4-+valuation+of+money +stocks +bonds
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Valuation of
Money,
Stock; and
Bonds
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The Time Value of Money
Simple and compound interest
Future Values and Compound Interest
As m approaches infinity, the factor
where e = 2.71828
,
1
mT
m T
rfv
m
= +
( )1m
rrm
e+
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The Time Value of Money Given continuous compounding, the future
value of a dollar n periods hence is
r TTfv e=
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The Time Value of Money
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The Time Value of Money
Annually Compounded Interest Rates
(Effective Interest Rate)
(EIR)
= (1+ i/m)m -1
VsAnnual Percentage Rates (i)
= Monthly rate X 12
= Quarterly rate X4
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The Time Value of Money
Present Values
Pv = Future Value After t Periods
(1+r)t
DF is always
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The Time Value of MoneyCompounding
Periodm Interest
per
period%
(1+i/m)mEIR
%
1 year 1 6 (1.06)1 6.00
Semiannually 2 3 (1.03)2 6.0900
Quarterly 4 1.5 (1.015)4 6.134
Monthly 12 0.5 (1.005)12 6.1678
Weekly 53 0.11538 (1.0011538)52 6.1800
Daily 365 0.01644 (1.0001644)365 6.1831
Continuous
Almost 0
(2.718)0.06 6.1837
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Calculating Present Value
Present Value for Single Period:
PV = C X DF
= C / (1+r)
Present Value for Several Periods of Varying
Amounts:PV (A+B+C+D.) = C1 + C2 + C3 + C4 +
(1+r) (1+r)2 (1+r)3 (1+r)4
= E [Ct]( 1+ r)t
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The Time Value of MoneyAnnuity
Fv = C[(1+ r/m)nm-1]r/m
Pv = C[(1+ r/m)nm-1] = 1 - 1
r/m (1+ r/m)nm r r (1+r)tGrowing Annuities= C
1C
1(1 g) C
1(1 g)2 C
1(1 g) t-1
(1 r) (1 r)2 (1 r)3 (1 r)t
= C1 1 (1 g)t
(r g) (r g) (1 r)t
+
++
+++++ +.+
+
+
+
-- -
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Types of Annuities
Ordinary Annuity
Annuity DuePV AnnuityDue =PV Ordinary Annuity for (t-1) Payments
+
Initial Payment
Annuity Deferred
PV AnnuityDeferred = PV OrdinaryAnnuity
(1 + r)n-1
Where n = No. of years annuity delayed
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General Annuity when compoundingperiods and payment periods do not
coincide
There are two ways that we can do it:
1) Convert the interest rate to the effective
rate for the payment period2) Convert the payments to EACs that
correspond with the compounding periods
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Convert the interest rate to the effective ratefor the payment period
R=100 per month for 03 yearsi = 14% pa, being compounded daily
Effective daily rate = .14/365 =0.0003836 or0.03836%
(For 28 days a month and for calendar months?)
Effective monthly rate = (1+ 0.0003836)28 -1
= 0.010796 = 1.0796%
PV of loan = 100 1 - 10.010796 0.010796(1+ 0.010796)39
= 3169.28
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Convert the payments to EACs that correspondwith the compounding periods
R=100i = 14% pa, being compounded daily
Effective daily rate = .14/365 =0.0003836 or 0.03836%EAC =?
100 =EAC 1 - 10.0003836 0.0003836(1+ 0.0003836)28
EAC = 3.553 per day
PV of loan =3.353 1 - 1
0.0003836 0.0003836(10.0003836)1092
= 31169.28
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Relationship between Present Value, Future
Value and Equivalent Annuity Cash Flows
PV
EACEAC
FV
Multiply PV byinverse of present
value annuity factor
Multiply EAC byFuture value
annuity factor
Multiply EAC bypresent valueannuity factor
Multiply FV byinverse of Future
value annuity factor
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Perpetuities
Pv of Perpetuity = C/r
Pv of Growing Perpetuity = C1
C2(1 g) C
3(1 g)2
(1 r) (1 r)2 (1 r)3
= C1
(r - g)
+ ++ +
+ + +.
+
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Declining Discount Factor?
Basis of Capital Market A dollar tomorrow has value lower than
a dollar today
There is no money machine (arbitrage)
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The Time Value of Money
Net Present Value of Project
= Pv ofCash flows outRequired Investment
Remember:A risky dollar is worth less than a
safe one
Other Names:
Present Value or just Pv
Discounted ValueMarket Price
Market Value
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The Time Value of Bonds
Bonds and Notes Bond (10% bonds of 22 etc.) Coupon Face Value/Par Value/ Maturity Value (US$1000) Coupon Rate (%age of par value)
Current Yield Yield to Maturity (equates bonds price to present
value) Rate of Return on Bonds The %age rate of
actual income upon investment by holding a bondfor a specific period
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Effects of interest rates on bonds market
values/Prices
PV of bondpayments
Interest rate, %
Th Ti V l f B d
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The Time Value of Bonds
Reading the Financial Pages
Treasury Bonds
Maturity AskRate Mo/Yr Bids Ask Chg Yld
7 June 93n 100:10 100:12 - 1 2.08
111 /2 Nov 95 116:17 116:21 - 3 4.27
CouponRate p.a
It is in32nds
YTM ofInvestorIt is quoted in 32nds
Prices are100 + 10/32 = 100.315% of Face value
(Spread of 2/32)&
116+ 17/32 = 116.375%of Face value
(Spread of 4/32)
Note
Th Ti V l f B d
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The Time Value of BondsReading the Financial Pages
Other Bonds
Cur. Net
Bonds Yld Vol Close Chg
AT&T 81 /8 22 7.7 84 105 1/4
CouponRate p.a.
of 1 % offace vale
Interest income as percentage of theBonds price
Interest income =81.25 (8 1/8% of 1000)Bonds Price =1052.50
Cur. Yld = 81.25/1052.50 = 0.077 =7.7%
Year ofmaturity
105 of the facevalue
How to show pricebelow par value?
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The Time Value of Bonds
Bond Price as %age of its face value e.g. 115.87%[A 5- year 9% Bond with prevailing interest rate @5.3%- 15.87% above the face value]
Yield to Maturity Internal rate of return Thediscount rate that makes the present value of a
bonds payments equal to its price (Diff in facevalue and present value of bonds)
Bonds Rating
Investment Bonds and Junk Bonds (Bbb/BBB &
above and Ba/BA& below) Coupon rate and rate of return
Standard
& Poors
Moodys
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YTM %
Abnormal Yield Curve
Normal Yield Curve
Maturity Period
The Term Structure of Interest Rates(Relationship between time to maturity and yield to maturity)
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The Term Structure of Interest Rates
Short term yield is lower than long termyield
Why not the investors go for only long
term bonds?
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Nominal and Real Interest Rate(Inflation and The Time Value of Money)
Consumer Price Index
%age increase in CPI is the measurement of Inflation Current or nominal dollar and constant or Real dollar
US CPI 1947-1993
0
200
400
600
800
Years
Years (1947-1992)
ConsumerPrice
Index
(1947=100)
Series1
Purchasing
Power
adjusted value
of dollar
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Real Future value of investment
= Investment X (1+ nominal interest rate)
(1+ Inflation rate)Real rate of interest
= (1+ nominal interest rate) - 1
(1+ inflation rate)Approximately it is the diff of NIR and IR if Nos. are
smallReal Cash Flow = Nominal Cash Flow
(1+Inflation rate)
Current cash flows to be discounted by the nominal interestrate and the real cash flows by the real interest rate
Rate at which the
purchasing power
of an investment
increases
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Price of the bond and its valuation
Price/PV (Bond)= PV (Coupon Payments)+ PV(Principal)
= (coupon X n-years annuity factor)
+ (final payment X n- year discount factor)
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Example: A 07 years 7.5% annual Coupon
rate treasury bond with face value of $1000
{Suppose the prevailing rate on a similar risk security(TB) is 5.41%}
Price/ Market value /PV=75 1 - 1
0.0541 0.0541 (1.0541)7
= 427.60 + 691.56
= $1119.56
What if the required rate of return is differentin different years?
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Valuation of Common Stock
Common Stock and Preferred Stock
Primary and Secondary markets
Stock Exchange Working
Reading The Stock Market Listing
Dividend
Price Earning Ratio
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Valuation of Common Stock
Book Value and Market Value
Liquidation Value
Going Concern Value The diff. betweenBook value and liquidation value- Can be
traced to:Extra Earning Power
Intangible assets; and
Value of Future Investments
V l i C St k
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Valuing Common Stock
Valuing Common Stock
Pay offs to owners comes in two ways
a) Cash Dividend
b) Capital Gains or Losses
Hence:
Expected Return =
r = DIV1 + ( P1 P0)P0
= Expected Dividend + Expected CapitalYield appreciation
Todays Price = P0 = DIV1 + P1
1+ r
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Valuing Common StockP1 is dependent upon all future cash flows
P1 = P2 + DIV21+ r
P2 = P3 + DIV3
(1+r)2
P0 = DIV1 + DIV2 + DIV3 +..+ DIVH +PH
(1+r) (1+r)2 (1+r)3 (1+r)H
The Dividend Discount Model
..
Valuing Common Stock
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Valuing Common Stock
Dividend dependent upon:a) EPS
b) Dividend Policy
The present value of all dividends =PD0
= DIV1 + DIV2 + DIV3 +.. + DIVt
1+ r (1+ r)2 (1+ r)3 (1+ r)t
If constant dividend:
PD0 = DIV 1 + 1 + 1 + + 1
1+ r (1+ r)2 (1+ r)3 (1+ r)t
If t
PD0 = DIVr
V l i C St k
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Valuing Common Stock
The Dividend Discount Model with No Growth(100% Dividend Pay out Ratio )
If the company pays a constant dividend a
Perpetuity
PD0= DIV
1
r
PD0 = EPS1r
Valuing Common Stock
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Valuing Common Stock
The Dividend Discount Model with Constant Growth
(Gordon Growth Model)
DIV1 = 3
DIV2 = 3(1+ g) = 3 (1+.08) = 3.24 (Suppose g = 8%)
DIV3 = 3(1+g)2 = 3 (1+.08)2= 3.50
PD0= D1 + D1(1+g)2 + D1(1+g)
3 +. = DIV1
1+r (1+r)2 (1+r)3 r g
= DIV0 + DIV1 (If the immediate dividend is included)
r g
= (Because DIV1 = DIV0 X g)
Provided: a) g is constant b) g < r
c) Dividend is paid d) There is no loss
DIV0 (1+g)
r - g
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Valuing Common Stock
The expected rate of return
PD0 = DIV1 =r g
r = DIV1+ g = The Market Capitalization rate
PD0
= Dividend Yield + Growth Rate
DIV1+ g = r = Expected rate of return offered by other, equally risky stocks
PD0
Given DIV
1and g So that subject company offers an
Investors set the stock Price adequate expected rate of return r
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Valuing Common Stock
Growth Stocks and Income Stocks
o Payout Ratioo Plowback Ratio
o g = Return on Equity X Plowback ratio
o Present Value of Growth opportunities (PVGO)
o Sustainable Growth Rates (g)o Methods to alter Profits
Depreciation Methods Inventory Evaluation Methods
Treating R&D Expense as current rather than Investment Methods of Reporting Tax Liabilities
1-POR(b)
EPS/BVPS
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Some Warnings about Constant-Growth Model
r is only for a single share . A large number of
equal risk securities may be taken intoaccount
Resist applying it to firms with high currentrates of growth Difficult to sustain these
rates Avoid using it in inflationary period Do not use simple constant-growth formula to
test whether the market is correct in itsassessment of shares value. You may bemaking poor dividend forecast
Remember:There are no mechanical rules toforecasting future
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