1 Chapter 16 The Analysis and Valuation of Bonds Innovative Financial Instruments Dr. A. DeMaskey 2 The Fundamentals of Bond Valuation
The present-value model
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 3 The 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. 4 Computing Bond Yields
Yield Measure Purpose
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 5 Computing 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 6 Calculating Future Bond Prices
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
7 Yield Adjustments for Tax-Exempt Bonds The fully taxable equivalent yield (FTEY) takes into account the bonds tax exemptions
T amount and type of tax exemption
8 Holding 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
9 What Determines Interest Rates?
Inverse relationship with bond prices
Forecasting interest rates
Fundamental determinants of interest rates
i RFR I RP
RFR real risk-free rate of interest
I expected rate of inflation
RP risk premium
10 Loanable Funds Theory
Interest rates are the price for loanable funds
Supply of loanable funds
Demand for loanable funds
11 Fundamental Determinants of Interest Rates
i f (Economic Forces Issue Characteristics)
Effect of economic factors
real growth rate
tightness or ease of capital market
or supply and demand of loanable funds
Impact of bond characteristics
term to maturity
indenture provisions collateral, call feature, sinking fund
foreign bond risk exchange rate risk and country risk
12 Term Structure of Interest Rates
Maturity of a security
Relationship between yields and maturity
13 Types of Yield Curves
Normal or upward sloping yield curve
Inverted or downward sloping yield curve
Horizontal or flat yield curve
Humped yield curve
14 Theories of Term Structure of Interest Rates
Liquidity preference hypothesis
Segmented market hypothesis
15 Expectations 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
16 Liquidity Preference Hypothesis
Preference for short-term rather than long-term securities
Shape of yield curve
17 Segmented Markets Hypothesis
Strong preference for securities of a particular maturity
Preferred habitat, or institutional, or hedging pressure theory
Shape of yield curve
18 Trading 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
19 Yield 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
20 Bond 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).
21 Malkiel 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.
22 Price-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
23 Trading 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
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.
25 Duration Measures
26 The Macaulay Duration
Developed by Frederick R. Macaulay, 1938
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
27 Characteristics 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.
28 Modified 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 29 Modified 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 30 Trading 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.
31 Bond 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.
32 Price-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
33 Desirability 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
34 Modified 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.
35 Determinants 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.
36 Determinants 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 (-)
37 Modified Duration-Convexity Effects
Changes in a bonds price resulting from a change in yield are due to
Bonds modified duration
The relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change.
38 Duration 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.
39 Option-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
40 Convexity of Callable Bond
Noncallable bond has positive convexity
Callable bond has negative convexity
41 Limitations 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.
42 Effective and Empirical 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.
Actual percent change for an asset in response to a change in yield during a specified time period.