Perpetuity:
P = C / r
Growing Perpetuity:
P = C / rg
Annuity:
C ( 1/r ) ( 1 – 1/(1+r)
T
)
Growing Annuity :
C / rg (1 (1g)/(1+r)
T
)
(1+r) (1+inf) = (1+n)
n = nominal
r = real
inf = inflation
EAY v. APR
EAY = Effective Annual Yield
APR = Annual % Rate
EAY = APR (annual Compounding)
EAY > APR (More Freq. Compounding)
Conversion
FV = (1+ APR/n)
nT
= (1 + EAY)
T
Continuous Compounding
FV = e
APR(T)
Mortgage Payments (Monthly)
Term = 20 yrs * 12 = 240 Pds
APR = 12% / 12 = 1%
r
=
1%
Each Payment:
Balance x r = Interest Payment
Monthly Payment – Interest Payment =
Principle Reduction
Balance A  Principle Reduction=Balance B
Comparing Assets w/ Unequal Lives
1. Find NPV for 1 life cycle
2. Find Cost / yr
PV = x * A
3. Compare Ax and Bx
Determining Cash Flow
Cash = (Revenue – Cash Costs)(1  .34) + .
34(Depreciation)
What happens to Cash Flow…
1. Sales rise by $1? Down .34 tax Up .66
2. Materials costs rise by $1?
Pay $1
Retain .40
Down .60
Depreciation expense rises by $1?
Income Down $1
Tax down .34
Cash up .34
Calculate:
NPV of Project: 30 + 21/1.12 + 21/1.12^2
IRR
: 30 + 21 / (1+r) + 21 / (1+r) ^2 = 0
Cash Effects of Purchase and Sale of Assets
1. Initial cash outlay:
1000 @ C 0
(Cannot be expensed)
2. Depreciation Expense = 1000
Depreciation expense each Year:
D1 = .200 (1000) = 200
C1 = 68
D2 = .320 (1000) = 320
C2 = 108.8
D3 = .192 (1000) = 192
C3 = 65.28
D4 = .115 (1000) = 115
C4 = 39.10
D5 = .115 (1000) = 115
C5 = 39.10
D6 = .058 (1000) = 58
C6 = 19.72
A.
D * T (.34)
= C
B.
Discount All Cash Flows
C. Sum to Get NPV of
Depreciation Tax Shield
* For ever 1$ Depreciation…
Pre Tax Income Down 1$
3. Resale of Used Asset
Resale Price
500K
Book Value

0

Taxable gain
500K
C5 = 500 ( 1  .34) = 330
Stock Valuation:
Constant Dividends:
Div1 / r
Constant Growth:
Div1 / rg
To determine g:
A. Next years earnings will be the same unless
a NET INVESTMENT is made
A. Earnings Next Yr. = R.E. This yr
+R.E. This yr x ROE
B. Divide all by earnings this year
So…
g = Retention Ratio x ROE
Earnings = 2M
Retention Ratio = 1 – Payout ratio = .4
Historical ROE = .16
1.
Will Retain: Retention Ratio
x
Earnings
.4
x
2M
2.
This yrs Earnings = Earnings x ROE
2M
x
.16 = 128k
3. g = Change in Earnings
=
128k
= .064
Total Earnings
2M
4. This Implies Earnings in 1 yr will be:
Total earnings
x
(1 + g) =
2M
x
1.064
=
1,128,000
Stock: 1M Outstanding (P = 10)
Payout Ratio = .6
Earnings = 1,128,000
Solve For Div:
Payout = Div / Earnings
Div = $1.28
P (growth)= Div / rg
Now… Solve for r
Finally, r =.192 or 19.2%
NVPGO EXAMPLE
1.
Earnings constant
2.
EPS = Div (Cash Cow
Value of a share = EPS / r
=
Div / r
Now Firm retains entire dividend @ Date 1 to
invest in a single project
Date 0:
NPV of Project
= NPVGO
# of Shares
Thus, Value of a Share after commit to Project:
EPS / r
+ NPVGO
Earnings / yr = 1M
Stock: 100k outstanding
EPS = 10
A. @ Date 1 Firm can spend 1M on a Mktg
Campaign which will increase subsequent
earnings by 210, 000 or increase EPS by 21%
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 Spring '10
 jaffe
 Time Value Of Money, Annual Percentage Rate, Annuity, Net Present Value, Perpetuity, Spot rate, Internal rate of return, EAY

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