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SEMINAR ON BONDS

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					                 SEMINAR ON BONDS
Day 1
• Some Preliminaries
• The Time Value of Money
• Compounding
• Types of Bonds
• Yields & Pricing
• Yield Curves
• Duration (Macauley’s duration , Modified Duration, PVBP)
Day2
• Effective Duration. Callable/ Putable Bonds
• Convexity
• Bond Price Volatility
• Inflation Linked Bonds



                     George Stylianopoulos, Fixed Income     1
              Present Value and Future Value
                   (One Interest period)

• For securities with remaining maturity < 1 year (one cash flow)
                                         FV
                            PV 
                                           days 
                                   1  y       
                                           base 

PV= present value ( i.e. the cash amount we pay to buy the security)
FV = future value ( i.e. the redemption amount of the security including
   coupon payment, if any)
Days= no of days from purchase to maturity
Base = 360, or 365, or actual, depending on the market convention
Y= yield annualized.


                       George Stylianopoulos, Fixed Income                 2
     Present Value and Future Value(…continued)


Example: Investing 100 € now (PV) for one year at a yield of 10% we get
   on maturity 110 € (FV) assuming 365 days and 365 base



   Investing though, today 90 € (PV) for one year and getting on maturity
   100 €,(FV) we realize a yield of ?

           100                     11.11%
               1
            90




                       George Stylianopoulos, Fixed Income                3
                   Present Value and Future Value
                      Multiple interest periods

                     The effect of compounding
Example: We invest 100 € on a 3 years security paying an annual coupon 3%
   once a year each year. What will be the future value of our 100 € investment?
    FV  PV * (1  Y ) n ,
    FV  100 * (1  3%)3  109.2727


This is because each coupon payment we assume we reinvest at the initial
   yield.
The same security paying a semiannual coupon yields us a future value of:
                     y m n
     FV  PV  (1     )
                     m
                     3% 6
     FV  100  (1      )  109.3443
                      2


                                 George Stylianopoulos, Fixed Income               4
                     Interest Calculations

• Periodic vs. Continuous compounding
Periodic Compounding:                                 m n
                                       R
                          FV  PV  1  
                                       m
FV=Future Value (Principle+interest)
PV=Present Value (Principle)
m=interest frequency per year
n=years
Continuous compounding: the limit of periodic compounding with m   
i.e.            FV  PV  e Rn
e=2.71828


                      George Stylianopoulos, Fixed Income          5
                Interest Calculations (…continued)

Converting periodic to continuously compounding interest rate
Let R1= continuously compounded interest rate
    R2= periodic compounded interest rate
                                                              m n
                                             R 
   FV  PV  e R1 n            FV  PV  1  2 
                                             m 
                        m
                R 
  e   R1
            1  2 
                m
                    R                           R1 m 
   R1  m  ln  1  2                R2  m   e  1
                    m                                




                            George Stylianopoulos, Fixed Income      6
     Interest Calculations (examples)
Example 1: The effect of increasing the compounding frequency
Interest Rate : 5%
Compoundi
      ng                 FV at end
                PV
 Frequency                of year 1
     (m)
       1        100        105.00
       2        100        105.06
       4        100        105.09
      12        100        105.12
      52        100        105.12
     365        100        105.13

Example 2: Converting periodic interest rates to continuously compounded




                        Periodic Continuously          Periodic Continuously
  Period    Frequency
                         Rates Compounded               Rates Compounded


Overnight      365       2.10%       2.10%             10.00%      10.00%
  1 week       52        2.10%       2.10%             10.00%      9.99%
 1 month       12        2.12%       2.12%             10.00%      9.96%
 2 months       6        2.14%       2.13%             10.00%      9.92%
 3 months       4        2.18%       2.17%             10.00%      9.88%
 6 months       2        2.22%       2.20%             10.00%      9.76%
12 months       1        2.33%       2.31%             10.00%      9.53%




                         George Stylianopoulos, Fixed Income                   7
                    Definitions and Concepts

 • BOND: a financial obligation for which the issuer promises to pay the
   bondholder a specified stream of future cash flows, including periodic
   interest payments (coupons) and a principal repayment.

                              Default or Credit Risk
BONDHOLDER
   RISK                       Market (interest rates risk)

                               Liquidity Risk




                        George Stylianopoulos, Fixed Income                 8
                          Why buy bonds?

Investors have traditionally held bonds in their portfolios for three
   reasons:
• Income. Most bonds provide their holders with fixed income. On a
   set schedule, annually or semiannually or quarterly the issuer sends the
   bondholder a fixed payment
• Diversification. Although diversification does not ensure against loss,
   an investor can diversify a portfolio across different asset classes that
   perform independently in market cycles to reduce the risk of low, or
   even negative, returns.
• Protection against Economic Slowdown or Deflation.



                        George Stylianopoulos, Fixed Income                9
                                Types of Bonds

•       Bonds are differentiated according to the issuer, and according to the type of their
        cash flows.
•       The major bond markets are:
 1) sovereign bonds (issued by governments)
 2) corporate bonds (issued by corporates)
There are many types of bonds, depending on their cash flows among which are:
1)      Fixed rate bonds (bonds paying periodically fixed coupon)
2)      Floating rate notes (bonds that their coupons are linked to an index, e.g Libor)
3)      Callables: I.e. bonds giving the right to the issuer to buy them back before their
        maturity (e.g. mortgages)
4)      Bonds with Sinking Funds: for example a 7% 10 years corporate bond, paying down
        10% of the principal amount annually beginning in year 3
5)      Zero Coupon bonds: bonds that are just redeemed on maturity without any other
        periodic payment of interest


                              George Stylianopoulos, Fixed Income                          10
                            Market conventions

•   Bonds are quoted in the secondary market as a percentage of their face value (Clean or
    Flat Price) without including the accruals (if any).
Example:
Market quote for Hell Rep 3.10% due 20 April 10 on 7th Oct 2005 with settlement 12th Oct
    2005 was:
101.12 - 101.14 This means that to buy Face value 1 million Euro of this bond we have to
    pay 1,011,400 Euro + 175 days (days from 20 April 2005 to 12 Oct 2005) accrued
    interest (3.10%*175*1,000,000/365) 14863.01 Euro = total 1,026,263.01 Euro
• Dirty or Full Price of the Bond = Clean Price + Accruals.
• Coupon frequency (annual or semiannual), and the days count convention (e.g. 30/360,
    Act/360, Act/Act, etc..) is predetermined by the issuer.
• Most Euro zone government bonds have annual coupons and Actual/Actual year fraction
• US Treasuries, Italian BTPs, U.K. Gilts have semiannual coupons and Actual/Actual
    year fractions.




                             George Stylianopoulos, Fixed Income                        11
                                       Yields

The main types of yields are the following:
• Current Yield
                                                    c
Yc= current yield
                                            Yc 
c = annual coupon payment                           P
P= Clean price
• Simple Yield to maturity (or Japanese Yield):
Ys=simple yield to maturity                                 RP
                                                        c
R=redemption value (most cases 100)                        Tsm
                                                   Ys         365
P=Clean Price
                                                             P
Tsm=days from settlement to maturity
c=annual coupon
• Yield to maturity: is the value of the discount rate in the bond equation that
    equates the present value of all future cash payments of the bond to the current
    market price



                            George Stylianopoulos, Fixed Income                        12
             Yield to Maturity. The Bond Equation

B= dirty price of the bond = Present Value of the bond’s cash flows discounted by the yield
    to maturity
Ci= cash flow of the bond at time ti with 1<i<n
Y=yield to maturity
                                  n
                     B   ci  (1  y ) ti
                              i 1


Analytically. Having a bond paying an annual coupon c for n years and redeems at par (100),
   yielding to maturity y its present value is given by:


                              c        c2      c             c      100
                       B                    3 3    n n 
                            (1  y) (1  y) (1  y)
                                      1    2
                                                          (1  y) (1  y) n



                                  George Stylianopoulos, Fixed Income                     13
Yield to Maturity. The Bond Equation (…continued)

• For any coupon frequency the present value of the bond ( i.e. its dirty
  price is given by the formula:

B= dirty price=PV                                                     c
                                                              n
                                               R
R= redemption value (usually 100)     B               DSC
                                                                    m
                                                                              DSC
                                                 n 1      i 1        i 1
                                           Y          E         Y          E
Y= Yield to maturity                      1                   1  
m= coupon frequency                        m                    m
DSC= days from settlement to next coupon payment
E= no of days between coupon dates




                               George Stylianopoulos, Fixed Income                  14
                              Yields (…continued)

 Example:
 Bond Hell Rep. 3.10% due on 20th April 10. Market price 101.14 as at 7th Oct 2005,
    settlement 12th Oct 2005. Calculate its current yield, simple yield and yield to maturity.

Setlement:   12-Oct-05          Current Yield:         3.065%
Maturity:    20-Apr-10          Simple YTM:            2.816%
Coupon:       3.10%             Yield to Maturity:     2.825%
Price:        101.14

Tsm                1651 days from settlement to maturity
Tsm/365      4.5232877 remaining years to maturity
R-P            -1.14    premium/discount of the bond




                                George Stylianopoulos, Fixed Income                         15
Current Yield, Simple Yield, Yield to Maturity Examples
                     Settlement:                    1-Nov-05
         Maturity:           26-Mar-08 20-Apr-08   21-Jun-08
         Coupon:              8.60%      3.50%      2.90%
       Clean Price:           113.45    101.79      100.32
           Tsm:                  876       901        963
         Tsm/365                2.400    2.468       2.638
           R-P                 -13.45     -1.79      -0.32
      Current Yield:          7.580%    3.438%      2.891%
  Simple (Japanese Yield)     2.641%    2.726%      2.770%
    Yield To Maturity:        2.729%    2.734%      2.769%




                               George Stylianopoulos, Fixed Income   16
Bloomberg Yield Analysis screen for GGB 3.60% JUL 16




              George Stylianopoulos, Fixed Income      17
         Current vs. Japanese and Yield to Maturity

•   Current yield is the quickest check of the yield of a bond but not reliable as it ignores
    any change in the value of the capital invested. Investors today do not rely on current
    yield.

•   Simple Yield to Maturity although more accurate than current yield is still not the ideal
    measure of yield because 1)it assumes constant capital gain (for bonds trading in
    discount) or capital loss (for bonds trading at premium), and 2) it ignores the time value
    of money. It is simple.

•   Yield to Maturity is the most accurate measure of a bond’s yield as it explicitly
    recognizes the importance of points in time at which different cash payments from a
    bond are to be received. Implicit in the definition of the yield to maturity is the
    assumption that the investor will be able to reinvest all coupon payments at at a rate
    equal to yield to maturity at which he bought his bond. This risk is known as the
    reinvestment risk.


                              George Stylianopoulos, Fixed Income                               18
                      ZERO COUPON BONDS


•  Zero coupon bonds are bonds that pay no coupons, or bonds of which their coupons
   have been striped
Cash flows wise are the simplest fixed rate bonds as they have a single cash flow on
   maturity. The bond equation is simplified as there are no coupon payments to:

B=Bond price (dirty=clean)
R=redemption value (usually 100)                        R
Y= yield to maturity                         B              DSM
DSM= days from settlement to maturity             (1  Y )         E


E= base of the year (360,365)




                            George Stylianopoulos, Fixed Income                        19
       ZERO COUPON BONDS (examples)


A. Calculating the price of a zero coupon bond from its yield to maturity
Settlement:     12-Oct-05
Maturity:       20-Apr-10
Yield:             5%                Price: 80.1964
DSM:              1651
Base:             365
Redemption:       100


B. Calculate what is the yield to maturity of the following zero coupon bond
Settlement:    12-Oct-05
Maturity:      12-Oct-10                       Yield: 5.159%
Price:          77.75
DSM:             1826
Base:             365
Redemption:       100




                              George Stylianopoulos, Fixed Income              20
                      The Price:Yield Function


                                  Price vs Yield




        180.00
        150.00
Price




        120.00
         90.00
         60.00
         30.00
                 0%          5%              10%               15%   20%
                                             Yield



                         George Stylianopoulos, Fixed Income               21
                                    Yield Curves
•   A yield curve plots the yields to maturity of a series of bonds of the same quality ( i.e.
    same credit) against their respective terms to maturity.

                      A yield curve can exhibit the following 4 shapes:

•   Normal or positively sloped yield curve: A curve in which short-term interest rates are
    lower than longer term interest rates
•   Inverted yield curve or negatively sloped: A curve in which long -term interest rates are
    lower than that of shorter term interest rates.
•   Flat yield curve: Short-term and long term interest rates are roughly equal.
•   Humped yield curve: A humped yield curve is positively sloped from the short maturity
    sector to the intermediate sector, but negatively sloped from the intermediate to the long
    sector.




                               George Stylianopoulos, Fixed Income                               22
                           Normal Yield Curve


              4
             3.8
             3.6
             3.4
Yields (%)




             3.2
              3
             2.8
             2.6
             2.4
             2.2
              2
                   0   5     10          15          20          25   30   35

                                      Time to maturity

                           George Stylianopoulos, Fixed Income             23
                           Inverted Yield Curve

              8
             7.5
              7
Yields (%)




             6.5
              6
             5.5
              5
             4.5
              4
                   0   5          10            15            20   25   30

                                       Time to maturity

                            George Stylianopoulos, Fixed Income         24
                            Flat Yield Curve

             5

            4.5

             4
Yields(%)




            3.5

             3

            2.5

             2
                  0   5         10            15            20   25   30

                                     Time to Maturity

                          George Stylianopoulos, Fixed Income         25
                           Humped Yield Curve


             4.5


              4
Yields (%)




             3.5


              3


             2.5


              2
                   0   5          10            15            20   25   30

                                       Time to maturity

                            George Stylianopoulos, Fixed Income         26
          Theories explaining the yield curve shape

•   Liquidity preference theory. This theory states that that the yield curve will be upward
    sloping because of the preference of investors for liquidity. Liquidity here is defined as
    the ability to recover the principal of the bond in a reasonably short period of time.

•   Market segmentation theory. This theory views the fixed income market as a series of
    distinct markets, segregated by maturity. Individual investors and issuers are restricted
    to specific maturity sectors. Thus investors and issuers do not have complete maturity
    flexibility.

•   Expectations theory. According to the expectations theory the shape of the yield curve
    reflects the market consensus forecast of future interest rate levels.




                              George Stylianopoulos, Fixed Income                            27
           Summing up theories of the yield curve shape


•   Yield curves tend to exhibit a modestly positive slope over long periods of
    time, reflecting the market participants desire for liquidity. Liquidity
    preference
•   Market segmentation shows up as an influence on yield curve shape,
    particularly over short term horizons (e.g. auction periods) and within specific
    issuer sectors. Imbalances of supply and demand create bumps on the yield
    curve at various maturity points for specific periods of time.
•   Finally there are an adequate number of investors with maturity flexibility (e.g.
    mutual funds) to validate a degree of expectations reflected in the yield curve
    shape. Inflation fears tend to steepen the slope of the yield curve while
    disinflation expectations act to flatten or invert the yield curve.




                           George Stylianopoulos, Fixed Income                     28
             The higher credit quality, the flatter the yield curve


                             German, Italian and Greek Curve

              4.5

             4.25
               4

             3.75

              3.5
Yields (%)




                                                                       Germany (AAA)
             3.25                                                      Italy (AA)
               3                                                       Greece (A)

             2.75

              2.5
             2.25

               2
                    0   5    10         15           20    25     30
                                  Maturity (years)



                            George Stylianopoulos, Fixed Income                        29
Germany (AAA)- Belgium (AA+) – Greece (A)




            George Stylianopoulos, Fixed Income   30
      Determinants of the absolute level of the yield curve


A nominal interest rate can be dissected into three basic components .


Nominal interest rate = real interest rate + inflation premium + risk premium
1.    The real interest rate is the compensation to the investor for deferring consumption
      to a future period. Even if inflation is stable at 0% a risk less investment (e.g US
      Treasury) must offer a positive rate of return.
2.    The inflation premium is intended to preserve the purchasing power of the investor
      over time. This premium reflects an expectation of the future inflation level over the
      lifespan of the investment.
3.    The risk premium protects the investor against all other potential negatives,
      including a) credit or default risk, b) call or early redemption risk, c) liquidity or
      marketability risk, d) risk of unexpected changes in inflation (i.e. the degree of
      unpredictability in assessing future inflation.



                             George Stylianopoulos, Fixed Income                           31
                                   DURATION

•   The weighted average maturity of a bond’s cashflows, where the present values of the
    cash flows serve as weights
•   the term to maturity of the equivalent zero coupon bond
•   the balancing point of a bond’s cash flow stream, where the cash flows are expressed in
    terms of present value




    Cash
    flows

            0           Time in years

                                                Duration




                             George Stylianopoulos, Fixed Income                          32
Calculation of duration of a 10 year bond with coupon 4%
      priced at par to yield 4% on maturity


                                           (4)=(3)/Price
     (1)       (2)             (3)                         (5)= (4)x(1) PV
                                            Weighting
    Years   Cashflow   PV=(2)/(1+4%)^(1)                     Weighted t
                                              Factor
     1        4.00          3.846             0.0385           0.0385
     2        4.00          3.698             0.0370           0.0740
     3        4.00          3.556             0.0356           0.1067
     4        4.00          3.419             0.0342           0.1368
     5        4.00          3.288             0.0329           0.1644
     6        4.00          3.161             0.0316           0.1897
     7        4.00          3.040             0.0304           0.2128
     8        4.00          2.923             0.0292           0.2338
     9        4.00          2.810             0.0281           0.2529
     10      104.00        70.259             0.7026           7.0259
                           100.000            1.0000           8.4353
                          Bond Price                          Duration




                 George Stylianopoulos, Fixed Income                         33
                 Factors influencing Duration

•   Term to maturity
•   Coupon rate
•   Accrued interest
•   Market yield level
•   Sinking fund features
•   Call provisions
•   Passage of time




                        George Stylianopoulos, Fixed Income   34
The influence of each factor on duration
                                                Duration Behavior

    Factor                       Low Duration                 High Duration

Term to maturity                     Shorter                        Longer

 Coupon Rate                          Higher                        Lower

Accrued Interest                      Large                         Small

 Market yield                         Higher                        Lower

 Sinking funds                        Many                           Few

Call provisions                       Many                           Few


                   George Stylianopoulos, Fixed Income                       35
     Factors influencing duration- A rule of thumb

As a rule of thumb:




  Long Maturity, Low Coupon, Low Yield= High Duration




                      George Stylianopoulos, Fixed Income   36
Duration of 7% coupon bonds of various maturities
      Each bond is priced at par YTM: 7%

Term to      Duration                Term to        Duration
Maturity                             Maturity
   1              1                    11             8.82
   2            1.94                   12             9.49
   3            2.83                   13            10.14
   4            3.67                   14            10.78
   5            4.48                   15            11.42
   6            5.26                   16            12.04
   7            6.01                   17            12.65
   8            6.74                   18            13.26
   9            7.45                   19            13.86
  10            8.15                   20            14.46

              George Stylianopoulos, Fixed Income              37
The duration:term to maturity relationship for
           coupon bearing bonds
              15


              10
   Duration




              5


              0
                   1       15              30                40   50
                             Term to maturity (years)


                       George Stylianopoulos, Fixed Income             38
 The duration:term to maturity relationship for zero
                  coupon bonds

           45


           30
Duration




           15


            0
                0   10               20              30        40   50
                               Term to maturity (years)

                         George Stylianopoulos, Fixed Income         39
The durations of 15 year Govt. Bonds for several
     coupon rates. Yield environment 7%

Coupon rate   Duration               Coupon rate    Duration

    0            15.00                       10       9.10
    1            13.34                       13       8.69
    2            12.24                       15       8.49
    3            11.46                       16       8.40
    4            10.87                       17       8.32
    5            10.41                       18       8.25
    6            10.04                       19       8.19
    7             9.75                       20       8.13
              George Stylianopoulos, Fixed Income            40
                The duration:coupon rate relationship


           30
           25
DURATION




           20
           15
           10

            5
            0
                0      5               10              15        20   25
                                   COUPON RATE(%)

                           George Stylianopoulos, Fixed Income         41
  The Duration:accrued interest relationship
  A bond’s duration is inversely related to the amount of
  accrued interest attached to the bond.



Hellenic Republic due 20/5/13, 7.50%
price: 111, settlement 19/5/99, YTM:6.29%, Duration: 8.80
price: 111, settlement: 20/5/99, YTM:6.29%, Duration:9.39




                     George Stylianopoulos, Fixed Income    42
The Duration:market yield relationship

The Durations of 15 year Govt. bonds priced at par in a variety of
                      yield environments

Yield environment (%) Duration Yield environment (%) Duration
        1              14.00                  11          7.98
        2              13.11                  12          7.63
        3              12.30                  13          7.30
        4              11.56                  14          7.00
        5              10.90                  15          6.72
        6              10.29                  16          6.47
        7               9.75                  17          6.23
        8               9.24                  18          6.01
        9               8.79                  19          5.80
        10              8.37                  20          5.61


                    George Stylianopoulos, Fixed Income              43
                    The duration-market yield relationship

           26

           21

           16
Duration




           11

            6

            1

                0          5              10               15        20   25
           -4

                                         Market Yield (%)

                               George Stylianopoulos, Fixed Income        44
       Calculation of the duration of a 7% coupon, 10-year corporate
       bond with a sinking fund paying down10%of the principal
       amount annually, beginning in year 3. The bond is priced at par
       to yield 7% to maturity
 (1)    Redeem           (2)        (3) PV    (4)=(3)/Price     (5)= (1) x (4)
               Coupon
Year      ed          Cashflow        CF         Weight         PV weighted
 1          0       7        7       6.54         0.0654           0.0654
 2          0       7        7       6.11         0.0611           0.1223
 3        10%       7       17      13.88         0.1388           0.4163
 4        10%      6.3     16.3     12.44         0.1244           0.4974
 5        10%      5.6     15.6     11.12         0.1112           0.5561
 6        10%      4.9     14.9      9.93         0.0993           0.5957
 7        10%      4.2     14.2      8.84         0.0884           0.6190
 8        10%      3.5     13.5      7.86         0.0786           0.6286
 9        10%      2.8     12.8      6.96         0.0696           0.6266
 10       30%      2.1     32.1     16.32         0.1632           1.6318
         100%                       100.00        1.0000            5.76

                                   Bond Price                     Duration




                          George Stylianopoulos, Fixed Income                    45
   The relative contributions to the price of a 7% coupon, 10year corporate
  bond with (1) no sinking fund provisions and (2) a 70% sinker. Each bond
                   is priced at par to yield 7% to maturity.



                 Percentage of Bond Price Attributable to:    Bond
  Issue          Coupon Payments           Principal Payment Duration
No Sinker              49.17                      50.83        7.52
70% Sinker              37.8                       62.2        5.76



    Sinking fund provisions lower the duration of a bond by reducing
            the average maturity of the principal repayment.


                        George Stylianopoulos, Fixed Income               46
          The duration:passage of time relationship

•   As time passes a bond’s duration falls
                                                                                         13
    at an increasing rate. The duration
                                                                                         12
    decline is a natural consequence of the                                              11
    progressively smaller set of remaining                                               10
    coupon cash flows and the approaching                                                 9




                                                  Duration
    principal repayment.                                                                  8
                                                                                          7
                                                                                          6
                                                                                          5
                                                                                          4
                                                                                          3
                                                                                          2
                                                                                          1
                                                                                          0
                                                             0   5   10   15   20   25        30

                                                             Remaining Time to Maturity

                             George Stylianopoulos, Fixed Income                              47
                        Modified Duration

• Macaulay’s duration can be used as a measure of a bond’s risk. A
  longer duration implies a higher degree of price sensitivity and
  therefore, greater market risk.
• Macaulay’s duration in order to be more accurate as a measure of bond
  risk requires a modification. This revised version of duration is called
  modified duration and is calculated as follows:

MD=modified duration                         D
                                       MD 
D=duration                                  1 Y
Y=yield to maturity



                       George Stylianopoulos, Fixed Income              48
              Modified Duration (..continued)

• Modified duration calculates the percentage change in a bond price for
  one basis point change in yield.

                                  B
                                       MD  Y
                                   B


• B=dirty price of the bond
• Y=yield to maturity




                       George Stylianopoulos, Fixed Income             49
     Price Value of a Basis Point (PVBP,or PV01)

• Price Value of a Basis point is a measure that shows how much the
  price of the bond (or a portfolio) will change for a shift of 1 b.p. in
  yield . It is simply the Modified Duration times the dirty price of the
  bond
• PV01 is widely used as it shows with good approximation how much
  money a bond or a portfolio of bonds will profit (loose) from a
  favorable (adverse) shifts in yield(s).




                        George Stylianopoulos, Fixed Income                 50
                                  Mathematics of Duration
                     n
               B   ci  (1  y ) ti
                    i 1
                      n

                    t      i    ci  (1  y ) ti   n ci  (1  y ) ti 
               D    i 1
                                               ti                      
                              B                  i 1          B          
               B         n
                                                      B                D
                     t i  ci  (1  y ) ti 1         B 
               y       i 1                          y           (1  y )
                  D                 B                       B
                         MD             B  MD                 MD  y
               (1  y )             y                         B


•   B=dirty price of the bond
•   ci= cash flow of the bond at time ti, 1< i < n
•   y=yield to maturity
•   D= Duration, MD= Modified Duration



                                      George Stylianopoulos, Fixed Income       51
            …continued Mathematics of Duration
                      B
                           MD  y ,
                       B
                          y  1
                           B  PV01 ,
                      PV01   MD  B

•   B=dirty price of the bond,
•   MD=Modified Duration,
•   PV01=Present Value of a basis point.




                            George Stylianopoulos, Fixed Income   52
Duration, Modified Duration, PV01 (example)

Bond Data
Settlement: 15-Oct-05                                a) Acrruals? , Dirty Price?, Yield? , Duration?, M D?, PVBP?
Maturity:    15-Apr-08                               b) PVBP for a 5 million Eur Face Value?
Coupon:       6.00% Annual, 30/360
Clean Price:    105

Accruals:     3.0000                  Dirty Price:                      108.00      Yield:   3.85%

Duration Calculation
                                        Price
   Dates     Cashflows PV=C/(1+y)^n    Weighting     PVWeighted t
                                        factor
 15-Oct-05
 15-Apr-06     6.00       5.888         0.05451         0.02726
 15-Apr-07     6.00       5.669         0.05249          0.0787
 15-Apr-08    106.00     96.446         0.89299          2.2325
                         108.00         1.00000         2.33848
                                                        Duration
Modified Duration        2.25177
PVBP                     2.4319
Face Value:             5,000,000     Market Value:                  5,400,000.00
PVBP                      1,216       Market Value for +1bp=         5,398,784.87
                                      Loss:                            -1,215.13




                                  George Stylianopoulos, Fixed Income                                               53
                         Effective Duration
                            Effective Duration is…
•   For an option-free bond the bond’s modified duration
•   A measure of the average maturity of a bond’s cash flows. For a
    callable bond the average maturity of the cash flows is shortened by
    the possibility of early redemption of the issue.
•   A sophisticated weighted average of the modified durations that a
    callable (putable) bond can have.
•   A simple weighted average of the modified duration of the option-free
    component and the modified duration of the option component
•    For a callable (putable) bond effective duration lies between the
    modified duration to call (put) and the modified duration to maturity. It
    approaches the MD to call(put) as yields fall(rise), and the MD to
    maturity as yields rise(fall).

                         George Stylianopoulos, Fixed Income               54
                              Callable Bonds
•A callable bond is a bond that gives the right to the issuer to buy it back at a
pre-specified price over a predetermined period
•The price of a callable bond is calculated as the price of a an equivalent non
callable bond of similar structure less the value of the call option(s) attached
•The call option value is subtracted because the bondholder implicitly sells the
call to the issuer of the bond.

                                    Noncallable                  Call Option
     Callable Bond Price    =                          -
                                    Bond Price                     Value

•The crossover price is the price at which the Yield to Call and the
yield to Maturity are equal. The yield level is termed crossover yield




                           George Stylianopoulos, Fixed Income                 55
The price yield curves for a callable bond and for two non callable
counterparts: a non callable maturing on the first call date and the a non
callable maturing on the final maturity date
                     M                                           MM=Maturity Date Bond
                                                                  C1C1=Call Date Bond
                                                                  C2C2=Callable Bond


                         Crossover Yield, Crossover Price
  Bond price




                C1



               C2

                                                            C1



                                                             M

                                                            C2




                         Market Yield (%)

                         George Stylianopoulos, Fixed Income                            56
      Assessing the Duration of a Callable Bond

There are 3 ways of assessing a callable bond duration


1.   Calculate the duration to call (DTC) and duration to maturity (DTM).
     Use DTC if the bond’s market price exceeds the crossover price, and
     use DTM if the bond’s market price exceeds the crossover price
2.   Calculate a weighted average duration based on a subjective assessment
     of the probability of call.
3.   Calculate an effective, or option- adjusted duration by using an option
     valuation model.



                       George Stylianopoulos, Fixed Income              57
Factors influencing a callable bond’s duration
A.   Selecting either DTC or DTM
Market price > Crossover price                           Modified DTC
Market price<=Crossover price                        Modified DTM


 Under this (more naive) approach the factors influencing duration
 are the market price and the bonds crossover price




                   George Stylianopoulos, Fixed Income                  58
Factors influencing duration of a callable bond

B. The weighted average duration approach
1.   The modified DTC and the modified DTM that form the lower and upper
     boundaries respectively of the bond’s modified duration.
2.   The current yield environment. A low yield environment increases the
     probability of future calls.
3.   The expected trend in interest rates. A trend to lower rates increases the
     probability of early redemption
4.   The expected variability in interest rates. The greater the variability the more
     probable it is that the bond will be redeemed early.
        DTC                                                     DTM
                                AVGDUR




                     George Stylianopoulos, Fixed Income                      59
 Factors influencing the duration of a callable bond
C. Effective duration approach
1.   Call date(s). An early call date reduces a bond’s effective duration.
2.   Maturity date. A short remaining term to maturity lowers a bond’s effective
     duration.
3.   Call (i.e. strike) price. A low call price shortens effective duration.
4.   Market price. A high market price (i.e. low yield) decreases duration.
5.   Market yield volatility. A high degree of yield volatility reduces duration.




                           George Stylianopoulos, Fixed Income                      60
                    Example of Calculating either DTC or DTM

A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00.
At market price 106.00 the modified duration of a non callable bond maturing in 10 years is: 7.48
At the same price (106.00) the modified duration of a non callable bond with maturity 5 years and redeeming
      at 103.00 is: 4.26




   Yield         Price to   Price to
Environment      Maturity   Call
    5.21%         106.00       105.71

   5.30%          105.32       105.32                       Crossover price and Yield

     6%           100.00       102.24


 Market price (106.00) > than crossover price (105.32) therefore the duration of the callable bond is
 4.26


                                 George Stylianopoulos, Fixed Income                                    61
                  Example of calculating the weighted average duration
  A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00

  mod ified ( AVGDUR )  mod ifiedDTC  Pc   mod ifiedDTM  (1  Pc ) 


    Pc = probability of call             Yield           Bond Price       Probability of
                                      Environment                             Call
    DTC= 4.26
                                            2%             135.93               1
    DTM = 7.48
                                            3%             125.59              0.9
                                            4%             116.22              0.8
                                            5%             107.72              0.7
                                          5.21%            106.00             0.600
                                            6%             100.00              0.4
                                            7%              92.98              0.3
                                            8%              86.58              0.2
                                            9%              80.75              0.1
                                           10%              75.42               0
AVGDUR=4.26X0.60+7.48X0.40=5.55

                                      George Stylianopoulos, Fixed Income                  62
                   Example of calculating the effective duration

    A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00

Effective Duration=(modified duration of a noncallable X Wb)+( modified duration of the call option X Wc)
Wb = the market value weight of the bond component (expressed in decimal form)
Wc = 1-Wb = the market value weight of the call option (expressed in decimal form)
Wb + Wc =1
A non- callable bond with same maturity of the callable trades at 110.00 yielding 4.72% and having a modified
duration of 7.56
The value of the call option is:
Callable bond price = noncallable bond price – call option value
106 = 110 – call option value, therefore call option value = - 4
An option is extremely price sensitive and therefore has a large duration. If the call option has a modified
duration of 50 then:
Effective duration of the callable bond =7.56X110/106+50X(-4/106)=7.56X1.037736-50X1.88679= 5.96




                                   George Stylianopoulos, Fixed Income                                  63
                              Putable Bonds
•A putable bond is a bond that gives the right to the holder to sell it back at a
pre-specified price on a predetermined put date.
•The price of a putable bond is calculated as the price of a an equivalent non
putable bond of similar structure plus the value of the put option attached
•The put option value is added because the bondholder implicitly buys the put
option from the issuer of the bond.

   Putable Bond Price = Option-free bond price + put option value


 •A putable bond is attractive because the holder (not the issuer) has the
 discretion to exercise the put option.
 •In a bull market it behaves like an option – free bond, (with significant
 price gains) whilst in a bear market downside price losses are limited.


                          George Stylianopoulos, Fixed Income                 64
        Comparison of callable and putable bonds
          Feature                             Callable                             Putable

     Option definition           A call option gives its holder the   A put option gives its holder the
                                 right to buy the bond at a pre-      right to sell a bond at a pre-
                                 specified price on a pre-specified   specified price at a pre-specified
                                 date.                                date.
       Option holder                            Issuer                             Investor

     Market availability                     Widespread                        Limited Supply

    Option Strike Price                Typically at a premium                  Typically at par

      Option Exercise               A series of call dates and call           A single put date
                                                prices
    In the money when                 Market price > call price           Market price < put price

    Out the money when                Market price < call price              Market > put price

Yield vs. an option-free bond               Higher yield                         Lower yield


                                George Stylianopoulos, Fixed Income                                   65
         The Limitations of Modified Duration

1.   Instantaneous yield change. Although it is possible to experience
     sizable intraday, yield shifts in turbulent financial markets, yield
     shifts typically occur over time.
2.   Small change in yield. Modified duration is a better approximation
     of the price behavior of the bond for small yield changes (10 bps or
     less)
3.   Parallel shifts in yield. It would be a rarity to find all bond yields
     moving in tandem. Short-term bond yields fluctuate more than long-
     term bond yields. Nonparallel yield shifts often occur.




                       George Stylianopoulos, Fixed Income               66
The Limitations of Modified Duration (example)

                                                   Yld     Duration Predicted Actual Bond
         Bond Data                    YTM(%)                                                Error
                                               Change(bps)    Bond Price         Price
Settlement:       15/10/2005            0.00      -400          132.44          140.00      7.56
Maturity:         15/10/2015            1.00      -300          124.33          128.41      4.08
Coupon:               4%                2.00      -200          116.22          117.97      1.74
Price:               100                3.00      -100          108.11          108.53      0.42
Yield to Maturity   4.00%               3.50       -50          104.06          104.16      0.10
Modified Duration   8.111               3.90       -10          100.81          100.81      0.00
                                        3.99        -1          100.08          100.08      0.00
                                        4.00        0             100             100       0.00
                                        4.01        1            99.92           99.92      0.00
                                        4.10        10           99.19           99.19      0.00
                                        4.50        50           95.94           96.04      0.10
                                        5.00      100            91.89           92.28      0.39
                                        6.00      200            83.78           85.28      1.50
                                        7.00      300            75.67           78.93      3.26
                                        8.00      400            67.56           73.16      5.60




                               George Stylianopoulos, Fixed Income                                  67
                          CONVEXITY

        210.00

        180.00
                                      Positive Convexity
        150.00                              Region
Price




        120.00

         90.00

         60.00

         30.00                                               Modified Duration
          0.00
                 0%         5%                         10%              15%

                                        Yield


                      George Stylianopoulos, Fixed Income                  68
                   Definitions of Convexity

• Convexity is the second derivative of the price:yield function and it
  shows the rate of change of modified duration as yields shift
• Convexity can be defined as the difference between the actual bond
  price and the bond price predicted by the modified duration line.
• The term convexity arises from the fact that price:yield curve is convex
  to the origin of the graph. This curvature creates the convexity effect
• Convexity enhances a bond’s price performance in both bull and bear
  markets but not in a uniform manner.
• The larger the changes in yield the greater the convexity effect.
• A decline in yields creates stronger convexity impacts than does an
  equivalent rise in yields.


                       George Stylianopoulos, Fixed Income              69
             Convexity and the Price:Yield function

The nonlinear price:yield function can be analyzed as a Taylor series of
   derivatives

       dP         1 d P                1 d 3P                       1 d nP           
                           2
 P      ( Y )          ( Y ) 2         ( Y ) 3               ( Y ) n 
       dY          2! dY
                             2                    3                              n
                                          3! dY                        n! dY            
               dP
 mduration        dY
                  P
                2
              d P
 convexity         dY 2
                  P
 P  dP 1             1 d P 1              2   1 d 3P 1             3           1 d nP 1            
                                2
           ( Y )             ( Y )                ( Y )                  ( Y ) n 
       dY P            2! dY
                                  2                         3                                 n
  P                                   P          3! dY        P                    n! dY     P         
 P
     mdurationY    convexityY   residuals
                         1                2

  P                      2




                                    George Stylianopoulos, Fixed Income                                        70
               Factors influencing Convexity

1.   Duration. Convexity is positively related to the duration of the
     underlying bond. It is also an increasing function of duration

2.   Cash flow distribution. Convexity is positively related to the degree
     of dispersion in a bond’s cash flows.

3.   Market yield volatility. High Volatility in interest rates creates large
     convexity effects.

4.   Direction of yield change. Convexity is more positively influenced
     by a downward movement in yields than by an upward surge in
     yields
                        George Stylianopoulos, Fixed Income                 71
                   The Convexity:Duration relationship

Price,yield curves for 3,10, 30 years bonds with a coupon 7% and price at par to yield 7%

                   350

                   300

                   250
                                 P30, 30 year bond                              P3
                   200
           Price




                                                                                P10
                   150               P10, 10 year bond                          P30
                                                              P3, 3year bond
                   100

                   50

                    0
                         0   5       10       15         20       25       30

                                    Market Yield(%)


                                 George Stylianopoulos, Fixed Income                        72
Convexity:cash flow distribution relationship
The price:yield curves of a 30year Bond with a coupon 7% priced
at par to yield 7% and a duration of 13.25 and a zero coupon bond
of same duration(I.e. 13.25 years to maturity) at 7% yield


        350

        300

        250       P30,30years coupon bond
        200
Price




                                                                    P0
        150                                                         P 30
        100                           P0,zero coupon bond
        50

         0
              0     5         10         15         20        25

                         MarketYield (%)




                        George Stylianopoulos, Fixed Income                73
              The convexity:yield volatility relationship

        High Volatility in interest rates creates large convexity effects

             2.50



             2.00
Volatility




             1.50



             1.00



             0.50



             0.00
                0.00   20.00      40.00       60.00       80.00   100.00

                                    Convexity




                         George Stylianopoulos, Fixed Income                74
        The Convexity:direction of yield change relationship

•   Convexity effects are greater in declining yield environment than in rising yield
    environment.


                                                 35

                                                 30

                                                 25
       Convexity




                                                 20

                                                 15

                                                 10

                                                  5

                                                  0
                   -400   -300   -200     -100        0      100       200   300        400

                                        Yield Change (in bps)

                                 George Stylianopoulos, Fixed Income                          75

				
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