Title: The Analysis and Valuation of Bonds
1Chapter 16
The Analysis and Valuation of Bonds
Innovative Financial Instruments
Dr. A. DeMaskey
2The Fundamentals of Bond Valuation
Where V0 the current market price of the
bond n the number of years to maturity Ci the
annual coupon payment for bond i i the
prevailing yield to maturity for this bond
issue M the par value of the bond
3The Yield Model
- The expected yield on the bond may be computed
from the current market price V0.
Where i is the discount rate that will discount
the cash flows to equal the current market price
of the bond.
4Computing Bond Yields
Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held to
maturity
Promised yield to call
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.
Realized (holding period) yield
5Computing Bond Yields
- Yield Measure Calculation
Nominal Yield
Current yield
Promised yield to maturity
Coupon Bond
Discount Bond
Promised yield to call
Realized (holding period) yield
6Calculating Future Bond Prices
- Where
- Vf estimated future price of the bond
- Ci annual coupon payment
- n number of years to maturity
- hp holding period of the bond in years
- i expected semiannual rate at the end of the
holding period
7Yield Adjustments for Tax-Exempt Bonds
The fully taxable equivalent yield (FTEY) takes
into account the bonds tax exemptions
- Where
- T amount and type of tax exemption
8Holding Period Yield With Differential
Reinvestment Rates
- Estimate potential reinvestment rates
- Calculate the ending wealth position
- Find the ending value of the coupons
- Find the future price of the bond
- Compute the estimated holding period yield by
equating the initial investment to the ending
wealth position
9What Determines Interest Rates?
- Inverse relationship with bond prices
- Forecasting interest rates
- Fundamental determinants of interest rates
- i RFR I RP
- where
- RFR real risk-free rate of interest
- I expected rate of inflation
- RP risk premium
10Loanable Funds Theory
- Interest rates are the price for loanable funds
- Supply of loanable funds
- Federal Reserve
- Domestic saving
- Foreign saving
- Demand for loanable funds
- Government
- Corporations
- Consumers
11Fundamental Determinants of Interest Rates
- i f (Economic Forces Issue Characteristics)
- Effect of economic factors
- real growth rate
- tightness or ease of capital market
- expected inflation
- or supply and demand of loanable funds
- Impact of bond characteristics
- credit quality
- term to maturity
- indenture provisions collateral, call feature,
sinking fund - foreign bond risk exchange rate risk and country
risk
12Term Structure of Interest Rates
- Maturity of a security
- Relationship between yields and maturity
13Types of Yield Curves
- Normal or upward sloping yield curve
- Inverted or downward sloping yield curve
- Horizontal or flat yield curve
- Humped yield curve
14Theories of Term Structure of Interest Rates
- Expectations hypothesis
- Liquidity preference hypothesis
- Segmented market hypothesis
15Expectations Hypothesis
- Investor expectations of future interest rates
- Long-term interest rates are the geometric
average of current and future 1-year interest
rates - Shape of yield curve
- Rising
- Declining
- Horizontal
- Humped
16Liquidity Preference Hypothesis
- Preference for short-term rather than long-term
securities - Shape of yield curve
- Upward sloping
- Downward sloping
17Segmented Markets Hypothesis
- Strong preference for securities of a particular
maturity - Preferred habitat, or institutional, or hedging
pressure theory - Shape of yield curve
- Upward sloping
- Downward sloping
18Trading Implications of the Term Structure
- The shape of the yield curve alone may contain
information that is useful in predicting interest
rates - A downward sloping yield curve may indicate
strong expectations of falling interest rates
19Yield Spreads
- Difference in promised yields between two issues
or segments of the market - There are four major yield spreads
- Segments government bonds and corporate bonds
- Sectors high and low grade municipal bonds
- Coupons or seasoning within a segment or sector
- Maturities within a given market segment or
sector - Magnitudes and direction of yield spreads can
change over time
20Bond Price Volatility
- Interest rate sensitivity refers to the effect
that yield changes have on the price and rate of
return for different bonds. - A bonds percentage price change is
- The market price of a bond is a function of its
par value (M), coupon (PMT), time to maturity
(N), and prevailing market rate (i).
21Malkiel Bond Theorems
- Bond prices move inversely to bond yields
(interest rates). - For a given change in yields, longer maturity
bonds post larger price changes. Thus, bond
price volatility is directly related to maturity. - Price volatility increases at a diminishing rate
as term to maturity increases. - Price movements resulting from equal absolute
increases or decreases in yield are not
symmetrical. - Higher coupon bonds show smaller percentage price
fluctuations for a given change in yield. Thus,
bond price volatility is inversely related to
coupon.
22Price-Yield Relationships
- The price volatility of a bond for a given change
in yield is affected by - The maturity effect
- The coupon effect
- The yield level effect
23Trading Strategies
- If a decline in interest rates is expected
- Long-maturity bond with low coupons
- 30-year zero-coupon bond
- If an increase in interest rates is expected
- Short-maturity bond with high coupons
- 1-year 8 coupon bond
24Duration
- Price volatility of a bond varies
- inversely with its coupon and
- directly with its term to maturity
- A composite measure, which considers both
variables would be beneficial. - Duration is a measure of the bonds interest rate
sensitivity.
25Duration Measures
- Macaulay Duration
- Modified Duration
- Effective Duration
- Empirical Duration
26The Macaulay Duration
- 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 - i yield to maturity on the bond
27Characteristics of Duration
- The duration of a coupon bond is always less than
its term to maturity because duration gives
weight to these interim payments. - A zero-coupon bonds duration equals its
maturity. - There is an inverse relation between duration and
coupon. - There is a positive relationship between term to
maturity and duration, but duration increases at
a decreasing rate with maturity. - There is an inverse relationship between YTM and
duration. - Sinking funds and call provisions can have a
dramatic effect on a bonds duration.
28Modified Duration
- An adjusted measure of duration can be used to
approximate the price volatility of a bond. - Modified duration is defined as
Where m number of payments a year YTM
nominal YTM
29Modified Duration and Bond Price Volatility
- Bond price movements will vary proportionally
with modified duration for small changes in
yields. - An estimate of the percentage change in bond
prices equals the change in yield times modified
duration.
Where ?P change in price for the bond P
beginning price for the bond Dmod the modified
duration of the bond ?i yield change in basis
points divided by 100
30Trading Strategies Using Modified 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. - Note that the duration of your portfolio is the
market-value-weighted average of the duration of
the individual bonds in the portfolio.
31Bond Convexity
- Modified duration is a linear approximation of
bond price changes for small changes in market
yields. - Price changes are not linear, but a curvilinear
(convex) function.
32Price-Yield Relationship for Bonds
- 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. - Modified duration is the percentage change in
price for a nominal change in yield
33Desirability of Convexity
- The greater the convexity of a bond, the better
its price performance. - Based on the convexity of the price-yield
relationship - As yield increases, the rate at which the price
of the bond declines becomes slower - As yield declines, the rate at which the price of
the bond increases becomes faster
34Modified Duration
- For small interest rate changes, this will give a
good estimate. - For larger changes, it will underestimate price
increases and overestimate price decreases. - This misestimate arises because the modified
duration line is a linear estimate of a
curvilinear relationship.
35Determinants of Convexity
- Convexity is a measure of the curvature of the
price-yield relationship. - Since modified duration is the slope of the curve
at a given yield, convexity indicates changes in
duration. - Thus, convexity is the second derivative of price
with respect to yield (d2P/di2) divided by price. - Specifically, convexity is the percentage change
in dP/di for a given change in yield.
36Determinants of Convexity
- The lower the coupon, the higher the convexity
(-) - The longer the maturity, the higher the convexity
() - The lower the yield to maturity, the higher the
convexity (-)
37Modified Duration-Convexity Effects
- Changes in a bonds price resulting from a change
in yield are due to - Bonds modified duration
- Bonds convexity
- The relative effect of these two factors depends
on the characteristics of the bond (its
convexity) and the size of the yield change.
38Duration and Convexity for Callable Bonds
- The call provision is an example of an embedded
option. - Option-adjusted duration is an estimate of
duration based on the probability that the bond
will be called. - If interest rates gt coupon rate, call is unlikely
- If interest rates lt coupon rate, call is likely
- A callable bond is a combination of a noncallable
bond plus a call option that was sold to the
issuer. - The option has a negative value to the bond
investor. - Thus, any increase in value of the call option
reduces the value of the callable bond.
39Option-Adjusted Duration
- Based on the probability that the issuing firm
will exercise its call option - Duration of the non-callable bond
- Duration of the call option
40Convexity of Callable Bond
- Noncallable bond has positive convexity
- Callable bond has negative convexity
41Limitations of Macaulay and Modified Duration
- Percentage change estimates using modified
duration are good only for small-yield changes. - Difficult to determine the interest-rate
sensitivity of a portfolio of bonds when there is
a change in interest rates and the yield curve
experiences a nonparallel shift. - Callable bond duration depends on market
conditions.
42Effective and Empirical Duration
- Effective Duration
- A direct measure of the interest rate sensitivity
of a bond where it is possible to estimate price
changes for an asset using a valuation model. - Empirical Duration
- Actual percent change for an asset in response to
a change in yield during a specified time period.