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					Debt Investments

1. The best technique to analyze the ability of a corporation to pay its debt service
      is:
A. Traditional ratio analysis
B. A forecast of the firm’s profitability
C. The DuPont model
D. An analysis and projection of discretionary cash flow.
1. D :Ultimately, a firm must service its debt from cash
2. In which of the following cases would estimated changes to portfolio’s value,
         which are based on the portfolio’s effective duration only, be most likely to be
         appropriate?
I. There is a small, parallel shift in the yield curve.
II. On average interest rates change very little, with long-term rates rising 50 basis
         points and short-term rates falling 50 basis points.
III. Interest rates fall substantially in parallel fashion over the next year.
2. A :For III the convexity effect may be important. Neither duration nor convexity deals with nonparallel
shifts (II).

A. I only
B.   II only
C.   III only
D.    I and II
(七)、修正的存續期間與價格存續期間:

    除了 Macaulay 的存續期間用來衡量債券價格的利率彈性之外,還有

 修正的存續期間與價格存續期間

 1.修正的存續期間(Modified Duration, Dmod)

                                                P
  公式: Dm od                 D       mac
                                              P
                               y               y
                            1                
                             m                m


   表示當殖利率微小變動時,債券價格變動的比例;即利率微小變動時,
   債券價格的變動幅度。
 2.價格存續期間(Dollar Duration, Ddol)
                             P
  公式: Ddol                            P
                              y  Dm od
                             
                             m
  由此公式可以計算出殖利率微小變動時,對債券價格「金額」上的影響
  程度。
(八)、存續期間之應用:

  例子:有一張 3 年期的零息債券,每年付息一次,殖利率為 8%,面額

  100 萬。若殖利率下降 10 個基本點,則債券之價格波動比例為何?波動

  的金額為何?




  解答:796041 – 793832=2209.18
  步驟 1:求算 Dmac
                     0 1                  0 2           0  3  PV 1,000 ,000   3
  D   mac
                3
                                      3
                                                                   3

                 PV
                t 1
                             t        PV
                                      t 1
                                                  t                PV
                                                                   t 1
                                                                            t
                           
                 1,000,000   3                       P
                
                        
                  1.08 3 
                                  
                                  3             3     P
                    1,000,000                           y
                                                       
                            1.08     3
                                                       m
                                                          y
                                                     1  
                                                      m
         *由此計算結果亦可得知:零息債券的存續期間等於到期期間。


         步驟 2:求算 Dmod

                        D   mac
                                   
                                         3
                                             2.78
          D   mod
                           y         1.08
                        1  
                         m
         故當殖利率下降 0.1%時,債券價格將上漲的比例=2.78×0.1%=0.278%。
                      y
                     
          P                      0.001
              3     3 
                       m
                                           .00278
           P           y      8% 
                   1        1      
                    m              1 

                                               1,000,000
         步驟 3:原債券價格 P                                          793,832.24
                                             10.08      3




         步驟 4:Ddol=Dmod×P=2.78×$793,832.24=$2,206,853
              0.1%×$2,206,853=$2,206
         故當殖利率下降 0.1%時,債券價格將上漲 2,206 元。
         P
             .00278  P  P  .00278  793842 .00278  2206
          P
         793832  796041P  2209.18
3. A ten-year 8% coupon bond issued by ABC Corp. is selling at 95.00, with a yield
    to maturity of 8.76%. Your bond valuation model indicates that a 50 bp
    decrease in rates will result in an increase in the price from 95.00 to 97.75,
    whereas a 50 bp increase in rates will result in a decrease in the price from 95.00
    to 92.75. What is the price value of a basis point change in this bond?
A. $5.26
B. $1.05
C. $0.05
3. C:Duration = (97.75-92.75)/ (2*95*50 bp) = 5.2632,PVBP = 5.26 * 1bp *95 = 0.05
                                                                                    P
             PB  PS   97.75  92.25   50000
DMO                                           5.2632, Dadj                    P 
          2  PM  Y 2  95  50bp 2  95  50                                    Y
                                                                                     1
            P
                                Y
Dadj     P  95  P  Dadj      P  PVBP  P WHEN Y  1BP
            Y                   1
             1




                                                               C.   $0.06

4. Given the 2-year spot rate of 4% and knowing the market expects a 3-year
    security two years from now to yield 5%. Calculate the current 5-year spot
    rate.
A. 2.3%
B. 4.6%
4. B : (1+S5)5 = (1+S2)2 * (1+ 2f3)3 ,S5 = 4.6%
1.04^2=                        1.0816
1.05^3=                      1.157625
1.04^2*1.05^3=              1.2520872
1.252087^0.2=           1.045988513
1.0459-1=                    0.0459


5. Ken Cambie owns a barbell portfolio consisting of equal investments in two
    bonds: A 5-year bond with a modified duration of 4.0 and a 10-year bond with
    modified duration of 8.0. If the yield curve inverts so that the yields on 5-year
    maturities increase 100 basis points, and the yields on 10-year maturities
    decrease by 100 basis points, the return on Cambie’s portfolio will be:
                  P                                  P
                      P                               P  P  8  100BP,
       D A  4  P      4  100BP, DB  8 
                  Y   P                              Y    P
                   1                                   1
5. C : price change= -WA * DA * Interest rate change A – WB * DB * Interest rate change B
   = -0.5 * 4 * 100 bp – 0.5* 8 * (-100 bp) = 2%
C. +2%
D. –2%

                       P
                           P
DMO  Dadj           P       Dadj  Y
                      Y    P
                        1

        P
D4 :         Dadj  Y  4  (.01)  .04
         P

        P
D8 :         Dadj  Y  8  (.01)  .08
         P
.5D4  .5D8  .02
      95.42029   (4.57971)   -0.045797072

     109.47130     9.47130   0.094713045

0.5*B1+0.5*B2= 2.445798672      0.024458




0.5*C1+0.5*C2 0.024457987 0.5*C1+0.5*C2




       99.5045     (.4955)

      100.5025       .5025




0.5*B1+0.5*B2= . 00350246
6. What is the fair price to pay for a 5.25% coupon, 2-year corporate bond, callable
at 100.25 consistent with the binomial interest rate tree for this corporate issuer shown
below? (Assume annual compounding) 6. a-
   NH = 105.25/ 1.0597 = 99.32
   NL = 105.25/ 1.0489 = 100.34 > 100.25 = call price
   N0 = 0.5* (99.32+5.25)/ 1.054 + 0.5* (100.25+5.25)/ 1.054 = 99.65


                                                100.00 price
                                      NHH       5.25 coupon
                                                5.25%
               NH        5.97%
                                                100.00 price
N0        5.4%                        NHL       5.25 coupon
                                                5.7%
               NL        4.89%
                                                100.00 price
                                      NLL       5.25 coupon
                                                4.67%
A.    99.65
B.    99.69
C.    99.98
D.    100.25

  EXHIBIT 14-10 Finding the One-Year Forward Rates for Year 11Using
   EXHIBIT 14-10 Finding the One-Year Forward Rates for Year Using
               the Two-Year 4% On-the-Run:First Trial
                the Two-Year 4% On-the-Run:First Trial

                                                        V=100
                                                   ● V=100
                                                    ● C=4.00
                                                   NHH C=4.00
                                                    NHH r2,HH=?
                                                          r2,HH=?

                                V=98.582
                          ●       V=98.582
                           ●    C=4.00
                          NH      C=4.00
                           NH   r1,H=5.496%
                                  r1,H=5.496%
       V=99.567
         V=99.567                                       V=100
     ● C=0                                         ● V=100
      ● C=0                                         ● C=4.00
     N r0r=3.500                                   NHL C=4.00
      N 0=3.500                                     NHL r2,HL=?
       %                                                  r2,HL=?
         %
                                V=99.522
                          ●       V=99.522
                           ●    C=4.00
                          NL      C=4.00
                           NL   r1,L=4.500%
                                  r1,L=4.500%

                                                        V=100
                                                   ● V=100
                                                    ● C=4.00
                                                   NLL C=4.00
                                                    NLL r2,LL=?
                                                          r2,LL=?
 3a. The bond’s value two years from now must be
      determined. In our example this is simple. We are
      using a two-year bond, so the bond’s value is its
      maturity value ($100) Plus its final coupon
      payment ($4). Thus, it is $104.
 3b. Calculate the present value of the bond's value
      found in 3a using the higher rate. In our example
      the appropriate discount rate is the one-year
      higher forward rate, 5.496%. The present value is
     $98.582 (= $104/1.05496). This is the value of VH
     that we referred to earlier.
 PVH=(VH+C)/(1+r1,H)=(100+4)/ (1+5.496%)
       =$98.582
 3c. Calculate the present value of the bond’s value
         found in 3a using the lower rate. The discount rate
         used is then the lower one-year forward rate,
         4.5%. The value is $99.522(= $104/1.045) and is
     the value of VL.
 PVL=(VL+C)/(1+r1,L)=(100+4)/(1+4.5%)
       =$99.522
 3d. Add the coupon to VH and VL to get the cash flow
      at NH and NL, respectively. In our example we
      have $102.582 for the higher rate and $103.522
      for the lower rate.
 3e. Calculate the present value of the two values
      using the one-year forward rate using r*. At this
         point in the valuation, r* is the root rate, 3.50%.
EXHIBIT 14-10 Finding the One-Year Forward Rates for Year 1 Using
 EXHIBIT 14-10 Finding the One-Year Forward Rates for Year 1 Using
             the Two-Year 4% On-the-Run:First Trial
               the Two-Year 4% On-the-Run:First Trial

                                               V=100
                                           ●     V=100
                                             ● C=4.00
                                           NHH C=4.00
                                            NHHr2,HH=?
                                                 r2,HH=?
                         V=98.582
                     ●     V=98.582
                      ● C=4.00
                     NH    C=4.00
                      NH r1,H=5.496%
                           r1,H=5.496%
    V=99.567
      V=99.567                                 V=100
 ● C=0                                     ●     V=100
  ● r =3.500
      C=0                                    ● C=4.00
 N 0                                       NHL C=4.00
  N r0=3.500                                NHLr2,HL=?
    %                                            r2,HL=?
      %
                         V=99.522
                     ●     V=99.522
                      ● C=4.00
                     NL    C=4.00
                      NL r1,L=4.500%
                           r1,L=4.500%
                                                V=100
                                            ●     V=100
                                              ● C=4.00
                                            NLL C=4.00
                                             NLLr2,LL=?
                                                  r2,LL=?
 在N點:
             VH  C $102 .582
 PVH                         $99 .113
              1  r*   1.035

       VL  C $103 .522
 PVL                   $100 .021
        1  r*   1.035

Step 4: Calculate the average present value of the
two cash flows in step 3. This is the value we referred to

earlier as



                      1 V H  C  V C
  value at a node =              L      
                      2  1  r*   1  r* 
value at a node = 1/2【($99.113 + $100.021)】= $99.567
 Step 5: Compare the value in step 4 with the bond's
           market value. If the two values are the same,
           the r1 used in this trial is the one we seek.
  This is the one-year forward rate that would then be
  used in the binomial interest-rate tree for the lower
  rate, and the corresponding rate would be for the
  higher rate. If, instead, the value found in step 4 is not
  equal to the market value of the bond, this means that
  the value r1 in this trial is not the one-period forward
  rate that is consistent with:
In this example, when r1 is 4.5%
we get a value of $99.567 in step
4, which is less than the observed
market value of $l00. Therefore,
4.5% is too large and the five
steps must be repeated, trying a
lower value for r1. Let's jump right
to the correct value for r1 in this
example and rework steps l
through 5. This occurs when r1,L is
4.074%; The corresponding
binomial interest-rate tree is
shown in Exhibit 14-11.
  99.113
 100.021
 199.134
  99.567
       EXHIBIT 14-10 Finding the One-Year Forward Rates for Year 11Using
        EXHIBIT 14-10 Finding the One-Year Forward Rates for Year Using
                    the Two-Year 4% On-the-Run:First Trial
                     the Two-Year 4% On-the-Run:First Trial

                                                       V=100
                                                  ● V=100
                                                   ● C=4.00
                                                  NHH C=4.00
                                                   NHH r2,HH=?
                                                        r2,HH=?

                                 V=98.582
                             ● V=98.582
                              ● C=4.00
                             NH C=4.00
                              NH r1,H=5.496%
                                  r1,H=5.496%
             V=99.567
              V=99.567                                 V=100
        ●
        ●    C=0
              C=0                                 ● V=100
                                                   ● C=4.00
        NN   r0=3.500
              r0=3.500                            NHL C=4.00
                                                   NHL r2,HL=?
             %                                          r2,HL=?
              %
                                 V=99.522
                             ● V=99.522
                              ● C=4.00
                             NL C=4.00
                              NL r1,L=4.500%
                                  r1,L=4.500%

                                                       V=100
                                                  ● V=100
                                                   ● C=4.00
                                                  NLL C=4.00
                                                   NLL r2,LL=?
                                                        r2,LL=?


EXHIBIT 14-11 One-Year Forward Rates for Year 1 Using
 EXHIBIT 14-11 One-Year Forward Rates for Year 1 Using
             the Two-Year 4% On-the-Run:Issue
              the Two-Year 4% On-the-Run:Issue

                                                       V=100
                                                        V=100
                                                  ●● C=4.00
                                                  NHH C=4.00
                                                   NHH r2,HH=?
                                                        r2,HH=?

                             V=99.071
                              V=99.071
                         ●● C=4.00
                         NH   C=4.00
                          NH r1,H=4.976%
                              r1,H=4.976%

    V=100
     V=100                                              V=100
                                                         V=100
 ● C=0                                             ●● C=4.00
 ● C=0
 N                                                 NHL C=4.00
                                                    NHL r2,HL=?
  N r0=3.500%
     r0=3.500%                                           r2,HL=?

                             V=99.929
                              V=99.929
                         ●● C=4.00
                         NL   C=4.00
                          NL r1,L=4.074%
                              r1,L=4.074%

                                                        V=100
                                                         V=100
                                                   ●● C=4.00
                                                   NLL   C=4.00
                                                    NLL r2,LL=?
                                                         r2,LL=?
99.071   103.071 99.58551
99.929   103.929 100.4145
           1.035      200
           1.035 100
 Step 1:In this trial we select a value of 4.074% for r     1,L.



 Step 2:The corresponding value for the higher
                 one-year forward rate is 4.976%(=
                 4.074% e2*0.10).
                       r1,H= r1,L*e2= 4.074%*e2*0.1=4.976

 Step 3:The bond’s value one year from now is
             determined as follows:
       3a. The bond’s value two years from now is
             $104, just as in the first trial.
       3b. The present value of the bond's value
             found in 3a for the higher rate, VH, is
             $99.071(= $104/1.04976).
PVH=(VH+C)/(1+r1,H)=(100+4)/(1+4.976%)
      =$99.071
3c. The present value of the bond's value found in 3a
     for the lower rate, VL, is $99.929(= $104/1.04074).
PVL=(VL+C)/(1+r1,L)=(100+4)/(1+4.074%)
   =$99.929
3d. Adding the coupon to VH and VL, we get
     $103.071 as the cash flow for the higher rate and
     $103.929 as the cash flow for the lower rate.
3e. The present value of the two cash flows using the
     one-year forward rate at the node to the left,
     3.5%, gives
在N點:
             VH  C    $103 .071
 PV H                           $99 .586
              1  r*    1.035
             VL  C $103 .929
  PVL                        $100 .414
              1  r*   1.035

Step 4: The average present value is $100, which is
                     1 V  C
   the value at the node  V1 rC 
                       
                            H
                     2  1 r   *
                                    L

                                        *




the value at the node

                             1 VH  C VL  C 
   value at a node =                           
                             2  1  r*   1  r* 




  value at a node = 1/2【($99.586 + $100.414)】= $100

Step 5: Because the average present value is equal
                       to the observed market value of

$100, r1 is 4.074%.
7. Comparing expected yield and risk on agency and non-agency mortgage
    passthroughs, you would expect:
A. Agencies to have the highest yield and risk
B. Agencies to have the lowest yield and risk
C. Agencies to have the lowest yield and highest risk
D. Agencies to have the highest yield and lowest risk


7. b-


8. If you expect interest rates to rise, how would IO and PO prices respond?
     http://www.ginniemae.gov/investors/strips_press.asp?Section=Investors
A. Both will drop
B. Both will rise
C. The PO will drop and the IO would rise or fall
D. The IO will rise and the PO could rise or fall


8. c-
The interest rate rise reduces the PV of the cash flows for the PO and its price drops.
This is also true for the IO, but prepays also slow, so total cash flows increase and the
price could rise or fall.
9. As the economy slips into a recession, it is typical for the yield curve to:
A. Shift upward in a parallel manner
B. Shift downward in a parallel manner
C. Shift downward and flatten or invert
D. Shift downward and steepen
9. d –
As the economy slips into recession, the yield curve usually shift downward because
rates will be cut to spur activity, but long-term rates won’t fall as much as investors
will still need a premium to invest in long-term assets.
10. The advantage of using implied yield volatility to forecast future interest rate
    volatility is:
A. It is calculated from the long-term history of interest rate moves
B. It puts more weight on recent interest rate movement
C. It best captures the market consensus for future interest rate movement
D. It is computed from an equal weighting of each of these items.
10. c-
a and b are historical methods. Since implied volatility is based on the current prices
of bond and options, it should reflect the market’s expectations of future interest rate.
11. As interest rates become more volatile, the price of a putable bond will tend to:
a. Decrease
b. Increase
c. Remain unchanged
d. It could increase or decrease
12. All of the following would tend to affect mortgage prepayment 預付款項 speeds 提前還
   本速度 except:
a. The cost of refinancing 重新貸款 existing mortgages
b. The pace of homes sales
c. The age of the mortgages
d. The call premium paid by the mortgage servicer.
12. d-
及時償付本息風險(Timely Payment of Interest and Principal)、提前還本風險(Prepayment Risk)

     提前還本風險:不論何種房貸轉付證券,投資人都必須承擔提前還本風險。




13. For a new 30-year $10,000,000 passthrough with 6% coupon, calculate the
    expected prepayment 預付款項 amount for the first month. The CPR is 10%
a. 0
b. $59,955
c. $87,313
d. $99,925
13. c-
  PMT = 59,955 ( N =30*12 = 360; I = 6/12 = 0.5; PV = 10,000,000; comp PMT =




  -59,955.05)
14. During the lockout period on a credit card asset-backed securities:
a. The investor receives coupon interest but no principal.
b. The credit card holder may not pay down their principal.
c. The investor may not resell their holding.
d. All of the above are true.
14. a-


15. Internal credit enhancements for an asset-backed security can be provided with a:
I. Third-party guarantee
II. Cash reserve set aside
III. Over collateralization(超額擔保)
IV. Senior subordinate structure
V. Bank line of credit


a. All of the above
b. II, III, IV, V
c. II, III, IV
d. I, V


15. c- I and V are external credit enhancements


16. Which of the following statements is correct?
a. Option-free bonds must be evaluated on both their OAS and zero-volatility spread
b. Asset-backed securities whose prepayments vary with the level of interest rates
    should be evaluated on their Z-spreads
c. Asset-backed securities whose prepayments are affected by interest rates and
    “burnout” should be evaluated with Monte Carlo- based OAS
d. Asset-backed securities whose prepayments are affected by interest rates and
    “burnout” can be evaluated with Monte Carlo or binomial based OAS
16. c- For ABSs which cashflow are interest rate path dependent, Monte Carlo- based
OAS should be used.


Answers for Bonus Questions: Debt Investments
1. d- Ultimately, a firm must service its debt from cash


2. a- For III the convexity effect may be important.       Neither duration nor convexity
deals with nonparallel shifts (II).
3. c-
   Duration = (97.75-92.75)/ (2*95*50 bp) = 5.2632
   PVBP = 5.26 * 1bp *95 = 0.05


4. b-
(1+S5)5 = (1+S2)2 * (1+ 2f3)3
  S5 = 4.6%


5. c-
  price change
  = -WA * DA * Interest rate change A – WB * DB * Interest rate change B
  = -0.5 * 4 * 100 bp – 0.5* 8 * (-100 bp) = 2%


6. a-
  NH = 105.25/ 1.0597 = 99.32
  NL = 105.25/ 1.0489 = 100.34 > 100.25 = call price
  N0 = 0.5* (99.32+5.25)/ 1.054 + 0.5* (100.25+5.25)/ 1.054 = 99.65


7. b-


8. c-
The interest rate rise reduces the PV of the cash flows for the PO and its price drops.
This is also true for the IO, but prepays also slow, so total cash flows increase and the
price could rise or fall.


9. d –
As the economy slips into recession, the yield curve usually shift downward because
rates will be cut to spur activity, but long-term rates won’t fall as much as investors
will still need a premium to invest in long-term assets.


10. c-
a and b are historical methods. Since implied volatility is based on the current prices
of bond and options, it should reflect the market’s expectations of future interest rate.


11. b-
Vputable bond = V option-free bond + Vput option
Interest rate volatility     Vput option                 Vputable bond
12. d-


13. c-
  PMT = 59,955 ( N =30*12 = 360; I = 6/12 = 0.5; PV = 10,000,000; comp PMT =
  -59,955.05)
  Interest portion = 10,000,000*6%/12 = 50,000
  Principal portion = 59,955 – 50,000 = 9,955
  CPR = 10%, SMM = 1- (1-CPR)1/12 = 0.874%
  Expected prepayment = 0.874%*(10,000,000 – 9,955) *0.874% = 87,313


14. a-


15. c- I and V are external credit enhancements


16. c- For ABSs which cashflow are interest rate path dependent, Monte Carlo- based
OAS should be used.
 98.588+5.25=103.838   1.04976 103.838/1.04976=98.91594
 99.732+5.25=104.982   1.04976 104.982/1.04976=100.0057
                                               99.46083
             104.982   1.04074 104.982/1.04074=100.8725
100.689+5.25=105.939   1.04074 105.939/1.04074=101.7920
                                               101.3322



         Application to Valuing an
           Option-Free Bond
                 無選擇權債券評價
To illustrate how to use the binomial interest-rate
tree,

     consider a 5.25% corporate bond that has two years

     remaining to maturity and is option-free.

Exhibit 14-13 shows the various values in the
     discounting process and produces a bond value of

     $102.075.

This clearly demonstrates that the valuation model is
     consistent with the standard valuation model for an

     option-free bond.
EXHIBIT 14-13 Valuing Option-Free Corporate Bond with Three
 EXHIBIT 14-13 Valuing Option-Free Corporate Bond with Three
             Years to Maturity and aaCoupon Rate of 5.25%
              Years to Maturity and Coupon Rate of 5.25%

                                                V=98.588
                                           ● V=98.588
                                            ● C=5.25
                                           NHH C=5.25
                                            NHH r2,HH=6.757%
                                                 r2,HH=6.757%

                         V=99.461
                     ● V=99.461
                      ● C=5.25
                     NH C=5.25
                      NH r1,H=4.976%
                          r1,H=4.976%
    V=102.07
     V=102.07                                   V=99.732
 ● 55 C=0
 ●        C=0                              ● V=99.732
                                            ● C=5.25
 N r0=3.500
  N r0=3.500                               NHL C=5.25
                                            NHL r2,HL=5.532%
    %                                            r2,HL=5.532%
     %
                         V=101.333
                     ● V=101.333
                      ● C=5.25
                     NL C=5.25
                      NL r1,L=4.074%
                          r1,L=4.074%

                                                V=100.689
                                           ● V=100.689
                                            ● C=5.25
                                           NLL C=5.25
                                            NLL r2,LL=4.530%
                                                 r2,LL=4.530%
 The valuation process proceeds in the same fashion as in
 the case of an option-free bond.
 One exception: When the call option may be exercised by
 the issuer, the bond value at a node must be changed to
 reflect the lesser of its value if it is not called.
 For example, consider a 5.25% corporate bond with three
 years remaining to maturity that is callable in one year at
 $100.
 The discounting process is identical to that shown in Exhibit
14-13 except that at two nodes, NL and NLL, the values from
 the recursive valuation formula ($101.3322 at NL and
 $100.689 at NLL) exceed the call price ($100) and therefore
 have been struck out and replaced with $100.
 The value for this callable bond is $101.432.
EXHIBIT 14-14 Valuing a Callable Corporate Bond with Three
              Years to Maturity, a Coupon Rate of 5.25%
              and Callable in One Year at 100
                                                    V=98.588
                                                ●
                                                    C=5.25
                                                NHH
                                                    r2,HH=6.757%

                          V=99.461
                       ●
                          C=5.25
                       NH
                          r1,H=4.976%

   V=101.432                                        V=99.732
 ●                                              ●
   C=0                                              C=5.25
 N                                              NHL
   r0=3.500%                                        r2,HL=5.532%

                          V=100(100.001
                       ●
                          )    C=5.25
                       NL
                          r1,L=4.074%

                                                    V=100(100.689)
                                                ●
                                                    C=5.25
                                                NLL
                                                    r2,LL=4.530%
  Value of a call option = value of a noncallable bond –
  value of a callable bond.


  Value of a noncallable bond ($102.075)
- ) Value of a callable bond ($101.432)
       Value of a call option ($0.643)
 
 
  The value of the noncallable bond is $102.075 and the
 value of the callable bond is $101.432, so the value of
 the call option is $0.643.
          Extension to
    Other Embedded Options
     其他嵌入式選擇權的延伸
Other embedded options, such as put options, caps
   and floors on floating-rate notes.

For example, let's consider a putable bond. Suppose
   that a 5.25% corporate bond with three years
   remaining to maturity is putable in one year at par $100.

The bond values altered at two nodes (N    HHand NLH)
   because the bond values at these two nodes exceed
   $100, the value at which the bond can be put. The value
   of this putable bond is $102.523.
EXHIBIT 14-15 Valuing a Putable Corporate Bond with Three
 EXHIBIT 14-15 Valuing a Putable Corporate Bond with Three
             Years to Maturity, a Coupon Rate of 5.25%
              Years to Maturity, Year at 100
             and Putable in One a Coupon Rate of 5.25%
              and Putable in One Year at 100
                                             V=100(98.588
    價值不足100                               ●      V=100(98.588
     價值不足100                                ●)
                                          NHH )
                                                      C=5.25
                                                        C=5.25
  時                                          r
                                           NHH 2,HH=6.757%
   時                                             r2,HH=6.757%
    以100折現              V=100.261
                                         價值不足100
     以100折現 ●              V=100.261
                        C=5.25            價值不足100
                      ● C=5.25
                     NH r1,H=4.976     時
                      NH r1,H=4.976
                        %               時
                           %             以100折現
    V=102.52
      V=102.52                            以100折現
                                            V=100(99.732
 ● 3      C=0                             ●      V=100(99.732
 N● r0=3.500
      3     C=0                             ●)
                                          NHL
                                                      C=5.25
                                              r        C=5.25
                                                 ) =5.532%
   N% r0=3.500                             NHL 2,HL
                                                 r2,HL=5.532%
      %                 V=101.461
                           V=101.461
                     ● C=5.25
                      ● C=5.25
                     NL r1,L=4.074
                      NL r1,L=4.074
                        %
                           %
                                               V=100.689
                                           ●      V=100.689
                                             ● C=5.25
                                           NLL    C=5.25
                                            NLLr2,LL=4.530%
                                                  r2,LL=4.530%
 Value of a put option = value of a nonputable bond –
 value of a putable bond.

      Value of a noncallable bond
  - ) Value of a callable bond
      Value of a call option
 The value of the putable bond is $102.523 and the
value of the corresponding nonputable bond is
$102.075, the value of the put option is -$0.448.
 The negative sign indicates that the issuer has sold
the option, or equivalently, the investor has
purchased the option.
           EXHIBIT 14-15 Valuing a Putable Corporate Bond with Three
                         Years to Maturity, a Coupon Rate of 5.25%
                         and Putable in One Year at 100

                                                                                             V=100(98.588)
                      v 價值不足 100 時                                                     ●
                                                                                             C=5.25
                                                                                       NHH
                                                                                             r2,HH=6.757%
                                以 100 折現
                                                                                    v 價值不足 100 時
                                                                      V=100.261
                                                                 ●
                                                                      C=5.25
                                                                 NH                    以 100 折現
                                                                      r1,H=4.976%

                        V=102.523^                                                           V=100(99.732)
                  ●                                                                    ●
                        C=0                                                                  C=5.25
                  N                                                                    NHL
                        r0=3.500%                                                            r2,HL=5.532%


                                                                      V=101.461
                                                                 ●
                                                                      C=5.25
                                                                 NL
                                                                      r1,L=4.074%
EXHIBIT 14-13 Valuing Option-Free Corporate Bond with Three
 EXHIBIT 14-13 Valuing Option-Free Corporate Bond with Three
             Years to Maturity and a Coupon Rate of 5.25%
              Years to Maturity and a Coupon Rate of 5.25%

                                           ●
                                               V=98.588
                                                  V=98.588                                   V=100.689
                                             ●C=5.25
                                           NHH
                                               r C=5.25
                                             NHH2,HH=6.757%
                                                                                       ●
                                                  r2,HH=6.757%                               C=5.25
                      ●
                          V=99.461
                             V=99.461
                                                                                       NLL
                          C=5.25
                      NH● C=5.25
                        NH 1,H=4.976%
                          r
                                                                                             r2,LL=4.530%
                             r1,H=4.976%
    V=102.07
                                               V=99.732
 ● 5 V=102.07
          C=0                              ●      V=99.732
 N ●r0=3.500
       5    C=0                              ●C=5.25
                                           NHL
   N% r0=3.500                                 r C=5.25
                                             NHL2,HL=5.532%
                                                  r2,HL=5.532%
       %
                          V=101.333
                      ●      V=101.333
                          C=5.25
                      NL● C=5.25
                        NLr1,L=4.074%
                             r1,L=4.074%
                                               V=100.689
                                           ●      V=100.689
                                               C=5.25
                                           NLL● C=5.25
                                               r
                                             NLL2,LL=4.530%
                                                  r2,LL=4.530%
    EXHIBIT 14-15 Valuing a Putable Corporate Bond with Three
     EXHIBIT 14-15 Valuing a Putable Corporate Bond with Three
                 Years to Maturity, a Coupon Rate of 5.25%
                  Years to Maturity, Year at 100
                 and Putable in One a Coupon Rate of 5.25%
                  and Putable in One Year at 100
                                                          V=100(98.588
                                                      ●      V=100(98.588
                                                        ●)
                                                      NHH )
                                                                  C=5.25
                                                                    C=5.25
                                                          r
                                                       NHH 2,HH=6.757%
                                                             r2,HH=6.757%
                                V=100.261
                                  V=100.261
                             ● C=5.25
                              ● r C=5.25
                             NH 1,H=4.976
                              NH r1,H=4.976
                                %
        V=102.52                  %
          V=102.52                                        V=100(99.732
     ● 3      C=0                                     ●      V=100(99.732
      ● r =3.500
     N 0
          3    C=0                                      ●)
                                                      NHL )
                                                                  C=5.25
                                                                   C=5.25
      N %r0=3.500                                      NHLr2,HL=5.532%
                                                             r2,HL=5.532%
          %                     V=101.461
                                   V=101.461
                             ● C=5.25
                              ● C=5.25
                             NL r1,L=4.074
                              NL r1,L=4.074
                                %
                                   %
                                                           V=100.689
                                                       ●      V=100.689
                                                         ● C=5.25
                                                       NLL    C=5.25
                                                        NLLr2,LL=4.530%
                                                              r2,LL=4.530%




                    105.25       1.04976                     100.2610
                    105.25       1.04976                     100.2610
                                                             100.2610
         98.588+5.25+100=        1.04976       105.25/1.04976=100.261
         98.588+5.25+101=        1.04976       105.25/1.04976=100.261
                                                               100.261
                    105.25       1.04074                  101.1299652 105.25/1.04074=   105
      100.689+5.25=105.939       1.04074                  101.7919942 105.939/1.04074
                   105.939                                101.4609797
100.261+5.25=                                             101.4609797
                   105.511         1.035                     101.9430
                   106.711         1.035                     103.1024
101.461+5.25=
0.5(101.943+103.1024)=                                     ^102.5227

				
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