Formula Sheet 2021 - Formelblad PDF

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FORMULA SHEET

Basic Formulas n = period, C = cash flow, ฀ ฀ = growth rate Present value

฀฀

฀฀฀฀ = �

฀฀=0

Future value

Net present value

฀฀฀฀ (1 + ฀฀)฀฀

฀฀฀฀฀ ฀ = (1 + ฀฀)฀฀ ฀฀

฀฀฀฀ ฀฀฀฀฀฀ = −฀฀฀฀฀฀฀฀ + � (1 + ฀฀)฀฀ ฀฀=0

Perpetuity

฀฀฀฀ =

Growing perpetuity ฀฀฀฀ =

฀฀ ฀฀

฀฀ ฀฀ − ฀฀

Annuity 1 1 ฀฀฀฀ = ฀ ฀ × �1 − � (1 + ฀฀)฀฀ ฀฀ 1 ฀฀฀฀ = ฀ ฀ × ((1 + ฀฀)฀ ฀ − 1) ฀฀ Growing annuity

฀฀ 1 + ฀฀ ฀฀ � � ฀฀฀฀ = �1 − � 1 + ฀฀ ฀฀ − ฀฀

Solve for the cash payments C=

P 1 1 1 − r  (1 + r) N

  

1

IRR with two cash flows

฀฀฀฀฀฀ = (฀฀฀฀/฀฀)1/฀ ฀ − 1

Interest Rates Formulas r = discount rate, n = year, k = compounding intervals per year ฀฀฀฀฀฀ = (1 + ฀฀)฀ ฀ − 1 Converting an APR to and EAR APR   1 + EAR = 1 +  k  

k

APR = ( (1 + EAR) 1 /k − 1 ) × k

Real interest rate (r = nominal rate, i = inflation rate) ฀฀ − ฀฀ ฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀ = 1 + ฀฀ Basic Bond and Stock Valuation Formulas Bond (y = YTM, FV = face value) ฀฀=

฀฀฀฀฀฀ ฀฀฀฀฀฀ + ฀฀฀฀ ฀฀฀฀฀฀ + + ⋯ + (1 + ฀฀฀฀ )฀ ฀ 1 + ฀฀1 (1 + ฀฀2 )2 or

1 1 ฀฀฀฀ ฀ ฀ = ฀฀฀฀฀฀ × �1 − �+ ฀฀ (1 + ฀฀)฀฀ ฀฀ (1 + ฀฀) The YTM for a zero-coupon bond ฀฀฀฀฀฀฀ Stock

฀฀฀฀ =� � ฀฀

฀

฀฀

฀฀0 = � Constant growth (฀฀) case

฀฀=0

1/฀ ฀

−1

฀฀฀฀฀฀฀฀ (1 + ฀฀฀฀ ฀฀ )฀฀

฀฀0 =

฀฀฀฀฀฀1 ฀฀฀ ฀ − ฀฀ 2

฀฀ = Retention Rate × Return on New Investment

Constant long-term growth (฀฀) case ฀฀0 =

฀฀฀฀฀฀1

1 ฀฀฀฀฀฀฀฀+1 ฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀2 + � � + ⋯ + + ฀ ฀ ฀ ฀ (1 + ฀฀฀฀ ) (1 + ฀฀฀฀ ) ฀฀฀ ฀ − ฀฀ 1 + ฀฀฀ ฀ (1 + ฀฀฀฀ )2

Portfolio Formulas Mean and variance of one or two assets Mean

R=

1 T

T

∑R

t

t =1

⁄ 1 ฀฀ −1 ฀฀฀฀฀฀฀฀ = [(1 + ฀฀1 ) × (1 + ฀฀2 ) × … × (1 + ฀฀฀฀ )]

Sample variance

Var( R) =

1 T ( Rt − R ) 2 T −1 ∑ t =1

Standard deviation

SD( R) = Var ( R) Covariance

Cov (Ri , R j ) =

1 T (Ri ,t − R i )(R j ,t − R j ) T −1∑ t= 1

Correlation Corr (Ri , R j ) =

Cov(Ri, R j )

SD(R i ) SD(R j )

Mean and variance of a portfolio with proportion Xi in i and proportion Xj in j (i.e. two assets) Mean

[ ]

E R p = X i Ri + X j R j Variance 2

2

Varp = X i σ i2 + X j σ 2j + 2 X i X j Cov (Ri, R j ) Expected mean and variance of a portfolio with multiple assets i and probabilities Mean

฀฀

฀฀�฀฀฀฀ � = � ฀฀฀ ฀ × ฀฀฀฀ ฀฀=1

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Variance

฀฀

฀฀฀฀฀฀�฀฀฀฀ � = � ฀฀฀ ฀ × (฀฀฀ ฀ �−)2฀฀ ฀฀=1

The volatility of a large portfolio The variance of an equally weighted portfolio (p)

1 Var (Rp )= ( Average Variance of the Individual Stocks ) n  1 +  1 −  ( Average Covariance between the Stocks )  n SD�฀฀฀฀ � = �฀฀฀฀฀฀(฀฀฀฀ ) The volatility of a large portfolio results from the common risk between the stocks in the portfolio. Each security (i) contributes to the volatility of the portfolio (p) according to its total risk scaled by its correlation with the portfolio. T

SD( R p ) = ∑ xi × SD( Ri ) × Corr( Ri , R p ) t= 1

Risk-free Saving and Borrowing ฀฀[฀฀฀฀ ] = ฀฀฀ ฀ + ฀฀฀฀ �฀฀ (฀฀฀฀ ) − ฀฀฀฀ � SD(฀฀฀฀ ) = ฀฀฀฀ ฀฀฀฀�฀฀฀฀ �

฀฀�฀฀฀฀ � − ฀฀฀฀ ฀฀ℎ฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀ = ฀฀฀฀(฀฀฀฀ ) CAPM ฀฀[฀฀฀฀ ] = ฀฀฀ ฀ + ฀฀฀฀ �฀฀ (฀฀฀฀ ) − ฀฀฀฀ �

Market risk premium = ฀฀(฀฀฀฀ ) − ฀฀฀฀

฀฀฀ ฀ =

฀฀฀฀�฀฀฀฀ � ฀฀฀฀฀฀฀฀�฀฀฀฀, ฀฀฀฀฀฀฀฀฀฀�฀฀ � ฀฀, ฀฀฀฀ � = ฀฀฀฀(฀฀฀฀ ) ฀฀฀฀฀฀ (฀฀฀฀ )

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Key Financial Ratios Profitability Ratios Gross Margin = Gross Profit / Sales Operating Margin = Operating Income / Sales EBIT Margin = EBIT / Sales Net Income Sales Net Profit Margin = Net Income / Sales Liquidity Ratios Current Ratio = Current Assets / Current Liabilities Quick Ratio = (Cash & Short-term Investments + Accounts Receivable) / Current Liabilities Cash Ratio = Cash / Current Liabilities Working Capital Ratios Accounts Receivable Days = Accounts Receivable / Average Daily Sales Accounts Payable Days = Daily Cost of Sales Inventory / Average Daily Cost of Sales Inventory Days = Inventory / Average Daily Cost of Sales Interest Coverage Ratios EBIT / Intere st Coverage = EBIT / Interest Expense EBITDA / Intere st Coverage = EBITDA / Interest Expense Leverage Ratios Debt-Equity Ratio (book) = Total Debt / Book Value of Equity Debt-Equity Ratio (market) = Total Debt / Market Value of Equity Debt-to-Capital Ratio = Total Debt / (Total Equity + Total Debt) Debt-to-Enterprise Value Ratio = Net Debt / Enterprise Value Equity Multiplier (book) = Total Assets / Book Value of Equity Equity Multiplier (market) = Enterprise Value / Market Value of Equity Enterprise Value = Market Value of Equity + Debt - Cash Valuation Ratios Market-to-Book Ratio = Market Value of Equity / Book Value of Equity Price-Earnings Ratio = Share Price / Earnings per Share Enterprise Value to Sales = Enterprise Value / Sales Enterprise Value to EBIT = Enterprise Value / EBIT Enterprise Value to EBITDA = Enterprise Value / EBITDA Operating Returns Asset Turnover = Sales / Total Assets Return on Equity (ROE) = Net Income / Book Value of Equity Return on Assets (ROA) = (Net Income + Interest Expense) / Book Value of Assets Return on Invested Capital (ROIC) = EBIT (1 - Tax Rate) / (Book Value of Equity + Net Debt)

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