Fins1613 Formulas

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FINS1613 Formulas

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FINS1613 FORMULAS
Compounding:

interest rate
100
Value t +n=Value t ×(1+ r)n

Interest= principal ×
Discounting:

Value t=

Value t +n
n

(1+r )

Annual percentage rate (APR)
 APR = Per Period IR x Number of Compounding Periods per Year

APR =r × n
Effective annual rate (EAR)
 The total amount of interest that will be earned at the end of one year with
compounding.
n

EAR=(1+ r) −1
Converting APR to EAR

EAR=(1+

APR n
) −1
n

Converting EAR to APR
1
n

APR =n ×[( 1+ EAR ) −1]
Annuities
 A stream of cash flows arriving at a regular interval over a specified time period.
Constant annuity:

1
1
Annuity Value t =C × (1−
)
r
( 1+r )n

The annuity value formula gives the total time-t value of all n cash flows
beginning at t+1
Annuity factor:

PV ( C for n periods at interest rate r )=C ×

1
1
1−
r
( 1+r )n

(

)
¿ C × annuity factor ( n ,r )

1
1
Annuity Factor ( n , r ) = (1−
)
r
( 1+r )n
Growing annuity value:

1+ g
1+ r
¿
1−( ¿ ¿ n )
Growing Annuity Value t =C ×

1
¿
r −g

The growing annuity value formula gives the total time-t value of all n growing cash
flows beginning at t + 1

Perpetuities
 A stream of cash flows that occur at regular intervals and makes payments
forever
Constant perpetuity:

Perpetuity Valuet =

C
r

The perpetuity value formula gives the total time-t value of the infinite cash
flows beginning at t + 1
Growing perpetuity:

Growing Perpetuity Value t=

C
r−g

The growing perpetuity value formula gives the total time-t value of the
infinite growing cash flows beginning at t + 1
A constant annuity is a growing annuity with no growth:

1+ g
1+r
¿
1+r
¿
¿
(¿ n¿)
1
1− ¿
1
1−( ¿¿ n ) with g=0 C × ¿
r

1

¿
r−g
A perpetuity is an annuity that makes an infinite number of payments (provided g < r):

1+ g
1+ r
¿
C
as
1−( ¿¿ n ) n →∞
r−g

1

¿
r −g
Scaling of cash flows and valuation
If the time t value of the cash flows is:

Value t ( Ct +Ct +1 +C t+2 +… )= X
Then the time t value of M times the cash flows is:

Value t ( M × Ct + M ×C t +1+ M ×C t +2+ … ) =M × X
Adding and subtracting cash flows
If the time t value of the cash flows A and B are:

Value t ( A t + A t +1+ A t+2 + … )= X
Value t ( B t + Bt +1+ Bt +2 +… )=Y
Then the time t value of the combined cash flows is:

Value t ( A t + Bt + A t +1+ Bt +1 + At +2 + Bt +2 … )=X + Y
The values X and Y must be at the same reference time and use the same
discount rate.
Delayed & accelerated cash flows
If the time t value of the cash flows

Value t ( Ct +Ct +1 +C t+2 +… )= X

Delays by s periods

Value t ( Ct +Ct +1 +C t+2 +…+ delayed s periods )=
Delayed cash flows are received later.
They are worth “less”, so divide
(discount)

Accelerated by u periods

XValue t ( Ct +Ct +1 +C t+2 +…+ accelerated u periods )=X ×(1+r )
(1+r )s
Accelerated cash flows are received
earlier. They are “more” valuable, so
multiply (compound)

Bond Valuation

Bond=

c
1
FV
1−
+
C= per− period coupon pa yment
n
r
( 1+ r )
( 1+ r )n
n=number of periods
PV =face value

(

)

r= per −period yield

Couponrate × face value
number of payments per year
Bond value=PV ( coupons )+ PV ( face value )
Bond value=PV ( annuity ) + PV ( single cash flow)
Coupon payment=

Zero coupon bonds

Bond value=

FV
( 1+r )n

r= per −period yield

Fisher effect

n=number of periods

PV =face value

1+ Nominal Rate
1+ Inflation Rate
Growth of money
¿
Growth of prices

Growth∈ purchasing power=1+ Real Rate=

Real rate=

Nominal rate−Inflation rate
=Nominal rate−Inflation rate
1+ Inflation rate

Total return from equity ownership can be separated into two components:
 Dividend yield: a share’s expected cash dividend divided by its current price
 Capital gain: the amount by which the selling price of an asset exceeds it initial
purchase price.
 Capital gain rate: the change in stock price as a percentage of the initial price
 Total Return = Dividend yield + Capital gain rate

¿1 + P1
−1
P0
¿
P −P0
¿ 1 ( dividend yield ) + 1
(captial gain yield )
P0
P0
r E=

A one-year investor
 Value of the stock today: P0
 Expects to receive a dividend of Div1 in one year and sell the stock for P1

P0=
An n-year investor
 Value of the stock today: P0

Div 1 + P1
1+r E



Expects to receive a dividend of Div1 each year through to time n. At time n, will
receive a final dividend and sell the stock for P n

1+r E
¿
¿
¿2
¿
1+r E
¿
¿
¿
Di v 1 ¿2
P 0=
+
1+r E ¿
An infinite horizon investor

1+r E
¿
¿
¿2
¿
1+r E
¿
¿
¿3
¿
¿1
¿2
P 0=
+¿
1+r E
Constant dividend growth model

P 0=

¿1
r

P 0=

¿1
r E−g
r E=

¿1
+g
P0

Retention rate: The fraction of earnings that a firm retains for new investment.
 Return on new investment: Measures the ability of a firm to turn investment into
earnings. It is the ratio of new earnings to new investment

dividend payout rate+ retention rate=1
Estimating dividend growth

¿t =


Earningst
Dividends pai d t
×
Shares outstanding t
Earnings t

¿ EPS t × Divident payout ratet

Assume the dividend payout rate is constant, then

¿t +1 – ¿t
¿t
EPS t +1 × Dividend pa yout rate−EPS t × Dividend payout rate
¿
EPS t × Dividend payout rate
Dividend growth ( g )=

¿

EPSt +1 × EPS t
EPSt

An estimate for earnings growth is an estimate for dividend growth
  Assuming dividends are paid from earnings, estimating dividend growth requires
estimating earnings growth.
 Use accounting measures to:
- Determine the amount of earnings retained for new investments
- Determine the return on this new investment

g=



EPS t +1−EPS t
EPS t

New earningst +1
New Investment t New Earnings t +1
¿
×
EPS t
EPS t
New Investment t
¿ Retention Ratio × Return on New Investment
¿

Assuming the dividend payout rate is constant, the dividend growth rate can be
expressed as follows:

Divident growth=retentionrate × RONI

¿ ( 1−dividend payout rate ) × RONI

RONI: return on new investment
Growth in dividends:

Growth∈dividends=

¿t +1 – ¿t
¿t

¿

EPS × ( 1−payout ) × RONI × payout
EPS × Payout

¿ ( 1− payout ) × RONI

¿ retention × RONI
Total payout
  The total amount paid by the firm to shareholders through dividends and share
repurchases. Total payout is expressed as a dollar amount for the firm and NOT
normalised by the number of shares outstanding.

Total payout=Dividends+Share repurchases
As share repurchases changes the number of equity shares outstanding, total
payout per share would not be a meaningful measure.

Total Payout Model
The dividend discount model can be easily adapted to allow for share repurchases:
 Estimate total payouts (dividends + share repurchases) to equity. Do not
normalise by the number of shares.
 Use total payouts as cash flows to equity in valuation model.
 Discount payouts at the cost of equity to the market value of equity.
 Divide market value by current number of shares outstanding to find current
share price.

P 0=

PV (Future total dividends∧repurchases)
Shares outstanding

Equivalent annual annuities
 The constant annual cash flow that has the same present value as the actual
cash flows of a project

Equivalent annual annuity cash flow=

present value
1
1
(1−
)
r
( 1+r )n

Profitability index
 The net present value of a project per unit of resource consumed

Profitability index=

NPV
resource consumed

Capital budgeting

Assets=Liabilitites+ Shareholder s ' equity
Incremental Earnings=Inc remental EBIT x (1−tax rate)
Net working capital = working capital assets – working capital liabilities
Net working capital = accounts receivable + inventories – accounts payable
The indirect method

IncrementalCF=incremental earnings+depreciation−CapEX + after tax salvage−change∈net working capital
FCF=Incremental EBIT × ( 1−tax rate ) + depreciationi−CapE X−change ∈NWC
¿ ( Revenues−Costs −Depreciation) × ( 1−Taxrate )+ depreciation−CapEX −change∈ NWC
¿ ( Revenus−Costs ) × ( 1−Tax rate )−CapEX −Change∈NWC + Depreciation×Tax
Depreciationtax shield=Depreciation× Tax rate
After tax earnings= ( revenues−costs−depreciation ) × ( 1−tax rate )
After tax salvage=salvage price ( sale price )−taxrate × capital gain(sale price−book price)

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