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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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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

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.

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.

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

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

 Callable bonds are more advantageous
to issuing companies

INTERNATIONAL FINANCIAL INVESTMENT

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