Financial Market Theory I:
Rate of Returns on Bonds
Dr. J. D. Han
King’s College
University of Western Ontario
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1. What are Bonds?
• Definition: Fixed Income Securities consist of
bonds(debts secured with collaterals) and
debentures(unsecured debts).
• Examples:
1) Government Bonds: All Levels
Treasury Bills
Short-term, Medium Term, and Long-term Bonds
2) Corporate Bonds
Commercial Papers
Mortgage Backed Securities
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• Coupon Rate: Nominal Interest Rate
2.Rate of Returns on Bond
The most important concept of the rate of returns on bond is the Yield To
Maturity or YTM:
YTM is, “What is the annual average rate of return you will get if you hold
the bond until it matures?”
It is the ‘one’ ‘average’ ‘constant’ rate of return for this year, and the next
year and so on, until the maturity time:
Thus one YTM of multiple years contains not only
1) the financial market’s evaluation of the individual borrower’s risk
characteristics, but also 2) the current financial market expectations of
some relevant macro variables of the future (period up to the
corresponding maturity time).
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• One corollary is that
when there is a revision of the financial
market’s evaluation of the individual
borrower’s risk, or/ad its expectations of
future macroeconomic variables, there will
be a change in YTM.
4
• The YTM of different terms at any given
time is called the ‘Term Structure’.
• Its graph is called ‘Yield Curve’.
• If we see the changing Term Structure over
time, then we may able to unravel the
market expectations of the future macro
variables.
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So let’s get the YTM
• Rate of returns on Bond come
from Coupons as well as
capital gains or loss.
-“Fixed Income” may be a misnomer
-What you see(Coupon) is not all that you
get”
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*YTM 1:
Annual Payment of Coupons
Face Value FV(100), Market Price MP(% to FV),
Semiannual Coupon Payment C(%= coupon rate
times 100), Maturity periods n (years), and
Annual YTM r (in 0.0x):
C 
1 
MP   1 

r  (1  r )n 
1
 FV 
(1  r )n
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How have we got this formula?
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*YTM 2:
Semiannual Payment of Coupons
Face Value FV(100), Market Price MP(% to
FV), Semiannual Coupon Payment C/2(%),
Maturity periods 2n (half years), and
Annual YTM r(in 0.0x):
1

1  (1  r / 2) 2 n
MP  C / 2  
r/2







1
 FV 
(1  r / 2) 2 n
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Approximation Formula to get
YTM I directly
Coupon Payment  Annual Changes in Price
Average Price
Coupon Payment C(%) = coupon rate c(0.00)
times FV(=100)
Average Price = (FV + MP)/2;
And Annual Changes in Prices = (F-MP) / n
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FV  MP
C
n
FV  MP
2
where FV = 100 at all times
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*Example
• suppose that newspaper on March 1, 2004
Issue
ABC Co.
Coupon Rate
10%
Maturity Date
Bid/Ask
1 March 08/09
92
Yield
?
Yield to Maturity = (10 + 8/4) / 96 x 100 = 11.46%
* ‘/09’ means that the bonds are extendable for a year.
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Practice Question
• Get the YTM of different terms of bonds
n
YTM(%)
issued byC theMP(%
Canadian
government:
to FV)
1 year
-
97
2 years A
4%
100
3 years
5%
97
4 years
5%
94
5 years A
4%
90
5 years B
10%
116
10 years
8%
100
discount
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Discount or Premium?
• Market Price < Face Value: Discount
• Market Price > Face Value: Premium
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For one borrower, there is one
equilibrium market interest rate i.
For a market-determined/given r(YTM), depending on what Coupon
Rate the borrower chooses, there might be discount or premium.
• Discount(MP < FV) will happen when C < r
•
Premium(MP> FV) will happen when C> r :
If he chooses the coupon rate which is higher than his YTM, then MP
will be higher than the face value. So the bond will be sold at
premium.
Note that r is exogenously set for the borrower; and C and MP become
endogenous.
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Recall Approximation Formula for
Yield to Maturity I
FV  MP1
FV  MP2
C1 
C2 
n
n
r
FV  MP1
FV  MP2
2
2
-FV = 100;
-C1 and C2 should be in % terms;
- i= only one YTM determined for the borrower in the
financial market
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Suppose that Borrower A has to pay 5% per annum on 5 year term in the
financial market.
-> This is the market equilibrium interest rate for this borrower or the
YTM for a 5 year-term bond.
Now it offers two different bonds of 5 year-term: Bond A with coupon
rate 3%, and Bond B with the coupon rate =10%.
Equilibrium in the financial market leads to the equalization of YTMs
on these two bonds.
This bond should have –5% on the capital gains so that the total effective
rate of return = 5%.
* Of course, all bonds command differing risk premiums, and thus the
discount and premium vary for each bond.
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Approximation Formula for
Yield to Maturity I
100  MP1
100  MP2
3
10 
5
5
 0.05 
100  MP1
100  MP2
2
2
MP1 should be less than 100 (Discount)
MP2 should be more than 100 (Premium)
*Only when the units of inputs are right, the
formula works.
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We find that
• All other things being equal,
the coupon rate and the market price of bond
are positively related.
However, the YTM does not change unless
there is a change in borrowing terms or
other characteristics.
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Why do a borrower choose different coupon
rates even for the same term?
The answer is to manage the overtime features of cash flows
(in from the funded/invested project and out from the
interest payment)
In the above case, from the viewpoint of the borrower’s cash flows,
• Bond A has smaller interim interest rate/coupon payment compared to
Bond B.
• Bond A is suitable for financing a project which may take more time
for completion and thus the returns from it come later:
• The borrower does not have too high a burden of interest payment
while there is no return from the project.
eg) Raising funds for Highway toll road as opposed to Massive Hydro
dam
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Time Trends of Discount and Premium on Bond:
they decrease as the maturity period come near.
• Market Price
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* Zeros (Zero Coupon Bonds)
• Bonds which are stripped of coupons.
• Zeros are all sold in the market below the
face value:
FV
MP 
;
n
(1  r )
1
FV  n

r
 1
 P 
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Example)
• Different Zero coupon bonds are sold as
follows. Calculate the YTM or effective rate
of return:
(market price quote has been standardized as percentage to the face value)
Maturity
1 year
2 year
3 year
4 year
Market Price
96.62
92.45
87.63
83.06
YTM
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Answers)
• YTM1 = (100/96.62) -1 = 3.5 %
• YTM2= r2= (100/92.45)1/2 -1 = 4.00% because 92.45 = 100/(1+r2)2
Maturity
1 year
2 year
3 year
4 year
Market Price
96.62
92.45
87.63
83.06
YTM
3.5
4.00
4.50
4.75
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