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					333FF2 – Bond Pricing & Bond Pricing Theorems   1




  BOND PRICING THEOREMS



1. Bond prices move inversely to
 interest rate changes.


2. The longest the maturity of the
 bond, the more sensitive it is to
 changes in interest rates.


3. The price changes resulting
 from equal absolute increases in
 YTM are not symmetrical.




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4. The lower a bond’s coupon, the
 more sensitive its price will be to
 given changes in interest rates.




These 4 principles were first derived and
proven from the basic bond valuation
equation in:


Burton G. Malkiel, “Expectations, Bond
Prices and the Term Structure of Interest
Rates”, Quarterly Journal of Economics,
1962 (p. 197-218)




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THEOREM 1: Bond prices move
inversely to interest rate changes.


    When y  P
    When y  P




Proof:


C = £20p.a., F = £100, N = 1.5 years,
y = 10% p.a.


Price of the bond = ?




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From bond valuation model:


P = 10/(1+0.05) + 10/(1+0.05)2 +
    110/(1+0.05)3


P = £113.616


Assume that interest rates rise and let y
= 20% p.a.


With higher interest rates, the price of
the bond falls:


P = £100.00




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        THEOREM 2: The longest the
        maturity of the bond, the more
        sensitive it is to changes in interest
        rates.


        Proof:


      Original annual YTM 10% for all bonds


                     Bond A       Bond B              Bond C

Term to maturity      3 yrs         6 yrs              9 yrs

Annual Coupon          £10           £10                £10

Current price         £100          £100               £100




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          Assume that interest rates fall by 2%,
         so that YTM becomes 8%.



                          Bond A       Bond B         Bond C

Term to maturity           3 yrs        6 yrs            9 yrs

Current price            £105.24      £109.38         £112.66
Price sensitivity
                          + £5.24      + £9.38        + £12.66
(in £)




         A 2% fall in YTM causes a higher price
         increase (+ £12.66) for the 9 year bond.




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         Assume that interest rates increase
          by 2%, so that YTM becomes 12%.



                        Bond A      Bond B           Bond C

Term to maturity         3 yrs        6 yrs           9 yrs

Current price           £95.08       £91.62          £89.17

Price sensitivity       - £4.92      - £8.38         - £10.83
(in £)




        A 2% increase in YTM causes a higher
        price fall (- £12.66) for the 9 year bond.




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     THEOREM 3: The price changes
     resulting from equal absolute
     increases   in YTM  are   not
     symmetrical.


     Proof:

                     Bond A       Bond B           Bond C


                              Price changes (in £)


YTM falls by 2% + £5.24           + £9.38          + £12.66


YTM rises by 2% - £4.92            - £8.38         - £10.83




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For any given maturity, a x%
decrease in YTM causes a price rise
that is larger than the price loss
resulting from an equal x% increase
in YTM.




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THEOREM 4: The lower a bond’s
coupon, the more sensitive its price
will be to given changes in interest
rates.


Proof:
 Assume annual YTM 10% for all
  bonds.

             Bond A Bond B Bond C               ZCB

Term to                        3 years
maturity

Annual         £30         £15          £5       £0
Coupon

Current £150.76 £112.69 £87.31                  £74.62
price



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    Assume that interest rates increase by
     2%, so that YTM becomes 12%.



               Bond A Bond B Bond C Z C B

Term to                        3 years
maturity

Annual            £30        £15         £5        £0
Coupon

Current        £144.26 £107.38 £82.79 £70.50
price




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     The change in the price of the bond as a
     percentage of the initial price is:



              Bond A       Bond B       Bond C        Zero
                                                     Coupon
                                                      Bond

Annual          £30           £15          £5          £0
Coupon

% Change      - 4.31%      - 4.71%      - 5.18% - 5.52%
in Price



     Notice that zero coupon bonds are the
     most sensitive to changes in interest
     rates.



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ADDITIONAL FACTORS
AFFECTING BOND VALUE


1.Taxation


2.Marketability


3.Call and Put Option provisions




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1. TAXATION


There is no Capital Gains Tax (CGT) on
most UK bonds.


Most investors pay tax on interest
income (coupon payments).


For the latter reason, investors in high
tax brackets prefer low coupon bonds.


For purely taxation purposes and for
maintaining market share, high coupon
bonds usually offer higher YTM to
compensate investors for tax losses.




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2. MARKETABILITY
(or LIQUIDITY)


Marketability or liquidity of a bond
depends on the ability to be bought or
sold    without    significant   price
concessions.


In general, the larger the size of a bond
issue, the greater its marketability.




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Institutional investors usually require
bond liquidity; liquid bonds tend to
trade at a premium.


The more marketable a bond is, the
lower its yield.


Credit ratings provide a good indication
of a bond’s marketability.




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3. CALL AND                PUT       OPTION
PROVISIONS


Bonds often are issued with an option
provision.


Callable bonds (i.e. bonds with a call
option provision) give issuer the right to
redeem bond prior to maturity at a
specified price.




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Putable bonds (i.e. bonds with a put
option provision) give investor the right
to force issuer to redeem the bond prior
to maturity at a specified price.


Option provisions are exercised when
market interest rates and coupon rates
differ.




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Call provisions are more valuable when
interest rates fall or become more
volatile.


Callable bonds are redeemed when
market interest rates drop below coupon
rates.


This is because the bond price
increases. In this case, the issuer, in
order to avoid paying the appreciation
in the bond’s value, exercises the call
option.




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Callable bonds restrict investors’ capital
gains. Therefore, callable bonds are
expected to offer high yields.


Low coupon bonds with call option
provisions attached are unlikely to be
called.


Callable bonds are more advantageous
to issuing companies.


Callable bonds are usually subject to
low credit ratings.




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Summary


 The 4 bond pricing theorems have as
 follows:


    1. Bond prices move inversely to
      interest rate changes.


    2. The longest the maturity of the
      bond, the more sensitive it is to
      changes in interest rates.


    3. The price changes resulting from
      equal absolute increases in YTM
      are not symmetrical.




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    4. The lower a bond’s coupon, the
      more sensitive its price will be to
      given changes in interest rates.


 Additional factors affecting bond
  value are taxation, marketability and
  option provisions.


 There is no Capital Gains Tax (CGT)
  on most UK bonds. Most investors pay
  tax on interest income (coupon
  payments).


 Marketability or liquidity of a bond
  depends on the ability to be bought or
  sold   without     significant   price
  concessions.

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 The more marketable a bond is, the
  lower its yield.


 Institutional investors usually require
  bond liquidity; liquid bonds tend to
  trade at a premium.


 Option provisions are exercised when
  market interest rates and coupon rates
  differ.


 Callable bonds are more advantageous
  to issuing companies




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