Chapter 4
Understanding Interest
Rates
Purposes
In this chapter we introduce the
yield to maturity, the most accurate
measure of interest rates, and
alternative less accurate ways in
which interest rates are quoted.
We also discuss the rate of return
of holding bonds, and distinguish
between the real and nominal
interest rate.
Measuring Interest Rates
According to the method of
repaying the debt and interests, the
debt instruments can be classified
into four types: simple loan, fixed-
payment loan, coupon bond, and
discount bond.
1. Simple Loan
An amount of funds (principal) that
must be repaid to the lender at the
maturity date along with an
additional amount as an interest
payment.
2. Fixed-payment loan
An amount of funds that is to be
repaid by making the same payment
every month, consisting of part of the
principal and interest during the
lending periods.
3. Coupon bond
It pays the owner of the bond a fixed
interest payment (coupon payment)
every year until the maturity date,
when a specified final amount (face
value or par value) is repaid.
A Coupon bond specifies three
elements: issuer, maturity date, and
coupon rate.
4. Discount bond
(zero-coupon bond) a bond bought at
a price below its face value, and the
face value is repaid at the maturity
date.
These four types of instruments
require payments at different times.
So their present values of future
payments will be different too.
How to compare these four types of
instruments in terms of their
interest rates?
Present Value
Present value is based on the notion
that a dollar paid to you in the
future is less valuable to you than a
dollar paid to you today.
The actual values of the payments
paid at different dates in the future
should be compared by their
present values.
Yield to Maturity
Yield to maturity is the interest
rate that equates the present value
of payments received from a debt
instrument with its value today.
Yield to maturity is the most
accurate measure of interest rates.
Simple loan
For simple loan, its simple interest
rate equals the yield to maturity.
Fixed payment loan
Three interesting facts about
the coupon bonds
1. When the coupon bond is priced at its
face value, the yield to maturity equals
the coupon rate.
2. The price of a coupon bond and the
yield to maturity are negatively related.
3. The yield to maturity is greater than
the coupon rate when the bond price is
lower than its face value.
Consol or perpetuity
Consol is a perpetual bond with no
maturity date and no repayment of
principal that makes fixed coupon
payments of $C forever.
P = C / i
i = C / P
Discount bound
P = F / ( 1 + i )
i = ( F – P ) / P
F = face value of the discount bond
P = current price of the discount bond
Current bond prices and
interest rates are negatively
related.
Other Measures of Interest
Rates
The yield to maturity is sometimes
difficult to calculate, two of other
measures that are commonly used:
the current yield and the discount
yield.
The Current Yield
The current yield is an approximation
of the yield to maturity on coupon
bonds
i = C / P
i = current yield
P = price of the coupon bond
C = yearly coupon payment
Features of the current yield
1. 1. When a coupon bond has a long term to When a coupon bond has a long term to
maturity, it is very much like a , it is very much like a consol.
2. The current yield is a rather close 2. The current yield is a rather close
approximation of the yield to maturity for a approximation of the yield to maturity for a
long-term coupon bond. When the time to long-term coupon bond. When the time to
maturity of the coupon bond shortens, the maturity of the coupon bond shortens, the
approximation becomes becomes worse.
3. When the bond price equals the par value of 3. When the bond price equals the par value of
the bond, the yield to maturity is equal to the the bond, the yield to maturity is equal to the
coupon rate. The yield to maturity is also equal coupon rate. The yield to maturity is also equal
to the current the current yield.
4. When the bond price is nearer to the
bond’s par value, the better the current
yield will approximate the yield to
maturity.
5. The current yield is negatively related to
the price of the bond.
6. The current yield moves in the same
direction as the yield to maturity.
Yield on a Discount Basis (
Discount Yield)
The yield on a discount basis is
used to quote the yield to maturity
on Treasury bills.
i = ( F – P ) / F * ( 360 / days to
maturity )
Two peculiarities in the
calculation of discount yield
1. It uses the percentage gain on the face
value of the bill rather than the
percentage gain on the purchase price of
the bill.
2. It puts the yield on an annual basis by
taking the year to be 360 days long rather
than 365 days.
Features of the discount yield
1. Therefore the discount yield understates
the interest rate on bills measured by the
yield to maturity.
2. This understatement becomes more
severe the longer the maturity of the
discount bond
3. The discount yield is negatively related
to the price of the bond.
4. The discount yield and the yield to
maturity always move together.
The Distinction Between
Interest Rates and Returns
The rate of return on a security is defined
as the payments to the owner plus the
change in its value, expressed as a fraction
of its purchase price.
The return on a bond will not necessarily
equal the interest rate on that bond.
RET = ( C + P (t+1) – P (t) ) / P (t)
RET = C / P + (P (t+1) – P (t)) / P (t)
RET = current yield + capital gain
RET = return from holding the
bond from time t to time t + 1
P (t) = price of bond at time t
P (t+1) = price of bond at time t + 1
C = coupon payment
The features of the return of
coupon bonds
1. The only bond whose return equals the
initial yield to maturity is one whose time
to maturity is the same as the holding
period.
2. A rise in interest rates is associated with
a fall in bond prices, resulting in capital
losses on bonds whose terms to maturity
are longer than the holding period.
3. The more distant a bond’s maturity, the
greater the size of the price change
associated with a interest-rate change.
4. The more distant a bond’s maturity, the
lower the rate of return that occurs as a
result of the increase in the interest rate.
5. Even though a bond has a substantial
initial interest rate, its return can turn
out to be negative if interest rates rise.
TABLE 2 One -Year Returns on Different-Maturity 10 percentTABLE 2 One -Year Returns on Different-Maturity 10 percent
Coupon Rate Bonds When Interest Rates RiseCoupon Rate Bonds When Interest Rates Rise
(1)(1) (2)(2) (3)(3) (4)(4) (5)(5) (6)(6) (7)(7) (8)(8)
Years toYears to
MaturityMaturity
When When
Bond IsBond Is
PurchasedPurchased
InitialInitial
Yield toYield to
MaturityMaturity
(%)(%)
InitialInitial
PricePrice
($)($)
Yield toYield to
MaturityMaturity
NextNext
YearYear
(%)(%)
PricePrice
NextNext
Year*Year*
($)($)
InitialInitial
CurrentCurrent
YieldYield
(%)(%)
Rate of Rate of
CapitalCapital
GainGain
(%)(%)
Rate ofRate of
ReturnReturn
(6+7)(6+7)
(%)(%)
3030 1010 10001000 2020 503503 1010
2020 1010 10001000 2020 516516 1010
1010 1010 10001000 2020 597597 1010
55 1010 10001000 2020 741741 1010
22 1010 10001000 2020 917917 1010 ++
11 1010 10001000 2020 10001000 1010 ++
Maturity and Volatility of Bond
Returns: Interest Rate Risk
Prices and returns for long-term bonds
are more volatile than those for shorter
-term bonds.
Interest-rate risk: The riskiness of an
asset’s return that results from interest
rate changes.
The rate of return is equal to the yield
to maturity only when the holding
period and the maturity of the bond
are identical.
Bonds whose term to maturity is
longer than the holding period are
subject to interest-rate risk.
Changes in interest rates lead to
capital gains and losses.
Distinction Between Real and
Nominal Interest Rates
Real interest rate reflects the true
cost of borrowing and the return on
lending.
Fisher equation
i = r + expected rate of inflation
i = nominal interest rate
r = real interest rate
When the real interest rate is low,
there are greater incentives to
borrow and fewer incentives to
lend.
Indexed bonds
The . Treasury issues Treasury
inflation protection security (TIPS),
where their interest and principal
payments are adjusted for changes
in the price level.
The interest rate on these bonds
provides a direct measure of a real
interest rate.
