Chapter 12
FIXED-INCOME ANALYSIS
Chapter 12 Questions
• What different bond yields are important to
investors?
• How are the following major yields on bonds
computed: current yield, yield to maturity,
yield to call, and compound realized (horizon)
yield?
• What factors affect the level of bond yields at
a point in time?
• What economic forces cause changes in the
yields on bonds over time?
Chapter 12 Questions
• When yields change, what characteristics of
a bond cause differential price changes for
individual bonds?
• What do we mean by the duration of a bond,
how is it computed, and what factors affect it?
• What is modified duration and what is the
relationship between a bond’s modified
duration and its volatility?
Chapter 12 Questions
• What is the convexity for a bond, what
factors affect it, and what is its effect on
a bond’s volatility?
• Under what conditions is it necessary to
consider both modified duration and
convexity when estimating a bond’s
price volatility?
The Fundamentals of Bond
Valuation
• Like other financial assets,the value of
a bond is the present value of its
expected future cash flows:
Vj = SCFt/(1+k)t
The Fundamentals of Bond
Valuation
• To incorporate the specifics of bonds:
P = S(Ci/2)/(1+Ym/2)t + Pp /(1+Ym/2)2n
• This is the present value model where:
– P is the current market price of the bond
– n is the number of years to maturity
– Ci is the annual coupon payment
– Ym is the yield to maturity of the bond
– Pp is the par value of the bond
Bond Price/Yield
Relationships
• Bond prices change as yields change, and
have the following relationships:
– When yield is below the coupon rate, the bond will
be priced at a premium to par value
– When yield is above the coupon rate, the bond will
be priced at a discount from its par value
– The price-yield relationship is not a straight line,
but rather convex (This is convexity)
• As yields decline, prices increase at an increasing rate
• As yield increase, prices fall at a declining rate
The Yield Model
The yield on the bond may be computed
when we know the market price
n
Where:
1
P   Ct
t
(1  Y )
t i
P = the current market price of the bond
Ct = the cash flow received in period t
Y = the discount rate that will discount the cash flows to
equal the current market price of the bond
Computing Bond Yields
Yield Measure
Purpose
Coupon rate
Measures the coupon rate or the percentage
of par paid out annually as interest
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held
to maturity
Measures expected rate of return for bond held
to first call date
Measures expected rate of return for a bond
likely to be sold prior to maturity. It considers
specified reinvestment assumptions and an
estimated sales price. It can also measure the
actual rate of return on a bond during some past
period of time.
Promised yield to call
Realized (horizon) yield
Current Yield
• Similar to dividend yield for stocks, this
measure is important to income oriented
investors
CY = C/P
• where:
– CY = the current yield on a bond
– C = the annual coupon payment of the bond
– P = the current market price of the bond
Promised Yield to Maturity
• Widely used bond yield figure
• Assumes
– Investor holds bond to maturity
– All the bond’s cash flow is reinvested at the
computed yield to maturity
n
1
P   Ct
t
(1  Ym)
t i
Solve for Y that will equate the
current price to all cash flows
from the bond to maturity, similar
to IRR
Promised Yield to Maturity
• For zero coupon bonds, the only cash
flow is the par value at maturity. This
simplifies the calculation of yield.
P = 1,000/(1+Ym/2)2n
– Where n is the number of years to maturity.
Promised Yield to Call
• When a callable bond is likely to be
called, yield to call is the more
appropriate yield measure than yield to
maturity
– As a rule of thumb, when a callable bond is
selling at a price equal to par value plus
one year of interest, the value should be
based on yield to call
Calculating Promised Yield to
Call
2 nc
Ct / 2
Pc
P

t
2 nc
(1  Yc / 2)
t 1 (1  Yc / 2)
Where:
P = market price of the bond
Ct = annual coupon payment
nc = number of years to first call
Pc = call price of the bond
Realized Yield
• The horizon yield measures yield when
the investor expects to sell the bond (for
a price of Pf in hp time periods) prior to
maturity or call
2 hp
Pf
Ct / 2
P 

t
2 hp
(1  YR / 2)
t 1 (1  YR / 2)
Calculating Future Bond
Prices
• Expected future bond prices are an
important calculation in several
instances:
– When computing horizon yield, we need
an estimated future selling price
– When issues are quoted on a promised
yield, as with municipals
– For portfolio managers who frequently
trade bonds
Calculating Future Bond
Prices
2 n  2 hp
Ci / 2
Pp
Pf  

t
2 n  2 hp
(1  Ym / 2)
t 1 (1  Ym / 2)
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
Ym = expected semiannual rate at the end of the holding
period
Adjusting for Differential
Reinvestment Rates
• The yield calculations implicitly assume
reinvestment of early coupon payments at
the calculated yield
• If expectations are not consistent with this
assumption, we can compound early cash
flows at differential rates over the life of the
bond and then find the yield based on an
“Ending wealth” measure, which is calculated
from the differential rates
Yield Adjustments for TaxExempt Bonds
• In order to compare taxable and taxexempt bonds on an “equal playing
field” for an investor, we calculate the
fully taxable equivalent yield (FTEY) for
tax-free bonds based on their returns
FTEY = Tax-Free Annual Return/(1-T)
• Where T is the investor’s marginal tax
rate
What Determines Interest
Rates?
• Inverse relationship with bond prices
– Changes in interest rates have an impact
on bond portfolios, in particular rising
interest rates
– It is therefore important to learn about what
determines interest rates and to gain some
insight as to forecasting future interest
rates
Forecasting interest rates
• Interest rates are the cost of borrowing
money, or the cost of “loanable funds”
• Factors that affect the supply of loanable
funds (through saving) and the demand for
loanable funds (borrowing) affect interest
rates
– The goal is to monitor these factors, and to
anticipate changes in interest rates and to be wellpositioned to either benefit from the forecast or at
least be protected from adverse changes in rates
Determinants of Interest Rates
• Nominal interest rates (i) can be broken down
into the following components:
i = RFR + I + RP
where:
– RFR = real risk-free rate of interest
– I = expected rate of inflation
– RP = risk premium
• The key is to anticipate changes in any of
these factors
Determinants of Interest Rates
• Alternatively, we can break down interest rate
factors into two groups of effects:
– Effect of economic factors
•
•
•
•
real growth rate
tightness or ease of capital market
expected inflation
supply and demand of loanable funds
– Impact of bond characteristics
•
•
•
•
credit quality
term to maturity
indenture provisions
foreign bond risk (exchange rate risk and country risk)
Determinants of Interest Rates
Term structure of interest rates
– One important source of interest rate variability is
the time to maturity
– The yield curve shows the relationship between
bond yields and time to maturity at a point in time
• Yield curve shapes
–
–
–
–
–
Rising curve (common) when rates are modest
Declining curve when rates are relatively high
Flat curves can happen any time
Humped when high rates are expected to decline
Note: usually relatively flat beyond 15 years
Determinants of Interest Rates
Term Structure Theories (what explains the
changing shape of the yield curve?)
• Expectations hypothesis
– The shape of the yield curve depends on
expected future interest rates and inflation rates
– An upward-sloping curve indicates expectations of
higher rates in the future
– We can use this hypothesis to compute implied
future (forward) interest rates
– Yields of different maturities continually adjusting
to estimates of future interest rates
Determinants of Interest Rates
Term Structure Theories
• Liquidity preference hypothesis
– Indicates that long term rates have to pay a
premium over short term rates because:
• Investors need a premium to compensate for the added
price risk associated with long-term bonds
• Borrowers are willing to pay higher rates on long-term
debt to avoid refinancing risk
– Works well in combination with the expectations
hypothesis to explain the normal upward slope of
the yield curve
Determinants of Interest Rates
Term Structure Theories
• Segmented market hypothesis
– Asserts that different investors, in
particular institutions, have different
maturity needs, so have “preferred
habitats” along the yield curve
– Interest rates in differentiated maturity
markets are determined by unique supply
and demand factors in those markets
Determinants of Interest Rates
• Term Structure and Trading
– Knowledge of the term structure can aid in
bond market trading strategies
• For example, if the yield curve is sharply
downward sloping, rates are likely to fall –
lengthen bond maturities to take the most
advantage of price appreciation as interest
rates fall in the future
Determinants of Interest Rates
Yield Spreads
• Bond investing strategies can focus on
predicting various changing yield spreads,
which exist between:
– Segments: government bonds, agency bonds,
and corporate bonds
– Sectors: prime-grade municipal bonds versus
good-grade municipal bonds, AA utilities versus
BBB utilities
– Different coupons within a segment or sector
– Maturities within a given market segment or sector
Bond Price Volatility
• As interest rates and bond yields change, so
do bond prices (that’s we we’ve been talking
about interest rates!)
• What determines how much a bond’s price
will change as a result of changing yields
(interest rates)?
• Percentage Change = (EPB/BPB) – 1
– EPB = Ending Price of the Bond
– BPB = Beginning Price of the Bond
Determinants of Bond Price
Volatility
Four factors determine a
bond’s price volatility
to changing interest
rates:
1. Par value
2. Coupon
3. Years to maturity
4. Prevailing level of
market interest rate
Determinants of Bond Price
Volatility
Malkiel’s five bond relationships:
1. Bond prices move inversely to bond yields (interest
rates)
2. For a given change in yields, longer maturity bonds post
larger price changes, thus bond price volatility is directly
related to maturity
3. Price volatility increases at a diminishing rate as term to
maturity increases
4. Price movements resulting from equal absolute
increases or decreases in yield are not symmetrical
5. Higher coupon issues show smaller percentage price
fluctuation for a given change in yield, thus bond price
volatility is inversely related to coupon
Determinants of Bond Price
Volatility
• The maturity effect
– The longer the time to maturity, the greater
a bond’s price sensitivity
– Price volatility increases at a decreasing
rate with maturity
• The coupon effect
– The greater the coupon rate, the lower a
bond’s price sensitivity
Determinants of Bond Price
Volatility
• The yield level effect
– For the same change in basis point yield,
there is greater price sensitivity of lower
yield bonds
• Some trading implications
– If our interest rate forecast is for lower
rates, invest in bonds with the greatest
price sensitivity, and do the opposite if we
expect higher interest rates
Determinants of Bond Price
Volatility
• The Duration Measure
– Since price volatility of a bond varies
inversely with its coupon and directly with
its term to maturity, it is necessary to
determine the best combination of these
two variables to achieve your objective
– A composite measure considering both
coupon and maturity would be beneficial,
and that’s what this measure provides
Determinants of Bond Price
Volatility
n
Ct (t )

t
t 1 (1  Ym )
D n

Ct

t
t 1 (1  Ym )
n
 t  PV (C )
t
t 1
Price
Developed by Frederick R. Macaulay,1938
Where:
t = time period in which the coupon or principal payment occurs
Ct = interest or principal payment that occurs in period t
Ym = yield to maturity on the bond
Determinants of Bond Price
Volatility
• Characteristics of Macaulay Duration
– Duration of a bond with coupons is always less
than its term to maturity because duration gives
weight to these interim payments
• A zero-coupon bond’s duration equals its maturity
– There is an inverse relation between duration and
the coupon rate
– A positive relation between term to maturity and
duration, but duration increases at a decreasing
rate with maturity
Determinants of Bond Price
Volatility
• Characteristics of Macaulay Duration
– There is an inverse relation between YTM
and duration
– Sinking funds and call provisions can have
a dramatic effect on a bond’s duration
Duration and Bond Price
Volatility
• An adjusted measure of duration can be
used to approximate the price volatility
of a bond
Macaulay duration
Modified duration 
Ym
Where:
1
m
m = number of payments a year
Ym = nominal YTM
Duration and Bond Price
Volatility
Bond price movements will vary
proportionally with modified duration for
small changes in yields:
P
100   Dmod  Ym
P
Where:
P = change in price for the bond
P = beginning price for the bond
Dmod = the modified duration of the bond
Ym = yield change in basis points divided by 100
Trading Strategies Using
Duration
• Longest-duration security provides the
maximum price variation
– If you expect a decline in interest rates, increase
the average duration of your bond portfolio to
experience maximum price volatility
– If you expect an increase in interest rates, reduce
the average duration to minimize your price
decline
• Duration of a portfolio is the market-valueweighted average of the duration of the
individual bonds in the portfolio
Bond Convexity
• The percentage price change formula using
duration is a linear approximation of bond
price change for small changes in market
yields
P
100   Dmod  Ym
P
• Price changes are not linear, but a curvilinear
(convex) function
Bond Convexity
• The graph of prices relative to yields is not a straight
line, but a curvilinear relationship
– This can be applied to a single bond, a portfolio of bonds, or
any stream of future cash flows
• The convex price-yield relationship will differ among
bonds or other cash flow streams depending on the
coupon and maturity
– The convexity of the price-yield relationship declines slower
as the yield increases
• Modified duration is the percentage change in price
for a nominal change in yield
Bond Convexity
– The convexity is the measure of the
curvature and is the second derivative of
price with resect to yield (d2P/di2)
– Convexity is the percentage change in
dP/di for a given change in yield
2
d P
2
dY
Convexity 
P
Bond Convexity
• Determinants of Convexity
– Inverse relationship between coupon and
convexity
– Direct relationship between maturity and
convexity
– Inverse relationship between yield and
convexity
Modified Duration-Convexity
Effects
• Changes in a bond’s price resulting from a
change in yield are due to:
– Bond’s modified duration
– Bond’s convexity
• Relative effect of these two factors depends
on the characteristics of the bond (its
convexity) and the size of the yield change
• Convexity is desirable
– Greater price appreciation if interest rates fall,
smaller price drop if interest rates rise