Risk & Return
𝑯𝒐𝒍𝒅𝒊𝒏𝒈 𝑷𝒆𝒓𝒊𝒐𝒅 𝑹𝒆𝒕𝒖𝒓𝒏𝒔 (𝑯𝑷𝑹) =
𝐸𝑛𝑑𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 + 𝐶𝑎𝑠ℎ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
Beta
Beta measures the responsiveness of a security to movements in the market portfolio, i.e., systematic risk
𝐶𝑜𝑣(𝑅𝑎 , 𝑅𝑏 )
𝜷𝒊 = (
)
𝑉𝑎𝑟(𝑅𝑒𝑡𝑢𝑟𝑛𝐴 − 𝑅𝑒𝑡𝑢𝑟𝑛𝐵
Cov= correlation*stdA*stdB
Cost of Capital
Cost of Equity, RE
𝐷1
)+𝑔
𝑃0
𝑺𝑴𝑳 𝑨𝒑𝒑𝒓𝒐𝒂𝒄𝒉 = 𝑅𝐸 = 𝑅𝑓 + 𝛽𝐸 ∗ (𝑅𝑀 − 𝑅𝑓 )
D1 – Expected dividend in one period; g – Dividend growth rate; P0 – Current stock price; RE – Risk-free rate
Rm – Expected return on the overall market; βE – Systematic risk of the equity
𝑫𝒊𝒗𝒊𝒅𝒆𝒏𝒅 𝑮𝒓𝒐𝒘𝒕𝒉 𝑴𝒐𝒅𝒆𝒍 = 𝑅𝐸 = (
Cost of Debt, RD
For a firm with public debt, the cost of debt is the yield to maturity on outstanding debt. If the firm has no public debt, use the yield to maturity of similarly rated bonds.
Weighted Average Cost of Capital, WACC
𝐸
𝐷
𝑾𝑨𝑪𝑪 = ( ) ∗ 𝑅𝐸 + ( ) ∗ 𝑅𝐷 ∗ (1 − 𝑇𝐶 )
𝑉
𝑉
T C – Corporate tax rate; E – Market value of the firm’s equity; D – Market value of the firm’s debt; V = E+D
TVL  TVU  PVGL  PVFD  PVAC
where TVU = Total Value of an Unlevered (i.e., no debt) Firms
PVGL = Present Value of Gains from Leverage
PVFD = Present Value of Financial Distress Costs
PVAC = Present Value of Agency Costs
Debt
Preferred
Retained Earnings
New Common Equity
Proportion
%
%
%
%
Cost
RD
Rpf
RE
RNE
Weighted Costs
(%)(RD)
(%)(Rpf)
(%)(RE)
(%)(RNE)
WACC
After-Tax Cost of Debt
Cost of Retained Earnings
First calculate before-tax cost of debt incorporating fixed and variable floatation costs
(i.e., floatation costs are fees charged by the issuing investment bank) such that:
The component cost of retained earnings can be derived in a number of ways.
$ Amount of Debt Issued
-$ Amount of Fixed Floatation Cost
-$ Amount of Variable Floatation Cost
$ Amount of Debt Proceeds Received by Firm





DCF (Discounted Cash Flow or Gordon) Model
Dollar amount of debt proceeds received by firm is present value of new debt (PV).
Dollar amount of debt issued is future value of new debt (FV).
Payment (PMT) is FV (i.e., book value) of new debt times new rate on funding
divided by number of payment periods per year.
Time (n) is number of repayment periods until maturity on the new debt.
Enter all this information into a financial calculator and press i to find YTM (yield-tomaturity), or discount rate, on new debt.
After-tax component cost of debt can be calculated using the following formula:
RD  YTM (1  tc )
where
Two common techniques include the DCF (discounted cash flow or Gordon) Model and
the CAPM (Capital Asset Pricing Model).
RD = after-tax cost of bonds
YTM = floatation-adjusted yield-to-maturity
tc = corporate tax rate
RE 
D1
g
P0
where D1 =
P0 =
g =
the expected future dividend
current market price of common stock
constant, long-term dividend growth rate
CAPM (Capital Asset Pricing Model)
RE  R f   [ E ( Rm )  R f ]
where Rf = risk-free rate of return
 = firm’s beta coefficient
Rm = return on the market
Cost of New Common Stock (rne)
Terminal Growth Rate, g
Must be slightly higher than the Cost of Retained Earnings or existing common stock (re)
in order to cover flotation costs (again, the fees charged by an underwriting investment
bank) associated with the issuance of new common stock.
There are at least three ways to estimate long-term growth (g)
(1) Sustainable Growth Model
g  (1 
The formula is:
Div
)( ROE )
EPS
where
Flotation Cost as a Percent/Share
RNE 
=
=
=
=
dividend per share
earnings per share
dividend payout ratio
return on equity
(2) Growth in EPS (Earning Per Share)
D1
g
P0 (1  F )
EPS 
Flotation Cost as a Dollar Amount/Share
RNE 
Div
EPS
Div/EPS
ROE
NetIncomeAvailableToCommonStockholders
CommonSharesOuts tan ding
GrowthRateEPS 
EPS2008  EPS2007
EPS2007
(3) Growth in Dividends
D1
g
P0  F
GrowthRateDiv 
Div 2008  Div 2007
Div 2007
𝐸
𝐷
𝑾𝑨𝑪𝑪𝒘 𝒊𝒕𝒉𝒐𝒖𝒕 𝑷𝒓𝒆𝒇𝒆𝒓𝒓𝒆𝒅 𝑺𝒕𝒐𝒄𝒌 = ( ) 𝑥 𝑅𝐸 + ( ) ∗ 𝑅𝐷 ∗ (1 − 𝑇𝐶 )
𝑉
𝑉
𝐸
𝑃
𝐷
𝐸
𝐷
𝑾𝑨𝑪𝑪 𝒘𝒊𝒕𝒉 𝑷𝒓𝒆𝒇𝒆𝒓𝒓𝒆𝒅 𝑺𝒕𝒐𝒄𝒌 = ( ) 𝑥 𝑅𝐸 + ( ) ∗ 𝑅𝑃 + ( ) ∗ 𝑅𝐷 ∗ (1 − 𝑇𝐶 )
𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝑭𝒍𝒐𝒕𝒂𝒕𝒊𝒐𝒏 𝑪𝒐𝒔𝒕 = 𝐹𝐴 = ( ) 𝑥 𝐹𝐸 + ( ) ∗ 𝐹𝐷
𝑨𝒇𝒕𝒆𝒓𝒕𝒂𝒙 𝑹𝒂𝒕𝒆 = 𝑅𝑑 ∗ (1 − 𝑇𝐶 )
𝑉
𝑉
𝑉
𝑉
𝑉
𝑹𝑬 = 𝑅𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒 + (𝛽 ∗ 𝑀𝑎𝑟𝑘𝑒𝑡 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚); Rd = Normal interest rate; Rp = Cost of preferred stock; FE= Equity flotation cost; FD = Debt flotation cost
𝐴𝑓𝑡𝑒𝑟𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑𝑇𝑎𝑥𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔𝑃𝑟𝑜𝑓𝑖𝑡𝑠
𝑁𝑂𝑃𝐿𝐴𝑇
𝑹𝒆𝒕𝒖𝒓𝒏 𝒐𝒏 𝑰𝒏𝒗𝒆𝒔𝒕𝒆𝒅 𝑪𝒂𝒑𝒊𝒕𝒂𝒍 (𝑹𝑶𝑰𝑪) =
=
𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑒𝑑𝐼𝑛𝑊𝐶 & 𝑃𝑃&𝐸
𝐼𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
𝑬𝒄𝒐𝒏𝒐𝒎𝒊𝒄 𝑷𝒓𝒐𝒇𝒊𝒕 = 𝐼𝑛𝑣𝑒𝑠𝑡𝑒𝑑𝐶𝑎𝑝𝑖𝑡𝑎𝑙 ∗ (𝑅𝑂𝐼𝐶 − 𝑊𝐴𝐶𝐶)
DCFValue0 
FCFn
FCF1
FCF2

 ... 
(1  WACC )1 (1  WACC ) 2
(1  WACC ) n
 A
L
AFN     S     S  p  S1  RR
S
0
 
 S0 
Where Delta are the assets tied directly to sales
L are the liabilities tied directly to sales; S0 is this year’s sales; ∆S is the change in sales
S1 is next year’s projected sales; p is the profit margin; RR is the retention ratio, or (1 – dividend payout ratio)
Sales
p  (1  d )  (1  DE )

Sales T  p  (1  d )  (1  DE )
p
Debt
TotalAsset s
NetIncome
Dividends DE 
T
d
Equity
Sales
Sales
NetIncome
NetIncome
t
Sales
 ROE (1  d ) ROE  Shareholde rs ' Equity
t 1 d  DividendPa yments  ShareRe purchases
Sales
After-tax cash flows from operations may be calculated as follows:
REV
- VC
-FCC
-DEP
NOI
-kdD
EBT
- T
NI
If assume free cash flows, after an initial period of non-constant growth (say 3 to 7
years), becomes a zero-growth firm, can use formula for a perpetuity to determine
present value of the infinite zero-growth free cash flow stream:
ATCF  NOI  t c ( NOI )  DEP
Revenues
Variable Costs of Operations
Fixed Cash Costs (SG&A, or Selling, General & Administrative) From the pro forma income statement, we know:
Non-Cash Charges (e.g., depreciation, amortization, depletion)
NOI  REV  VC  FCC  DEP
Net Operating Income
Interest on Debt (interest rate on debt x $ amount of Debt)
Therefore we can rewrite NOI less taxes as follows:
Earnings Before Taxes
Taxes (corporate tax rate, tc, x EBT)
ATCF  ( REV  VC  FCC  DEP)(1  t c )  DEP
Net Income
PV perp 
E ( ZeroGrowthFCF )
WACC
If assume free cash flows, after an initial period of non-constant growth, becomes a
constant growth firm, can use Gordon Model (or constant growth model) to determine
present value of the infinite constant-growth free cash flow stream:
PVCons tan tGrowth 
E ( NextPeriodFCF )
WACC  g
where g = the constant growth rate
This gives us the after tax free cash flow (FCF) available for payment to creditors and
shareholders:
FCF  ( REV  VC  FCC  DEP)(1  t c )  DEP  I
Since we are interested in future free cash flows as opposed to historic flows, we must
prepare pro forma (or expected) free cash flow statements:
E ( FCF )  E[( REV  VC  FCC  DEP)(1  t c )  DEP  I
Sales (Revenues from operations)
- COGS (Cost of goods sold-labor, material, book depreciation)
- SG&A (Selling, general administrative costs)
EBIT (Earnings before interest and taxes or Operating Earnings)
- Taxes (Cash taxes)
EBIAT (Earnings before interest after taxes)
+ DEP (Book depreciation)
- CAPX (Capital expenditures)
- ChgWC (Change in working capital)
= C (Free cash flows)
Capital Budgeting
Other Capital Budgeting Techniques
𝐴𝑣𝑒𝑟𝑎𝑡𝑒 𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒
𝐴𝑐𝑡𝑢𝑎𝑙 𝐶𝑎𝑠ℎ 𝐼𝑛𝑓𝑙𝑜𝑤
𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕𝒆𝒅 𝑷𝒂𝒚𝒃𝒂𝒄𝒌 𝑷𝒆𝒓𝒊𝒐𝒅 =
(1 + 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒)𝑛
𝐶𝐹1
𝐶𝐹2
𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒍 𝑹𝒂𝒕𝒆 𝒐𝒇 𝑹𝒆𝒕𝒖𝒓𝒏 = 𝑁𝑃𝑉(0) = 𝐶𝐹0 + (
)+(
)+⋯ =0
1 + 𝐼𝑅𝑅
(1 + 𝐼𝑅𝑅)2
𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒊𝒏𝒈 𝑹𝒆𝒕𝒖𝒓𝒏 (𝑨𝑹𝑹) =
𝑛
𝑴𝒐𝒅𝒊𝒇𝒊𝒆𝒅 𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒍 𝑹𝒂𝒕𝒆 𝒐𝒇 𝑹𝒆𝒕𝒖𝒓𝒏 (𝑴𝑰𝑹𝑹) = √(
𝐹𝑉(𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝐶𝐹, 𝐶𝑜𝑠𝑡𝑜𝑓𝐶𝑎𝑝𝑖𝑡𝑎𝑙)
)−1
𝑃𝑉(𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑂𝑢𝑡𝑙𝑎𝑦𝑠, 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑛𝑔 𝐶𝑜𝑠𝑡)
Discounting Approach – Discount negative CF to PV at RRR and add to initial cost.
Reinvestment Approach = Discount pos + neg CF to PV at RRR
Combination – Neg CF discounted to PV, Pos CF compounded to FV
𝑃𝑉 𝑜𝑓 𝐹𝑢𝑡𝑢𝑟𝑒 𝐶𝐹𝑠
𝑷𝒓𝒐𝒇𝒊𝒕𝒂𝒃𝒊𝒍𝒕𝒚 𝑰𝒏𝒅𝒆𝒙 (𝑷𝑰) = (
)
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
Random Stuff
𝑻𝒐𝒕𝒂𝒍 𝑪𝑭 = 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝐶𝐹 − 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑁𝑊𝐶 − 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔
Basic Components of Discount Rate
Risk-free rate - Time preferences suggest a positive component to all discount rates.
Risk Premium - Risk aversion suggests an additional component representative of the asset's risk
Risk Premia - Variation in discount rates across assets
Stock Returns
𝑫𝒐𝒍𝒍𝒂𝒓 𝑹𝒆𝒕𝒖𝒓𝒏 = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐼𝑛𝑐𝑜𝑚𝑒 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝐺𝑎𝑖𝑛(𝐿𝑜𝑠𝑠)
𝐷𝑜𝑙𝑙𝑎𝑟 𝑅𝑒𝑡𝑢𝑟𝑛
𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆 𝑹𝒆𝒕𝒖𝒓𝒏𝒔 =
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒
𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝐺𝑎𝑖𝑛(𝐿𝑜𝑠𝑠)
𝑷𝒆𝒓𝒄𝒆𝒏𝒕 𝑹𝒆𝒕𝒖𝒓𝒏𝒔 = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐼𝑛𝑐𝑜𝑚𝑒 + (
)
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑺𝒕𝒂𝒏𝒅𝒂𝒍𝒐𝒏𝒆 𝑹𝒊𝒔𝒌 / 𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑽𝒂𝒓𝒊𝒂𝒕𝒊𝒐𝒏 (𝑪𝑽) = ( 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 )
æ T -1 ö
æ N -T ö
R(T ) = ç
÷ ´ GeometricAverage + ç
÷ ´ ArithmeticAverage
è N -1 ø
è N -1 ø
Where T is the forecast horizon and N is the number of years of historical data we are working with
T must be less than N
E ( Ra ) 
Expected Return =
S
 p(state)  R(state)
state1
a
Re wardToRisk Ratio 
E(Rp )  R f
p
𝐶𝑂𝑉(𝑅𝑎 , 𝑅𝑏 )
)
𝜎𝑎 𝜎𝑏
𝑵𝒆𝒕 𝑾𝒐𝒓𝒌𝒊𝒏𝒈 𝑪𝒂𝒑𝒊𝒕𝒂𝒍 = 𝐶𝐴 − 𝐶𝐿
𝑪𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 = 𝜌 = (
7
Arithmetic Average - Return earned in an average period over multiple periods
Geometric Average - Average compound return per period over multiple periods. The geometric average will be less than the arithmetic average unless all the returns are equal
𝑺𝒕𝒐𝒄𝒌 𝒘𝒊𝒕𝒉 𝒊𝒏𝒅𝒆𝒇𝒊𝒏𝒊𝒕𝒆 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒆 𝒊𝒏 𝑫𝒊𝒗𝒊𝒅𝒆𝒏𝒅 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑆𝑡𝑜𝑐𝑘 𝑃𝑟𝑖𝑐𝑒 +
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐴𝑚𝑜𝑢𝑛𝑡 ∗ (1 + 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 % 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑)
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒 − 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 % 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑