional future obligation. Amount of the future obligation, based on the plan’s formula, must be estimated in the current period.

Eventual benefits are specified. The amount of the future obligation must be estimated in the current period.

Sponsoring Company’s Pre-funding of Its Future Obligation

Not applicable.

Companies typically pre-fund the DB plans by contributing funds to a pension trust. Regulatory requirements to pre-fund vary by country.

Companies typically do not pre-fund other post-employment benefit obligations.

Estimated annual payment = (Estimated final salary × Benefit formula) × Years of service Final year’s salary = Current salary × [(1 + Annual compensation increase)years until retirement]

Annual unit credit = Value at retirement / Years of service

A company’s pension obligation will increase as a result of:  Current service costs.

 Interest costs.  Past service costs.  Actuarial losses.

A company’s pension obligation will decrease as a result of:  Actuarial gains.

 Benefits paid.

Priodic pension cost = Ending funded status  Beginning funded status + Employer contributions

Periodic pension cost = Current service costs + Interest costs + Past service costs

+ Actuarial losses  Actuarial gains  Actual return on plan assets Reconciliation of the Pension Obligation:

Pension obligation at the beginning of the period + Current service costs

+ Interest costs + Past service costs + Actuarial losses – Actuarial gains – Benefits paid

Pension obligation at the end of the period

The fair value of assets held in the pension trust (plan) will increase as a result of:  A positive actual dollar return earned on plan assets; and

 Contributions made by the employer to the plan. The fair value of plan assets will decrease as a result of:  Benefits paid to employees.

Reconciliation of the Fair Value of Plan Assets:

Fair value of plan assets at the beginning of the period + Actual return on plan assets

+ Contributions made by the employer to the plan  Benefits paid to employees

Fair value of plan assets at the end of the period

Funded status = Pension obligation  Fair value of plan assets  If Pension obligation > Fair value of plan assets:

Plan is underfunded  Positive funded status  Net pension liability.  If Pension obligation < Fair value of plan assets:

Plan is overfunded Negative funded status  Net pension asset. Balance Sheet Presentation of Defined Benefit Pension Plans

Calculating Periodic Pension Cost

Under the corridor method, if the net cumulative amount of unrecognized actuarial gains and losses at the beginning of the reporting period exceeds 10% of the greater of (1) the defined benefit obligation or (2) the fair value of plan assets, then the excess is amortized over the expected average remaining working lives of the employees participating in the plan and included as a component of periodic pension expense on the P&L. EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED

Components of a Company’s Defined Benefit Pension Periodic Costs IFRS Component Service costs Net interest income/ expense Remeasurements: Net return on plan assets and actuarial gains and losses

IFRS Recognition

Recognized in P&L.

Recognized in P&L as the following amount: Net pension liability or asset × interest rate(a)

Recognized in OCI and not subsequently amortized to P&L.

 Net return on plan assets = Actual return  (Plan assets × Interest rate).  Actuarial gains and

losses = Changes in a company’s pension obligation arising from changes in actuarial assumptions.

U.S. GAAP Component

Current service costs Past service costs

Interest expense on pension obligation Expected return on plan assets

Actuarial gains and losses including differences between the actual and expected returns on plan assets

U.S. GAAP Recognition

Recognized in P&L. Recognized in OCI and subsequently amortized to P&L over the service life of employees.

Recognized in P&L. Recognized in P&L as the following amount: Plan assets × expected return.

Recognized immediately in P&L or, more commonly, recognized in OCI and subsequently amortized to P&L using the corridor or faster recognition method.(b)

 Difference between expected and actual return on assets = Actual return  (Plan assets × Expected return).

 Actuarial gains and losses = Changes in a company’s pension obligation arising from changes in actuarial assumptions.

(a) The interest rate used is equal to the discount rate used to measure the pension liability (the yield on high-quality corporate bonds.)

(b) If the cumulative amount of unrecognized actuarial gains and losses exceeds 10 percent of the greater of the value of the plan assets or of the present value of the DB obligation (under U.S. GAAP, the projected benefit obligation), the difference must be amortized over the service lives of the employees.

Impact of Key Assumptions on Net Pension Liability and Periodic Pension Cost Assumption Higher discount rate Higher rate of compensation increase Higher expected return on plan assets

Impact of Assumption on Net Pension Liability (Asset) Lower obligation

Higher obligation

No effect, because fair value of plan assets are used on balance sheet

Impact of Assumption on Periodic Pension Cost and Pension Expense Pension cost and pension expense will both typically be lower because of lower opening obligation and lower service costs

Higher service and interest costs will increase periodic pension cost and pension expense.

Not applicable for IFRS

No effect on periodic pension cost under U.S. GAAP

Lower periodic pension expense under U.S. GAAP

M

ULTINATIONAL

O

PERATIONS

 The presentation currency (PC) is the currency in which the parent company reports its financial statements. It is typically the currency of the country where the parent is located. For example, U.S. companies are required to present their financial results in USD, German companies in EUR, Japanese companies in JPY, and so on.

 The functional currency (FC) is the currency of the primary business environment in which an entity operates. It is usually the currency in which the entity primarily generates and expends cash.

 The local currency (LC) is the currency of the country where the subsidiary operates. Table 1

Export sale Import purchase

Loss Gain Asset (account receivable)

Liability (account payable)

Foreign Currency

Gain Loss

Transaction Type of Exposure Strengthens Weakens

Methods for Translating Foreign Currency Financial Statements of Subsidiaries Functional Currency Local Currency Local Currency Presentation Currency Presentation Currency Temporal Method Current Rate Method Current Rate/ Temporal Method Functional Currency Functional Currency Presentation Currency Local Currency T T CR CR = =

 The current rate is the exchange rate that exists on the balance sheet date.  The average rate is the average exchange rate over the reporting period.

 The historical rate is the actual exchange rate that existed on the original transaction date.

Income Statement Component Sales

Cost of goods sold Selling expenses Depreciation expense Amortization expense Interest expense Income tax

Net income before translation gain (loss)

Translation gain (loss) Net income

Less: Dividends

Change in retained earnings

Current Rate Method FC = LC

Temporal Method FC = PC Exchange Rate Used

Average rate Average rate Average rate Average rate Average rate Average rate Average rate N/A

Computed as Rev – Exp Historical rate Computed as NI – Dividends Used as input for translated

B/S Average rate Historical rate Average rate Historical rate Historical rate Average rate Average rate Computed as Rev – Exp

Plug in Number Computed as RE +

Dividends Historical rate

From B/S Rules for Foreign Currency Translation

Balance Sheet Component Exchange Rate Used

Cash

Accounts receivable Monetary assets Inventory

Nonmonetary assets measured at current value

Property, plant and equipment Less: Accumulated depreciation Nonmonetary assets measured at historical cost

Accounts payable Long-term notes payable Monetary liabilities Nonmonetary liabilities: Measured at current value Measured at historical cost Capital stock

Retained earnings

Cumulative translation adjustment

Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Historical rate From I/S Plug in Number Current rate Current rate Current rate Historical rate Current rate Historical rate Historical rate Historical rate Current rate Current rate Current rate Current rate Historical rate Historical rate To balance Used as input for

translated I/S N/A MULTINATIONAL OPERATIONS

Balance Sheet Exposure Net asset

Net liability

Balance Sheet Exposure

Foreign Currency (FC) Strengthens

Positive translation adjustment Negative translation adjustment

Weakens

Negative translation adjustment Positive translation adjustment

Effects of Exchange Rate Movements on Financial Statements

Foreign currency strengthens relative to parent’s presentation currency Foreign currency weakens relative to parent’s presentation currency Temporal Method, Net Monetary Liability Exposure Revenues Assets Liabilities  Net income  Shareholders’ equity Translation loss  Revenues  Assets  Liabilities Net income Shareholders’ equity Translation gain Temporal Method, Net Monetary Asset Exposure Revenues Assets Liabilities Net income Shareholders’ equity Translation gain  Revenues  Assets  Liabilities  Net income  Shareholders’ equity Translation loss

Current Rate Method

Revenues Assets Liabilities Net income Shareholders’ equity Positive translation adjustment  Revenues  Assets  Liabilities  Net income  Shareholders’ equity Negative translation adjustment

Measuring Earnings Quality

Aggregate accruals = Accrual-basis earnings – Cash earnings

Balance Sheet Approach Net Operating Assets (NOA)

NOAt = [(Total assetst Casht) (Total liabilitiest Total debtt)]

Aggregate Accruals

Aggregate accruals_tb/s = NOA_tNOA_t1 Aggregate Ratio

A Financial Statement Analysis Framework:

Phase Sources of Information Examples of Output

Define the purpose and context of the analysis.

Collect input data.

Process input data, as required, into analytically useful data.

Analyze/interpret the data.

The nature of the analyst’s function, such as evaluating an equity or debt investment or issuing a credit rating. Communication with client or supervisor on needs and concerns.

Institutional guidelines related to developing specific work product.

Financial statements, other financial data, questionnaires, and industry/economic data. Discussions with management, suppliers, customers, and competitors.

Company site visits (e.g., to production facilities or retail stores)

Data from the previous phase.

Input data and processed data                   

Statement of the purpose or objective of analysis.

A list (written or unwritten) of specific questions to be answered by the analysis. Nature and content of report to be provided.

Timetable and budgeted resources for completion.

Organized financial statements. Financial data tables.

Completed questionnaires, if applicable.

Adjusted financial statements. Common-size statements. Forecasts. Analytical results 1. 2. 3. 4.

ROE = Tax Burden × Interest burden × EBIT margin × Total asset turnover × Financial leverage DuPont Analysis Develop and communicate conclusions and recommendations (e.g., with an analysis report). Follow-up. 5. 6.

Analytical results and previous reports

Institutional guidelines for published reports

Information gathered by periodically repeating above steps as necessary to determine whether changes to holdings or recommendations are necessary







Analytical report answering questions posed in Phase 1 Recommendations regarding the purpose of the analysis, such as whether to make an investment or grant credit. Update reports and recommendations 





I

NTEGRATION

O

F

F

INANCIAL

S

TATEMENT

A

NALYSIS

T

ECHNIQUES INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES

C

APITAL

B

UDGETING

Expansion Project

Initial investment outlay for a new investment = FCInv + NWCInv

NWCInv = Non-cash current assets – Non-debt current liabilities

Annual after-tax operating cash flows (CF)

CF = (S – C – D) (1 – t) + D or CF = (S – C) (1 – t) + tD Terminal year after-tax non-operating cash flow (TNOCF):

TNOCF = Sal_T + NWCInv – t(Sal_T – BV_T)

Replacement Project Investment outlays:

Initial investment for a replacement project = FCInv + NWCInv – Sal₀ + t(Sal₀ – BV₀) Annual after-tax operating cash flow:

CF = (S – C) (1 – t) + tD

Terminal year after-tax non-operating cash flow:

TNOCF = Sal_T + NWCInv – t(Sal_T – B_T)

Mutually Exclusive Projects with Unequal Lives  Least Common Multiple of Lives Approach

In this approach, both projects are repeated until their ‘chains’ extend over the same time horizon. Given equal time horizons, the NPVs of the two project chains are compared and the project with the higher chain NPV is chosen.

 Equivalent Annual Annuity Approach (EAA)

This approach calculates the annuity payment (equal annual payment) over the project’s life that is equivalent in present value (PV) to the project’s NPV. The project with the higher EAA is chosen.

SML

Economic Income

Economic income = After-tax operating cash flow + Increase in market value

Economic income = After-tax operating cash flow + (Ending market value – Beginning market value) Economic income = After-tax operating cash flow – (Beginning market value – Ending market value) Economic income = After-tax cash flows – Economic depreciation

Economic Profit

Economic profit = [EBIT (1 – Tax rate)] – $WACC Economic profit = NOPAT – $WACC

NOPAT = Net operating profit after tax

$WACC = Dollar cost of capital = Cost of capital (%) × Invested capital

Under this approach, a project’s NPV is calculated as the sum of the present values of economic profit earned over its life discounted at the cost of capital.



 _{(1 + WACC)}t

EPt

NPV = MVA =

Residual Income

Residual income = Net income for the period – Equity charge for the period

Equity charge for the period = Required return on equity × Beginning-of-period book value of equity The RI approach calculates value from the perspective of equity holders only. Therefore, future residual income is discounted at the required rate of return on equity to calculate NPV.

NPV =





(1 + r_E)t

RI_t

Claims Valuation

 First, we separate the cash flows available to debt and equity holders  Then we discount them at their respective required rates of return.

o Cash flows available to debt holders are discounted at the cost of debt, o Cash flows available to equity holders are discounted at the cost of equity.

 The present values of the two cash flow streams are added to calculate the total value of the company/asset. CAPITAL BUDGETING

C

APITAL

S

TRUCTURE

r_WACC = r_D(1  t) + r_E The Capital Structure Decision

r_WACC =

( ) ( )

r_D + r_E = r₀

MM Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity

Independent variable Intercept

Dependent variable Slope r_E =r0 +(r0  rD)

The systematic risk (ß) of the company’s assets can be expressed as the weighted average of the systematic risk of the company’s debt and equity.

=

_A

( ) ( )

+

This formula can also be expressed as: rD = Marginal cost of debt

rE = Marginal cost of equity

t = Marginal tax rate

D = Market value of the company’s outstanding debt E = Market value of shareholders’ equity

V = D + E = Value of the company

Company’s cost of equity (rE) under MM Proposition II without taxs is calculated as:

The total value of the company is calculated as:

V = + r_E EBIT  Interest r_D Interest CAPITAL STRUCTURE

= + tD

Relaxing the Assumption of no Taxes

r_WACC = r_D(1  t) + r_E The WACC is then calculated as:

rE =r0+(r0  rD) (1  t)

And the cost of equity is calculated as:

Modigilani and Miller Propositions

Without Taxes With Taxes

= r_E =r₀ +(r₀  r_D) + tD = r_E =r₀+(r₀  r_D) (1  t) Proposition I Proposition II

The Optimal Capital Structure: The Static Trade-Off Theory VL = VU + tD – PV(Costs of financial distress)

D

IVIDENDS AND

S

HARE

R

EPURCHASE

Pw = Share price with the right to receive the dividend

PX = Share price without the right to receive the dividend

D = Amount of dividend TD = Tax rate on dividends

TCG = Tax rate on capital gains

Double Taxation System

ETR = CTR + [(1 – CTR) × MTRD]

ETR = Effective tax rate CTR = Corporate tax rate

MTRD = Investor’s marginal tax rate on dividends

Split-Rate Tax System

ETR = CTRD + [(1 – CTRD) × MTRD]

CTRD = Corporate tax rate on earnings distributed as dividends.

Stable Dividend Policy

The expected increase in dividends is calculated as:

Expected dividend increase = Increase in earnings × Target payout ratio × Adjustment factor Adjustment factor = 1/N

N = Number of years over which the adjustment is expected to occur

Analysis of Dividend Safety

Dividend payout ratio = (dividends / net income)

Dividend coverage ratio = (net income / dividends)

FCFE coverage ratio = FCFE / [Dividends + Share repurchases]

PW PX= D

The expected decrease in share price when it goes ex-dividend can be calculated using the following equation: DIVIDENDS AND SHARE REPURCHASE

Industry Life Cycle Stage Pioneering development Rapid accelerating growth Mature growth Stabilization and market maturity Deceleration of growth and decline           Industry Description Low but slowly increasing sales growth. Substantial development costs. High profit margins. Low competition. Decrease in the entry of new competitors. Growth potential remains. Increasing capacity constraints Increasing competition. Overcapacity. Eroding profit margins. Types of Merger Conglomerate Horizontal Conglomerate Horizontal Horizontal Vertical Horizontal Horizontal Vertical Conglomerate           Motives for Merger

Younger, smaller companies may sell themselves to larger firms in mature or declining industries to enter into a new growth industry.

Young companies may merge with firms that allow them to pool management and capital resources. To meet substantial capital

requirements for expansion.

To achieve economies of scale, savings, and operational efficiencies.

To achieve economies of scale in research, production, and marketing to match low costs and prices of competitors.

Large companies may buy smaller companies to improve management and provide a broader financial base. Horizontal mergers to ensure survival. Vertical mergers to increase efficiency and profit margins.

Conglomerate mergers to exploit synergy.

Companies in the industry may acquire companies in young industries.          

Source: Adapted from J. Fred Weston, Kwang S. Chung, and Susan E. Hoag, Mergers, Restructuring, and Corporate Control (New York: Prentice Hall, 1990, p.102) and Bruno Solnik and Dennis McLeavy, International Investments, 5th edition (Boston: Addison Wesley, 2004, p. 264 – 265).

Mergers and the Industry Life Cycle

M

ERGERS AND

A

CQUISITION MERGERS AND ACQUISITION

Major Differences of Stock versus Asset Purchases Payment Approval Tax: Corporate Tax: Shareholder Liabilities Stock Purchase

Target shareholders receive compensation in exchange for their shares.

Shareholder approval required. No corporate-level taxes. Target company’s shareholders are taxed on their capital gain. Acquirer assumes the target’s liabilities.

Asset Purchase

Payment is made to the selling company rather than directly to shareholders.

Shareholder approval might not be required.

Target company pays taxes on any capital gains.

No direct tax consequence for target company’s shareholders.

Acquirer generally avoids the assumption of liabilities.



n i

(

Sales or output of firm i

Total sales or output of market100

)

2

Herfindahl-Hirschman Index (HHI)

Post-Merger HHI Less than 1,000 Between 1,000 and 1,800 More than 1,800 Concentration Not concentrated Moderately concentrated Highly concentrated Change in HHI Any amount 100 or more 50 or more Government Action No action Possible challenge Challenge

HHI Concentration Levels and Possible Government Response

FCFF is estimated by:

Net income

+ Net interest after tax = Unlevered income + Changes in deferred taxes

= NOPLAT (net operating profit less adjusted taxes) + Net noncash charges

– Change in net working capital – Capital expenditures (capex) Free cash flow to the firm (FCFF)

Net interest after tax = (Interest expense – Interest income) (1 – tax rate)

Working capital = Current assets (excl. cash and equivalents) – Current liabilities (excl. short-term debt)

Comparable Company Analysis

TP = Takeover premium DP = Deal price per share

SP = Target’s stock price per share TP =

SP (DP  SP)

Bid Evaluation

Target shareholders’ gain = Takeover premium = P_T – V_T Acquirer’s gain = Synergies – Premium

= S – (PT – VT)

S = Synergies created by the merger transaction

The post-merger value of the combined company is composed of the pre-merger value of the acquirer, the pre-merger value of the target, and the synergies created by the merger. These sources of value are adjusted for the cash paid to target shareholders to determine the value of the combined post-merger company.

V_A* = V_A + V_T + S – C

V_A* = Value of combined company C = Cash paid to target shareholders MERGERS AND ACQUISITION

Perceived mispricing:

Perceived mispricing = True mispricing + Error in the estimate of intrinsic value. V_E – P = (V – P) + (V_E – V)

V_E = Estimate of intrinsic value P = Market price

V = True (unobservable) intrinsic value

E

QUITY

V

ALUATION

: A

PPLICATIONS

A

ND

P

ROCESSES

R

ETURN

C

ONCEPTS

Holding period return = PH – P0 + DH P₀

Intrinsic Value = Next year’s expected dividend

Required return – Expected dividend growth rate

V₀ = D1 k_e – g

ke (IRR) = +g

D₁ P0

If the asset is assumed to be efficiently-priced (i.e. the market price equals its intrinsic value), the IRR would equal the required return on equity. Therefore, the IRR can be estimated as:

Required return (IRR) = + Expected dividend growth rate Market price

Next year’s dividend Holding Period Return

P_H = Price at the end of the holding period P₀ = Price at the beginning of the period D_H = Dividend

Required Return

 The difference between an asset’s expected return and its required return is known as expected alpha, ex ante alpha or expected abnormal return.

o Expected alpha = Expected return – Required return

 The difference between the actual (realized) return on an asset and its required return is known as realized alpha or ex post alpha.

o Realized alpha = Actual HPR – Required return for the period

When the investor’s estimate of intrinsic value (V₀) is different from the current market price (P₀), the investor’s expected return has two components:

1. The required return (rT) earned on the asset’s current market price; and

2. The return from convergence of price to value [(V₀ – P₀)/P₀].

Internal Rate of Return RETURN CONCEPTS

Equity Risk Premium

The required rate of return on a particular stock can be computed using either of the following two approaches. Both these approaches require the equity risk premium to be estimated first.

1. Required return on share i = Current expected risk-free return + ß_i(Equity risk premium)

 A beta greater (lower) than 1 indicates that the security has greater-than-average (lower-than-average) systematic risk.

2. Required return on share i = Current expected risk-free return + Equity risk premium ± Other risk premia/discounts appropriate for i

  This method of estimating the required return is known as the build-up method. It is discussed later in the reading and is primarily used for valuations of private businesses.

GCM equity risk premium estimate = D1 + g – r_LTGD P0

Gordon Growth Model (GGM) Estimates

Macroeconomic Model Estimates

Equity risk premium = {[(1 + EINFL) (1 + EGREPS) (1 + EGPE) – 1] + EINC} – Expected RF

Expected inflation =

1 + YTM of 20-year maturity TIPS 1 + YTM of 20-year maturity T-bonds

– 1

The Captial Asset Pricing Model (CAPM)

Required return on i = Expected risk-free rate + Beta_i (Equity risk premium)

The Fama-French Model

r_i = RF +_imktRMRF +_isizeSMB+ _ivalueHML ß_mkt = Market beta

ß_size = Size beta ß_value = Value beta

The Pastor-Stambaugh model (PSM)

r_i = R_F+_imktRMRF+_isizeSMB+_ivalueHML+_iliqLIQ

BIRR model

r_i = T-bill rate + (Sensitivity to confidence risk × Confidence risk) + (Sensitivity to time horizon risk × Time horizon risk) + (Sensitivity to inflation risk × Inflation risk)

+ (Sensitivity to business cycle risk × Business cycle risk) + (Sensitivity to market timing risk × Market timing risk) Build-up method

r_i = Risk-free rate + Equity risk premium + Size premium + Specific-company premium For companies with publicly-traded debt, the bond-yield plus risk premium approach can be used to calculate the cost of equity:

BYPRP cost of equity = YTM on the company’s long-term debt + Risk premium

Adjusting Beta for Beta Drift

Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0)

Estimating the Asset Beta for the Comparable Publicly Traded Firm:

where:

D/E = debt-to-equity ratio of the comparable company. t = marginal tax rate of the comparable company.

To adjust the asset beta of the comparable for the capital structure (financial risk) of the project or company being evaluated, we use the following formula:

where:

D/E = debt-to-equity ratio of the subject company. t = marginal tax rate of the subject company.

B_ASSET reflects only business risk of the comparable company. Therefore it is used as a proxy for business risk of the project being studied. B_EQUITY reflects business and financial risk of comparable company. ß_ASSET = ß_EQUITY

)

1

(

1 + (1 - t) D E B_PROJECT reflects business and financial risk of the project. B_ASSET reflects business risk of project. ß_PROJECT = ß_ASSET 1 + (1 - t) D E

Country Spread Model RETURN CONCEPTS

Weighted Average Cost of Capital (WACC) WACC = + MVCE r MVD + MVCE r_d (1 – Tax rate ) MVD MVD + MVCE

MVD = Market value of the company’s debt r_d = Required rate of return on debt

MVCE = Market value of the company’s common equity r = Required rate of return on equity

Justified leading P/E ratio = P0 E₁ D₁/E₁ r  g = = (1 b) r  g P D/E D _{(1 g)} / E _{(1 b)(1 g)} P/E ratio

Present value of Growth Opportunities

V₀ = E1 + PVGO r

V0 = The value of the stock today (t = 0)

P₁ = Expected price of the stock after one year (t = 1)

D1 = Expected dividend for Year 1, assuming it will be paid at the end of Year 1 (t = 1)

r = Required return on the stock One-Period DDM

Multiple-Period DDM

D

ISCOUNTED

D

IVIDEND

V

ALUATION

V₀ = D1 + = (1 + r)1 P₁ (1 + r)1 D₁ P₁ (1 + r)1 + V₀ = D1 + (1 + r)1 P_n (1 + r)n D_n (1 + r)n + + ... V₀ = Dt + (1 + r)t P_n (1 + r)n



n t = 1

Expression for calculating Value of a share of stock

V₀ = Dt (1 + r)t



 t = 1 V₀ = D0 (1 + g) (r – g) , orV0 = D₁ (r – g) Gordon Growth Model

g_S = Short term supernormal growth rate g_L = Long-term sustainable growth rate r = required return

n = Length of the supernormal growth period Two-Stage Dividend Discount Model

g_S = Short term high growth rate gL = Long-term sustainable growth rate

r = required return

H = Half-life = 0.5 times the length of the high growth period

The H-model equation can be rearranged to calculate the required rate of return as follows: The H-Model

r = [(1 + gL) + H(gs – gL)] + gL

D₀ P₀

)

(

The Gordon growth formula can be rearranged to calculate the required rate of return given the other variables.

r = D1 + g P₀ V0 =

D r

Value of Fixed-Rate Perpetual Preferred Stock

V₀ =



n t = 1 D₀ (1 + g_S)t (1 + r)t + D₀ (1 + g_S)n(1 + g_L) (1 + r)n(r – g_L) V₀ = D0 (1 + gL) r – g_L + D₀H(g_s – g_L) r – g_L

Sustainable growth rate (SGR)

b = Earnings retention rate, calculated as 1 – Dividend payout ratio g = b × ROE

ROE can be calculated as:

ROE = Net income × ×

Sales Total assets

Sales Total assets

Shareholders’ equity

g = Net income × ×

Sales Total assets

Sales Total assets

Shareholders’ equity Net income - Dividends

Net income ×

PRAT model

g = Profit margin × Retention rate × Asset turnover × Financial leverage DISCOUNTED DIVIDEND VALUATION

FCFF = NI + NCC + Int(1  Tax Rate)  FCInv  WCInv Computing FCFF from Net Income

FCInv = Capital expenditures  Proceeds from sale of long-term assets Investment in fixed capital (FCInv)

WCInv = Change in working capital over the year

Working capital = Current assets (exc. cash)  Current liabilities (exc. short-term debt) Investment in working capital (WCInv)

WACC= MV(Debt)

MV(Debt) + MV(Equity) rd(1  Tax Rate)

MV(Equity) MV(Debt) + MV(Equity) r + = Firm Value



 t=1 FCFF_t (1+WACC)t

Equity Value=Firm Value  Market value of debt

Equity Value ₌



 t=1 FCFE_t (1 + r)t FCFF/FCFE

F

REE

C

ASH

F

LOW

V

ALUATION

Table: Noncash Items and FCFF Noncash Item

Depreciation

Amortization and impairment of intangibles Restructuring charges (expense)

Restructuring charges (income resulting from reversal) Losses Gains Adjustment to NI to Arrive at FCFF Added back Added back Added back Subtracted Added back Subtracted