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					Valuation of
  Money,
Stock; and
  Bonds
          The Time Value of Money

• Simple and compound interest
• Future Values and Compound Interest
                              mT
                       r 
        fvm ,T    1    
                       m
• As m approaches infinity, the factor

         1  
                      m        r
                 r        e
                 m
  where e = 2.71828
     The Time Value of Money
• Given continuous compounding, the future
  value of a dollar n periods hence is

                        rT
          fvT  e
The Time Value of Money
    The Time Value of Money
• Annually Compounded Interest Rates
        (Effective Interest Rate)
                  (EIR)
    = (1+ i/m)m -1
                  Vs
    Annual Percentage Rates (i)

    = Monthly rate X 12
    = Quarterly rate X4
           The Time Value of Money
Present Values
 Pv = Future Value After “t” Periods
                (1+r)t
DF is always <1 and keeps decreasing with increasing “t”
        The Time Value of Money
Compounding     m     Interest                   EIR
   Period               per        (1+i/m)m       %
                       period
                         %
    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
        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
             The Time Value of Money
Annuity
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)t
Growing Annuities =     C1   + C1 (1 + g) + C1(1 + g)2 …. + C1 (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
                Types of Annuities
 Ordinary Annuity
 Annuity Due
  PV Annuity Due = PV Ordinary Annuity for (t-1) Payments
                                    +
                                 Initial Payment

 Annuity Deferred
   PV Annuity Deferred = PV Ordinary Annuity
                              (1 + r)n-1
Where n = No. of years annuity delayed
General Annuity – when compounding
  periods 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 period
  2) Convert the payments to EACs that
     correspond with the compounding periods
 Convert the interest rate to the effective rate
           for the payment period
R=100 per month for 03 years
i = 14% pa, being compounded daily
Effective daily rate = .14/365 =0.0003836 or
0.03836%
(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    -       1
                   0.010796 0.010796(1+ 0.010796)39
            = 3169.28
Convert the payments to EACs that correspond
        with the compounding periods
R=100
i = 14% pa, being compounded daily
Effective daily rate = .14/365 =0.0003836 or 0.03836%
EAC =?
    100 =EAC       1    -             1
               0.0003836      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
   Relationship between Present Value, Future
     Value and Equivalent Annuity Cash Flows

 Multiply EAC by                           Multiply PV by
  present value              PV          inverse of present
  annuity factor                        value annuity factor



                       EAC        EAC



                             FV
                                         Multiply EAC by
   Multiply FV by                          Future value
 inverse of Future                        annuity factor
value annuity factor
                   Perpetuities
Pv of Perpetuity = C/r
Pv of Growing Perpetuity = C1      C2 (1 + g)   C3(1 + g)2
                                 +            +               ….
                          (1 +r)     (1 + r)2     (1+ r)3 +


                         =      C1
                             (r - g)
       Declining Discount Factor?

Basis of Capital Market
• A dollar tomorrow has value lower than
  a dollar today
• There is no money machine (arbitrage)
         The Time Value of Money

Net Present Value of Project
= Pv of Cash flows out – Required Investment

Remember: A risky dollar is worth less than a
          safe one
Other Names:
Present Value or just Pv
Discounted Value
Market Price
Market Value
       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 bond’s price to
  present value)
• Rate of Return on Bonds – The %age rate of
  actual income upon investment by holding a
  bond for a specific period
Effects of interest rates on bond’s market
values/Prices


PV of bond
 payments




             Interest rate, %
           The Time Value of Bonds
          Reading the Financial Pages
Treasury Bonds
       Maturity                                              Ask
Rate    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
                Note
                                                  It is in    YTM of
Coupon             It is quoted in 32nds                     Investor
                                                  32nds
Rate p.a                  Prices are
           100 + 10/32 = 100.315% of Face value
                       (Spread of 2/32)
                              &
                  116+ 17/32 = 116.375%
                        of Face value
                       (Spread of 4/32)
                The Time Value of Bonds
               Reading the Financial Pages
Other Bonds
                    Cur.                                     Net
Bonds               Yld          Vol          Close         Chg
AT&T 81/8 22        7.7          84           105 ¼         1/4
              Year of
              maturity
Coupon                         105 ¼ of the face      ¼ of 1 % of
Rate p.a.                           value
                                                        face vale
     Interest income as percentage of the
                  Bond’s price
    Interest income =81.25 (8 1/8% of 1000)        How to show price
             Bond’s Price =1052.50                  below par value?
    Cur. Yld = 81.25/1052.50 = 0.077 =7.7%
         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 – The
  discount rate that makes the present value of a
  bond’s payments equal to its price (Diff in face
  value 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
                               Moody’s       Standard
                                             & Poor’s
The Term Structure of Interest Rates
(Relationship between time to maturity and yield to maturity)


   YTM %
                            Abnormal Yield Curve




                            Normal Yield Curve




                            Maturity Period
  The Term Structure of Interest Rates

 Short term yield is lower than long term
  yield
 Why not the investors go for only long
  term bonds?
                         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
                                                                      Purchasing
                         800
                                                                         Power
      Index (1947=100)
       Consumer Price




                         600
                                                                     adjusted value
                         400
                                                                        of dollar
                         200
                                                           Series1
                          0
                               Years
                                       Years (1947-1992)
  Real Future value of investment
  = Investment X (1+ nominal interest rate)
             (1+ Inflation rate)
  Real rate of interest             Rate at which the
                                    purchasing power
  = (1+ nominal interest rate) - 1 of an investment
         (1+ inflation rate)            increases
 Approximately it is the diff of NIR and IR if
Nos. are small
Real Cash Flow = Nominal Cash Flow
                    (1+Inflation rate)
Current cash flows to be discounted by the nominal
interest rate and the real cash flows by the real interest
rate
• 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)
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 different
 in different years?
        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
    Valuation of Common Stock
• Book Value and Market Value
• Liquidation Value
• Going Concern Value – The diff. between
  Book value and liquidation value- Can be
  traced to:
   Extra Earning Power
   Intangible assets; and
   Value of Future Investments
          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 Capital
              Yield               appreciation

     Today’s Price = P0 = DIV1 + P1
                             1+ r
          Valuing Common Stock
P1 is dependent upon all future cash flows
P1 = P2 + DIV2
        1+ 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
•   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 = DIV
                                 r
          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 =   DIV1
                 r
                PD0 = EPS1
                        r
               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
    = DIV0 (1+g) (Because DIV1 = DIV0 X g)
            r-g
Provided:       a) g is constant       b) g < r
                c) Dividend is paid    d) There is no loss
                     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 DIV1 and g                     So that subject company offers an
Investors set the stock Price         adequate expected rate of return r
             Valuing Common Stock
Growth Stocks and Income Stocks
  o   Payout Ratio            EPS/BVPS          1-POR
  o   Plowback Ratio                               (b)
  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
Some Warnings about Constant-Growth Model
• r is only for a single share . A large number of
  equal risk securities may be taken into
  account
• Resist applying it to firms with high current
  rates 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 its
  assessment of share’s value. You may be
  making poor dividend forecast
  Remember: There are no mechanical rules to
  forecasting future

				
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posted:7/7/2010
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Description: valuation of bonds and other inventories