Analyst Dojo

Learning Module 6 of Equities 11–14% of the exam

Discounted Cash Flow and Growth Models

7 lessons. About 45 minutes in cram mode, 210 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary Practice this module

Lessons

  1. 1
  2. 2
  3. 3
  4. 4
    Two-stage (multistage) growth models

    calculate 7 min cram, 30 min full

  5. 5
    Discount rates and terminal values

    calculate 7 min cram, 32 min full

  6. 6
  7. 7
    Where DCF and growth models break down

    explain 5 min cram, 24 min full

What the exam expects you to do

  1. Contrast the inputs used in dividend discount, free cash flow to equity (FCFE), free cash flow to the firm (FCFF), and residual income models, and explain the process for using present value models to value equity.Dividend discount and free-cash-flow-to-equity models
  2. Calculate and interpret the intrinsic value of an equity security based on present value models that assume constant cash flow growth or multistage cash flow growth.Constant growth models, implied growth and the sustainable growth rate
  3. Explain the shortcomings of constant and multistage cash flow growth assumptions in present value models.Where DCF and growth models break down
  4. Calculate the intrinsic value of a non-callable, non-convertible preferred stock and describe how contingency features affect intrinsic value.Valuing preferred stock, and how call, put and conversion features change value

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

How deeply to learn each part
Know the idea
Understand it in general. It frames other topics, but you will not calculate with it.
Use it
Understand it well enough to solve a question with it. Recognize when it applies.
Memorize
Learn it by heart. The exam will not give it to you.
1

Dividend discount and free-cash-flow-to-equity models

Known

MemorizeFCFE three ways

RouteFormula
From cash flow from operationscash flow from operations − capital expenditures + net debt issuance
From the cash it fundedΔcash + dividends + repurchases − share issuance
With a target debt rationet income − (1 − DR) × net investment

Memory hook Where the cash comes from = where it goes: operations, investment and borrowing on one side; payouts and cash kept on the other.

Know it cold Cash flow from operations = net income + depreciation − Δworking capital; net investment = (Δgross long-term assets − depreciation) + Δworking capital.

MemorizeDDM against FCFE

DDMFCFE
Cash flowdividends paidcash available, paid or not
Point of viewnon-controllingcontrolling
Can be negative?noyes
Discount atrer_erer_e (never WACC)

Memory hook Outsiders get dividends; insiders decide what FCFE becomes.

Know it cold IVt=∑E(CFt+i)/(1+re)i+E(Pt+n)/(1+re)nIV_t = \sum E(CF_{t+i})/(1+r_e)^i + E(P_{t+n})/(1+r_e)^n; the horizon is the analyst's forecast horizon, not an investor's holding period.

Key terms

  • generalized discounted cash flow model Memorize
  • forecast horizon Use it
  • terminal value Use it
  • required return on equity (cost of equity) Know the idea
  • dividend discount model (DDM) Memorize
  • non-controlling shareholder perspective Know the idea
  • free cash flow to equity (FCFE) Memorize
  • cash flow from operations Use it
  • net debt issuance Use it
  • change in working capital Use it
  • target debt ratio Use it
  • net investment Use it
  • positive and negative FCFE Know the idea
  • controlling investor perspective Know the idea
Dividends or FCFE?
Cash flow to shareholdersdiscounted at r_e
  • Dividend discount model (DDM) Memorize dividends actually paid; never negative
    • Best for stable, high, consistent payout; REITs, regulated utilities
    • Point of view non-controlling shareholder
    • Weak for ignores buybacks, share issues, retained cash and debt changes
  • FCFE model Memorize cash available, paid out or not; can be negative
    • Best for irregular or no dividends; buybacks matter
    • Point of view controlling investor
    • Weak for large capital needs; big capital structure changes
Building FCFE
FCFEcash available to shareholders
  • Cash flow from operations Memorize net income + depreciation − Δworking capital
  • minus Capital expenditures Memorize the change in gross long-term assets
  • plus Net debt issuance Memorize additional borrowing − debt repayment

Formulas

Generalized discounted cash flow model Memorize
IVt=∑i=1nE(CFt+i)(1+Required rate of return)i+E(Pt+n)(1+Required rate of return)nIV_t = \sum_{i=1}^{n} \dfrac{E(CF_{t+i})}{(1 + \text{Required rate of return})^{i}} + \dfrac{E(P_{t+n})}{(1 + \text{Required rate of return})^{n}}
Quick example

Dividends of 2.10 and 2.25 and an expected price of 55.00 at the end of year 2, required return 7%: 2.10 ÷ 1.07 + 57.25 ÷ 1.07² = 51.97.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓2.1ENTER↓↓57.25ENTER
  3. NPV7ENTER↓CPT 51.967

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · NPV net present value · CPT compute

  1. fCLEAR REG
  2. 0gCF02.1gCFj57.25gCFj7i
  3. fNPV 51.967

f gold shift key · CLEAR REG clear all registers · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · i interest rate per period · NPV net present value

Free cash flow to equity Memorize
FCFE=Cash flow from operations−Capital expenditures+Net debt issuance\text{FCFE} = \text{Cash flow from operations} - \text{Capital expenditures} + \text{Net debt issuance}
Quick example

Cash flow from operations 275, capital expenditures 150, borrowed 55, repaid 20: 275 − 150 + 35 = 160.

On your calculator
  1. 275−150+(55−20)= 160
  1. 275ENTER150−55ENTER20−+ 160

ENTER enter (on the HP, puts the number on the stack)

FCFE from the change in cash Use it
FCFE=ΔCash+Dividends+Stock repurchases−Stock issuance\text{FCFE} = \Delta\text{Cash} + \text{Dividends} + \text{Stock repurchases} - \text{Stock issuance}
Quick example

Cash rose 8, dividends 25, buybacks 30, new shares 5: 8 + 25 + 30 − 5 = 58.

On your calculator
  1. 8+25+30−5= 58
  1. 8ENTER25+30+5− 58

ENTER enter (on the HP, puts the number on the stack)

FCFE with a target debt ratio Use it
FCFE=Net income−(1−DR)×Net investmentNet investment=(ΔGross long-term assets−Depreciation)+ΔWorking capitalΔDebt=DR×Net investment\begin{aligned} \text{FCFE} &= \text{Net income} - (1 - DR) \times \text{Net investment} \\ \text{Net investment} &= (\Delta\text{Gross long-term assets} - \text{Depreciation}) + \Delta\text{Working capital} \\ \Delta\text{Debt} &= DR \times \text{Net investment} \end{aligned}
Quick example

Net income 90, capital expenditures 50, depreciation 20, Δworking capital 10, DR 25%: net investment 40, FCFE = 90 − 0.75 × 40 = 60.

On your calculator
  1. 50−20+10=×0.75=+|-+90= 60

+|- change sign

  1. 90ENTER50ENTER20−10+0.75×− 60

ENTER enter (on the HP, puts the number on the stack)

FCFE valuation Use it
IVt=∑i=1nFCFEt+i(1+re)i+E(Pt+n)(1+re)nIV_t = \sum_{i=1}^{n} \dfrac{\text{FCFE}_{t+i}}{(1 + r_e)^{i}} + \dfrac{E(P_{t+n})}{(1 + r_e)^{n}}
Quick example

Free cash flow to equity per share of 0.95 and 1.05 and an expected price of 25.00 at the end of year 2, r_e = 12%: 0.95 ÷ 1.12 + 26.05 ÷ 1.12² = 21.62.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓0.95ENTER↓↓26.05ENTER
  3. NPV12ENTER↓CPT 21.6151

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · NPV net present value · CPT compute

  1. fCLEAR REG
  2. 0gCF00.95gCFj26.05gCFj12i
  3. fNPV 21.6151

f gold shift key · CLEAR REG clear all registers · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · i interest rate per period · NPV net present value

2

Dividends, FCFE, FCFF or residual income: which cash flow, which discount rate

Known

MemorizeWhich measure, which rate

MeasureBelongs toDiscount atNegative?
Dividendsshareholders (paid)rer_enever
FCFEshareholders (available)rer_ecan be
FCFFlenders and shareholdersWACC, then subtract debtcan be
EBITDAbefore interest, tax and investmenta pre-tax ratecan be
Residual incomeshareholders, above the equity chargerer_e (not a cash flow)can be

Memory hook Match the rate to the owner: shareholders r_e, everyone WACC, before tax a before-tax rate.

Know it cold FCFF = cash flow from operations + interest × (1 − t) − capital expenditures = FCFE + interest × (1 − t) − ΔDebt.

MemorizeFive steps and two formulas

StepDo
1select the measure
2forecast it (constant, multistage, detailed)
3set the matching rate
4set the terminal value
5point estimate or range against the price

Memory hook Measure, forecast, rate, terminal, compare.

Know it cold RIt=EPSt−re×(BVE/Share)t−1RI_t = EPS_t - r_e \times (BVE/\text{Share})_{t-1}; EBITDA = net income ÷ (1 − t) + interest + depreciation.

Key terms

  • present value metrics Know the idea
  • free cash flow to the firm (FCFF) Memorize
  • after-tax interest expense Use it
  • FCFF and FCFE relationship Memorize
  • earnings before interest, taxes, depreciation, and amortization (EBITDA) Use it
  • adjusted EBITDA Know the idea
  • pre-tax discount rate for EBITDA Know the idea
  • residual income Memorize
  • economic profit Know the idea
  • residual income per share Use it
  • clean surplus accounting Know the idea
  • matching the discount rate to the claim Memorize
  • five-step valuation process Memorize
  • forecast method Know the idea
  • point estimate and value range Use it
Which measure, whose money
Present value measuresmatch the discount rate to the owner
  • Dividends Memorize paid to all shareholders; never negative
  • FCFE Memorize available to shareholders; can be negative
  • FCFF Memorize available to lenders and shareholders; values the firm
  • Residual income Memorize profit above the equity charge; not a cash flow
  • EBITDA Know the idea before tax and investment; a pre-tax rate
Five steps of a present value valuation
Intrinsic value, IV_t
  • 1 Select the measure Memorize dividends, FCFE, FCFF, EBITDA or residual income
  • 2 Forecast it Memorize constant, multistage or detailed
  • 3 Set the rate Memorize r_e or WACC, matched to the owner
  • 4 Set the terminal value Memorize constant growth, multiples or book value
  • 5 Estimate and compare Memorize a point or a range against the price

Formulas

Free cash flow to the firm Memorize
FCFF=Cash flow from operations+Interest expense×(1−Tax rate)−Capital expenditures\text{FCFF} = \text{Cash flow from operations} + \text{Interest expense} \times (1 - \text{Tax rate}) - \text{Capital expenditures}
Quick example

Cash flow from operations 450, interest 50, tax 20%, capital expenditures 180: 450 + 40 − 180 = 310.

On your calculator
  1. 50×0.8+450−180= 310
  1. 450ENTER50ENTER0.8×+180− 310

ENTER enter (on the HP, puts the number on the stack)

FCFF from FCFE Memorize
FCFF=FCFE+Interest expense×(1−Tax rate)−ΔDebt\text{FCFF} = \text{FCFE} + \text{Interest expense} \times (1 - \text{Tax rate}) - \Delta\text{Debt}
Quick example

FCFE 120, interest 50, tax 20%, net borrowing 30: 120 + 40 − 30 = 130.

On your calculator
  1. 50×0.8+120−30= 130
  1. 120ENTER50ENTER0.8×+30− 130

ENTER enter (on the HP, puts the number on the stack)

EBITDA from net income Use it
EBITDA=Net income1−Tax rate+Interest+Depreciation\text{EBITDA} = \dfrac{\text{Net income}}{1 - \text{Tax rate}} + \text{Interest} + \text{Depreciation}
Quick example

Net income 60, tax 25%, interest 15, depreciation 25: 80 + 15 + 25 = 120.

On your calculator
  1. 60÷0.75+15+25= 120
  1. 60ENTER0.75÷15+25+ 120

ENTER enter (on the HP, puts the number on the stack)

FCFF from EBITDA Use it
FCFF=EBITDA(1−Tax rate)+Depreciation×Tax rate−ΔGross long-term assets−ΔWorking capital\text{FCFF} = \text{EBITDA}(1 - \text{Tax rate}) + \text{Depreciation} \times \text{Tax rate} - \Delta\text{Gross long-term assets} - \Delta\text{Working capital}
Quick example

EBITDA 300, tax 25%, depreciation 60, capital expenditures 90, Δworking capital 15: 225 + 15 − 90 − 15 = 135.

On your calculator
  1. 300×0.75=STO1
  2. 60×0.25+RCL1−90−15= 135

STO store · RCL recall a stored number

  1. 300ENTER0.75×60ENTER0.25×+90−15− 135

ENTER enter (on the HP, puts the number on the stack)

Residual income per share Memorize
RIt=EPSt−re×(BVEShare)t−1RI_t = EPS_t - r_e \times \left(\dfrac{BVE}{\text{Share}}\right)_{t-1}
Quick example

Earnings per share 2.10, opening book value per share 15.00, r_e 9%: 2.10 − 1.35 = 0.75.

On your calculator
  1. 0.09×15=+|-+2.1= 0.75

+|- change sign

  1. 2.1ENTER0.09ENTER15×− 0.75

ENTER enter (on the HP, puts the number on the stack)

3

Constant growth models, implied growth and the sustainable growth rate

Known

MemorizeConstant growth formulas

UseFormulaRate
DividendsIVt=Dt(1+g)/(re−g)=Dt+1/(re−g)IV_t = D_t(1+g)/(r_e - g) = D_{t+1}/(r_e - g)rer_e
FCFE (equity value)E(CF)t+1/(re−g)E(CF)_{t+1}/(r_e - g), then ÷ shares, no debt subtractedrer_e
FCFF (firm value)E(CF)t+1/(WACC−g)E(CF)_{t+1}/(\text{WACC} - g), then − market value of debtWACC
Implied growthg=r−E(CF)t+1/Ptg = r - E(CF)_{t+1}/P_tany

Memory hook One period ahead on top, the gap r − g underneath.

Know it cold g below the required return and no higher than the economy's nominal growth; g can be negative for a declining firm.

MemorizeSustainable growth three ways

BasisKept shareReturn
Dividends1 − dividends ÷ net incomeROE = net income ÷ opening BVE
FCFE1 − FCFE ÷ net incomeROE
FCFF1 − FCFF ÷ NOPATROC = NOPAT ÷ opening book debt + equity

Memory hook Growth = what you keep × what it earns.

Know it cold NOPAT = operating income × (1 − t). More cash out means a lower SGR.

Key terms

  • constant growth dividend discount model Memorize
  • constant growth model for any cash flow Memorize
  • no-growth perpetuity Know the idea
  • stable-growth firm Know the idea
  • company life cycle Know the idea
  • sensitivity to g Use it
  • nominal growth rate Know the idea
  • implied constant growth rate Use it
  • historical growth rate Know the idea
  • base effect Know the idea
  • sustainable growth rate (SGR) Memorize
  • FCFE sustainable growth rate Use it
  • FCFF sustainable growth rate Use it
  • net operating profit after tax (NOPAT) Use it
  • return on capital (ROC) Use it
When constant growth fits
Is constant growth appropriate?stable cash flow growing below the required return, forever
  • Good fit Use it mature, stable
    • Traits average risk; industry-average debt; return on capital near WACC
    • Cash lower capital spending, higher payout
  • Poor fit Use it use multistage growth or detailed forecasts
    • Startup little revenue, losses
    • Fast or changing growth growth above the required return or set to change
Where g comes from
Long-run growth g
  • Historical growth Know the idea watch the base effect
  • Sustainable growth rate Memorize share retained × ROE; FCFE and FCFF versions
  • The economy Know the idea nominal growth, about the risk-free rate

Formulas

Constant growth dividend discount model Memorize
IVt=Dt(1+g)re−g=Dt+1re−gIV_t = \dfrac{D_t(1 + g)}{r_e - g} = \dfrac{D_{t+1}}{r_e - g}
Quick example

Last dividend 2.40, g 2%, r_e 7%: D_{t+1} = 2.448 and 2.448 ÷ 0.05 = 48.96.

On your calculator
  1. 2.4×1.02÷(0.07−0.02)= 48.96
  1. 2.4ENTER1.02× D_{t+1}
  2. 0.07ENTER0.02− r_e − g
  3. ÷ 48.96

ENTER enter (on the HP, puts the number on the stack)

Constant growth model, any cash flow Memorize
IVt=E(CF)t+1Required rate of return−gIV_t = \dfrac{E(CF)_{t+1}}{\text{Required rate of return} - g}
Quick example

Next year's FCFE 34 million, r_e 12%, g 4%, 20 million shares: 34 ÷ 0.08 = 425 million, 21.25 per share.

On your calculator
  1. 34÷(0.12−0.04)=÷20= 21.25
  1. 34ENTER0.12ENTER0.04−÷20÷ 21.25

ENTER enter (on the HP, puts the number on the stack)

Implied constant growth rate Use it
gimplied=Required rate of return−E(CF)t+1Ptg_{implied} = \text{Required rate of return} - \dfrac{E(CF)_{t+1}}{P_t}
Quick example

Next year's dividend 1.80, price 45.00, r_e 8.5%: 8.5% − 4.0% = 4.5%.

On your calculator
  1. 1.8÷45=+|-+0.085= 0.045

+|- change sign

  1. 0.085ENTER1.8ENTER45÷− 0.045

ENTER enter (on the HP, puts the number on the stack)

Sustainable growth rate Memorize
SGRt=(1−DividendstNet incomet)×Net incometBVEt−1SGR_t = \left(1 - \dfrac{\text{Dividends}_t}{\text{Net income}_t}\right) \times \dfrac{\text{Net income}_t}{\text{BVE}_{t-1}}
Quick example

Payout 40%, ROE 12%: 0.60 × 12% = 7.2%.

On your calculator
  1. 1−0.4=×0.12= 0.072
  1. 1ENTER0.4−0.12× 0.072

ENTER enter (on the HP, puts the number on the stack)

FCFE and FCFF sustainable growth rates Use it
FCFE SGRt=(1−FCFEtNet incomet)×Net incometBVEt−1FCFF SGRt=(1−FCFFtNOPATt)×NOPATtBook value of total capitalt−1\begin{aligned} \text{FCFE SGR}_t &= \left(1 - \dfrac{\text{FCFE}_t}{\text{Net income}_t}\right) \times \dfrac{\text{Net income}_t}{\text{BVE}_{t-1}} \\ \text{FCFF SGR}_t &= \left(1 - \dfrac{\text{FCFF}_t}{\text{NOPAT}_t}\right) \times \dfrac{\text{NOPAT}_t}{\text{Book value of total capital}_{t-1}} \end{aligned}
Quick example

FCFE is 70% of net income and ROE is 15%: 0.30 × 15% = 4.5%. FCFF is half of NOPAT and ROC is 11%: 5.5%.

On your calculator
  1. 1−0.7=×0.15= 0.045
  1. 1ENTER0.7−0.15× 0.045

ENTER enter (on the HP, puts the number on the stack)

4

Two-stage (multistage) growth models

Known

MemorizeTwo-stage in four steps

StepWhat to do
1Grow CF_t at g_s to get CF_{t+1} … CF_{t+n}
2CF_{t+n+1} = CF_{t+n} × (1 + g_l)
3IV_{t+n} = CF_{t+n+1} ÷ (r_e − g_l), a value at period n
4IV_t = PV of CF_{t+1} … CF_{t+n} + IV_{t+n} ÷ (1 + r_e)^n

Memory hook The constant growth value sits one period before its first cash flow, so the cash flow you need is the one AFTER the fast stage.

Know it cold Discount IV_{t+n} for n periods, grow the first stable year at g_l, and divide total FCFE values by shares at the end.

MemorizeWhich suits which company

CompanyModel
Mature, stable growth below r_econstant growth
A spell of fast or slow growth, then stabletwo-stage
Path known year by yeara rate for each year
Startup without revenue, or transformationno growth model; detailed forecasts or multiples

Memory hook One stage per growth regime you can defend.

Know it cold The long-term growth rate must stay below r_e and near the economy's growth.

Key terms

  • multistage growth model Use it
  • two-stage growth model Memorize
  • short-term growth rate g_s Use it
  • long-term growth rate g_l Use it
  • terminal value at the end of the fast stage Memorize
  • first cash flow of the stable stage Memorize
  • revenue growth versus cash flow growth Know the idea
  • terminal value share of intrinsic value Use it
  • sensitivity to n, g_l and r_e Use it
Which growth model for which path
Choose by the growth pathstages you can defend
  • Constant growth Use it already stable; mature
  • Two-stage Use it a spell of above- or below-trend growth, then stable
    • Typical case advantage that fades as rivals arrive
    • Also fits a temporary slump before recovery
  • A rate for each year Know the idea path forecast year by year
  • No growth model Know the idea startups without revenue; transformational change

Formulas

Two-stage growth model Memorize
IVt=∑i=1nFCFEt(1+gs)i(1+re)i+IVt+n(1+re)nIV_t = \sum_{i=1}^{n} \dfrac{\text{FCFE}_t(1 + g_s)^{i}}{(1 + r_e)^{i}} + \dfrac{IV_{t+n}}{(1 + r_e)^{n}}
Quick example

D_t = 1.50, growth 12% for two years then 3%, r_e = 10%. D_{t+1} = 1.68, D_{t+2} = 1.8816, IV_{t+2} = 1.938048 ÷ 0.07 = 27.6864. IV_t = 1.68 ÷ 1.10 + (1.8816 + 27.6864) ÷ 1.10² = 25.96.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓1.68ENTER↓↓29.568ENTER
  3. NPV10ENTER↓CPT 25.9636

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · NPV net present value · CPT compute

  1. fCLEAR REG
  2. 0gCF01.68gCFj29.568gCFj10i
  3. fNPV 25.9636

f gold shift key · CLEAR REG clear all registers · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · i interest rate per period · NPV net present value

Terminal value at the end of the fast stage Memorize
IVt+n=FCFEt+n+1re−glIV_{t+n} = \dfrac{\text{FCFE}_{t+n+1}}{r_e - g_l}
Quick example

The last fast-stage dividend is 2.50 (year 3), g_l = 5%, r_e = 12%. D_{t+4} = 2.50 × 1.05 = 2.625 and IV_{t+3} = 2.625 ÷ 0.07 = 37.50.

On your calculator
  1. 2.5×1.05÷(0.12−0.05)= 37.5
  1. 2.5ENTER1.05× D_{t+4}
  2. 0.12ENTER0.05− r_e − g_l
  3. ÷ 37.5

ENTER enter (on the HP, puts the number on the stack)

First cash flow of the stable stage Memorize
FCFEt+n+1=FCFEt(1+gs)n(1+gl)\text{FCFE}_{t+n+1} = \text{FCFE}_t(1 + g_s)^{n}(1 + g_l)
Quick example

D_t = 1.20, three years at 15%, then 5%. D_{t+4} = 1.20 × 1.15³ × 1.05 = 1.9163.

On your calculator
  1. 1.2×1.15×1.15×1.15×1.05= 1.9163
  1. 1.2ENTER1.15×1.15×1.15×1.05× 1.9163

ENTER enter (on the HP, puts the number on the stack)

5

Discount rates and terminal values

Known

MemorizeRates and the firm-to-equity bridge

Cash flowRateGives
Dividends, FCFE, residual incomerer_eequity value
FCFFWACCfirm value; − market value of debt = equity
EBITDAa pre-tax rateneeds care

Memory hook WACC ≤ r_e: mixing in cheaper, tax-saving debt can only pull the average down.

Know it cold WACC=wd rd(1−t)+we re\text{WACC} = w_d\, r_d(1 - t) + w_e\, r_e with market-value weights; FCFE at WACC overstates value and mismatches the claim.

MemorizeTerminal values

QuestionAnswer
What is it?expected value at the end of the horizon
Two methodsconstant growth (present value) or multiples
When is it a bigger share?shorter forecast horizons
Usual g_lbetween expected inflation and nominal GDP growth, weighted by where the firm sells
Can g_l be negative?yes: slow decline, not a negative price

Memory hook Long-run, local, below r.

Know it cold Use long-run averages, not this year's rates; the base case sits inside a range.

Key terms

  • discount rate consistent with the claim Memorize
  • weighted average cost of capital (WACC) Memorize
  • market value weights Use it
  • after-tax cost of debt Use it
  • enterprise value from FCFF Use it
  • equity value equals firm value minus market value of debt Memorize
  • WACC at or below the cost of equity Memorize
  • consistency of FCFE and FCFF valuations Know the idea
  • implied FCFF growth Know the idea
  • terminal value Memorize
  • terminal value share and forecast horizon Use it
  • terminal growth between inflation and nominal GDP growth Memorize
  • geographic mix of growth Use it
  • negative terminal growth Use it
  • base case and value range Know the idea
Which rate for which cash flow
Discount ratematch the claim in the numerator
  • Required return on equity, r_e Memorize dividends, FCFE, residual income
  • WACC Memorize FCFF; then subtract the market value of debt
  • A pre-tax rate Know the idea EBITDA
Choosing long-run growth g_l
Long-run growth g_lbelow the discount rate
  • Sources Know the idea SGR, history, the economy
  • Usual range Memorize expected inflation to nominal GDP growth
  • Which economies Use it where the company sells, weighted
  • Declining firm Use it can be below inflation or negative

Formulas

Weighted average cost of capital Memorize
WACC=MVDebtMVDebt+MVEquity rd(1−Tax rate)+MVEquityMVDebt+MVEquity re\text{WACC} = \dfrac{MV_{Debt}}{MV_{Debt} + MV_{Equity}}\, r_d(1 - \text{Tax rate}) + \dfrac{MV_{Equity}}{MV_{Debt} + MV_{Equity}}\, r_e
Quick example

Debt 35% of market value, r_d 7%, tax 25%, r_e 11%: 0.35 × 5.25% + 0.65 × 11% = 1.84% + 7.15% = 8.99%.

On your calculator
  1. 0.35×7×0.75=STO1
  2. 0.65×11+RCL1= 8.9875

STO store · RCL recall a stored number

  1. 0.35ENTER7×0.75×0.65ENTER11×+ 8.9875

ENTER enter (on the HP, puts the number on the stack)

Equity value from FCFF Memorize
IVt=[∑i=1nFCFFt+i(1+WACC)i+E(Pt+n)(1+WACC)n]−MVDebtIV_t = \left[\sum_{i=1}^{n} \dfrac{\text{FCFF}_{t+i}}{(1 + \text{WACC})^{i}} + \dfrac{E(P_{t+n})}{(1 + \text{WACC})^{n}}\right] - MV_{Debt}
Quick example

Next year's FCFF 33, WACC 9.5%, g 2.5%, debt 180: 33 ÷ 0.07 = 471.43; equity 291.43.

On your calculator
  1. 33÷(0.095−0.025)=−180= 291.43
  1. 33ENTER0.095ENTER0.025−÷180− 291.43

ENTER enter (on the HP, puts the number on the stack)

6

Valuing preferred stock, and how call, put and conversion features change value

Known

MemorizePreferred stock values

ShareFormula
PerpetualIVt=D/rpIV_t = D / r_p
With a maturity date∑D/(1+rp)i+FV/(1+rp)n\sum D/(1+r_p)^i + FV/(1+r_p)^n
Implied return (perpetual)rp=D/Ptr_p = D / P_t

Memory hook A bond with a dividend: fixed payments, par back at the end, no growth, no terminal value.

Know it cold rp<rer_p < r_e because preferred is paid first, for dividends and in liquidation; discount at r_p, never at the dividend rate.

MemorizeFeatures and dividends

FeatureWhose rightDividendValue at the same dividend
Callableissuerhigherlower
Putableinvestorlowerhigher
Convertibleinvestorlowerhigher

Memory hook Callable > plain > putable: whoever holds the option pays for it.

Know it cold Issuer-friendly features lower intrinsic value; investor-friendly features raise it.

Key terms

  • preferred stock Know the idea
  • priority claim Memorize
  • fixed preferred dividend D Use it
  • face or par value FV Use it
  • required return on preferred stock r_p Memorize
  • r_p below r_e Memorize
  • preferred stock with a maturity date Memorize
  • perpetual preferred stock Memorize
  • implied required return on preferred Use it
  • no terminal value for preferred Know the idea
  • contingency features Know the idea
  • callable preferred stock Memorize
  • putable preferred stock Memorize
  • convertible preferred stock Memorize
  • conversion ratio Know the idea
  • cumulative and participating dividends Know the idea
How features change preferred stock
Non-callable, non-convertible preferredthe base case: D ÷ r_p, or bond-style with FV
  • Callable by the issuer Memorize issuer's right: higher dividend, lower value at the same dividend
  • Putable by the investor Memorize investor's right: lower dividend, higher value
  • Convertible into common Memorize investor's right: lower dividend; conversion ratio
Preferred against common in a present value model
Dividend discount model
  • Common shares Know the idea growing dividends, r_e, a terminal value
  • Preferred shares Memorize fixed D, r_p < r_e, FV at maturity or a perpetuity; no growth, no terminal value

Formulas

Preferred stock with a maturity date Memorize
IVt=∑i=1nD(1+rp)i+FV(1+rp)nIV_t = \sum_{i=1}^{n} \dfrac{D}{(1 + r_p)^{i}} + \dfrac{FV}{(1 + r_p)^{n}}
Quick example

Par (FV) of 50.00, annual dividend of 2.75, eight years to maturity, r_p = 6.25%. Value = 47.69, valued like a bond.

On your calculator
  1. 2NDCLR TVM
  2. 8N6.25I/Y2.75PMT50FV
  3. CPTPV −47.69

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 8n6.25i2.75PMT50FV
  3. PV −47.69

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Perpetual preferred stock Memorize
IVt=Drp,D=dividend rate×parIV_t = \dfrac{D}{r_p}, \quad D = \text{dividend rate} \times \text{par}
Quick example

A 5.0% preferred share on par of 45.00 pays 2.25. At a required return of 7.5%, 2.25 ÷ 0.075 = 30.00.

On your calculator
  1. 0.05×45÷0.075= 30
  1. 0.05ENTER45×0.075÷ 30

ENTER enter (on the HP, puts the number on the stack)

Implied required return on preferred Use it
rp=DPtr_p = \dfrac{D}{P_t}
Quick example

Dividend 4.40, price 52.00: 4.40 ÷ 52.00 = 8.46%.

On your calculator
  1. 4.4÷52= 0.0846
  1. 4.4ENTER52÷ 0.0846

ENTER enter (on the HP, puts the number on the stack)

7

Where DCF and growth models break down

Known

MemorizeTwo kinds of limitation

KindHitsExamples
Bottom-upthe cash flow (numerator)dividends miss buybacks; base-year effects; hard-to-define margins and investment; transformation; sudden distress
Top-downthe cost of capital (denominator)changing risk-free rate; developing markets without a risk-free or long-term rate

Memory hook Bottom-up breaks the top of the fraction; top-down breaks the bottom.

Know it cold Best for stable firms with steady dividends and cash flows; weakest for non-payers, negative cash flow and life-cycle turning points.

MemorizeWhat to do about it

ProblemResponse
Possible failureprobability-weighted scenario, not just a negative g
Slow declinea negative long-run growth rate
Rates changinglong-run average risk-free rate matched to the horizon
Rising buybacksFCFE or FCFF, not dividends
Any single modeluse more than one

Memory hook Scenario, negative g, long-run rate, broader cash flow, second model.

Know it cold IVt=p×IVgoing concern+(1−p)×value in distressIV_t = p \times IV_{\text{going concern}} + (1 - p) \times \text{value in distress}.

Key terms

  • bottom-up (company-specific) limitations Memorize
  • top-down (macroeconomic and market) limitations Memorize
  • incomplete cash flow metric Use it
  • dividend versus FCFE valuation gap Use it
  • base-year effects Know the idea
  • static growth assumptions Know the idea
  • transformational growth Know the idea
  • going concern assumption Memorize
  • financial distress Know the idea
  • probability-weighted scenario value Use it
  • changing risk-free rate Use it
  • matching the risk-free rate to the forecast horizon Memorize
  • developing market discount rates Know the idea
  • using more than one model Know the idea
Why a growth model can mislead
Limitations of DCF growth models
  • Bottom-up (company-specific) Memorize the numerator
    • Incomplete measure dividends miss buybacks; payout and share-count shifts
    • Base year one-off years bias growth and forecasts
    • Static assumptions transformation, decline, sudden distress
  • Top-down (macro and market) Memorize the denominator
    • Changing risk-free rate moves r_d, r_e and WACC
    • Developing markets no truly risk-free or long-term rate

Formulas

Probability-weighted value with financial distress Use it
IVt=p×IVgoing concern+(1−p)×Value in distressIV_t = p \times IV_{\text{going concern}} + (1 - p) \times \text{Value in distress}
Quick example

80% chance of a going-concern value of 50, 20% chance of 5 in distress: 0.80 × 50 + 0.20 × 5 = 41.00.

On your calculator
  1. 0.8×50=STO1
  2. 0.2×5+RCL1= 41

STO store · RCL recall a stored number

  1. 0.8ENTER50×0.2ENTER5×+ 41

ENTER enter (on the HP, puts the number on the stack)

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