🔎 Formulas

Formulas will be added to this page as they are covered in class. The formulas are grouped by lecture, and each lecture has a link to the relevant lecture notes.

Press Ctrl-D to bookmark this page. A downloadable paper/Microsoft Word formula sheet can be found in my File Share.

Questions or comments? Please email robmgmte2700@gmail.com. Remember, your first reference is always the lectures and the homework. Feel free to download my materials, but please do not reupload them.

In some of the right-hand examples columns, I write formulas in "spreadsheet-style." '*' represents multiplication and '^' represents exponents.

Formulas will be added to this document after Bruce has introduced them in class.

L2 - Forms of Business Organization

	# of owners	Limited Liability	Owners Manage the Firm?	Ownership changes dissolve firm?	Taxation
Sole Proprietorship	One	No¹	Yes	Yes	Personal/ Passthrough
Partnership	Unlimited	No¹	Yes	Yes	Personal/ Passthrough
Limited Partnership	Unlimited²	GP - No¹ LP - Yes	GP - Yes LP - No	GP - Yes LP - No	Personal/ Passthrough
LLC	Unlimited	Yes	Yes	No³	Personal/ Passthrough
S Corp	At most 100	Yes	No⁴	No	Personal/ Passthrough
C corporation	Unlimited	Yes	No (but they may)	No	Double

1 Limited liability means that each investor's liability is limited to their investment. Without it, owners are liable for the firm's debts.
2 At least one general partner (GP). No limit on limited partners (LPs).
3 However, most LLCs require the approval of the other members to transfer your ownership.
4 but they may.

References: 2 Feb 3.ppt and L2-Notes

L3 - Accounting Concepts

Name

Equation

Balance Sheet

Balance Sheet Equation

Assets = Liabilities + Equity

Net Working Capital (NWC)

= Current Assets - Current Liabilities

Enterprise Value

= Market Value of Equity + Debt - Cash

Income Statement

Earnings Per Share (EPS)

= Net Income / Shares Outstanding

Fully Diluted EPS

= Net Inc./ (Shares + Stock Options + Convertibles)

Statement of Cash Flows (t = tax rate)

Effect of Depreciation on Cash Flow

ΔCF = ΔDepreciation * t

Effect of Depreciation on Net Income

ΔNI = - ΔDepreciation * (1-t)

Payout Ratio

= Dividends/ Net Income

Balance Sheet
Assets
Current Assets
(Cash, Acct. Rec., Inventories)
Long-Term Assets
(PP&E, Intangibles, other)
Liabilities
Current Liabilities
(Acct Pay, Notes & Short-term Debt due in 1 yr.)
Long-Term Debt
(Bonds & Loans due in >1 year)
Equity
Common Stock and Paid in Surplus
Retained Earnings

Income Statement
Net Sales - Cost of Sales
= Gross Profit
- SG&A - R&D - Dep & Amort
= Operating Income
+ Other Income
= EBIT
- Interest Inc. (Exp.)
= Pretax Inc.
- Taxes
= Net Income

Statement of Cash Flow
Operating Activities
Start w/ Net Income
Add Depreciation & Amort
Deduct increase in Accts Rec.
Add increase in Accounts Pay
Deduct increase in Inventory
(Instead of last three steps, you can simply deduct increases in Net Working Capital)

Investment Activity
Deduct Capex
Deduct other assets purchased

Financing Activity
Deduct dividends paid & share purchases.
Add increase in borrowing

References: 2 Feb 3.ppt and L2-Notes | 3 Feb 10.ppt and L3-Notes

Many more formulas used for Financial Statement and DuPont Analysis can be found at the end of this document.

L4/L5 - Capital Budgeting

Name

Equation

Example

Present Discounted Value (PDV)

= Future Value / ((1+ i)^n)

= 600/(1+.035)^1 = 579.71

Future Value

= Present Value * ((1+ i)^n)

= 600*(1.035)^1 = 621.00

Average Profits after Taxes

= Average annual operating cash inflows - Average annual depreciation

=32000 - 9000 = 21000

Payback Period

The amount of time it takes to pay back the initial investment.

Discounted Payback Period

Calc. Payback Period using PDV of cash flows

Free Cash Flow

= (Revenues - Costs - Depreciation) × (1 - Tax Rate) + Depreciation - CapEx - Change in NWC

= (13,000 - 8,300 - 1500) * (1-40%) + 1500 + 0 - 1125 = $2295

Net Present Value (NPV)

PV of inflows - PV of outflows

Use =NPV(Rate, CashFlow1, CashFlow2, T=3Cash Flow, ...)

Internal Rate of Return (IRR)

PV of inflows - PV of outflows = 0

Use IRR() in spreadsheet or IRR button in calculator.

NPV in spreadsheet

= NPV(Rate, T=1 Net CashFlow, T=2 Net CashFlow, T=3 Net CashFlow 3, ...) + (T=0 Net CashFlow)

=NPV(11%, 0, 1000) - 650 [invest $650 today and get $1000 in 2 years]

IRR in spreadsheet

= IRR(values)

=IRR(A1:A5)

References: 4 Feb 17.ppt and L4-Notes | 5 Feb 24.ppt

L6 - Project Analysis

Name

Equation

Example

Profitability Index

\frac{= \text{PV of Inflows}}{\text{PV of initial Outflows}}

2B - (1.6B/1.08) = .51B or 35% difference

Sensitivity Analysis

Critical Parameters (focus on 1 parameter)
Revenue, Unit Price, Market Share, Market Size, Unit Variable Cost, Fixed Costs

Revenue = \text{unit sales} × \text{unit price} \\ \text{Unit Sales} = \text{market share} × \text{size of market}

Scenario Analysis

Several simultaneous parameter assumptions

Break-even Analysis

Change in Single Factor that makes NPV = 0

Monte Carlo Simulation

Step 1: Modeling the Project
Step 2: Specify Probabilities
Step 3: Simulate Cash Flows
Step 4: Calculate Present Value

Real Options Analysis

A right but not obligated to exercise.

References: 6 Mar 3.ppt

L7 - WACC (Weighted Average Cost of Capital)

Name

Equation

Example

Weighted Average Cost of Cap (WACC)

1. (Total Value of Debt) + (Pref*Mkt Val) + (Market Cap of Equity) = Mkt Val of Firm
2. Weights = Percent of Mkt Val
3. Multiply Weights × Effective Rates
4. Add together for weighted average

r_{wacc} = r_E E\% + r_{pfd} P\% + r_D (1 - T_C)D\%

Example has no preferred equity.

A = 1.8B + 1B*13.2 = 15
D% = 1.8/15 = 12%,
E% = 13.2/15 = 88%
WACC=12%*7% + 88%*9% = 8.76%

Effective Annual Rate (EAR)

= (1+ period return) ^ (# periods in year) - 1

= (1+.0102)^2- 1 = 2.41%

Rate (Excel)

= RATE(nper, paymt, -P_B, F)

= RATE(30,10,-950,1000) = 1.20%

Eff. Rate Cost of Debt Capital

= Debt Yield × (1- Corp. Tax Rate)

= 3.5% * (1-.30) = 2.45% Eff. rate

Eff. Rate Cost of Preferred Stock Cap

= Preferred Dividend/Preferred Stock Price

= 3.5/66.67= 5.255% Eff. rate

Eff. Rate Cost of Common Stock:
1) Capital Asset Pricing Model (CAPM)
2) Constant Div. Growth Model (CDGM)

E(r) = r_f + β(E(rm) - r_f)
= r_f + β(Risk Prem on Mkt)
= (div (1yr)/ stock price) + est. growth rate

= .02 + 1.2(.07 -.02) = .08 or 8%
= 2.81/92+4% = .0705 or 7.05%

Valuing a project w/ perpetual growth

= \text{- Initial Outflow} + \frac{CF}{WACC - g}

= -200M + (100M/(.057-.03))
= 3.5B

Valuing a project with financing costs and perpetual growth

= - \text{ Initial Outflow} - \text{Fin. Costs} + \frac{CF}{WACC - g}

= -200-20 + (100/(.057-.03))
= 3.4B

rf = Risk Free Rate
Risk Premium = Expected Rate - Risk Free Rate
B = Beta
E(r_m) = Expected Return of the Market
E(r) = Expected Return of Stock

CF = Cash Flow
g = Growth Rate
P_B = Bond price
F = Face Value

References: 7 Mar 24.ppt and L7-Outline

L8 - Equity

Name

Equation

Example

Ownership %

= \frac{\text{\# Shares}}{\text{Total Outstanding Shares}}

= 500,000/1,550,000 = 32.26%

Post money valuation

= \text{Outstanding Shares} × \text{Last Investor's Price} \\ \text{Last Investor's Price} = \frac{\text{Cash Paid}}{\text{\#shares}}

= 1.55M * $15 = $23.25 Million
$15M/1M = $15 per share

Share price estimate with P/E ratio

\text{Industry Avg. P/E} × \text{EarningsPerShare}

= (18.3PE *$1.51EPS) = $27.63

Auction IPO Pricing

= \text{Highest Price to Sell All} × \text{Full Allotment}

= $7 * 500,000 = $3.5 Million

Seasoned Equity Offering Cash Offer

= \text{\#shares} × \text{current mkt. price}

=1M * $12 = 12M

Seasoned Equity Offering Right Offer

= \frac{\text{Total Outstanding Shares}}{\text{\#rights}} × price

= (100M/ 4) → 25 * $8 = $200M

Evaluate SEO

1. \frac{\text{New Capital}}{\text{New Share Total}} \\ 2. \text{Return on right} = \frac{\text{new price} - \text{old}}{\#rights} \\ 3. \text{Add new price} + \text{profit per share}

= 1.2B/ 125M = $9.60 per share
= (9.60-8.00)/4.00 = .40 profit
= $9.60 + $.40 = $10.00

References: 8 Mar 31.ppt and L8-Notes

L9 - Bonds

Purpose

Calculator Key

Excel Function

Number of Periods

= nper(rate, pmt, -pv, fv, type)

Periodic Interest rate

I/Yr

= rate(nper, pmt, - pv, fv, type, guess) ← YTM

Present Value

= pv(rate, nper, pmt, fv, type) ← Bond Pricing

Annuity Payment

PMT

= pmt(rate, nper, -pv, fv, type)

Future Value

= fv(rate, nper, pmt, pv, type)

APR to EAR

= (1+(APR/n))^n-1

EAR to APR

= n*((1+EAR)^(1/n)-1)

Rate = rate quoted
nper = # of period of coupon payments
pmt = period payment
fv = face value of bond (“future value”)

pv = price of the bond (“present value”), enter as a negative, because it is a cash outflow to pay for the bond
type = leave blank
guess = leave blank

References: 9 Apr 7.ppt and L9-Notes

L10/L11 - Capital Structure

Name

Equation

Example

MM Proposition 1

V_L = E + D = V_U

(in a perfect capital market)

$30,000 = $15,000 + $15,000

MM2: Cost of Capital (Equity)

r_E = r_U + \frac{D}{E} × (r_U - r_D)

15%+(6000/ 24000) * (15%-10%) = 17.5%

Debt to Value Ratio

= \frac{Debt}{\text{Equity} + \text{Debt}}

3M / (7M + 3M) = 30%

Cash Raised by Selling Equity

= \text{PDV of cash flows to equity}

$34,500 / (1+15%)^1 = $30,000

Interest Tax Shield

= \text{Corporate Tax Rate} × \text{Interest Payment}

35% * 300M = 105M

Cash Flows to Investors w/ Leverage

= (Cash Flows to Investors w/out Leverage) + (Interest Tax Shield)

Present Value of Tax Shield of Permanent Debt

= \text{Corporate Tax Rate} × \text{Market Value of Debt} \\ \text{PV(Interest Tax Shield)} = T_C × D

Tradeoff Theory

V^L = V^U + PV(\text{Interest Tax Shield}) - PV (\text{Financial Distress Costs})

WACC pre-tax

=\frac{D}{D+E} × r_D + \frac{E}{D+E} × r_E

(25/75) * 5% + (50/75) * 12% = 9.67%

WACC with Taxes

\% \text{ equity} × r_E + \% \text{ debt} × r_D × (1-t)

50% * 8% + 50% * (1-35%) * 4% = 5.3%

r_E = Return or Cost of Capital for Levered Equity
r_U = Return or Cost of Capital for Unlevered Equity
r_U = Return or Cost of Capital for Debt

E = Market Cap of Equity
D = Market Cap of Debt

References: 10 Apr 14.ppt and L10-Notes and L11-Notes

L12 - Mergers and Acquisitions

Name

Equation

Example

Market Cap

= Share Price × Total Shares Outstanding

= $80 * 2M = $160M

Combined Market Cap

C = Acquirer Mkt Cap + Target Mkt Cap + Synergies
C = A + T + S

= $160M + $100M + $10M = $270M

# New Shares = x

x = \frac{T(1+prem)}{P_A} \\ = \frac{\text{Target's Market Cap} (1+prem)}{\text{Acquirer's Share Price}}

= $100M * (1+0%) / $80 = 1.25 million shares

Calculating Exchange Ratio (ER)

ER = \frac{\text{\# New Shares}}{\text{Target's Shares Outst.}} = \frac{x}{N_T} \\ \text{Shortcut: ER} = \frac{\text{Target's Share Price}}{\text{Acquirer's Share Price}}(1+premium) \\

= 1.25M / 2M = .625 Exchange Ratio

Maximum ER to be profitable for acquirer

ER < \frac{P_T}{P_A} \left( 1+ \frac{S}{T}\right)

ER < $66/75(1+$40M/$145M)=1.123
The acquirer will only accept an ER < 1.123.

# Shares Outstanding (SO) after Merger

N_C = N_A + x = \text{Acquirer’s SO} + \text{\# New Shares} \\ = \text{Acquirer’s SO} + \text{Exch Ratio} × \text{Target SO} = N_A + ER × N_T

= 2M shares + 1.25M shares = 3.25M shares
= 2M shares + .625 * 2M = 3.25M shares

Acquirer Share Price after Announcement / Merger

P_C = \frac{C}{N_C} = \frac{A+T+S}{N_A + x} = \frac{\text{Combined Mkt Cap}}{\text{\#Shares after Merger}}

= $270M / 3.25M = $83.08

Target Share Price (After)

= Acquirer New Share Price * Exchange Ratio = P_C × ER

= $83.08 * .625 = $51.93

New Earnings Total

= Acquirer’s Earnings + Target’s Earnings

= 5M + 5M = 10M New Total Earnings

New Earnings Per Share

= New Total Earnings / New Shares Outstanding

= 10 M / 1.6 M = $6.25 EPS

New P/E Ratio

= New share price / New EPS
OR = Market Cap / Total Earnings"

= $100/ 6.25 = 16 P/E Ratio

Expected Earnings

= (Possible Earnings * Probability) + (Possible Earnings * Probability) +

= (15* 60%) + (27.5* 40%) = 20

Post-Tax Earnings

= Earnings * (1 - tax rate)

= 10* (1- 35%) = 6.5

A = Acquiring firm (before merger), T=Target Firm, C=Combined Firm (ie the Acquirer after merger)
For the Acquiring firm, A is Market Cap, P_A is stock price, and N_A is number of shares outstanding. T, C, P_T, P_C, N_T, and N_C are defined similarly. x = number of new acquirer shares exchanged for target shares.

References: 12 Apr 28.ppt and L12-Notes

Appendix: L3 - Financial Statement Analysis Formulas

References: 3 Feb 11.ppt and L3-Notes (for all of the Appendix: L3 tables)

Profitability

Profitability Ratios measure profit as a percentage of sales/net sales/revenue. The measures of profit are taken from the Income Statement.

\textbf{Gross Margin} = \frac{Gross Profit}{Sales}

\textbf{Gross Profit} = \text{Revenues (Net Sales)} - \text{Cost of Sales}

Reflects a firm’s ability to sell a product for more than the cost of production.
Ex:

\frac{\$33.3M}{\$186.7M} = 17.8\%

\textbf{Operating Margin} = \frac{Operating Income}{Sales}

\textbf{Operating Income} = \text{Gross Profit} - \text{Operating Expenses}

How much a company earns before interest and taxes from each dollar of sales.
Ex:

\frac{\$10.4M}{\$186.7M}= 5.57\%

\textbf{Net Profit Margin} = \frac{Net Income}{Sales}

\textbf{Net Income = Earnings} \\ = \text{EBIT}- \text{Interest} - \text{Taxes}

The fraction of each dollar in revenues that is available to equity holders after the firm pays its expenses, interest, and taxes
Ex:

\frac{\$2.0M}{\$186.7M} = 1.07\%

Liquidity

Liquidity Ratios compare Current Assets to Current Liabilities to assess whether a firm has enough liquid assets to meet its short term financial obligations. Later formulas include assets more selectively (ie they are more “stringent”)

\textbf{Current Ratio} = \frac{\text{Current Assets}}{\text{Current Liabilities}}

The ratio of current assets to current liabilities
Ex: $57M/$48M=1.19

\textbf{Quick Ratio} = \frac{\text{Current Assets} - \text{Inventory}}{\text{Current Liabilities}}

The ratio of only cash and “near cash” assets to current liabilities. More narrow definitions of “near cash” can also be used.
Ex:

\frac{\$57M-\$15.3M}{\$48M} = 0.87

\textbf{Cash Ratio} = \frac{Cash}{\text{Current Liabilities}}

The most stringent liquidity ratio. Only cash counts.
Ex: $21.2M/$48M=0.44

Asset Efficiency

Asset Efficiency Ratios tell you whether the firm is using its assets efficiently to generate sales.

\text{Total } \textbf{Asset Turnover} = \frac{Sales}{\text{Total Assets}}

A first broad measure of efficiency

\textbf{Fixed Asset Turnover} = \frac{Sales}{\text{Fixed Assets}}

Since total assets include assets that are not directly involved in generating sales, a manager might also look at fixed asset turnover

Working Capital

Working Capital Ratios help you assess whether the firm as enough or too much working capital. Working capital is expensive, so we generally want to reduce working capital.

\textbf{Accounts Receivable Days} = \frac{\text{Account Receivable}}{\text{Average Daily Sales}}

\text{Average Daily Sales} = \frac{\text{Sales Revenue}}{365}

The number of days’ worth of sales that accounts receivable represents. Measures the average number of days it takes a firm to collect its receivables. Typically want this low to avoid giving interest-free loans to customers.
Ex: Daily Sales=$186.7M/365=$0.51M and
AR Days=$18.5M/$0.51M=36 days worth of sales

\textbf{Accounts Payable Days} = \frac{\text{Accounts Payable}}{\text{Average Daily Cost of Goods Sold}}

Average Daily Cost of Goods Sold = Cost of Goods Sold/365

Credit the firm has received from its vendors in terms of days worth of costs. Generally prefer this higher because it suggests the firm is receiving low-cost loans from its vendors.
Ex: Average Daily Cost of Goods Sold=153.4/365=$0.42M and Accounts Payable Days=$29.2M/$0.42M=69.5 days

\textbf{Inventory Days} = \frac{Inventory}{\text{Average Daily Cost of Goods Sold}}

Inventory Days is the number of days’ cost of goods sold represented by inventory. Holding inventory can be expensive, so you want “just enough” inventory.

\textbf{Inventory Turnover} = \frac{\text{Cost of Goods Sold}}{Inventory}

Inventory Turnover tells how efficiently a company turns its inventory into sales. Specifically, it tells you how many times a company turns over its inventory per year.

Interest Coverage Ratios

Interest Coverage Ratios assess how easily a firm is able to cover its interest payments. Divide earnings by interest expense to get a “Times Interest Earned” (TIE) measure.

\textbf{EBIT Interest Coverage Ratio} = \frac{EBIT}{\text{Interest Expense}}

\textbf{EBITDA Interest Coverage Ratio} = \frac{EBITDA}{\text{Interest Expense}}

Leverage Ratios

How much leverage (debt) has the firm taken on? Too much leverage can put the firm at risk of default. However, as we will see later in the course, debt also can increase expected return and confers tax advantages.

\textbf{Debt To Equity Ratio} = \frac{\text{Total Debt}}{\text{Total Equity}}

Can be calculated using book or market values

\textbf{Debt To Capital Ratio} = \frac{\text{Total Debt}}{\text{Total Equity} + \text{Total Debt}}

Measures the fraction of the firm’s assets financed by debt

\textbf{Net Debt} \\ = \text{Total Debt} \\ - \text{Excess Cash \& Short Term Investment}

While leverage increases risk to equity holders, firms may also hold cash reserves and short term investment to reduce risk. Net these out when calculating net debt.

\textbf{Debt To Enterprise Value Ratio} \\ =\frac{\text{Net Debt}}{\text{Market Value of Equity}+ \text{Net Debt}} \\ =\frac{\text{Net Debt}}{\text{Enterprise Value}}

\textbf{Equity Multiplier} = \frac{\text{Total Asset}}{\text{Book Value of Equity}}

We used this in e2000

Valuation

Valuation Ratios assess whether a stock’s price is high or low relative to the firm’s actual earnings.

\text{Price/Earnings Ratio:} \\ \text{P/E Ratio} = \frac{\text{Market Capitalization}}{\text{Net Income}} \\ =\frac{\text{Share Price}}{\text{Earnings Per Share}}

\text{P/E to Growth (PEG)} \\ \text{PEG} = \frac{\text{PE Ratio}}{\text{Expected Earnings Growth Rate}}

It is the ratio of the firm’s P/E to its expected earnings growth rate The higher the PEG ratio, the higher the price relative to growth, so some investors avoid companies with PEG ratios over 1

\text{Market-to-Book Ratio} = \frac{\text{Market Value of Equity}}{\text{Book Value of Equity}}

\text{Enterprise Value Ratios} = \frac{\text{Enterprise Value}}{\text{EBIT,EBITDA,or Sales}}

The Enterprise Value Ratios are like the PE ratio, but both numerator and denominator take into account both equity and debt holders.

Operating Returns

Operating Return Ratios measure the profit a firm earns relative to the Assets or Equity that the firm has. They can be used to assess management performance.

\text{Return on Equity} = \frac{\text{Net Income}}{\text{Book Value of Equity}}

Evaluating the firm’s return on investment by comparing its income to its investment

\text{Return on Asset} = \frac{\text{Net Income}}{\text{Total Assets}}

Evaluating the firm’s return on investment by comparing its income to its assets.
Some sources use

\frac{\text{Net Income+Interest Expense}}{\text{Total Assets}}

or other formulas.

\text{Return on Invested Capital} \\ = \frac{\text{EBIT}(1-\text{tax rate})}{\text{Book Value of Equity} + \text{Net Debt}}

After-tax profit generated by the business, excluding interest, compared to capital raised that has already been deployed.

DuPont Analysis

DuPont Analysis breaks down ROE into three different ratios. This helps us analyze which factors are driving ROE.

ROE = \frac{\text{Net Income}}{\text{Sales}} × \frac{\text{Sales}}{\text{Total Assets}} × \frac{\text{Total Assets}}{\text{Total Equity}} \\ = \text{Profit Margin} × \text{Asset Turnover} × \text{Equity Multiplier}

Financial Statement use in examples, above

These financial statements and some of the verbiage above were taken from our textbook. Also, while I have used parentheses to indicate negative numbers below, you should generally use minus signs to enter negative numbers into MyFinanceLab.

Assets	2024
Current Assets
Cash	21.2
Accounts receivable	18.5
Inventories	15.3
Other current assets	2.0
Total current assets	57.0
Long-Term Assets
Land	22.2
Buildings	36.5
Equipment	39.7
Less accumulated depreciation	(18.7)
Net property, plant, and equipment	79.7
Goodwill and intangible assets	20.0
Other long-term assets	21.0
Total long-term assets	120.7
Total Assets	177.7

Liabilities & Stockholders’ Equity	2024
Current Liabilities
Accounts payable	29.2
Notes payable/short-term debt	3.5
Current maturities of long-term debt	13.3
Other current liabilities	2.0
Total current liabilities	48.0
Long-Term Liabilities
Long-term debt	99.9
Lease obligations	—
Total debt	99.9
Deferred taxes	7.6
Other long-term liabilities	—
Total long-term liabilities	107.5
Total Liabilities	155.5
Stockholders’ Equity	22.2
Total Liabilities & Stockholders’ Equity	177.7

Income Statement (in $ million)	2024
Total Sales	186.7
Cost of Sales	-153.4
Gross Profit	33.3
Selling, general, and administrative expenses	-13.5
Research and development	-8.2
Operating Income	10.4
Other income	---
Earnings Before Interest and Taxes (EBIT)	10.4
Interest income (expense)	-7.7
Pretax Income	2.7
Taxes	-0.7
Net Income	2.0
Earnings per share:	$0.556
Diluted earnings per share:	$0.526

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