Chapter 4
Interest Rates and Rates of
Return
Objectives
• Define Present Value
• Calculate Present Value for simple debt
instruments.
• Define Yield to Maturity and calculate YTM
for the same debt instruments.
• Link the relationship between yield-tomaturity and return.
Single Payment
Loans Simple Loan
Multiple Payment
Fixed Payment
Loan
Borrow money today
and make 1 single
Borrow money today and
repayment in T periods make a repayment, C,
every year for T periods
Bonds Discount Bond
Sell a bond today and
pay face value, FACE,
in T periods.
Coupon Bond
Sell a bond today and
make a coupon payment,
C, every year for T
periods and pay the
FACE value in T periods.
Debt & Discount
• Debt represents a promise to pay some
money at 1 or more designated periods of
time in the future.
• As bond market investors or bank
accountants, we will want to value these
income streams from the standpoint of
today.
• The value, today, of a future payment is less
than the nominal value of that payment.
– (If you had the money today you could lend it
at interest).
Time Deposit
• At a bank, you might be offered a fixed,
simple interest rate, (1+i), if you deposit
money in the bank for a number of years T.
• The interest rate is annualized. This means
that the final balance at time t+T is a
multiple of the initial deposit at time t.
Balancet T  (1  i ) Depositt
T
Savings Deposits and Present Value
• Consider if you could put $100K in a bank account
that pays 10% interest for 5 years.
– When we say X% interest rate we mean that i=X/100.
• After 5 years, you have (1.1)5 ×$100K = $161.05K
• How much is a promise to pay $161.05K, 5
years from now worth to you today?
– If you have $100K today, then you can get
$161.05K in 5 years by putting it in the bank.
– We say that $100K is the present value of a
promise to pay $ 161.05K in 5 years.
Present Value
• For any payoff at any future period, you can
conjecture how much you would need to
deposit today at some constant interest rate
(1+i) to generate that payoff.
PAYOFFt T
PV 
T
(1  i)
• Present value says that to calculate the value
today of a promise of a future payment, you
need to apply some discount. Define 1+ i as
the (annual) discount rate.
Definitions
• Present Value – A concept used to evaluate
credit market instruments by placing all
payments in terms of today’s dollars so that they
can be added together. The present value of a
payment is less than its nominal value because
a dollar today can be used to earn interest.
• Discount Factor: The rate used to discount the
value of future payments. Should be equivalent
to the interest that could be earned over the time
until the future payment is made.
Present value of multiple payments
• Many kinds of debt feature multiple
payments at different times.
• Present value of a group of payments is
equal to the sum of the present value of
each payment.
Simple Math Trick:
PV of Stream of Constant Payments
• You receive $C dollars next year, and
every following year until year T. The
present value of your payments would be
given by:
C
C
+
+
2
(1+i)(1+i) (1+i)
C
+....+
3
(1+i)
C
T

1
1 
T
1

i
 

=
C
i



PV
Loans
Single
Multiple Payment
Payment
Simple Loan Fixed Payment Loan
PAYOFFt T
(1  i )T
Bonds
Discount
Bond
FACEt T
(1  i)T
C
C
C
C



....


(1  i ) (1  i ) 2 (1  i )3
(1  i )T

1 
1

 (1  i )T 

C
i
Coupon Bond
C
C
C
FACE


...



(1  i ) (1  i ) 2
(1  i )T (1  i)T

1
1

 (1  i )T
C
i


  FACE
(1  i )T
Which Discount Factor?
• What discount rate should be used to calculate the
present value of a discount or coupon bond? Should
the same interest rate be used to calculate the
present value for all bonds?
• These are two difficult questions. Basically, the
discount rate that should be used to calculate present
value of a future payment is the interest rate that
could be obtained on some alternative asset with
approximately the same level of risk.
• This implies that a different discount factor should be
used for risky assets than for safe assets. In general,
people do not like risk and risky assets should have
higher discount factors. Calculating the exact risk of
debt and thus the appropriate discount factor is part
art and part science.
Yield to Maturity
• In the secondary market, discount and
coupon bonds have a price at which they
are sold.
• One thing that can be determined is the
discount factor that sets the price of the
bond equal to the present value. This rate
is the yield to maturity .
Interpretations of Yield to Maturity
• Sellers of a bond would not sell if they believed the
price was less than the present value. Buyers of a
bond would not sell if they believed the price more
than the present value.
• In a competitive active market, buyers and sellers
would be very close in their assessment of present
value. Thus, the price will be equal to the market’s
assessment of the present value.
• Since YTM is the discount factor that sets price equal
to the present value, we can interpret YTM as the
discount factor that the market uses to discount the
value of the future payments offered by the bond.
Interpreting YTM
• YTM is like the average interest rate that you
would collect if you held the debt for the life of
the instrument.
• A discount bond is like a bank account in
which you make an initial deposit at time t
equal to the bond price and are able to
withdraw the face value at time t + T.
• The interest rate on a bank account that
would allow you to deposit the initial bond
price and make a final withdrawal equal to the
face value would be equal to the yield to
maturity of the discount bond.
Yield to Maturity
• In secondary markets, coupon and discount bonds
are sold. The yield to maturity is the interest rate that
sets the price equal to the present value of the bond.
The maturity date of a bond is the date of its final
pay-off.
• The yield to maturity of a discount bond that matures
in T periods is easy to calculate.
Price
Discout Bond
FACE
FACE
T

 1 i 
T
(1  i )
PRICE
• Note: The price of the bond is inversely related to the
yield to maturity. If you can buy a bond for a very low
price relative to the face value, you can earn a very
high return.
Examples
• HKMA issues a 3 year discount Exchange
Fund note that with a face value of $100.
They sell the note for $70.
1 i 
T
FACE

PRICE
1
3
3
100  1 
    1.12624
70
 .7 
The yield to maturity of the note is 12.62%
Discount Bonds and Price
• IF the ytm stays constant, the price of a
discount bond rises as we get closer to the
maturity date.
• If the yield to maturity stays constant, we
would expect that in one year (when the
maturity of the note is T = 2)
1.12624 
2
100
100
 PRICE 
 $78.84
2
PRICE
1.12624 
• Price may also change because ytm will
change. Say, after 1 year, market offers
$80 for the bond.
1 i 
2
1
100
 1.25 2  1.118
80
• Note: A rise in the price is associated
with a fall in the yield to maturity. Since
the new owner has paid a high price for
a future payment, he is discounting the
future payment by less.
Yield to Maturity of Coupon Bond
• The YTM of a Coupon Bond sets the
price equal to the present value
PRICE 
C  BOND
C
C
C
C
FACE



....


(1  i) (1  i) 2 (1  i)3
(1  i)T (1  i)T
[1 
PRICEC  BOND 
1
]
T
FACE
(1  i )
C
i
(1  i )T
Yield
• Yield to Maturity: Discount Rate that sets
price equal to present value.
• Current Yield: Coupon Payment divided by
the current price of the bond.
• Coupon rate: Yearly Coupon payment
divided by the face value.
Example (5-9-05)
KCRC
Coupon
4.65 A
Maturity YTM
10-Jun-13
4.29%
• The Kowloon-Canton Railroad has a bond with maturity date
of about 8 years.
– The bond pays an annual coupon payment of 4.65 and has a yield
to maturity of 4.29.
1
[1 
]
T
The price is equal to
100
(1.0429)

4.65 
= 102.40
T
.0429
(1.0429)
• The coupon rate is 4.65%.
• The current rate is 100%*4.65/102.40 = 100%* 0.0454. =
4.54%
YTM vs. Coupon Rate vs. Current Yield
• If the price of the bond is equal to the face value, then the yield
to maturity is equal to the coupon rate and the current yield.
FACE 
C
C
C
C
FACE



....


(1  i ) (1  i ) 2 (1  i )3
(1  i )T (1  i )T

1 
1 
T 


1

i

1 


FACE  1 

C
T 
(
1

i
)
i


FACE 
C
C
C
i

i
FACE PRICE
• If the price of the bond is lower than the face value, the yield to
maturity is greater the current yield which is greater than the
coupon rate face value.
C
C
C
C
FACE



....


(1  i ) (1  i ) 2 (1  i ) 3
(1  i )T (1  i )T
PRICE  FACE
PRICE 

1 
1 
T 


1

i

1  
FACE 


PRICE  1 

PRICE


C

T 
T 
(
1

i
)
(
1

i
)
i

 

PRICE 
C
C
C
i

i
PRICE FACE
• If the price of the bond is higher than the face value, the yield to
maturity is less than the current rate which is less than the
coupon rate.
C
C
C
C
FACE



....


(1  i ) (1  i ) 2 (1  i)3
(1  i)T (1  i)T
C
C
C
C
PRICE
PRICE 


 .... 

2
3
T
(1  i ) (1  i ) (1  i)
(1  i)
(1  i)T
PRICE 

1
1 
T
1

i
 


1 
PRICE  1 
C
T 
i
 (1  i ) 
PRICE 
C
C
C
i

i
PRICE FACE



Intuition
• A discount bond is like a bank account in which
you deposit some money and leave it to earn
interest [The price is like the initial deposit, the
yield is like the interest rate, and the face value
is the final balance].
• A coupon bond is like a bank account in which
you initially deposit some money and
periodically make some withdrawals. [The price
is like the initial deposit, the yield is like the
interest rate, the coupon is the withdrawal, and
the face value is the final balance].
• If the amount that you withdraw every year
[the coupon] is exactly equal to the interest
rate [the yield], then the final balance is equal
to the initial deposit [ the price is equal to the
face value].
• If the amount that you withdraw every year is
greater than the interest [the coupon rate is
greater than the yield], the final balance will
be less than the deposit [the price is greater
than the face value].
• If the amount that you withdraw every year is
less than the interest [the coupon rate is less
than the yield], the final balance will be
greater than the deposit [the price is less than
the face value].
• Intuitively, if the coupon paid on a bond
is greater than the markets discount
factor (the yield), bond holders will be
willing to pay more than the face value
of the bond to own it today.
• If the coupon paid is small relative to the
markets discount factor, the bond will be
unattractive at face value and must be
sold at a discount.
Price and Discount Factors
• Typically, bond underwriters negotiate an
interest rate and price with initial buyers of
bonds so that the price is very close to the
face value. Thus, initially, the coupon rate
will be very close to the yield to maturity.
• However, in secondary markets, the price
may fluctuate in later periods. Why?
• Remember, the yield to maturity is the
discount factor used by the market. If the
markets discount factor changes, so will
the price of a bond.
Examples
• A bank issues a bond and a note,each with a face
value of $100. The note is issued with a 2 year
maturity and the bond is issued with a 10 year
maturity. The market has a discount factor such
that a yield to maturity of 10% would set the price
equal to present value. The bond issuer offers a
coupon rate of 10% and sells both instruments for
the face value.
10
10
100


(1.10) (1.10) 2 (1.10) 2
10
10
10
10
100




....


(1.10) (1.10) 2 (1.10) 3
(1.10)10 (1.10)10
100  PRICE NOTE 
100  PRICE BOND
• After 1 year, the first coupon payment is
made. If new bonds issued at that date
also have an interest rate of 10%, the
price of both instruments would remain at
100.
10
100

(1.10) (1.10)
10
10
10
10
100




....


(1.10) (1.10) 2 (1.10) 3
(1.10) 9 (1.10) 9
100  PRICE NOTE 
100  PRICE BOND
Bond Price Changes
• What if the discount factor next year is a 20% interest
rate. This might constitute a reasonable alternative
discount factor. What is the price of the note at this
time? What is the price of the bond at this time?
10
100

 91.67
(1.20) (1.20)
10
10
10
10
100
PRICE 



....



2
3
9
9
(1.20) (1.20) (1.20)
(1.20) (1.20)
PRICE 

1 
1  (1.20)9 
  100  59.69
10  
.20
(1.20)9
• The rise in the interest rate leads to a decline
in bond prices and low ex post returns.
• A given rise in the YTM has a much larger
impact on the price of the bonds with later
maturities because higher interest rates persist
over the longer life of the bond and result on
much larger discount of the final pay-off.
• Fluctuations in the price of bonds due to
changes in interest rates are known as interest
rate risk. If you have to sell the bond before
the maturity date, you can lose money if
interest rates rise.
Return
• Consider the pay-off of the bond to a person who had
bought a 2 year bond and a one year bond at the
initial date for 100, but sold each after the first
coupon payment was made. The gross return on the
two year bond is:
PAYOFF COUPON  PRICE 1
PAYOFF
RETURN 
PRICE
PAYOFF NOTE  COUPON  PRICE 1  10  91.67  101.67
RETURN
NOTE
COUPON  PRICE 1 101.67


 1.016667  1.1
PRICE
100
• The net return on an asset is the gross return
–1 . We see that the note holder received a
return of less than 2%, much less than the
anticipated yield to maturity.
• The bond holder actually lost money and
received a large negative return.
PAYOFF COUPON  PRICE 1
PAYOFF
PRICE
PAYOFF NOTE  COUPON  PRICE 1  10  59.69  69.69
RETURN 
RETURN
NOTE

COUPON  PRICE 1 69.69

 .6969
PRICE
100
• Net Return is -.303
Returns vs. YTM
• YTM is an ex ante calculation of average
returns of the bond over its remaining life.
• Returns are ex post calculation of payoff to
holding bonds for past period.
• If market discount factor/YTM rises, bond
prices will fall. This means that ex post
returns will be lower than original YTM, but
future returns will be higher than original
YTM
Interest Rate Risk
• Bond holders face a risk that after they
purchase a bond, the market interest rate
will change.
• If interest rates/market discount factors
rise, bond prices will fall and they will
suffer very low returns.
• If interest rates rise, they will enjoy high
returns.
YTM and Interest Rate Risk
• The yield is the average interest rate you will
earn on a bond if you buy it and hold it until
maturity. So when you buy a bond you know the
average return.
• However, because you cannot know future
interest rates, you cannot know the exact timing
of the returns.
• If market interest rates rise in one year after you
buy a bond, the return on the bond will be very
low. But in the subsequent years, the returns will
be high.
Interest Risk and the Bondholder
• If you buy a bond and hold it, you do not
care about the timing of the returns.
• But if there is some chance that you may
want to sell the bond next year,
fluctuations in the interest rate will
generate some risk for you.
Long-term Bonds are more volatile
than short-term bonds
Japan Debt Returns - 20th Century
.6
.4
Net Returns
.2
.0
-.2
-.4
-.6
00
10
20
30
40
50
Short Term Bills
60
70
80
90
Long Term Bonds
00
Floating Rate Notes and Interest Rate
Risk
• Since the 1980’s, issuers of long-term debt
instruments have insulated their creditors from
interest rate risk through floating rate notes.
• Floating rate notes allow the coupon payment to
change with the market interest rate.
• When the coupon payment is made in any given
period, the payment is made as a percentage of
the face value called the coupon rate.
• Under a floating rate bond, the coupon rate
changes when the pre-determined interest rate
changes. The coupon rate is expressed as a
pre-determined markup over the benchmark
interest rate.
• Flexible rate international bond coupon rates are
often represented as markups over LIBOR or
HIBOR (London/Hong Kong InterBank Offered
Rate).
• If LIBOR is 5%, and a flexible rate bond has a
LIBOR +5% rate, than the coupon payment will
be 10% of the face value.
• When interest rates go up, and the markets
discount factor goes up, so do the coupon
payments, which reduces the fall in the price
level.
Floating Rates
• Most mortgages in HK are floating rate
loans in which the repayment is pegged to
prime lending rate.
• Most corporate and government bonds are
currently fixed rate bonds.
– International interest rate environment is
thought to be relatively stable.
– Most flexible rate bonds are designed with
banks as customers in mind.
Inflation and Interest Rates
• The Fischer hypothesis suggests that
savers and borrowers base their decisions
on the so called real interest rate.
• Nominal return represents how much
money you will receive after 1 year for
giving up 1 dollar of money today
• Real return represents how many goods
you can buy if you give up the opportunity
to buy 1 good today.
• Nominal Return
• Inflation
• Real Return
$PAYOFF
 1 i
$PRICE
CPI 1
 1   1
CPI
$ PAYOFF
CPI 1
1 i
 1 r 
$ PRICE
1   1
CPI
• Fischer Equation
i  r 
A
Inflation Protected Securities
• Inflation might be especially pernicious.
• If inflation rate goes up, then interest rates rise
and price of bond goes down.
– If you sell the bond, you sell at a low price and earn a
low return.
– If you don’t sell, you get the original pay-off which has
a diminished purchasing power.
• US Treasury and UK Exchequer offer bonds
whose coupons and face value rise with the
consumer price index.
10 Year Yields
5.00
4.00
3.00
2.00
1.00
0.00
Fe
TIPS
10 Year
05
b
Ap
5
-r 0
J
-0
n
u
5
A
-0
g
u
5
Types of Risk
• Default Risk – The risk that a borrower will
fail to make a promised payment of interest or
principal.
• Liquidity Risk – Risk that you cannot find a
secondary market for your bond when you
would like to have cash.
• Interest Rate Risk - The risk that the value of
financial assets and liabilities will fluctuate in
response to changes in market interest rates.
• Inflation Risk – A rise in expected inflation
may impact interest rates. An unexpected rise
in inflation might reduce the real value of the
money that you receive.
Fischer Equation Rough Guide to
HK Interest Rates
Hong Kong Inflation & Nominal Interest Rates
16
12
8
4
0
-4
-8
80
82
84
86
88
CPI_Inflation
90
92
94
96
98
Time_Deposit_Rate
00
Loan Interest Rate: Fixed Payment
Loan
• The interest rate associated with a fixed
payment loan sets the present value of
the payments equal to the amount of
the original loan.
LOAN 
C
C
C
C



....

(1  i) (1  i) 2 (1  i)3
(1  i)T
• It is difficult to solve for the interest rate
on a fixed payment loan if you know the
original loan and each payment. This is
usually done by computer or special
calculator.
Example: Car Loan
• If you know the original loan, it is relatively easy to
calculate the payment level needed to generate any
given interest rate.
• Example: You are a loan officer at a bank. A potential
borrower wants to borrow $100K to buy a car and will
pay back the loan over 10 years. You decide to
charge a 10% interest rate. What will the payment
be?

1 
1 
1  i T 
i

LOAN  C
C 
LOAN
i

1 
1 
T 
 1  i  
C
.1

1 
1 
10 
 1.1 
100000 
10000 10000

 $16274.45
1
.
614
1
2.59
Math Appendix
Simple Math Trick:
PV of Stream of Constant Payments
• What is the present value of a constant
stream of payments? You receive $C dollars
next year, and every following year until year
T. The present value of your payments would
be given by:
C
C
C
C


 .... 

2
3
T
(1  i) (1  i) (1  i)
(1  i)
 1
1
1
1 
C


 .... 
2
3
T 
(1

i
)
(1

i
)
(1

i
)
(1

i
)


• Take the second part of this product and
multiply by
1 

1  1  i 
1   1
1
1
1

1





....

T
 1  i   (1  i ) (1  i) 2 (1  i)3
(1

i
)




 1
1
1
1   1
1
1
1 



....




....



 (1  i ) (1  i ) 2 (1  i )3
 (1  i ) 2 (1  i )3
T 
T
T 1 
(1

i
)
(1

i
)
(1

i
)

 

 1
1 
1 
1 


1

 (1  i ) (1  i )T 1  (1  i )  (1  i )T  




 1
1
1
1
C


 .... 
2
3
T
(1

i
)
(1

i
)
(1

i
)
(1

i
)



1 
1
1 
1 
T 
T
 1  i  
 1  i 
C
C
i

1 
(1  i ) 1 

1

i

 





