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Chapter 16

The Analysis and Valuation of Bonds

Innovative Financial Instruments

Dr. A. DeMaskey

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

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.

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

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

Calculating 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

Yield 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

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

What 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

Loanable 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

Fundamental 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

Term Structure of Interest Rates

- Maturity of a security
- Relationship between yields and maturity

Types of Yield Curves

- Normal or upward sloping yield curve
- Inverted or downward sloping yield curve
- Horizontal or flat yield curve
- Humped yield curve

Theories of Term Structure of Interest Rates

- Expectations hypothesis
- Liquidity preference hypothesis
- Segmented market hypothesis

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
- Rising
- Declining
- Horizontal
- Humped

Liquidity Preference Hypothesis

- Preference for short-term rather than long-term

securities - Shape of yield curve
- Upward sloping
- Downward sloping

Segmented 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

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

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

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).

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.

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

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

Duration

- 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.

Duration Measures

- Macaulay Duration
- Modified Duration
- Effective Duration
- Empirical Duration

The 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

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.

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

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

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.

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.

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

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

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.

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.

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 (-)

Modified 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.

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.

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

Convexity of Callable Bond

- Noncallable bond has positive convexity
- Callable bond has negative convexity

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.

Effective 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.

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