Analyst Dojo

CFA Level I Equities formulas

Margin and leverage, company analysis and forecasting, dividend discount and free cash flow models, and price multiples.

163 formulas from the Equities lessons (11–14% of the exam), 86 to memorize. Each says how deeply to learn it and opens to a quick worked example with the keystrokes for the TI BA II Plus and the HP 12C.

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.

Module 1 Equity Instrument Features

Common shares and preference shares

Dividend due on cumulative preference shares Use it
D=(n+1)×r×parD = (n + 1) \times r \times \text{par}
Quick example

6% cumulative preference shares, par USD 25, one year missed. (1 + 1) × 0.06 × 25 = USD 3.00 per share before any common dividend.

On your calculator
  1. 1+1=×0.06×25= 3
  1. 1ENTER1+0.06×25× 3

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

Public vs private equity: venture capital, buyouts and PIPEs

Value per share after a PIPE Use it
Vpost=Nold×Pmkt+Nnew×PPIPENold+NnewV_{\text{post}} = \dfrac{N_{\text{old}} \times P_{\text{mkt}} + N_{\text{new}} \times P_{\text{PIPE}}}{N_{\text{old}} + N_{\text{new}}}
Quick example

4.5 million shares at 40.00, plus 0.5 million new shares sold at 36.00: (180.0 + 18.0) ÷ 5.0 million = 39.60 a share.

On your calculator
  1. 4.5×40+(0.5×36)=÷5= 39.6
  1. 4.5ENTER40×0.5ENTER36×+5÷ 39.6

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

Module 2 Equity Jurisdictions, Classes, and Voting

Share classes and voting rights: dual class structures and differential voting shares

Share of votes Use it
Vote share=∑(shares held×votes per share)∑(shares outstanding×votes per share)\text{Vote share} = \dfrac{\sum (\text{shares held} \times \text{votes per share})}{\sum (\text{shares outstanding} \times \text{votes per share})}
Quick example

500 million one-vote shares and 80 million ten-vote shares, with insiders holding all of the ten-vote class. Their votes are 80 × 10 = 800 million out of 500 + 800 = 1,300 million: 61.54% of the votes for 80 ÷ 580 = 13.79% of the shares.

On your calculator
  1. 80×10=STO1
  2. +500=STO2
  3. RCL1÷RCL2= 0.6154

STO store · RCL recall a stored number

  1. 80ENTER10×ENTERENTER500+÷ 0.6154

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

Investing across jurisdictions: dual listings and depositary receipts

Depositary receipt price Use it
PDR=Q×PlocalXP_{DR} = \dfrac{Q \times P_{local}}{X}
Quick example

One share to four depositary receipts (Q = 0.25), shares at Japanese yen (JPY) 6,240, JPY 160.00 per US dollar (USD): 0.25 × 6,240 ÷ 160 = USD 9.75.

On your calculator
  1. 0.25×6240÷160= 9.75
  1. 0.25ENTER6240×160÷ 9.75

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

Implied exchange rate Use it
Ximplied=Q×PlocalPDRX_{implied} = \dfrac{Q \times P_{local}}{P_{DR}}
Quick example

One share to three American depositary receipts (ADRs), so Q = 1/3. Shares at euro (EUR) 54.00, ADR at US dollars (USD) 19.80. The formula gives EUR per USD; for the usual USD per EUR quote, flip it, PDR÷(Q×Plocal)P_{DR} \div (Q \times P_{local}): 3 × 19.80 ÷ 54.00 = USD 1.10 per EUR.

On your calculator
  1. 3×19.8÷54= 1.1
  1. 3ENTER19.8×54÷ 1.1

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

Module 3 Equity Issuance and Trading

Primary and secondary equity markets: issuing shares and why trading matters

IPO underpricing Use it
U=Pclose,1−PofferPofferU = \dfrac{P_{close,1} - P_{offer}}{P_{offer}}
Quick example

Offer price 26.00 Swiss francs (CHF), first-day close CHF 30.68: (30.68 − 26.00) ÷ 26.00 = 18.0%.

On your calculator
  1. 30.68−26=÷26= 0.18
  1. 30.68ENTER26−26÷ 0.18

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

Ownership after a follow-on offering Use it
Stake after=Shares heldShares outstanding+New shares\text{Stake after} = \dfrac{\text{Shares held}}{\text{Shares outstanding} + \text{New shares}}
Quick example

You hold 9 million of 120 million shares. A follow-on sells 15 million new and 24 million existing shares: 9 ÷ 135 = 6.67%.

On your calculator
  1. 9÷(120+15)= 0.06667
  1. 9ENTER120ENTER15+÷ 0.06667

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

Where shares trade: exchanges, off-exchange and over the counter

Bid-ask spread Memorize
Bid-ask spread=Lowest ask−Highest bid\text{Bid-ask spread} = \text{Lowest ask} - \text{Highest bid}
Quick example

Highest bid 36.20, lowest ask 36.30: 36.30 − 36.20 = 0.10.

On your calculator
  1. 36.3−36.2= 0.10
  1. 36.3ENTER36.2− 0.10

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

Bid-ask spread (%) Memorize
Bid-ask spread (%)=Lowest ask−Highest bidLowest ask\text{Bid-ask spread (\%)} = \dfrac{\text{Lowest ask} - \text{Highest bid}}{\text{Lowest ask}}
Quick example

Bid 4.50, ask 5.00: (5.00 − 4.50) ÷ 5.00 = 10.00%. Over the mid-market price (4.75) it would be 10.53%, the wrong base.

On your calculator
  1. 5−4.5=÷5= 0.10
  1. 5ENTER4.5−5÷ 0.10

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

Mid-market price Memorize
Mid-market price=Lowest ask+Highest bid2\text{Mid-market price} = \dfrac{\text{Lowest ask} + \text{Highest bid}}{2}
Quick example

Highest bid 36.20, lowest ask 36.30: (36.30 + 36.20) ÷ 2 = 36.25.

On your calculator
  1. 36.3+36.2=÷2= 36.25
  1. 36.3ENTER36.2+2÷ 36.25

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

Average execution price of a market order Use it
Pˉ=∑iqiPi∑iqi\bar{P} = \dfrac{\sum_i q_i P_i}{\sum_i q_i}
Quick example

Asks are 400 shares at 36.30 and 900 at 36.35. A market buy for 1,000 shares pays (400 × 36.30 + 600 × 36.35) ÷ 1,000 = 36.33.

On your calculator
  1. (400×36.3)+(600×36.35)=÷1000= 36.33
  1. 400ENTER36.3×600ENTER36.35×+1000÷ 36.33

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

Volume-weighted average price (VWAP) Memorize
VWAP=∑i=1nPricei×Volumei∑i=1nVolumei\text{VWAP} = \dfrac{\sum_{i=1}^{n} \text{Price}_i \times \text{Volume}_i}{\sum_{i=1}^{n} \text{Volume}_i}
Quick example

Trades of 1,000 shares at 24.50, 3,000 at 24.80 and 1,000 at 24.65: 123,550 ÷ 5,000 = 24.71. The simple average of the prices, 24.65, ignores the sizes.

On your calculator
  1. (1000×24.5)+(3000×24.8)+(1000×24.65)= 123,550
  2. ÷5000= 24.71
  1. 1000ENTER24.5×3000ENTER24.8×+1000ENTER24.65×+ 123,550
  2. 5000÷ 24.71

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

Liquidity: market float, average daily volume and turnover

Market float Memorize
Market float=Shares outstanding−Restricted shares\text{Market float} = \text{Shares outstanding} - \text{Restricted shares}
Quick example

A company has 300 million shares outstanding, of which 60 million are restricted: 300 − 60 = 240 million shares in the float.

On your calculator
  1. 300−60= 240
  1. 300ENTER60− 240

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

Average daily volume (ADV) Memorize
ADV=∑i=1nShares tradedin\text{ADV} = \dfrac{\sum_{i=1}^{n} \text{Shares traded}_i}{n}
Quick example

A share trades 0.62, 0.48, 0.55, 0.51 and 0.64 million shares over five business days: 2.80 ÷ 5 = 0.56 million shares a day.

On your calculator
  1. 0.62+0.48+0.55+0.51+0.64=÷5= 0.56
  1. 0.62ENTER0.48+0.55+0.51+0.64+5÷ 0.56

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

Turnover ratio Memorize
Turnover ratio=ADVMarket float\text{Turnover ratio} = \dfrac{\text{ADV}}{\text{Market float}}
Quick example

ADV of 0.56 million shares and a market float of 240 million: 0.56 ÷ 240 = 0.23% a day.

On your calculator
  1. 0.56÷240= 0.002333
  1. 0.56ENTER240÷ 0.002333

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

Days to sell a position Use it
Days=⌈Position÷ADVDaily limit (% of ADV)⌉\text{Days} = \left\lceil \dfrac{\text{Position} \div \text{ADV}}{\text{Daily limit (\% of ADV)}} \right\rceil
Quick example

A fund holds 0.5 million shares, ADV is 0.8 million and the daily limit is 15% of ADV: 0.5 ÷ (0.15 × 0.8) = 4.17, so 5 trading days.

On your calculator
  1. 0.5÷(0.15×0.8)= 4.1667, round up to 5
  1. 0.5ENTER0.15ENTER0.8×÷ 4.1667, round up to 5

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

Types of equity indexes

Equal weight Use it
wi=1Nw_i = \dfrac{1}{N}
Quick example

A 30-stock equal-weighted index gives every member 1 ÷ 30 = 3.33%, whatever its cap weight was.

On your calculator
  1. 1÷30= 0.0333
  1. 301/x 0.0333

1/x reciprocal

Multi-market fund return in your currency Use it
Rhome=(1+Rindex)(1+Rcurrency)−1R_{home} = (1 + R_{index})(1 + R_{currency}) - 1
Quick example

A fund tracks an index that gains 4% in euros while the euro loses 2% against your currency: 1.04 × 0.98 − 1 = 1.92%.

On your calculator
  1. 1.04×0.98−1= 0.0192
  1. 1.04ENTER0.98×1− 0.0192

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

Module 4 Sources of Equity Returns

Dividends, share repurchases, stock splits and the dividend timetable

Price after a split or stock dividend Use it
P1=P0k,k=new sharesold sharesP_1 = \dfrac{P_0}{k}, \quad k = \dfrac{\text{new shares}}{\text{old shares}}
Quick example

A 4-for-3 split (k = 1.3333) of a share at 80.00 gives 80.00 ÷ 1.3333 = 60.00. A 20% stock dividend is k = 1.20, and a 1-for-5 reverse split is k = 0.20.

On your calculator
  1. 80÷4×3= 60
  1. 80ENTER4÷3× 60

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

Shares repurchased Use it
Nrepurchased=cash distributedPN_{repurchased} = \dfrac{\text{cash distributed}}{P}
Quick example

A company with 40 million shares at INR 250 spends INR 1,000 million on a buyback: 1,000 ÷ 250 = 4 million shares, leaving 36 million. As a dividend it would be INR 25 a share, a realized return of 25 ÷ 250 = 10%.

On your calculator
  1. 1000÷250= 4 million shares
  2. 25÷250= 0.10
  1. 1000ENTER250÷ 4
  2. 25ENTER250÷ 0.10

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

Ex-dividend price Use it
Pex≈Pt−1−DP_{ex} \approx P_{t-1} - D
Quick example

A share closes at 36.00 the day before its ex-dividend date and the dividend is 0.60. The price should open near 36.00 − 0.60 = 35.40.

On your calculator
  1. 36−0.6= 35.4
  1. 36ENTER0.6− 35.4

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

Ex-date price change Use it
Pex−Pt−1Pt−1=−DPt−1+rmarket+rcompany\dfrac{P_{ex} - P_{t-1}}{P_{t-1}} = -\dfrac{D}{P_{t-1}} + r_{market} + r_{company}
Quick example

Prior close 30.00, dividend 0.45, ex-date close 29.37, index −0.80%: −2.10% = −1.50% − 0.80% + company-specific, so company-specific = +0.20%.

On your calculator
  1. 29.37−30=÷30= −0.021 total
  2. 0.45÷30= 0.015 dividend, so −2.10% + 1.50% + 0.80% = +0.20%
  1. 29.37ENTER30−30÷ −0.021
  2. 0.45ENTER30÷ 0.015

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

Price return, total return and the price of a share

Price return Memorize
rE=Pt+1−PtPtr_E = \dfrac{P_{t+1} - P_t}{P_t}
Quick example

A share bought at 36.00 that pays nothing and is sold at 38.70 returns (38.70 − 36.00) ÷ 36.00 = 7.50%.

On your calculator
  1. 38.7−36=÷36= 0.075
  1. 38.7ENTER36−36÷ 0.075

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

Total return Memorize
rE=Pt+1−Pt+Dt+1Ptr_E = \dfrac{P_{t+1} - P_t + D_{t+1}}{P_t}
Quick example

Buy at 25.00, receive a dividend of 0.75 and sell at 26.50: (1.50 + 0.75) ÷ 25.00 = 9.00% (a price return of 6.00%).

On your calculator
  1. 26.5−25+0.75=÷25= 0.09
  1. 26.5ENTER25−0.75+25÷ 0.09

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

Return with the dividend reinvested in shares Memorize
rE=Pt+1(1+DPPD)−PtPtr_E = \dfrac{P_{t+1}\left(1 + \dfrac{D}{P_{PD}}\right) - P_t}{P_t}
Quick example

Buy at 40.00. A dividend of 0.80 is reinvested at 42.00 and the share ends at 46.00. 46.00 × (1 + 0.80 ÷ 42.00) = 46.8762, so the return is (46.8762 − 40.00) ÷ 40.00 = 17.19%.

On your calculator
  1. 0.8÷42+1=×46= 46.8762
  2. −40=÷40= 0.1719
  1. 0.8ENTER42÷1+46× 46.8762
  2. 40−40÷ 0.1719

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

Return over several periods Use it
r=(1+r1)(1+r2)⋯(1+rn)−1r = (1 + r_1)(1 + r_2)\cdots(1 + r_n) - 1
Quick example

Yearly total returns of 6%, −2% and 9%: 1.06 × 0.98 × 1.09 − 1 = 13.23%. Adding them would give 13.00%.

On your calculator
  1. 1.06×0.98×1.09−1= 0.1323
  1. 1.06ENTER0.98×1.09×1− 0.1323

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

Price from next year's expected price Memorize
Pt=E(Pt+1)+E(Dt+1)1+reP_t = \dfrac{E(P_{t+1}) + E(D_{t+1})}{1 + r_e}
Quick example

An expected price of 30.00 and an expected dividend of 0.90 in a year, with a required return of 8%: (30.00 + 0.90) ÷ 1.08 = 28.61.

On your calculator
  1. 30+0.9=÷1.08= 28.6111
  1. 30ENTER0.9+1.08÷ 28.6111

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

Price from n years of dividends and a target Memorize
Pt=∑i=1nD(1+re)i+E(Pt+n)(1+re)nP_t = \sum_{i=1}^{n} \dfrac{D}{(1 + r_e)^i} + \dfrac{E(P_{t+n})}{(1 + r_e)^n}
Quick example

A dividend of 0.60 a year for 5 years, a target of 20.00 and a required return of 7%: 0.60 × 4.1002 + 20.00 ÷ 1.07⁵ = 2.46 + 14.26 = 16.72.

On your calculator
  1. 2NDCLR TVM
  2. 5N7I/Y0.6PMT20FV
  3. CPTPV −16.72

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. 5n7i0.6PMT20FV
  3. PV −16.72

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

Module 5 Introduction to Equity Valuation

Market price, intrinsic value and perceived value

Expected return if the gap closes Use it
r=IVt−PtPtr = \dfrac{IV_t - P_t}{P_t}
Quick example

A share that pays no dividend is valued at 62.40 and trades at 52.00. (62.40 − 52.00) ÷ 52.00 = 20.00%. Divide by the price paid, not by the value (16.67% is the trap).

On your calculator
  1. 62.4−52=÷52= 0.2000
  1. 62.4ENTER52−52÷ 0.2000

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

Scenario-weighted intrinsic value Use it
IVt=∑i=1nProbability(i)×Pt,iIV_t = \sum_{i=1}^{n} \text{Probability}(i) \times P_{t,i}
Quick example

Bear 38.00 (25%), base 50.00 (50%), bull 66.00 (25%): 9.50 + 25.00 + 16.50 = 51.00.

On your calculator
  1. (0.25×38)+(0.5×50)+(0.25×66)= 51.00
  1. 0.25ENTER38×0.5ENTER50×+0.25ENTER66×+ 51.00

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

Equity valuation models: present value, asset-based and relative

Present value model (general form) Use it
IVt=∑i=1nE(CFt+i)(1+r)i+E(Pt+n)(1+r)nIV_t = \sum_{i=1}^{n} \dfrac{E(CF_{t+i})}{(1 + r)^{i}} + \dfrac{E(P_{t+n})}{(1 + r)^{n}}
Quick example

Cash flows 1.50 and 1.60, expected price 30.00 at the end of year 2, required return 8%: 1.50 ÷ 1.08 + 31.60 ÷ 1.08² = 1.39 + 27.09 = 28.48.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓1.5ENTER↓↓31.6ENTER
  3. NPV8ENTER↓CPT 28.48

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. 0gCF01.5gCFj31.6gCFj8i
  3. fNPV 28.48

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 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 500, interest 40, tax 30%, capital expenditures 220: 500 + 28 − 220 = 308.

On your calculator
  1. 40×0.7+500−220= 308
  1. 500ENTER40ENTER0.7×+220− 308

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

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 500, capital expenditures 220, borrowing 60, repayment 100: 500 − 220 − 40 = 240.

On your calculator
  1. 500−220+60−100= 240
  1. 500ENTER220−60+100− 240

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

Residual income model Know the idea
IVt=BVEt+∑i=1nRIt+i(1+re)i+E(Pt+n)−BVEt+n(1+re)nIV_t = BVE_t + \sum_{i=1}^{n} \dfrac{RI_{t+i}}{(1 + r_e)^{i}} + \dfrac{E(P_{t+n}) - BVE_{t+n}}{(1 + r_e)^{n}}
Quick example

Book value 20.00 per share and positive expected residual income: value above 20.00, so price-to-book above 1.

Price-to-earnings ratio Memorize
P/E=PtEPStP/E = \dfrac{P_t}{EPS_t}
Quick example

Price 63.00, earnings per share 4.20: 63.00 ÷ 4.20 = 15.0.

Weighted value across models Use it
IVt=∑jwj×VjIV_t = \sum_j w_j \times V_j
Quick example

Discounted cash flow 32, comparables 28, residual income 45 with weights 50%, 30%, 20%: 16.0 + 8.4 + 9.0 = 33.40.

On your calculator
  1. (0.5×32)+(0.3×28)+(0.2×45)= 33.40
  1. 0.5ENTER32×0.3ENTER28×+0.2ENTER45×+ 33.40

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

Book value, market capitalization and enterprise value

Book value of equity Memorize
BVEt=Book value of assetst−Book value of liabilitiestBVE_t = \text{Book value of assets}_t - \text{Book value of liabilities}_t
Quick example

A firm reports assets of 880 and liabilities of 540 (in millions). Book value of equity = 880 − 540 = 340.

Book value per share Memorize
(BVEShare)t=(Shareholders’ equityShares outstanding)t\left(\dfrac{BVE}{\text{Share}}\right)_t = \left(\dfrac{\text{Shareholders' equity}}{\text{Shares outstanding}}\right)_t
Quick example

Shareholders' equity of 468 million (minority interest already excluded) and 36 million shares: 468 ÷ 36 = 13.00 per share.

On your calculator
  1. 468÷36= 13.00
  1. 468ENTER36÷ 13.00

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

Shareholders' equity components Know the idea
Shareholders’ equity=Share capital+Additional paid-in capital+Retained earnings−Treasury stock\text{Shareholders' equity} = \text{Share capital} + \text{Additional paid-in capital} + \text{Retained earnings} - \text{Treasury stock}
Quick example

Share capital 100, additional paid-in capital 240, retained earnings 310, treasury stock 40: 100 + 240 + 310 − 40 = 610.

On your calculator
  1. 100+240+310−40= 610
  1. 100ENTER240+310+40− 610

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

Market capitalization Memorize
Market capitalizationt=Pt×Shares outstandingt\text{Market capitalization}_t = P_t \times \text{Shares outstanding}_t
Quick example

A share price of 18.50 and 60 million shares: 18.50 × 60 = 1,110 million.

On your calculator
  1. 18.5×60= 1,110
  1. 18.5ENTER60× 1,110

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

Price-to-book ratio Memorize
(PB)t=(PBVE/Share)t\left(\dfrac{P}{B}\right)_t = \left(\dfrac{P}{BVE/\text{Share}}\right)_t
Quick example

Price 30.00 and book value per share 12.50: 30.00 ÷ 12.50 = 2.40.

On your calculator
  1. 30÷12.5= 2.40
  1. 30ENTER12.5÷ 2.40

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

Enterprise value Memorize
EV=Market value of debt+Market value of equity+Market value of preferred shares−Cash and short-term investmentsEV = \text{Market value of debt} + \text{Market value of equity} + \text{Market value of preferred shares} - \text{Cash and short-term investments}
Quick example

Market capitalization 2,000, debt 700, no preferred shares, cash 250: 2,000 + 700 − 250 = 2,450.

On your calculator
  1. 2000+700−250= 2,450
  1. 2000ENTER700+250− 2,450

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

EV multiple Memorize
EV multiple=Enterprise valueRevenue or an operating profit measure\text{EV multiple} = \dfrac{\text{Enterprise value}}{\text{Revenue or an operating profit measure}}
Quick example

Enterprise value 2,450 and EBITDA 350: 2,450 ÷ 350 = 7.0.

On your calculator
  1. 2450÷350= 7.0
  1. 2450ENTER350÷ 7.0

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

Module 6 Discounted Cash Flow and Growth Models

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

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

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

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)

Constant growth models, implied growth and the sustainable growth rate

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)

Two-stage (multistage) growth models

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)

Discount rates and terminal values

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)

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

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)

Where DCF and growth models break down

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)

Module 7 Relative Valuation with Multiples

Valuing with multiples: comparables and forecasted fundamentals

Value from a peer multiple Use it
V=peer multiple×company fundamentalV = \text{peer multiple} \times \text{company fundamental}
Quick example

A peer median price-to-earnings (P/E) ratio of 12.0 and company earnings per share (EPS) of 2.40 give 12.0 × 2.40 = 28.80 per share. Applied to total net income it gives a market capitalization instead.

On your calculator
  1. 12×2.4= 28.80
  1. 12ENTER2.4× 28.80

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

Justified P/B from fundamentals Memorize
PtBVEt=ROE×Payout ratiore−g=ROE−gre−g\dfrac{P_t}{BVE_t} = \dfrac{ROE \times \text{Payout ratio}}{r_e - g} = \dfrac{ROE - g}{r_e - g}
Quick example

Return on equity 17%, cost of equity 11%, growth 3%: (0.17 − 0.03) ÷ (0.11 − 0.03) = 0.14 ÷ 0.08 = 1.75.

On your calculator
  1. 0.17−0.03=÷(0.11−0.03)= 1.75
  1. 0.17ENTER0.03−0.11ENTER0.03−÷ 1.75

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

Sustainable growth rate Memorize
g=(1−Payout ratio)×ROEg = (1 - \text{Payout ratio}) \times ROE
Quick example

Payout ratio 40% and return on equity 15%: 0.60 × 15% = 9.0%.

On your calculator
  1. 1−0.4=×0.15= 0.09
  1. 1ENTER0.4−0.15× 0.09

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

PEG ratio Use it
PEG=P/EExpected EPS growth rate\text{PEG} = \dfrac{\text{P/E}}{\text{Expected EPS growth rate}}
Quick example

A P/E of 24 with expected EPS growth of 12% (entered as 12) gives 24 ÷ 12 = 2.00.

On your calculator
  1. 24÷12= 2.00
  1. 24ENTER12÷ 2.00

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

Regression-predicted multiple Know the idea
P/Bi=a+b1×ROEi+b2×gi+ei\text{P/B}_i = a + b_1 \times ROE_i + b_2 \times g_i + e_i
Quick example

Estimated a = −0.80, b₁ = 20, b₂ = 25; a peer with ROE 12% and growth 3% is predicted at −0.80 + 2.40 + 0.75 = 2.35.

On your calculator
  1. 0.8+|-+(20×0.12)+(25×0.03)= 2.35

+|- change sign

  1. 0.8CHSENTER20ENTER0.12×+25ENTER0.03×+ 2.35

CHS change sign · ENTER enter (on the HP, puts the number on the stack)

Price multiples: P/E, earnings yield, P/S, P/B and dividend yield

Price-to-earnings ratio Memorize
P/E=PtEPS,Trailing: EPSt,Forward: EPSt+1\text{P/E} = \dfrac{P_t}{\text{EPS}}, \quad \text{Trailing: EPS}_t, \quad \text{Forward: EPS}_{t+1}
Quick example

Price 54.00, trailing EPS 3.60 and forward EPS 4.00: trailing P/E 15.0 and forward P/E 13.5.

On your calculator
  1. 54÷3.6= 15.0
  2. 54÷4= 13.5
  1. 54ENTER3.6÷ 15.0
  2. 54ENTER4÷ 13.5

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

Price and PVGO Use it
Pt=EPSt+1re+PVGO,PVGOtPt=1−EPSt+1/PtreP_t = \dfrac{\text{EPS}_{t+1}}{r_e} + \text{PVGO}, \quad \dfrac{\text{PVGO}_t}{P_t} = 1 - \dfrac{\text{EPS}_{t+1}/P_t}{r_e}
Quick example

Price 60.00, forward EPS 3.30, cost of equity 11%: earnings yield 5.5%, so PVGO ÷ P = 1 − 0.50 = 50%.

On your calculator
  1. 1−(3.3÷60÷0.11)= 0.50
  1. 1ENTER3.3ENTER60÷0.11÷− 0.50

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

Justified trailing P/E Memorize
PtEPSt=Payout ratio×(1+g)re−g\dfrac{P_t}{\text{EPS}_t} = \dfrac{\text{Payout ratio} \times (1 + g)}{r_e - g}
Quick example

Payout 65%, cost of equity 11%, growth 3%: 0.65 × 1.03 ÷ 0.08 = 8.37.

On your calculator
  1. 0.65×1.03÷(0.11−0.03)= 8.37
  1. 0.65ENTER1.03×0.11ENTER0.03−÷ 8.37

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

Normalized EPS Use it
EPSnorm=EPS‾cycle  or  ROE‾cycle×BVPSt\text{EPS}_{norm} = \overline{\text{EPS}}_{cycle} \;\text{or}\; \overline{\text{ROE}}_{cycle} \times \text{BVPS}_t
Quick example

Average return on equity 11% and book value per share 20.00: normalized EPS = 0.11 × 20.00 = 2.20.

Justified P/S Use it
PtSalest=Net profit margin×Payout ratio×(1+g)re−g\dfrac{P_t}{\text{Sales}_t} = \dfrac{\text{Net profit margin} \times \text{Payout ratio} \times (1 + g)}{r_e - g}
Quick example

Margin 12%, payout 65%, cost of equity 11%, growth 3%: 0.12 × 8.37 = 1.00.

On your calculator
  1. 0.12×0.65×1.03÷0.08= 1.00
  1. 0.12ENTER0.65×1.03×0.08÷ 1.00

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

Price-to-book and price to tangible book Use it
P/B=PtBVPSt,Price to tangible book=PtBVPSt−intangibles per share\text{P/B} = \dfrac{P_t}{\text{BVPS}_t}, \quad \text{Price to tangible book} = \dfrac{P_t}{\text{BVPS}_t - \text{intangibles per share}}
Quick example

Price 35.00, book value per share 28.00 of which intangibles 8.00: P/B 1.25, price to tangible book 1.75.

Justified dividend yield Memorize
Trailing: DivtPt=re−g1+g,Leading: Divt+1Pt=re−g\text{Trailing: } \dfrac{Div_t}{P_t} = \dfrac{r_e - g}{1 + g}, \quad \text{Leading: } \dfrac{Div_{t+1}}{P_t} = r_e - g
Quick example

Cost of equity 12%, growth 5%: trailing 0.07 ÷ 1.05 = 6.67%, leading 7.00%.

On your calculator
  1. 0.12−0.05=÷1.05= 0.0667
  1. 0.12ENTER0.05−1.05÷ 0.0667

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

Dividend yield, payout and P/E Use it
DivtPt=DivtEPSt×EPStPt\dfrac{Div_t}{P_t} = \dfrac{Div_t}{\text{EPS}_t} \times \dfrac{\text{EPS}_t}{P_t}
Quick example

Payout 50% and P/E 12.5: dividend yield = 0.50 ÷ 12.5 = 4.00%, so the actual P/E equals the payout ratio ÷ the dividend yield.

Price to FCFE Know the idea
PtFCFEt+1=1re−g\dfrac{P_t}{\text{FCFE}_{t+1}} = \dfrac{1}{r_e - g}
Quick example

Cost of equity 12.5%, growth 4.5%: 1 ÷ 0.08 = 12.5.

Enterprise value multiples

Enterprise value Memorize
EV=MV of debt+MV of equity+MV of preferred shares−Cash and short-term investmentsEV = \text{MV of debt} + \text{MV of equity} + \text{MV of preferred shares} - \text{Cash and short-term investments}
Quick example

Market capitalization 900, debt 350, preferred shares 50, cash 140: 900 + 350 + 50 − 140 = 1,160.

On your calculator
  1. 900+350+50−140= 1,160
  1. 900ENTER350+50+140− 1,160

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

Firm value Know the idea
Firm value=MV of debt+MV of equity\text{Firm value} = \text{MV of debt} + \text{MV of equity}
Quick example

Debt 350 and equity 900 at market value: 1,250.

Justified EV/EBITDA Use it
EVEBITDA=(1−t)+Dep×tEBITDA−CapexEBITDA−ΔWCEBITDAWACC−g\dfrac{EV}{\text{EBITDA}} = \dfrac{(1 - t) + \dfrac{\text{Dep} \times t}{\text{EBITDA}} - \dfrac{\text{Capex}}{\text{EBITDA}} - \dfrac{\Delta \text{WC}}{\text{EBITDA}}}{\text{WACC} - g}
Quick example

Ratios to EBITDA: depreciation 0.20, capital expenditures 0.28, change in working capital 0.05; tax 25%; WACC 10.5%, g 3.5%: (0.75 + 0.05 − 0.28 − 0.05) ÷ 0.07 = 6.71.

On your calculator
  1. 0.75+(0.2×0.25)−0.28−0.05=÷0.07= 6.71
  1. 0.75ENTER0.2ENTER0.25×+0.28−0.05−0.07÷ 6.71

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

Value to book and EV to book Use it
Value to book=MV equity+MV debtBV equity+BV debt,EV to book=MV equity+MV debt−CashBV equity+BV debt−Cash\text{Value to book} = \dfrac{\text{MV equity} + \text{MV debt}}{\text{BV equity} + \text{BV debt}}, \quad \text{EV to book} = \dfrac{\text{MV equity} + \text{MV debt} - \text{Cash}}{\text{BV equity} + \text{BV debt} - \text{Cash}}
Quick example

Equity 700 and debt 450 at market; book equity 350, book debt 450; cash 90: value to book 1,150 ÷ 800 = 1.44, EV to book 1,060 ÷ 710 = 1.49.

Justified EV to book value of capital Use it
EVBook value of capital=ROC×(1−Reinvestment rate)WACC−g\dfrac{EV}{\text{Book value of capital}} = \dfrac{\text{ROC} \times (1 - \text{Reinvestment rate})}{\text{WACC} - g}
Quick example

Return on capital 11%, reinvestment rate 20%, WACC 8%, g 2.2%: 0.088 ÷ 0.058 = 1.52.

On your calculator
  1. 0.11×0.8÷(0.08−0.022)= 1.52
  1. 0.11ENTER0.8×0.08ENTER0.022−÷ 1.52

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

Value to FCFF Know the idea
VtFCFFt+1=1WACC−g\dfrac{V_t}{\text{FCFF}_{t+1}} = \dfrac{1}{\text{WACC} - g}
Quick example

WACC 8.5%, g 3.5%: 1 ÷ 0.05 = 20.0.

Choosing a peer group for multiples

Segment-weighted peer multiple Use it
M=∑jwj×MjM = \sum_j w_j \times M_j
Quick example

Revenue 70% in a business whose peers trade at 9.0 times and 30% in one at 15.0 times: 6.3 + 4.5 = 10.8.

On your calculator
  1. 0.7×9+(0.3×15)= 10.8
  1. 0.7ENTER9×0.3ENTER15×+ 10.8

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

CAPM (for factor-based peers) Know the idea
E(Ri)=Rf+βi [E(Rm)−Rf]E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]
Quick example

Risk-free rate 4%, beta 1.2, expected market return 9%: 4% + 1.2 × 5% = 10.0%.

On your calculator
  1. 1.2×5+4= 10.0
  1. 1.2ENTER5×4+ 10.0

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

Trailing, current and forward multiples

Trailing and forward P/E Memorize
Trailing P/E=PtEPSt,Forward P/E=PtEPSt+1\text{Trailing P/E} = \dfrac{P_t}{\text{EPS}_t}, \quad \text{Forward P/E} = \dfrac{P_t}{\text{EPS}_{t+1}}
Quick example

Price 54.00, trailing EPS 4.50 and consensus forward EPS 5.40: trailing P/E 12.0, forward P/E 10.0.

On your calculator
  1. 54÷4.5= 12.0
  2. 54÷5.4= 10.0
  1. 54ENTER4.5÷ 12.0
  2. 54ENTER5.4÷ 10.0

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

Present value with a terminal multiple Use it
IV0=∑t=1nE(CFt)(1+r)t+E(Pt+n)(1+r)n,E(Pt+n)=year-n fundamental×multipleIV_0 = \sum_{t=1}^{n} \dfrac{E(CF_t)}{(1 + r)^t} + \dfrac{E(P_{t+n})}{(1 + r)^n}, \quad E(P_{t+n}) = \text{year-}n\text{ fundamental} \times \text{multiple}
Quick example

Cash flows of 10 in year 1 and 12 in year 2, a terminal multiple of 10 on the year-2 figure, required return 12%: 10 ÷ 1.12 + (12 + 120) ÷ 1.2544 = 114.16.

On your calculator
  1. 10÷1.12+(132÷1.2544)= 114.16
  1. 10ENTER1.12÷132ENTER1.2544÷+ 114.16

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

Module 8 Financial Statement Forecasting

Why build a disaggregated model, and at what level of detail

Intrinsic value from forecast FCFF Know the idea
IVt=∑i=1nFCFFt+i(1+WACC)i+E(Pt+n)(1+WACC)nIV_t = \sum_{i=1}^{n} \frac{FCFF_{t+i}}{(1 + WACC)^i} + \frac{E(P_{t+n})}{(1 + WACC)^n}
Quick example

Free cash flow to the firm of 40 and 44 in two years, a terminal value of 900 at the end of year 2, and a weighted average cost of capital of 10%. 40 ÷ 1.10 + 44 ÷ 1.10² + 900 ÷ 1.10² = 816.53, the value of the firm before subtracting debt.

On your calculator
  1. 944÷1.21=STO1 780.17 (year 2 flow and terminal value)
  2. 40÷1.1+RCL1= 816.53

STO store · RCL recall a stored number

  1. 944ENTER1.21÷ 780.17
  2. 40ENTER1.1÷+ 816.53

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

Intrinsic value from an EV/FCFF multiple Know the idea
IVt=FCFFt+1×EV multipletIV_t = FCFF_{t+1} \times \text{EV multiple}_t
Quick example

Next-year free cash flow to the firm of 50 and an enterprise value multiple of 14 times that cash flow give a firm value of 50 × 14 = 700.

On your calculator
  1. 50×14= 700
  1. 50ENTER14× 700

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

Dividend growth when the payout ratio changes Use it
gDPS=(1+gEPS)×PayouttPayoutt−1−1g_{DPS} = (1 + g_{EPS}) \times \frac{\text{Payout}_t}{\text{Payout}_{t-1}} - 1
Quick example

Earnings per share grow 5% while the payout ratio goes from 20% to 25%. Dividends per share grow 1.05 × 25 ÷ 20 − 1 = 31.25%.

On your calculator
  1. 1.05×0.25÷0.2−1= 0.3125
  1. 1.05ENTER0.25×0.2÷1− 0.3125

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

NOPAT Memorize
NOPAT=(Revenue−COGS−SG&A−D&A)(1−t)NOPAT = (\text{Revenue} - \text{COGS} - \text{SG\&A} - \text{D\&A})(1 - t)
Quick example

Revenue 200, cost of goods sold 110, selling, general and administrative expenses 40, depreciation and amortization 10, tax rate 30%. Operating profit is 40, so net operating profit after taxes is 40 × 0.70 = 28.

On your calculator
  1. 200−110−40−10=×0.7= 28
  1. 200ENTER110−40−10−0.7× 28

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

Net investment Memorize
Net investment=ΔWorking capital+ΔNet long-term assets\text{Net investment} = \Delta \text{Working capital} + \Delta \text{Net long-term assets}
Quick example

Working capital rises by 5 and net long-term assets by 8, so net investment is 5 + 8 = 13.

FCFF from NOPAT Memorize
FCFF=NOPAT−Net investmentFCFF = NOPAT - \text{Net investment}
Quick example

Net operating profit after taxes of 28 less net investment of 13 leaves free cash flow to the firm of 15.

On your calculator
  1. 28−13= 15
  1. 28ENTER13− 15

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

Revenue models: historical, bottom-up and top-down

Revenue from market size and share Use it
R=M×sgR=M1s1M0s0−1R = M \times s \qquad g_R = \frac{M_1 s_1}{M_0 s_0} - 1
Quick example

Market 312 → 330 (billions) at a constant 4.5% share. Revenue 14.04 → 14.85, growth 330 ÷ 312 − 1 = 5.77%. If the share also rose, growth would be higher.

On your calculator
  1. 330÷312−1= 0.0577
  1. 330ENTER312÷1− 0.0577

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

Revenue from units and revenue per unit Use it
R1=R0 (1+gunits)(1+grev per unit)R_1 = R_0\,(1 + g_{units})(1 + g_{\text{rev per unit}})
Quick example

Revenue 80.0; volume +5%, average price +2%. 80.0 × 1.05 × 1.02 = 85.68, growth 7.1%, not 7.0%: the cross term is real.

On your calculator
  1. 80×1.05×1.02= 85.68
  1. 80ENTER1.05×1.02× 85.68

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

Top-down revenue across regions Use it
R=∑iUnits soldi×Revenue per unitiUnits soldi=Market unitsi×siR = \sum_{i} \text{Units sold}_i \times \text{Revenue per unit}_i \qquad \text{Units sold}_i = \text{Market units}_i \times s_i
Quick example

Region A: market 1,000,000 units, share 10%, price 50. Region B: market 500,000 units, share 8%, price 80. Units sold are 100,000 and 40,000, so revenue is 5.0 million + 3.2 million = 8.2 million.

On your calculator
  1. 1000000×0.1×50=STO1 5000000
  2. 500000×0.08×80+RCL1= 8200000

STO store · RCL recall a stored number

  1. 1000000ENTER0.1×50×500000ENTER0.08×80×+ 8200000

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

Expense models: margins, itemization and operating leverage

Expense growth from a regression Use it
ΔExpenset=a+b×ΔRevenuet\Delta \text{Expense}_t = a + b \times \Delta \text{Revenue}_t
Quick example

Intercept a = 0.010, slope b = 0.85, revenue growth 8%: 0.010 + 0.85 × 0.08 = 7.8%, so the margin rises.

On your calculator
  1. 0.85×0.08+0.01= 0.078
  1. 0.85ENTER0.08×0.01+ 0.078

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

Cost of sales implied by a gross margin Use it
COGSt+1=(1−mt+1)×Revenuet+1\text{COGS}_{t+1} = (1 - m_{t+1}) \times \text{Revenue}_{t+1}
Quick example

Revenue 400 → 432 (+8%), gross margin 30% → 32%. Cost of goods sold goes from 280 to 0.68 × 432 = 293.76: growth 4.9%.

On your calculator
  1. 432×0.68=÷280−1= 0.0491
  1. 432ENTER0.68×280÷1− 0.0491

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

Cost of sales with fixed and variable parts Use it
COGS=F+v×RevenueGross margin=1−COGS÷Revenue\text{COGS} = F + v \times \text{Revenue} \qquad \text{Gross margin} = 1 - \text{COGS} \div \text{Revenue}
Quick example

Fixed 100, variable 50% of revenue. Revenue 300: cost of goods sold 250, margin 16.7%. Revenue 360: cost 280, margin 22.2%.

On your calculator
  1. 0.5×360+100=÷360=+|-+1= 0.2222

+|- change sign

  1. 0.5ENTER360×100+360÷CHS1+ 0.2222

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Asset and financing models

Asset growth from a regression Use it
%ΔAssett=a+b×%ΔRevenuet\%\Delta \text{Asset}_t = a + b \times \%\Delta \text{Revenue}_t
Quick example

Intercept a = 0.03, slope b = 0.40, revenue growth 5%: asset growth is 0.03 + 0.40 × 0.05 = 5.0%. "Assets grow with revenue" means a = 0 and b = 1.

On your calculator
  1. 0.4×0.05+0.03= 0.05
  1. 0.4ENTER0.05×0.03+ 0.05

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

Financing needs Memorize
Financing needs=ΔOperating assets−ΔNon-debt liabilities−ΔRetained earnings\text{Financing needs} = \Delta \text{Operating assets} - \Delta \text{Non-debt liabilities} - \Delta \text{Retained earnings}
Quick example

Operating assets rise 60 and non-debt liabilities 14. Net income is 35 and dividends 10, so retained earnings rise 25. Financing needs are 60 − 14 − 25 = 21: money is required.

On your calculator
  1. 60−14−35+10= 21
  1. 60ENTER14−35−10+ 21

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

FCFE from net income Memorize
FCFE=Net income−(ΔGross long-term assets−Depreciation)−ΔWorking capital+ΔDebtFCFE = \text{Net income} - (\Delta \text{Gross long-term assets} - \text{Depreciation}) - \Delta \text{Working capital} + \Delta \text{Debt}
Quick example

Net income 20, gross long-term assets +30, depreciation 10, working capital +5, debt +12. FCFE = 20 − (30 − 10) − 5 + 12 = 7.

On your calculator
  1. 20−30+10−5+12= 7
  1. 20ENTER30−10+5−12+ 7

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

FCFE from the uses of cash Memorize
FCFE=ΔCash+Dividends+Share repurchases−Share issuanceFCFE = \Delta \text{Cash} + \text{Dividends} + \text{Share repurchases} - \text{Share issuance}
Quick example

Cash rises 3, dividends are 6, buybacks 2 and new shares 4. FCFE = 3 + 6 + 2 − 4 = 7, the same company as the card above.

On your calculator
  1. 3+6+2−4= 7
  1. 3ENTER6+2+4− 7

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

Scenario analysis with probability weights

Probability-weighted equity value Use it
Equity value=∑sps×max⁡(Vs,0),∑sps=1\text{Equity value} = \sum_{s} p_s \times \max(V_s, 0), \qquad \sum_s p_s = 1
Quick example

Three scenarios: 90 (probability 35%), 60 (40%) and −20 (25%). The negative value counts as zero, so the weighted value is 0.35 × 90 + 0.40 × 60 + 0.25 × 0 = 55.5.

On your calculator
  1. 0.35×90=STO1 31.5
  2. 0.4×60+RCL1= 55.5

STO store · RCL recall a stored number

  1. 0.35ENTER90×0.4ENTER60×+ 55.5

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

Model construction by company life cycle: from venture to decline

Value from cash flows and a terminal value Know the idea
IVt=∑i=1nE(CFt+i)(1+r)i+E(Pt+n)(1+r)nIV_t = \sum_{i=1}^{n} \frac{E(CF_{t+i})}{(1 + r)^i} + \frac{E(P_{t+n})}{(1 + r)^n}
Quick example

Expected cash flows of 10 and 12, a terminal value of 200 at the end of year 2, required return 12%. 10 ÷ 1.12 + (12 + 200) ÷ 1.12² = 177.93.

On your calculator
  1. 212÷1.2544=STO1 169.01
  2. 10÷1.12+RCL1= 177.93

STO store · RCL recall a stored number

  1. 212ENTER1.2544÷ 169.01
  2. 10ENTER1.12÷+ 177.93

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

Venture capital method Use it
IVt=Expected revenuet+n×P/S(1+r)n=Expected revenuet+n×P/SROIIV_t = \frac{\text{Expected revenue}_{t+n} \times P/S}{(1 + r)^n} = \frac{\text{Expected revenue}_{t+n} \times P/S}{ROI}
Quick example

Revenue of 40 at the horizon, a price-to-sales multiple of 3 and a target return of 10 times the money: 40 × 3 ÷ 10 = 12.0, the post-money value.

On your calculator
  1. 40×3÷10= 12
  1. 40ENTER3×10÷ 12

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

ROI and IRR Use it
ROI=(1+IRR)nROI = (1 + IRR)^n
Quick example

A target IRR of 30% a year for 4 years means an ROI of 1.30⁴ = 2.8561 times the original investment.

On your calculator
  1. 1.3y^x4= 2.8561

y^x power

  1. 1.3ENTER4y^x 2.8561

ENTER enter (on the HP, puts the number on the stack) · y^x power

Pre-money valuation Memorize
Pre-money=Post-money−New equity\text{Pre-money} = \text{Post-money} - \text{New equity}
Quick example

A post-money value of 80 with 20 of new equity means a pre-money value of 80 − 20 = 60.

Fractional ownership Memorize
Fractional ownership=New equityPost-money\text{Fractional ownership} = \frac{\text{New equity}}{\text{Post-money}}
Quick example

New equity of 20 into a post-money value of 80 buys 20 ÷ 80 = 25% of the company.

On your calculator
  1. 20÷80= 0.25
  1. 20ENTER80÷ 0.25

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

Growth-phase value from an earnings multiple Use it
IVt=Net incomet+n×P/EROIIV_t = \frac{\text{Net income}_{t+n} \times P/E}{ROI}
Quick example

Net income of 8 at the horizon, a price-to-earnings multiple of 12 and a target return of 4 times the money: 8 × 12 ÷ 4 = 24.0.

On your calculator
  1. 8×12÷4= 24
  1. 8ENTER12×4÷ 24

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

Value with a probability of distress Memorize
IVt=Going-concern value×(1−pdistress)+Distress sale value×pdistressIV_t = \text{Going-concern value} \times (1 - p_{distress}) + \text{Distress sale value} \times p_{distress}
Quick example

A going-concern value of 100, a 20% chance of distress and a distress sale worth 30 today: 0.80 × 100 + 0.20 × 30 = 86.

On your calculator
  1. 0.2×30=STO1 6
  2. 0.8×100+RCL1= 86

STO store · RCL recall a stored number

  1. 0.2ENTER30×0.8ENTER100×+ 86

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

From the forecast model to an equity value

Sustainable growth rate of FCFF Memorize
FCFF SGRt+n=(1−FCFFt+nNOPATt+n)×NOPATt+nBook value of total capitalt+n−1FCFF\,SGR_{t+n} = \left(1 - \frac{FCFF_{t+n}}{NOPAT_{t+n}}\right) \times \frac{NOPAT_{t+n}}{\text{Book value of total capital}_{t+n-1}}
Quick example

Free cash flow to the firm 30, net operating profit after taxes 40, capital at the start of the year 250. Reinvestment rate 1 − 30 ÷ 40 = 25%, return on capital 40 ÷ 250 = 16%, so growth is 0.25 × 0.16 = 4.0%.

On your calculator
  1. 30÷40=+|-+1=×40÷250= 0.04

+|- change sign

  1. 1ENTER30ENTER40÷−40×250÷ 0.04

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

Terminal firm value (constant growth) Memorize
Firm valuet+n=FCFFt+n×(1+FCFF SGRt+n)WACC−FCFF SGRt+n\text{Firm value}_{t+n} = \frac{FCFF_{t+n} \times (1 + FCFF\,SGR_{t+n})}{WACC - FCFF\,SGR_{t+n}}
Quick example

Final-year free cash flow to the firm of 30, growth of 4.0% and a weighted average cost of capital of 9%: 30 × 1.04 ÷ (0.09 − 0.04) = 624.

On your calculator
  1. 30×1.04÷(0.09−0.04)= 624
  1. 30ENTER1.04×0.09ENTER0.04−÷ 624

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

Implied terminal multiple Use it
Implied multiple=Terminal firm valueEBITt+n\text{Implied multiple} = \frac{\text{Terminal firm value}}{EBIT_{t+n}}
Quick example

A terminal firm value of 624 against final-year operating profit (EBIT) of 52 implies 624 ÷ 52 = 12.0 times EBIT. Compare that with peers.

On your calculator
  1. 624÷52= 12
  1. 624ENTER52÷ 12

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

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

Free cash flows of 10 and 12, a terminal value of 200 at the end of year 2, a weighted average cost of capital of 10% and debt of 50: 10 ÷ 1.10 + (12 + 200) ÷ 1.10² − 50 = 134.30.

On your calculator
  1. 212÷1.21=STO1 175.21
  2. 10÷1.1+RCL1−50= 134.30

STO store · RCL recall a stored number

  1. 212ENTER1.21÷ 175.21
  2. 10ENTER1.1÷+50− 134.30

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

Module 9 Industry and Competitive Analysis

Industry size, growth, profitability and concentration

Market share Memorize
s=company revenue from the industryindustry sizes = \frac{\text{company revenue from the industry}}{\text{industry size}}
Quick example

A producer earns EUR 1.65 billion from regional cement (of EUR 3.10 billion total revenue). Regional cement sales total EUR 5.50 billion. 1.65 ÷ 5.50 = 30.0%. Total revenue would give 56.4%, an overstatement.

On your calculator
  1. 1.65÷5.5= 0.30
  1. 1.65ENTER5.5÷ 0.30

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

Operating profit margin Memorize
Operating margin=Operating incomeTotal revenue\text{Operating margin} = \frac{\text{Operating income}}{\text{Total revenue}}
Quick example

Operating income EUR 0.88 billion on revenue of EUR 5.50 billion. 0.88 ÷ 5.50 = 16.0%.

On your calculator
  1. 0.88÷5.5= 0.16
  1. 0.88ENTER5.5÷ 0.16

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

Return on invested capital Memorize
ROIC=NOPATtAverage invested capital\text{ROIC} = \frac{\text{NOPAT}_t}{\text{Average invested capital}}
Quick example

Net operating profit after taxes (NOPAT) EUR 0.66 billion; invested capital (long-term liabilities plus equity) EUR 6.2 billion at the start and EUR 6.8 billion at the end. 0.66 ÷ 6.5 = 10.15%.

On your calculator
  1. 6.2+6.8=÷2=1/x×0.66= 0.1015

1/x reciprocal

  1. 0.66ENTER6.2ENTER6.8+2÷÷ 0.1015

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

Excess return over the cost of capital Use it
ROIC−WACCorROICWACC\text{ROIC} - \text{WACC} \quad \text{or} \quad \frac{\text{ROIC}}{\text{WACC}}
Quick example

ROIC 10.15% against WACC 8.00%: spread 2.15 points, ratio 10.15 ÷ 8.00 = 1.27. Above 0 (or above 1) means the industry earns more than investors require.

On your calculator
  1. 10.15÷8= 1.27
  1. 10.15ENTER8÷ 1.27

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

Herfindahl-Hirschman Index Memorize
HHI=∑i=1nsi2(si as whole numbers)HHI = \sum_{i=1}^{n} s_i^2 \quad (s_i \text{ as whole numbers})
Quick example

Five equal firms with 20% each. 5 × 20² = 2,000: above 1,800, so highly concentrated. A 50% share is typed as 50, never 0.50. Monopoly = 100² = 10,000.

On your calculator
  1. 20x²+20x²+20x²+20x²+20x²= 2000

x² square

  1. 20ENTER×20ENTER×+20ENTER×+20ENTER×+20ENTER×+ 2000, ENTER then × squares

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

Change in HHI from a merger Use it
ΔHHI=2 sA sB\Delta HHI = 2\,s_A\,s_B
Quick example

Firms with 25% and 12% shares merge. 2 × 25 × 12 = +600. Check: (25 + 12)² − 25² − 12² = 1,369 − 625 − 144 = 600. More than 100 points in a market above 1,800 draws closer scrutiny.

On your calculator
  1. 2×25×12= 600
  1. 2ENTER25×12× 600

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

Module 10 Company Analysis

Competitive strategy: cost leadership, differentiation and focus

Load factor Use it
Load factor=revenue passenger kilometers (RPK)available seat kilometers (ASK)\text{Load factor} = \dfrac{\text{revenue passenger kilometers (RPK)}}{\text{available seat kilometers (ASK)}}
Quick example

Revenue passenger kilometers (RPK) of 1,440 million on available seat kilometers (ASK) of 1,800 million: 1,440 ÷ 1,800 = 80.0%.

On your calculator
  1. 1440÷1800= 0.8
  1. 1440ENTER1800÷ 0.8

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

Revenue drivers and pricing power

Revenue, bottom-up Use it
Revenue=∑volume×price per unit\text{Revenue} = \sum \text{volume} \times \text{price per unit}
Quick example

Segment A sells 3.0 million units at 20; segment B sells 1.2 million units at 35. 3.0 × 20 + 1.2 × 35 = 60 + 42 = 102.

On your calculator
  1. (3×20)+(1.2×35)= 102
  1. 3ENTER20×1.2ENTER35×+ 102

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

Revenue growth from volume and price Memorize
grev=(1+gvol)(1+gprice)−1g_{rev} = (1 + g_{vol})(1 + g_{price}) - 1
Quick example

Volume +12%, average price −2%. 1.12 × 0.98 − 1 = 9.76%. Adding the rates (10%) is the trap.

On your calculator
  1. 1.12×0.98−1= 0.0976
  1. 1.12ENTER0.98×1− 0.0976

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

Market share Memorize
Market share=company revenuemarket size\text{Market share} = \frac{\text{company revenue}}{\text{market size}}
Quick example

Revenue 360 in a market of 4,500 (same units, same period). 360 ÷ 4,500 = 8.0%. Rearranged, market size = revenue ÷ share.

On your calculator
  1. 360÷4500= 0.08
  1. 360ENTER4500÷ 0.08

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

Revenue, top-down Use it
Revenue=obtainable market×share of the obtainable market\text{Revenue} = \text{obtainable market} \times \text{share of the obtainable market}
Quick example

An obtainable market of 800 that grows 5%, and a share rising to 15%: 800 × 1.05 × 0.15 = 126.0.

On your calculator
  1. 800×1.05×0.15= 126
  1. 800ENTER1.05×0.15× 126

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

Operating profitability and working capital

Profit margins Memorize
Net margin=net incomerevenue,Operating margin=operating incomerevenue,EBITDA margin=EBITDArevenue\text{Net margin} = \dfrac{\text{net income}}{\text{revenue}},\quad \text{Operating margin} = \dfrac{\text{operating income}}{\text{revenue}},\quad \text{EBITDA margin} = \dfrac{\text{EBITDA}}{\text{revenue}}
Quick example

Revenue 500, EBITDA 100, D&A 45, interest 25, tax 25%. EBITDA margin 20.0%; operating income 55, 11.0%; net income (55 − 25) × 0.75 = 22.5, 4.5%.

On your calculator
  1. 100−45=÷500= 0.11
  2. 55−25=×0.75=÷500= 0.045
  1. 100ENTER45−500÷ 0.11
  2. 55ENTER25−0.75×500÷ 0.045

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

Operating profit Memorize
Operating profit=[Q×(P−VC)]−FC\text{Operating profit} = [Q \times (P - VC)] - FC
Quick example

A firm sells 2.0 million units at 12 with variable cost of 7 and fixed costs of 6.0 million. 2.0 × (12 − 7) − 6.0 = 4.0 million.

On your calculator
  1. 2×(12−7)−6= 4
  1. 12ENTER7−2×6− 4

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

Cash conversion cycle Memorize
CCC=DOH+DSO−DPOCCC = DOH + DSO - DPO
Quick example

DOH 58, DSO 32, DPO 40. 58 + 32 − 40 = 50 days. Negative means suppliers fund the operating cycle.

On your calculator
  1. 58+32−40= 50
  1. 58ENTER32+40− 50

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

Days measures (from Financial Statement Analysis) Use it
DOH=inventorycost of sales×365,DSO=receivablesrevenue×365,DPO=payablescost of sales×365DOH = \dfrac{\text{inventory}}{\text{cost of sales}} \times 365,\quad DSO = \dfrac{\text{receivables}}{\text{revenue}} \times 365,\quad DPO = \dfrac{\text{payables}}{\text{cost of sales}} \times 365
Quick example

Inventory 1.2, receivables 1.5, payables 1.0; revenue 18.0, cost of sales 10.0. DOH 1.2 ÷ 10.0 × 365 = 43.8; DSO 1.5 ÷ 18.0 × 365 = 30.4; DPO 1.0 ÷ 10.0 × 365 = 36.5 days. Inventory and payables go with cost of sales, receivables with revenue.

On your calculator
  1. 1.2÷10×365= 43.8
  2. 1.5÷18×365= 30.4167
  3. 1÷10×365= 36.5
  1. 1.2ENTER10÷365× 43.8
  2. 1.5ENTER18÷365× 30.4167
  3. 1ENTER10÷365× 36.5

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

Capital investments and capital structure

Free cash flow to equity Memorize
FCFE=NI+D&A−ΔWC−ΔLT assets+ΔDebtFCFE = \text{NI} + \text{D\&A} - \Delta\text{WC} - \Delta\text{LT assets} + \Delta\text{Debt}
Quick example

Net income (NI) 60, depreciation and amortization (D&A) 30, increase in working capital 10, investment in long-term assets 55, net borrowing 15: 60 + 30 − 10 − 55 + 15 = 40.

On your calculator
  1. 60+30−10−55+15= 40
  1. 60ENTER30+10−55−15+ 40

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

Net capital expenditure Memorize
Net capital expenditure=capital expenditure−D&A\text{Net capital expenditure} = \text{capital expenditure} - \text{D\&A}
Quick example

Capital expenditure 2.0, D&A 1.2: 2.0 − 1.2 = 0.8, so capacity is growing. Negative means the asset base is shrinking.

Return on invested capital Memorize
ROIC=NOPATtAverage invested capitalROIC = \dfrac{\text{NOPAT}_t}{\text{Average invested capital}}
Quick example

Net operating profit after taxes (NOPAT) 54, invested capital 420 at the start and 480 at the end. 54 ÷ 450 = 12.0%.

On your calculator
  1. 420+480=÷2=STO1 450
  2. 54÷RCL1= 0.12

STO store · RCL recall a stored number

  1. 54ENTER420ENTER480+2÷÷ 0.12

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

Excess accounting return Memorize
Excess accounting return=ROIC−WACC\text{Excess accounting return} = ROIC - WACC
Quick example

ROIC 12.0%, weighted average cost of capital (WACC) 9.5%: 12.0 − 9.5 = +2.5 points. As a ratio, 12.0 ÷ 9.5 = 1.26, above 1.

On your calculator
  1. 0.12−0.095= 0.025
  1. 0.12ENTER0.095− 0.025

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

Degree of financial leverage Memorize
DFL=%Δnet income%Δoperating incomeDFL = \frac{\%\Delta \text{net income}}{\%\Delta \text{operating income}}
Quick example

Operating income rises 10% and net income rises 14%. 14 ÷ 10 = 1.4.

On your calculator
  1. 14÷10= 1.4
  1. 14ENTER10÷ 1.4

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

Degree of financial leverage with fixed interest Use it
DFL=EBITEBIT−IDFL = \frac{EBIT}{EBIT - I}
Quick example

Operating income (EBIT) 70, fixed interest 20. 70 ÷ (70 − 20) = 1.4. A shortcut that follows from the definition when interest is fixed and the tax rate constant.

On your calculator
  1. 70÷(70−20)= 1.4
  1. 70ENTER70ENTER20−÷ 1.4

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

Total debt to total assets Use it
total debttotal assets\dfrac{\text{total debt}}{\text{total assets}}
Quick example

Debt 10.0, total assets 22.0: 45.5%.

On your calculator
  1. 10÷21= 0.4762
  1. 10ENTER21÷ 0.4762

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

EBITDA to interest expense Use it
EBITDAinterest expense\dfrac{\text{EBITDA}}{\text{interest expense}}
Quick example

Earnings before interest, taxes, depreciation, and amortization (EBITDA) 4.2, interest 1.0: 4.2 times.

On your calculator
  1. 4.2÷1= 4.2
  1. 4.2ENTER1÷ 4.2

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

Module 11 Equity Research Reports

Equity research reports: contents, ratings and price targets

Expected appreciation to the price target Memorize
Expected appreciation=Target price−Current priceCurrent price\text{Expected appreciation} = \dfrac{\text{Target price} - \text{Current price}}{\text{Current price}}
Quick example

A share trades at 2,500 Japanese yen and the one-year target is 2,900 yen. (2,900 − 2,500) ÷ 2,500 = 400 ÷ 2,500 = 16.0%.

On your calculator
  1. 2900−2500=÷2500= 0.16
  1. 2900ENTER2500−2500÷ 0.16

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

Why two analysts using the same model get different values

Two-stage FCFE value with price set equal to value Use it
Pt=IVt=∑i=1nFCFEt+i(1+re)i+FCFEt+n+1/(re−gl)(1+re)nP_t = IV_t = \sum_{i=1}^{n} \dfrac{\text{FCFE}_{t+i}}{(1 + r_e)^i} + \dfrac{\text{FCFE}_{t+n+1} / (r_e - g_l)}{(1 + r_e)^n}
Quick example

Free cash flow to equity (FCFE) of 150, 260 and 380 over three years, 450 in year 4, cost of equity 10% and long-run growth 5%. 636.74 + (450 ÷ 0.05) ÷ 1.331 = 636.74 + 6,761.83 = 7,398.57.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓150ENTER↓↓260ENTER↓↓380ENTER
  3. NPV10ENTER↓CPTSTO2 636.74
  4. 2NDQUIT
  5. 1.1y^x3=STO1 1.331
  6. 450÷0.05÷RCL1+RCL2= 7,398.57

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 · STO store · QUIT leave the worksheet · y^x power · RCL recall a stored number

  1. fCLEAR REG
  2. 0gCF0150gCFj260gCFj380gCFj
  3. 10ifNPV 636.74
  4. 450ENTER0.05÷1.1ENTER3y^x÷+ 7,398.57

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 · ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied long-run growth rate Use it
gl=re−FCFEt+n+1(1+re)n[Pt−∑i=1nFCFEt+i(1+re)i]g_l = r_e - \dfrac{\text{FCFE}_{t+n+1}}{(1 + r_e)^n \left[ P_t - \sum_{i=1}^{n} \dfrac{\text{FCFE}_{t+i}}{(1 + r_e)^i} \right]}
Quick example

Market value 2,000, cost of equity 9%, FCFE of 60 and 80 in years 1 and 2, 95 in year 3. (2,000 − 122.38) × 1.09² = 2,230.80; yield 95 ÷ 2,230.80 = 4.26%; g_l = 9% − 4.26% = 4.74%.

On your calculator
  1. CF2NDCLR WORK
  2. 0ENTER↓60ENTER↓↓80ENTER
  3. NPV9ENTER↓CPTSTO2 122.38
  4. 2NDQUIT
  5. 1.09y^x2=STO1 1.1881
  6. 2000−RCL2=×RCL1= 2,230.80
  7. 1/x×95= 0.0426
  8. +|-+0.09= 0.0474

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 · STO store · QUIT leave the worksheet · y^x power · RCL recall a stored number · 1/x reciprocal · +|- change sign

  1. fCLEAR REG
  2. 0gCF060gCFj80gCFj
  3. 9ifNPV 122.38
  4. CHS2000+ 1,877.62
  5. 1.09ENTER2y^x× 2,230.80
  6. 1/x95× 0.0426
  7. CHS0.09+ 0.0474

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 · CHS change sign · ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

FCFE from net income Memorize
FCFE=Net income−Net investment+Net debt issued\text{FCFE} = \text{Net income} - \text{Net investment} + \text{Net debt issued}
Quick example

Net income 540, net investment 150, net debt issued 60. 540 − 150 + 60 = 450.

On your calculator
  1. 540−150+60= 450
  1. 540ENTER150−60+ 450

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

Equity reinvestment rate Memorize
Equity reinvestment rate=Net investment−Net debt issuedNet income=1−FCFENet income\text{Equity reinvestment rate} = \dfrac{\text{Net investment} - \text{Net debt issued}}{\text{Net income}} = 1 - \dfrac{\text{FCFE}}{\text{Net income}}
Quick example

Net income 540, net investment 150, net debt issued 60. (150 − 60) ÷ 540 = 16.67%, the same as 1 − 450 ÷ 540.

On your calculator
  1. 150−60=÷540= 0.1667
  1. 150ENTER60−540÷ 0.1667

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

Long-run FCFE growth Memorize
Long-run FCFE growth=Equity reinvestment rate×ROE\text{Long-run FCFE growth} = \text{Equity reinvestment rate} \times \text{ROE}
Quick example

Net income 200, net investment 130, net debt issued 30, return on equity (ROE) 14%. Reinvestment rate (130 − 30) ÷ 200 = 50%, and 50% × 14% = 7.0%.

On your calculator
  1. 130−30=÷200=×0.14= 0.07
  1. 130ENTER30−200÷0.14× 0.07

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

Module 12 The CAPM, the Market Model and Factor Models

Estimating the cost of equity with the CAPM and the market model

Capital asset pricing model (CAPM) Memorize
E[ri]=rf+βi(E[rm]−rf)E[r_i] = r_f + \beta_i\left(E[r_m] - r_f\right)
Quick example

Risk-free rate 2.0%, expected market return 8.0%, beta 1.25. The premium is 8.0 − 2.0 = 6.0%, so the cost of equity is 2.0 + 1.25 × 6.0 = 9.50%.

On your calculator
  1. 8−2=×1.25+2= 9.5
  1. 8ENTER2−1.25×2+ 9.5

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

Beta Memorize
βi=σi,mσm2=ρi,m σi σmσm2=ρi,m σiσm\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \rho_{i,m}\,\dfrac{\sigma_i}{\sigma_m}
Quick example

Correlation 0.80, stock volatility 24%, market volatility 16%. Beta = 0.80 × 24 ÷ 16 = 1.20.

On your calculator
  1. 0.8×24÷16= 1.2
  1. 0.8ENTER24×16÷ 1.2

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

Correlation from beta Memorize
ρi,m=βi σmσi\rho_{i,m} = \beta_i\,\dfrac{\sigma_m}{\sigma_i}
Quick example

Beta 0.85, stock volatility 34%, market volatility 20%. Correlation = 0.85 × 20 ÷ 34 = 0.50.

On your calculator
  1. 0.85×20÷34= 0.5
  1. 0.85ENTER20×34÷ 0.5

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

Excess-return regression Know the idea
ri,t−rf,t=αi+βi(rm,t−rf,t)+εtr_{i,t} - r_{f,t} = \alpha_i + \beta_i\left(r_{m,t} - r_{f,t}\right) + \varepsilon_t
Quick example

With α = 0 and β = 1.2, a week in which the market beats the risk-free rate by 1.0% predicts a stock excess return of 1.2 × 1.0 = 1.2%.

Market model Memorize
ri,t=αi+βi rm,t+εi,tr_{i,t} = \alpha_i + \beta_i\,r_{m,t} + \varepsilon_{i,t}
Quick example

α = 0.05% a week, β = 1.10, market return 0.80% in the week. The fitted stock return is 0.05 + 1.10 × 0.80 = 0.93%.

On your calculator
  1. 1.1×0.8+0.05= 0.93
  1. 1.1ENTER0.8×0.05+ 0.93

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

Adjusted beta Memorize
Adjusted beta=w1×Raw beta+w2×1.0,w2=1−w1\text{Adjusted beta} = w_1 \times \text{Raw beta} + w_2 \times 1.0, \quad w_2 = 1 - w_1
Quick example

Raw beta 1.60 with weights of two-thirds and one-third: (2/3)(1.60) + (1/3)(1.0) = 1.40.

On your calculator
  1. 1.6×2+1=÷3= 1.4, two parts raw plus one part 1.0, over three
  1. 1.6ENTER2×1+3÷ 1.4

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

Industry betas, the market risk premium and country risk

Unlevered beta Memorize
βu=βL,Comps1+(1−t) DMVE\beta_u = \dfrac{\beta_{L,Comps}}{1 + (1 - t)\,\dfrac{D}{MVE}}
Quick example

Comparables average a levered beta of 1.25, a 22% tax rate and D/MVE of 0.35. β_u = 1.25 ÷ [1 + 0.78 × 0.35] = 1.25 ÷ 1.273 = 0.98 (0.9819).

On your calculator
  1. 0.78×0.35+1=STO1 1.273
  2. 1.25÷RCL1= 0.9819

STO store · RCL recall a stored number

  1. 1.25ENTER0.78ENTER0.35×1+÷ 0.9819

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

Relevered beta Memorize
βL,firm=βu[1+(1−t) DfirmMVEfirm]\beta_{L,firm} = \beta_u\left[1 + (1 - t)\,\dfrac{D_{firm}}{MVE_{firm}}\right]
Quick example

Unlevered beta 0.95; the company has a 30% tax rate and D/MVE of 0.60. β_L,firm = 0.95 × [1 + 0.70 × 0.60] = 0.95 × 1.42 = 1.35 (1.349).

On your calculator
  1. 0.7×0.6+1=×0.95= 1.349
  1. 0.95ENTER0.7ENTER0.6×1+× 1.349

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

Historical market risk premium Memorize
MRP=rˉindex−rˉgovernmentMRP = \bar{r}_{index} - \bar{r}_{government}
Quick example

Arithmetic means: stocks 9.8%, long-term government bonds 5.1%, bills 3.2%. Against bonds the premium is 9.8 − 5.1 = 4.7%; against bills it would be 6.6%.

On your calculator
  1. 9.8−5.1= 4.7
  1. 9.8ENTER5.1− 4.7

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

Implied market risk premium Memorize
E[rm]=Div1,mP0,m+gl,m,Implied MRP=E[rm]−rfE[r_m] = \dfrac{Div_{1,m}}{P_{0,m}} + g_{l,m}, \qquad \text{Implied MRP} = E[r_m] - r_f
Quick example

An index stands at 3,000 with expected dividends of 69 next year, long-run dividend growth of 5.0% and a government yield of 3.8%. E[r_m] = 69 ÷ 3,000 + 5.0% = 7.3%; implied MRP = 7.3 − 3.8 = 3.5%.

On your calculator
  1. 69÷3000+0.05−0.038= 0.035
  1. 69ENTER3000÷0.05+0.038− 0.035

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

CAPM with a country risk premium Memorize
E[ri]=rf+βi(MRPdeveloped+CRP)E[r_i] = r_f + \beta_i\left(MRP_{developed} + CRP\right)
Quick example

Risk-free rate 3.2%, beta 1.15, developed MRP 5.0%, CRP 3.6% (a 2.0% spread × 1.8). 3.2 + 1.15 × (5.0 + 3.6) = 3.2 + 9.89 = 13.09%.

On your calculator
  1. 2×1.8+5=×1.15+3.2= 13.09
  1. 2ENTER1.8×5+1.15×3.2+ 13.09

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

Country risk premium Memorize
CRP=Sovereign bond yield spread×Annualized equity market standard deviationAnnualized bond market standard deviationCRP = \text{Sovereign bond yield spread} \times \dfrac{\text{Annualized equity market standard deviation}}{\text{Annualized bond market standard deviation}}
Quick example

Spread 2.0%, equity market 1.8 times as volatile as the bond market: CRP = 2.0 × 1.8 = 3.6%.

On your calculator
  1. 2×1.8= 3.6
  1. 2ENTER1.8× 3.6

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

Arbitrage pricing theory and multi-factor models

Arbitrage pricing theory (APT) Memorize
E[ri]=rf+βi,1E(f1)+βi,2E(f2)+⋯+βi,kE(fk)E[r_i] = r_f + \beta_{i,1}E(f_1) + \beta_{i,2}E(f_2) + \cdots + \beta_{i,k}E(f_k)
Quick example

Risk-free rate 2.5%; exposure 1.1 to a factor with a 4.0% premium and −0.6 to a factor with a 3.0% premium. 2.5 + 1.1 × 4.0 − 0.6 × 3.0 = 2.5 + 4.4 − 1.8 = 5.1%.

On your calculator
  1. 1.1×4−(0.6×3)+2.5= 5.1
  1. 1.1ENTER4×0.6ENTER3×−2.5+ 5.1

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

Four-factor model Memorize
E(ri)=rf+βi,MarketE(rm−rf)+βi,SizeE(SMB)+βi,ValueE(HML)+βi,MomentumE(UMD)E(r_i) = r_f + \beta_{i,Market}E(r_m - r_f) + \beta_{i,Size}E(SMB) + \beta_{i,Value}E(HML) + \beta_{i,Momentum}E(UMD)
Quick example

Risk-free rate 2.5%; betas 1.05, 0.30, −0.25, 0.15; annual premiums 5.2%, 1.5%, 3.0%, 4.0%. 2.5 + 5.46 + 0.45 − 0.75 + 0.60 = 8.26%.

On your calculator
  1. 1.05×5.2+(0.3×1.5)−(0.25×3)+(0.15×4)+2.5= 8.26
  1. 1.05ENTER5.2×0.3ENTER1.5×+0.25ENTER3×−0.15ENTER4×+2.5+ 8.26

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

Multi-factor cost of equity Use it
E(ri)=rf+∑kβi,k E(fk),E(fk)annual=12×E(fk)monthlyE(r_i) = r_f + \sum_{k}\beta_{i,k}\,E(f_k), \qquad E(f_k)_{annual} = 12 \times E(f_k)_{monthly}
Quick example

Monthly premiums 0.55%, 0.20%, 0.30%, 0.50% become 6.6%, 2.4%, 3.6%, 6.0%. With a 4.0% risk-free rate and betas 1.30, −0.30, −0.80, 0.10: 4.0 + 8.58 − 0.72 − 2.88 + 0.60 = 9.58%.

On your calculator
  1. 1.3×6.6−(0.3×2.4)−(0.8×3.6)+(0.1×6)+4= 9.58
  1. 1.3ENTER6.6×0.3ENTER2.4×−0.8ENTER3.6×−0.1ENTER6×+4+ 9.58

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

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