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== Dynamic investment indicators ==
== Dynamic investment indicators ==


'''Present Value PV:''' value of a future payment C<sub>t</sub> (in year t), discounted to year 0:
'''Present value PV:''' value of a future payment C<sub>t</sub> (in year t), discounted to year 0: [[Datei:Form_FV.png]]


'''Future value FV<sub>t</sub>:''' value of a present payment (in year 0), calculated by compounding to year t: [[Datei:Form_PV.png]]<br/>
: t year<br/>
: r [[Kalkulationszinssatz|discount rate]] (interest rate)<br/>
: [[Datei:Form_Disc_Fact.png]] discount factor with discount rate (interest rate) r for t years<br/>
: C<sub>t</sub> cash flow in year t<br/>
: C<sub>0</sub> initial investment of a project (for normal investment projects: C<sub>0</sub>  < 0)<br/>
: T number of years of the project<br/> 


'''Future Value FV<sub>t</sub>:''' value of a present payment (in year 0), calculated by compounding to year t:
The sum of several present values is a PV, too '''(additivity of present values)''':[[Datei:Form_PV_Sum.png]]
 
with t year
r discount rate (interest rate)
discount factor with discount rate (interest rate) r for t years
C<sub>t</sub> Cash flow in year t
C<sub>0</sub> initial investment of a project (for normal investment projects: C<sub>0</sub>  < 0)
T number of years of the project
The sum of several present values is a PV, too (additivity of present values):


'''Net present value NPV:''' PV of future payments (of a project or a company) plus the - usually negative - initial investment C<sub>0</sub>:
'''[[Kapitalwert|Net present value NPV]]:''' PV of future payments (of a project or a company) plus the - usually negative - initial investment C<sub>0</sub>: [[Datei:Form_NPV.png]]
 
 
'''Perpetuity (console):''' a periodic (annual) payment C that is received or paid forever (beginning with the first payment at the end of year 1):
 
 
'''Annuity:''' a payment of a level cash flow C during a specified number of years (from year 1 to n). Its present value can be calculated as difference between two perpetuities:
 
'''Annuity (recovery) factor:''' average payment at the end of n periods, corresponding to a present value PV and considering for interest rate r:
 
(Wiedergewinnungsfaktor WGFr,n)


'''Annuity present value factor:''' factor for the PV of n equal payments at the end of years 1 to n.
'''Perpetuity (console):''' a periodic (annual) payment C that is received or paid forever (beginning with the first payment at the end of year 1):[[Datei:Form_PV_Perp.png]]


(Rentenbarwertfaktor RBFr,n)
'''[[Annuität|Annuity]]:''' a payment of a level cash flow C during a specified number of years (from year 1 to n). Its present value can be calculated as difference between two perpetuities: [[Datei:Form_PV_Ann.png]]


'''Annuity (recovery) factor:''' average payment at the end of n periods, corresponding to a present value PV and considering for interest rate r:[[Datei:Form_Recov_factor.png]]


The annuity C over years 1 to n corresponding to a present value PV and discount rate r is:
'''Annuity present value factor:''' factor for the PV of n equal payments at the end of years 1 to n: [[Datei:Form_Ann_PV_factor.png]]


'''Growing perpetuity:''' a perpetuity starting with cash flow C<sub>1</sub> in year 1 and increasing by the annual growth rate g forever:
The annuity C for years 1 to n corresponding to a present value PV and discount rate r is: [[Datei:Form_Ann_PV.png]]
t = 1, 2, …, &#8734; (g < r)


'''Growing annuity:''' an annuity starting with cash flow C<sub>1</sub> in year 1 and increasing by the annual growth rate g for n years:
'''Growing perpetuity:''' a perpetuity starting with cash flow C<sub>1</sub> in year 1 and increasing by the annual growth rate g forever:[[Datei:Form_C_growth.png]] (for t = 1, 2, …, &#8734; ;g < r) and [[Datei:Form_PV_perp_growth.png]]


t = 1, 2, …, n
'''Growing annuity:''' an annuity starting with cash flow C<sub>1</sub> in year 1 and increasing by the annual growth rate g for n years: [[Datei:Form_PV_ann_growth.png]] (with t = 1, 2, …, n)


'''Internal rate of return (IRR):''' discount rate that results in NPV = 0:
'''[[Interner Zinsfuß|Internal rate of return (IRR)]]:''' discount rate that results in NPV = 0: [[Datei:Form_NPV_zero.png]]
'''Profitability index:''' ratio of NPV to investment of a project:
'''Profitability index:''' ratio of NPV to investment of a project: [[Datei:Form_PI.png]]


'''Equivalent annual cash flow (EAC):''' cash flow per year with the same present value as the actual cash flow of the project:
'''Equivalent annual cash flow (EAC):''' cash flow per year with the same present value as the actual cash flow of the project: [[Datei:Form_EAC.png]]
   
   


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:- EAR for a daily rate d: EAR = (1+d)<sup>360</sup> – 1    (for 360 days per year)
:- EAR for a daily rate d: EAR = (1+d)<sup>360</sup> – 1    (for 360 days per year)


'''Annual Percentage Rate (APR) or simple rate:''' annualized rate of shorter period interest rates (monthly, daily rates) using simple interest.
'''Annual percentage rate (APR) or simple rate:''' annualized rate of shorter period interest rates (monthly, daily rates) using simple interest.


:- APR for a monthly rate m: APR = 12 m
:- APR for a monthly rate m: APR = 12 m
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'''Effective annual rate with continuous compounding:''' effective annual rate for n → &#8734; shorter periods
'''Effective annual rate with continuous compounding:''' effective annual rate for n → &#8734; shorter periods
(being r the simple annual rate)
(being r the simple annual rate)
'''
 
Real rate of return:''' rate of return adjusted for inflation
'''Real rate of return:''' rate of return adjusted for inflation
→  with i = inflation rate
→  with i = inflation rate






== Valuing bonds: ==
== Valuing bonds ==
 
'''Price of a bond:'''




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: C<sub>t</sub> annual coupon interest payment<br/>
: C<sub>t</sub> annual coupon interest payment<br/>
: F face value (or principal)<br/>
: F face value (or principal)<br/>
: r discount rate (yield to maturity)<br/>
: r discount rate ([[Wertpapierrendite|yield to maturity]])<br/>
: N maturity<br/>
: N maturity<br/>


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== Valuing stocks: ==
== Valuing stocks ==


'''(Expected) Stock return r (equity cost of capital):'''
'''(Expected) [[Wertpapierrendite|Stock return r]] (equity cost of capital):'''
'''Stock price P<sub>0</sub> in the single-period case:'''
'''Stock price P<sub>0</sub> in the single-period case:'''
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== Risk and Return ==
== Risk and return ==


'''Risk premium of an asset:''' asset return – return of risk-free asset
'''Risk premium of an asset:''' asset return – return of risk-free asset


'''Variance:''' expected value of squared deviations of observations from their expected value (mean):
'''Variance:''' expected value of squared deviations of observations from their expected value (mean):
based on j observations
based on j observations


'''Standard deviation:''' a measure of volatility of expected stock returns:
'''Standard deviation:''' a measure of volatility of expected stock returns:
 
 
 
'''Expected portfolio return''' (with two assets):
'''Expected portfolio return''' (with two assets):
'''Expected portfolio return''' (with j = 1, …, n assets):
'''Expected portfolio return''' (with j = 1, …, n assets):
:x<sub>j</sub> weight of asset j in the portfolio
:x<sub>j</sub> weight of asset j in the portfolio<br/>


:r<sub>j</sub> (expected) return of asset j
:r<sub>j</sub> (expected) return of asset j<br/>
 
'''Expected return following the security market line''' equation (SML):
Expected return in event studies:
'''Abnormal return''' = actual return – expected return =


'''Variance of portfolio return (portfolio variance)''' in the case of two assets:
'''Variance of portfolio return (portfolio variance)''' in the case of two assets:
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'''Correlation coefficient''' between asset i and j:  
'''Correlation coefficient''' between asset i and j:  
'''Variance of portfolio return (portfolio variance)''' in the case of n assets:
'''Beta''' of the return of asset j to the market return (return of market portfolio m):


'''Expected return following the security market line''' equation (SML):
Expected return of a stock in '''event studies''':
'''Variance of portfolio return (portfolio variance)''' in the case of n assets:
'''Abnormal return''' = actual return – expected return =


'''Beta''' of the return of asset j to the market return (return of market portfolio m):






== Rates of return on capital and assets: ==
== Rates of return on capital and assets: ==
'''[[Rentabilität|Rates of return]]:'''


'''Weighted average cost of capital (WACC):'''
'''Weighted average cost of capital (WACC):'''
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'''Weighted average cost of capital (WACC)''' with a zero-tax rate:
'''Weighted average cost of capital (WACC)''' with a zero-tax rate:
'''Leverage-formula''' for return on equity: Return on equity increases with debt/equity-ratio
'''Leverage-formula''' for return on equity: return on equity increases with debt/equity-ratio


'''Leverage-formula''' for equity beta: Risk increases with debt/equity-ratio:
'''Leverage-formula''' for equity beta: risk increases with debt/equity-ratio:

Version vom 2. Dezember 2011, 18:09 Uhr

by Clemens Werkmeister


Dynamic investment indicators

Present value PV: value of a future payment Ct (in year t), discounted to year 0: Form FV.png

Future value FVt: value of a present payment (in year 0), calculated by compounding to year t: Form PV.png

t year
r discount rate (interest rate)
Form Disc Fact.png discount factor with discount rate (interest rate) r for t years
Ct cash flow in year t
C0 initial investment of a project (for normal investment projects: C0 < 0)
T number of years of the project

The sum of several present values is a PV, too (additivity of present values):Form PV Sum.png


Net present value NPV: PV of future payments (of a project or a company) plus the - usually negative - initial investment C0: Form NPV.png

Perpetuity (console): a periodic (annual) payment C that is received or paid forever (beginning with the first payment at the end of year 1):Form PV Perp.png

Annuity: a payment of a level cash flow C during a specified number of years (from year 1 to n). Its present value can be calculated as difference between two perpetuities: Form PV Ann.png

Annuity (recovery) factor: average payment at the end of n periods, corresponding to a present value PV and considering for interest rate r:Form Recov factor.png

Annuity present value factor: factor for the PV of n equal payments at the end of years 1 to n: Form Ann PV factor.png

The annuity C for years 1 to n corresponding to a present value PV and discount rate r is: Form Ann PV.png

Growing perpetuity: a perpetuity starting with cash flow C1 in year 1 and increasing by the annual growth rate g forever:Form C growth.png (for t = 1, 2, …, ∞ ;g < r) and Form PV perp growth.png

Growing annuity: an annuity starting with cash flow C1 in year 1 and increasing by the annual growth rate g for n years: Form PV ann growth.png (with t = 1, 2, …, n)

Internal rate of return (IRR): discount rate that results in NPV = 0: Form NPV zero.png

Profitability index: ratio of NPV to investment of a project: Form PI.png

Equivalent annual cash flow (EAC): cash flow per year with the same present value as the actual cash flow of the project: Form EAC.png


Interest and discount rates

Effective annual rate (EAR): annualized rate of shorter period interest rates (monthly, daily rates) using compound interest:

- EAR for a monthly rate m: EAR = (1+m)12 – 1
- EAR for a daily rate d: EAR = (1+d)360 – 1 (for 360 days per year)

Annual percentage rate (APR) or simple rate: annualized rate of shorter period interest rates (monthly, daily rates) using simple interest.

- APR for a monthly rate m: APR = 12 m

Given an annual percentage rate (APR) of r, the corresponding EAR with respect to n shorter periods of equal length is:

Effective annual rate with continuous compounding: effective annual rate for n → ∞ shorter periods (being r the simple annual rate)

Real rate of return: rate of return adjusted for inflation → with i = inflation rate


Valuing bonds

Price of a bond:


with

Ct annual coupon interest payment
F face value (or principal)
r discount rate (yield to maturity)
N maturity

Duration of a bond with maturity N: weighted average period of bond payments.

Modified duration: a measure of volatility (elasticity) of bond prices:


Valuing stocks

(Expected) Stock return r (equity cost of capital):

Stock price P0 in the single-period case:

Dividend discount model for the stock price P0 in the multi-period case until time horizon H:

Stock price P0 with specific dividends until time horizon H and growing dividends after H:

Stock price for a perpetual stream of dividends:

Stock price for a perpetual stream of growing dividends:

Return (Equity cost of capital) of a perpetual stream of dividends with growth:

Return on Equity with market values:

Payout ratio: fraction of earnings paid out as dividends:

Plowback ratio: fraction of earnings retained by the firm:

Present value of growth opportunities (PVGO): net present value of a firm's future investments

Sustainable growth rate: rate at which a firm can steadily grow:

Discounted cash flow (DCF): value of the free cash flows that are available to investors plus company value at the planning horizon, all discounted to present:


Risk and return

Risk premium of an asset: asset return – return of risk-free asset

Variance: expected value of squared deviations of observations from their expected value (mean):

based on j observations

Standard deviation: a measure of volatility of expected stock returns:


Expected portfolio return (with two assets):

Expected portfolio return (with j = 1, …, n assets):

xj weight of asset j in the portfolio
rj (expected) return of asset j

Variance of portfolio return (portfolio variance) in the case of two assets:

Covariance between asset i and j with

Correlation coefficient between asset i and j:

Variance of portfolio return (portfolio variance) in the case of n assets:

Beta of the return of asset j to the market return (return of market portfolio m):


Expected return following the security market line equation (SML):

Expected return of a stock in event studies:

Abnormal return = actual return – expected return =



Rates of return on capital and assets:

Rates of return:



Weighted average cost of capital (WACC):

rD interest rate on debt resp. debt cost of capital
rE return on equity resp. equity cost of capital
Tc corporate tax rate

Weighted average cost of capital (WACC) with a zero-tax rate:

Leverage-formula for return on equity: return on equity increases with debt/equity-ratio

Leverage-formula for equity beta: risk increases with debt/equity-ratio: