Valuation of Common Stocks and Bonds by w3OacgU

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									Valuation of Common
 Stocks and Bonds


   How to apply the PV
        concept



          FIN 819: lecture 3   1
Today’s plan
   Review what we have learned in the last
    lecture
   Valuing stocks
    • Some terms about stocks
    • Valuing stocks using dividends
    • Valuing stocks using earnings
    • Valuing stocks using free cash flows


                       FIN 819: lecture 3    2
Today’s plan (Continue)
   Bond valuation and the term-structure of
    interest rates
    • Terminology about bonds
    • The valuation of bonds
    • The term structure of interest rates
    • Use duration to measure the volatility of the
      bond price




                        FIN 819: lecture 3        3
What have we learned in the
last lecture?
   Payback rule
    • Shortcomings
   IRR rule
    • Shortcomings
   Free-cash flow calculation




                     FIN 819: lecture 3   4
Some specific questions in the
calculation of cash flows
   Include all incidental effects
   Do not forget working capital requirements
   Forget sunk costs
   Include opportunity costs
   Be careful about inflation
   Depreciation
   Financing



                      FIN 819: lecture 3     5
Free cash flows calculation
   Free cash flows = cash flows from
    operations + cash flows from the change
    in working capital + cash flows from
    capital investment and disposal




                   FIN 819: lecture 3   6
Calculating cash flows from
operations
   Method 1
    • Cash flows from operations =revenue –cost
     (cash expenses) – tax payment
   Methods 2
    • Cash flows from operations = accounting
     profit + depreciation
   Method 3
    • Cash flows from operations =(revenue –
     cost)*(1-tax rate) + depreciation *tax rate

                       FIN 819: lecture 4          7
A summary example 2
   Now we can apply what we have
    learned about how to calculate cash
    flows to the IM&C’s Guano Project (in
    the textbook), whose information is
    given in the following slide.




                   FIN 819: lecture 4   8
IM&C’s Guano Project
Revised projections ($1000s) reflecting inflation




                           FIN 819: lecture 4       9
IM&C’s Guano Project
Cash flow analysis ($1000s)




                         FIN 819: lecture 4   10
IM&C’s Guano Project
   NPV using nominal cash flows
                 1,630 2,381        6,205 10,685 10,136
NPV  12,600                                 
                  1.20 1.20  2
                                   1.20 1.20 1.205
                                         3      4

         6,110     3,444
                         3,519 or $3,519,000
        1.20 1.20
              6         7




                      FIN 819: lecture 4        11
New formula
 In chapter 4, it is argued that
 FCF=earnings –net investment
 Net investment = total investment -
                        depreciation
 Do you agree with this formula? Why?




                 FIN 819: lecture 4   12
Example
   A project costs $2,000 and is expected
    to last 2 years, producing cash income of
    $1,500 and $500 respectively. The cost
    of the project can be depreciated at
    $1,000 per year. If the tax rate is 50%,
    what are the free cash flows?



                    FIN 819: lecture 4   13
One more question
   Mr. Pool is now 40 years old and plans
    to invest some fraction of his current
    annual income of $40,000 in an account
    with an annual real interest rate of 5%,
    starting next year until he retires at the
    age of 70 to accumulate $500,000 in real
    terms. If the real growth rate of his
    income is 2%, what fraction of his
    income must be invested?

                     FIN 819: lecture 4   14
Some terms about stocks

Book Value – The value of the stocks
  according to the balance sheet.
Liquidation Value - Net proceeds that
  would be realized by selling the firm’s
  assets and paying off its creditors.
Market Value Balance Sheet - Financial
  statement that uses market value of
  assets and liabilities.

                   FIN 819: lecture 4   15
Some terms about stocks

Secondary Market - market in which already
  issued securities are traded by investors.
Dividend - Periodic cash distribution from the
  firm to the shareholders.
P/E Ratio - Price per share divided by earnings
  per share.
Dividend yield – Dividends per share over the
  price of per share


                    FIN 819: lecture 4    16
Example
   IBM has a trading price of $70 per share.
    Its annual earnings per share is $5. Its
    annual dividend per share is $3.5. What
    are the P/E and the dividend yield?
   P/E=70/5=14
   Dividend yield=3.5/70 or 5%



                    FIN 819: lecture 4   17
Valuing Common Stocks using
dividends (first approach)
Dividend Discount Model - Computation of
  today’s stock price which states that share
  value equals the present value of all expected
  future dividends plus the selling price of the
  stock.
       Div1      Div 2          Div H  PH
P0                      ...
     (1  r ) (1  r )
             1         2
                                 (1  r ) H


H - Time horizon for your investment.

                     FIN 819: lecture 4     18
Example
   George has bought one IBM share in the
    beginning of this year and decides to
    hold this share until next year. The
    expected dividend this year is $10 per
    share and the stock is expected to sell at
    $110 per share in the end of the year. If
    the cost of the capital is 10%, what is the
    current stock price?

                     FIN 819: lecture 4   19
Solution


   P0=(110+10)/(1+0.1)=$109.1




                  FIN 819: lecture 4   20
Valuing common stocks using
dividends
Example
  Current forecasts are for XYZ Company to pay
  dividends of $3, $3.24, and $3.50 over the next
  three years, respectively. At the end of three
  years you anticipate selling your stock at a
  market price of $94.48. What is the price of the
  stock given a 12% expected return?




                     FIN 819: lecture 4     21
  Solution


       3.00        3.24                 3.50  94.48
P0                       
     (1  .12) (1  .12)
              1          2
                             (1  .12) 3

P0  $75.00




                   FIN 819: lecture 4           22
Valuing common stocks using
dividends
 If we forecast no growth, and plan to hold out
 stock indefinitely, we will then value the stock
 as the PV of a PERPETUITY.
                        Div1 EPS1
PV ( perpetuity)  P0      or
                         r     r
   Assumes all earnings are
    paid to shareholders.

                       FIN 819: lecture 4    23
Example
   Suppose that a stock is going to pay a
    dividend of $3 every year forever. If the
    discount rate is 10%, what is the stock
    price for the following cases:
    • (a) you invest and hold it forever?
    • (b) you invest and hold it for two years?
    • (c) you invest and hold it for 20 years?

                       FIN 819: lecture 4         24
Solution
 (a) P0=3/0.1=$30
 (b)P0=PV (annuity) + PV( the stock price at
        year 2)
       = 3/1.1 + 3/1.12+(3/0.1)/1.12
       = 3/0.1=$30
 (c) P0=PV (annuity of 20 years) +
         PV (the stock price at the year of 20)
       =$30


                     FIN 819: lecture 4      25
Valuing Common Stocks

Gordon Growth Model: A version of the
  dividend growth model in which dividends
  grow at a constant rate (Gordon Growth
  Model).
Stocks can be valued as a perpetuity with a
  growth rate, if you want to hold this stock
  forever, that is
                       Div1
                  P0 
                       rg
                     FIN 819: lecture 4    26
Example
   Suppose that a stock is going to pay a
    dividend of $3 next year. Dividends grow
    at a growth rate of 3%. If the discount
    rate is 10%, what is the stock price for
    the following cases:
    • (a) you buy and hold it forever?
    • (b) you buy and hold it for two years?
    • (c) you buy and hold it for 20 years?
                       FIN 819: lecture 4      27
Solution
 (a) P0=3/(0.1-0.03)=$42.86
 (b)P0=PV (annuity) + PV( the stock price at
        year 2)
       = 3/1.1 + 3*1.03/1.12+(3*1.032/(0.1-
        0.03))/1.12
       = 3/(0.1-0.03)=$42.86
 (c) P0=PV (annuity of 20 years) +
         PV (the stock price at the year of 20)
       =$42.86


                     FIN 819: lecture 4      28
Capitalization rate

Expected Return - The percentage yield that an
  investor forecasts from a specific investment
  over a set period of time. Sometimes called
  the market capitalization rate.


                       Div1  P1  P0
 Expected Return  r 
                             P0

                     FIN 819: lecture 4    29
Example

If Fledgling Electronics is selling for $100 per
   share today and is expected to sell for $110
   one year from now, what is the expected return
   if the dividend one year from now is forecasted
   to be $5.00?




                     FIN 819: lecture 4     30
Solution

According to the formula,


                   5  110  100
 Expected Return                 .15
                        100



                  FIN 819: lecture 4   31
Capitalization rate

The formula for the capitalization rate can
 be broken into two parts.

Capital. Rate = Dividend Yield + Capital Appreciation



                        Div1 P1  P0
  Expected Return  r      
                         P0     P0
                      FIN 819: lecture 4       32
Using dividends models to
derive the capitalization rate
 Capitalization Rate can be estimated using the
 perpetuity formula, given minor algebraic
 manipulation.
                   Div1
              P0 
                   rg
                  Div1
              r       g
                   P0
                   FIN 819: lecture 4     33
Valuing Common Stocks

Example-
  If a stock is selling for $100 in the stock
  market, the cost of capital is 12% and the next
  year dividend is $3, what might the market be
  assuming about the growth in dividends?

                     $3.00
             $100 
                    .12  g
             g .09

                     FIN 819: lecture 4     34
Some terms about dividend
growth rates
 If a firm elects to pay a lower dividend, and
  reinvest the funds, the stock price may
  increase because future dividends may be
  higher.
Payout Ratio - Fraction of earnings paid out as
  dividends
Plowback Ratio - Fraction of earnings retained by
  the firm.


                     FIN 819: lecture 4    35
Deriving the dividend growth
rate g
 Growth can be derived from applying the
 return on equity to the percentage of
 earnings plowed back into operations.
       ROE  Return on Equity
                      EPS
       ROE 
             Book Equity Per Share
g = return on equity X plowback ratio

                   FIN 819: lecture 4   36
Example

 Our company forecasts to pay a
 $5.00 dividend next year, which
 represents 100% of its earnings.
 This will provide investors with a
 12% expected return. Instead, we
 decide to plow back 40% of the
 earnings at the firm’s current return
 on equity of 20%. What is the
 value of the stock before and after
 the plowback decision?
                     FIN 819: lecture 4   37
Solution
   Without growth
                                        5
                                 P0        $41.67
                                      0.12
   With growth

          g  0.4 * 0.2  0.08
                 5 * 0.6
          P0               $75
               0.12  0.08
                     FIN 819: lecture 4           38
Example (continued)

 If the company did not plowback some
 earnings, the stock price would remain at
 $41.67. With the plowback, the price rose to
 $75.00.

 The difference between these two numbers
 (75.00-41.67=33.33) is called the Present
 Value of Growth Opportunities (PVGO).


                   FIN 819: lecture 4     39
Valuing common stocks using
earnings
   We often use earnings to value stocks
    as
               EPS1
          P0        PVGO
                r
   What is the relationship between this
    formula and the dividend growth
    formula?

                    FIN 819: lecture 4   40
Example
   Firm A has a market capitalization rate of 15%. The
    earnings are expected to be $8.33 per share next year.
    The plowback ratio is 0.4 and ROE is 25%. Every
    investment in year i is to yield a simple perpetuity starting
    in year (i+1) with each cash flow equal to total investment
    times ROE. All the investments have the same
    capitalization rate.
    •   (a) Using the formula P=ESP1/r + PVGO to calculate the
        stock price
    •   (b) If ROE is increased, what will happen to the stock price?
        Why?
    •   (c) Use the dividend model to calculate the stock price?
    •   (d) What have you found?
    •   (e) Think about why you have this kind of result?


                              FIN 819: lecture 4              41
Simple Solution
(a)
 g=10%, EPS1/r=8.33/0.15=$55.56
 PVGO=NPV1/(r-g)=2.22/(0.15-
  0.1)=$44.44, P=$100
(b) The price will be increased
(c) P=Div1/(r-g)=5/(0.15-0.1)=$100


                 FIN 819: lecture 4   42
Valuing common stocks using
FCF (free cash flows)

    The value of a business or stock is usually
    computed as the discounted value of FCF out to
    a valuation horizon (H).
   The horizon value is sometimes called the
    terminal value .

      FCF1       FCF2             FCFH          PVH
PV                      ...             
     (1  r ) (1  r )
             1         2
                                 (1  r ) H
                                              (1  r ) H

                       FIN 819: lecture 4         43
 FCF and PV


      FCF1       FCF2             FCFH          PVH
PV                      ...             
     (1  r ) (1  r )
             1         2
                                 (1  r ) H
                                              (1  r ) H

              PV (free cash flows)             PV (horizon value)




                          FIN 819: lecture 4             44
FCF and PV
   Free Cash Flows (FCF) should be the
    theoretical basis for all PV calculations.
   FCF is a more accurate measurement of
    PV than either Div or EPS.
   The market price does not always reflect
    the PV of FCF.
   When valuing a business for purchase,
    always use FCF.
                     FIN 819: lecture 4   45
  FCF and PV

  Example
     Given the cash flows for Concatenator Manufacturing
     Division, calculate the PV of near term cash flows, PV
     (horizon value), and the total value of the firm. r=10% and
     g= 6%
                                                             Year
                  1     2      3     4        5         6           7      8        9   10
Asset Value      10.00 12.00 14.40 17.28 20.74 23.43                26.47 28.05 29.73 31.51
Earnings          1.20 1.44 1.73 2.07 2.49 2.81                      3.18 3.36 3.57 3.78
Investment        2.00 2.40 2.88 3.46 2.69 3.04                      1.59 1.68 1.78 1.89
Free Cash Flow    - .80 - .96 - 1.15 - 1.39 - .20 - .23              1.59 1.68 1.79 1.89
.EPS growth (%) 20     20     20    20       20         13          13     6        6   6


                                   FIN 819: lecture 4                          46
FCF and PV

Solution

                        1  1.59 
 PV(horizon value)        6             22 .4
                     1.1   .10  .06 
            .80   .96     1.15 1.39    .20     .23
PV(FCF)  -                              
            1.1 1.1 1.1 1.1 1.1 1.16
                      2       3    4       5


         3.6


                       FIN 819: lecture 4      47
FCF and PV


PV(busines s)  PV(FCF)  PV(horizon value)
               -3.6  22.4
               $18.8




                  FIN 819: lecture 4   48
How to estimate the horizon
value?
   It is very difficult to forecast or estimate
    the horizon value. There are several
    ideas that may be used to estimate the
    horizon value.
    • Competition
    • Constant growth rate


                      FIN 819: lecture 4     49
Another example
Imagine Corporation has just paid a
  dividend of $0.40 per share. The
  dividends are expected to grow at 30%
  per year for the next two years and at
  5% per year thereafter. If the required
  rate of return in the stock is 15% (APR),
  calculate the current stock price.


                   FIN 819: lecture 4   50
Solution
   Answer:
   First: visualize the cash flow pattern;
    • C1, C2 and P2
   Then, you know what to do?
   P0 = [(0.4 *1.3)/1.15] + [(0.4 *
    1.3^2)/(1.15^2)] +
    [(0.4 * 1.3^2*1.05)/((1.15^2 * (.15 -.05))]
    = $6.33

                      FIN 819: lecture 4      51
Another cash flow problem!
Firm Excellent has an old packaging machine that can be used for another 3 years.
The remaining book value for this old machine is $15,000, which can be
depreciated in the next three years by using the straight-line depreciation approach.
If the old packaging machine is used, the annual total cost of operation, labor and
maintenance is $10,000. In the market, the new packaging machine is available
now at the price of $50,000, which will increase by 5% annually. Based on
Excellent’s experience, the new packaging machine will be used forever. The
capital investment cost of the new package machine will be depreciated in the next
10 years by using the straight-line depreciation approach. The annual total cost of
operation, labor and maintenance will be $8,000 when the new package machine is
used.
Questions:
a. What is the valuation horizon used in this problem?
b. Should Excellent invest in the new packaging machine now or
   wait three years later?



                                    FIN 819: Lecture 2
    Bonds

   Bond – a security or a financial instrument that
    obligates the issuer (borrower) to make
    specified payments to the bondholder during
    some time horizon.
   Coupon - The interest payments made to the
    bondholder.
   Face Value (Par Value, Principal or Maturity
    Value) - Payment at the maturity of the bond.
   Coupon Rate - Annual interest payment, as a
    percentage of face value.
                        FIN 819: lecture 4    53
Bonds
   A bond also has (legal) rights attached to
    it:
    • if the borrower doesn’t make the required
        payments, bondholders can force bankruptcy
        proceedings
    •   in the event of bankruptcy, bond holders get
        paid before equity holders




                        FIN 819: lecture 4     54
An example of a bond
   A coupon bond that pays coupon of 10%
    annually, with a face value of $1000, has a
    discount rate of 8% and matures in three
    years.
    •   The coupon payment is $100 annually
    •   The discount rate is different from the coupon rate.
    •   In the third year, the bondholder is supposed to get
        $100 coupon payment plus the face value of $1000.
    •   Can you visualize the cash flows pattern?



                            FIN 819: lecture 4          55
Bonds

WARNING
 The coupon rate IS NOT the discount
 rate used in the Present Value
 calculations.
 The coupon rate merely tells us what cash flow
 the bond will produce.
 Since the coupon rate is listed as a %, this
 misconception is quite common.

                   FIN 819: lecture 4     56
Bond Valuation

 The price of a bond is the Present Value
 of all cash flows generated by the bond
 (i.e. coupons and face value) discounted
 at the required rate of return.

          cpn             cpn                  1,000  cpn
 PV                                 ... 
        (1  r )1       (1  r ) 2                (1  r ) N



                             FIN 819: lecture 4                57
Zero coupon bonds

   Zero coupon bonds are the simplest type of bond
    (also called stripped bonds, discount bonds)
   You buy a zero coupon bond today (cash outflow)
    and you get paid back the bond’s face value at
    some point in the future (called the bond’s maturity )
    How much is a 10-yr zero coupon bond worth today
    if the face value is $1,000 and the effective annual Face
    rate is 8% ?                                         value
                              PV

                          Time=0                           Time=t


                           FIN 819: lecture 4         58
Zero coupon bonds (continue)
   P0=1000/1.0810=$463.2
   So for the zero-coupon bond, the price is
    just the present value of the face value
    paid at the maturity of the bond
   Do you know why it is also called a
    discount bond?



                    FIN 819: lecture 4   59
Coupon bond

 The price of a coupon bond is the
 Present Value of all cash flows
 generated by the bond (i.e. coupons and
 face value) discounted at the required
 rate of return.
          cpn             cpn                   (cpn  par)
 PV                                 .... 
        (1  r )1       (1  r ) 2                (1  r )t
        1       1      
    cpn                par  PV (annuity)  PV ( par)
         r r (1  r )t  (1  r )t
                       

                                       FIN 819: lecture 4     60
Bond Pricing

Example
  What is the price of a 6 % annual coupon
  bond, with a $1,000 face value, which matures
  in 3 years? Assume a required return of 5.6%.




                    FIN 819: lecture 4    61
Bond Pricing

Example
  What is the price of a 6 % annual coupon
  bond, with a $1,000 face value, which matures
  in 3 years? Assume a required return of 5.6%.
          60       60         1,060
  PV          1
                        2
                           
       (1.056 ) (1.056 )     (1.056 )3
  PV  $1,010 .77


                    FIN 819: lecture 4    62
Bond Pricing

Example (continued)
  What is the price of the bond if the required
  rate of return is 6 %?
             60       60        1,060
      PV         1
                          2
                                      3
           (1.06 ) (1.06 )     (1.06 )
      PV  $1,000



                      FIN 819: lecture 4     63
Bond Pricing

Example (continued)
  What is the price of the bond if the required
  rate of return is 15 %?
             60       60        1,060
      PV         1
                          2
                                      3
           (1.15 ) (1.15 )     (1.15 )
      PV  $794 .51


                      FIN 819: lecture 4     64
Bond Pricing

Example (continued)
  What is the price of the bond if the required
  rate of return is 5.6% AND the coupons are
  paid semi-annually?




                      FIN 819: lecture 4     65
Bond Pricing

Example (continued)
  What is the price of the bond if the required
  rate of return is 5.6% AND the coupons are
  paid semi-annually?
        30       30                 30       1,030
PV          1
                      2
                          ...            
     (1.028 ) (1.028 )                   5
                                 (1.028 ) (1.028 )6
PV  $1,010 .91


                      FIN 819: lecture 4     66
Bond Pricing

Example (continued)
Q: How did the calculation change, given semi-
  annual coupons versus annual coupon
  payments?




                    FIN 819: lecture 4     67
Bond Yields

   Current Yield - Annual coupon
    payments divided by bond price.
   Yield To Maturity (YTM)- Interest rate
    for which the present value of the
    bond’s payments equal the market
    price of the bond.
                 cpn             cpn                   (cpn  par )
          P                                .... 
               (1  y )1       (1  y ) 2                (1  y )t


                                  FIN 819: lecture 4                  68
An example of a bond
   A coupon bond that pays coupon of 10%
    annually, with a face value of $1000, has
    a discount rate of 8% and matures in
    three years. It is assumed that the
    market price of the bond is the present
    value of the bond at the discount rate of
    8%.
    • What is the current yield?
    • What is the yield to maturity.
                        FIN 819: lecture 4   69
My solution
   First, calculate the bond price
   P=100/1.08+100/1.082+1100/1.083
     =$1,051.54
   Current yield=100/1051.54=9.5%
   YTM=8%




                  FIN 819: lecture 4   70
Bond Yields

Calculating Yield to Maturity (YTM=r)
 If you are given the market price of a
 bond (P) and the coupon rate, the yield
 to maturity can be found by solving for r.

           cpn             cpn                    (cpn  par)
    P                                  ....
                 1                  2
         (1  r )        (1  r )                   (1  r )t


                                    FIN 819: lecture 4          71
Bond Yields

Example
  What is the YTM of a 6 % annual coupon
  bond, with a $1,000 face value, which matures
  in 3 years? The market price of the bond is
  $1,010.77
            60       60         1,060
    PV                     
         (1  r ) (1  r )
                 1         2
                               (1  r ) 3
    PV  $1,010.77

                    FIN 819: lecture 4      72
Bond Yields

   In general, there is no simple formula
    that can be used to calculate YTM
    unless for zero coupon bonds
   Calculating YTM by hand can be very
    tedious. We don’t have this kind of
    problems in the quiz or exam
   You may use the trial by errors
    approach get it.

                    FIN 819: lecture 4   73
Bond Yields (3)

     Can you guess which one is the
      solution?
(a)   6.6%
(b)   7.1%
(c)   6.0%
(d)   5.6%
     My solution is (d).

                    FIN 819: lecture 4   74
The rate of return on a bond
                 Coupon income + price change
Rate of return =
                   investment or bond price
                                   profit
         Rate of return =
                            cost of investment

Example: An 8 percent coupon bond has a
price of $110 dollars with maturity of 5 years
  and a face value of $100. Next year, the
  expected bond price will be $105. If you
 hold this bond this year, what is the rate of
                   return?

                            FIN 819: lecture 4   75
My solution
   The expected rate of return for holing the
    bond this year is (8-5)/110=2.73%
    • Price change =105-110=-$5
    • Coupon payment=100*8%=$8
    • Profit=8-5=$3
    • The investment cost or the initial price=$110


                       FIN 819: lecture 4      76
Some new terms
   So far, we consider one discount rate for all the
    cash flows
   In fact, the discount rate for one period cash
    flows can be different from the discount rate for
    two-period cash flows.
   Spot interest rate: the actual interest rate
    available today (t=0)
   Future interest rate: the spot rate in the future
    (t>0)


                       FIN 819: lecture 4      77
Example
   Spot rates (r)
    Investment Horizon                  r
           1                           6%
           2                           6.5%
           3                           7%
          4                            7.2%


                   FIN 819: lecture 4          78
The Yield Curve

Term Structure of Interest Rates: is the
  relationship between the spot rates and
  their maturity dates
Yield Curve - Graph of the term structure.




                  FIN 819: lecture 4   79
The term structure of interest
rates (Yield curve)




              FIN 819: lecture 4   80
Value the bond (revisit)
   If we are given the term structure of interest
    rates, we know the discount rates for cash
    flows in different time periods.
   Then
                   cpn        cpn               (cpn  par )
           PV                         .... 
                 (1  r1 ) (1  r2 )
                          1          2
                                                  (1  rt ) t
   Here r1, r2, …, rt are spot rates for period 1, 2,
    …, t, respectively.


                            FIN 819: lecture 4           81
Question
   Which kind of the yield curve can make
    you use a single discount rate for the
    bond valuation?
   For what kind of bonds, YTM is the same
    as spot rates?




                   FIN 819: lecture 4   82
Example
   Please use the following information to
    value a 10%, four years coupon bond, if
    the spot rates are:
                         Year            Spot rate

                           1               5%

                           2               5.4%

                           3               5.7%

                           4               5.9%

                    FIN 819: lecture 4      83
Solution
   The interest payment is $100 every year.
          100         100              100           (100  1,000)
PV                                            
        (1  .05)1 (1  .054) 2 (1  .057)3           (1  .059) 4
      $1,144.5




                            FIN 819: lecture 4                       84
A problem
   A 6 percent six-year bond yields 12%
    and a 10 percent six year bond yields 8
    percent. Please calculate six-year spot
    rate.




                    FIN 819: lecture 4   85
Forward rate
Forward Rate - The interest rate, fixed
 today, on a loan made in the future at a
 fixed time according to the term structure
 of the interest rates.
The forward rate is implied by the term
 structure of interest rates and doesn’t
 exist in a financial market


                  FIN 819: lecture 4   86
Forward rate
   Another way to look at the forward rate is
    that bonds can be priced in the following
    way:
                      cpn             cpn                (cpn  par )
            PV                               .... 
                    (1  r1)1 (1  r2 ) 2                  (1  rt )t


                    cpn            cpn                      (cpn  par )
           PV                                 .... 
                  (1  r1)1 (1  r1)(1  f 2 )          (1  rt 1)t 1(1  ft )




                                    FIN 819: lecture 4                             87
Forward rate calculation
   One period forward rate can be calculated by
    using the spot rates as follows:
                                             n
                                 (1  rn )
                   1  fn 
                                             n 1
                              (1  rn 1)
   Where fn is the forward rate from period (n-1)
    to period n, rn is the n-period spot rate and rn-1
    is the spot rate for the (n-1)-period spot rate.

                        FIN 819: lecture 4          88
Example
   Using the information for spot rates
    given in the previous example, what is
    the forward rate(f2) from year 1 to year
    2?




                     FIN 819: lecture 4   89
Solution
   Here n=2,
                (1  r2 ) 2 1.0542
       1  f2                     1.058
                 (1  r1)1   1.05
        f 2  5.8%




                         FIN 819: lecture 4   90
What moves the interest rate?
   Nominal interest rate
    •   What is it?
   Real interest rate
    •   What is it?
   Inflation
    •   What is it?
   Can the nominal interest rate be less than
    zero?
   Can the real interest rate be less zero?


                         FIN 819: lecture 4      91
What moves the interest rate?
   Irving Fisher’s theory
    •   Nominal r = (1+Real r)(1+ expected inflation)-1
    •   Real r is theoretically somewhat stable
    •   Change in inflation drives the change in the interest
        rate
   This theory doesn’t work bad in the past 50
    years.
   Why do we care about the movement of the
    interest rate?

                            FIN 819: lecture 4           92
Bond price volatility
   When the interest rate changes, what
    will happen to the bond price?
                  cpn        cpn                  ( cpn  par )
           P                          .... 
                (1  y )1 (1  y ) 2                (1  y )t

   Then what decides the sensitivity of the
    bond price to the interest rate change?



                             FIN 819: lecture 4                   93
Duration
   To capture the sensitivity of the bond
    price to the interest rate change,
    financial economists have defined a
    measure called Duration (or Macaulay
    Duration)
   Duration is a weighted average of the
    maturity for the cash flows of the bond



                    FIN 819: lecture 4   94
Example
A two-year 5% coupon bond with YTM of
 10%.
Maturity           Cash flows
   1                   50
    2                 50+1000




                FIN 819: lecture 4   95
Duration
                                             t
   Mathematically     Duration   iwi
                                            i 1
                            PV (Ci )       t
                       wi           and  wi  1
                              P          i 1

   Volatility (percent) =Duration/(1+YTM)
   Change in bond price = volatility*change in
    interest rates.
   If YTM is small, Change in bond price =
    Duration*change in interest rates.

                       FIN 819: lecture 4          96
Example
   Calculate the Duration, volatility and the
    change of the bond price for 1% change
    in the interest rate for a bond that is 6
    7/8%, 5-year coupon bond with 4.9%
    YTM.




                     FIN 819: lecture 4   97
    Solution

i      Ci      PV(Ci)        wi         __    _i* wi _______
1      68.75    65.54       .060              0.060
2      68.75    62.48       .058              0.115
3      68.75    59.56       .055              0.165
4      68.75    56.78       .052              0.209
5      68.75    841.39      .775              3.875
               1085.74      1.00              4.424


                         FIN 819: lecture 4             98
Solution
   Volatility=4.424/(1.049)=4.22
   Price change=4.22*1%=4.22%




                   FIN 819: lecture 4   99
Example (practice question 8)
   Please use the following information to
    answer the questions in the next slide.
          Year                 Spot rate

           1                       5%

           2                       5.4%

           3                       5.7%

           4                       5.9%

           5                       6.0%

                    FIN 819: lecture 4     100
Questions
   (a) What are the discount factors for
    each date?
   (b) What is the forward rates for each
    period?
   (c) Calculate the PV of 5 percent and 10
    percent five-year note?
   (d) Which note is going to have a higher
    YTM?

                    FIN 819: lecture 4   101
Solution
(a)df1=1/1.05=0.95
    df2=1/1.0542=0.90
(b) f2=1.0542/1.05-1=5.8%
    f3=1.0573/1.0542-1=6.3%
(c) PV(5%)=$959.34
    PV(10%)=$1171.43
(d) 5% note has a higher YTM.

                 FIN 819: lecture 4   102

								
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