Chapter 18

Chapter 18

FIXED-INCOME ANALYSIS

Chapter 18 Questions

How is the value of a bond determined, based on the present value formula?What alternative 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?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?


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 = CFt/(1+k)t

The Fundamentals of Bond Valuation

To incorporate the specifics of bonds:

Pm = (Ci/2)/(1+Ym/2)t + Pp /(1+Ym/2)2n This is the present value model where:Pm is the current market price of the bondn is the number of years to maturityCi is the annual coupon payment Ym is the yield to maturity of the bondPp 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

t

n

itt Y

CP)1(

1

Where:

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 YieldsYield Measure PurposeCoupon 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

Promised yield to call Measures expected rate of return for bond held to first call date

Realized (horizon) yield 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.

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 maturityAll the bond’s cash flow is reinvested at the

computed yield to maturity

tm

n

itt Y

CP)1(

1

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 maturityAs 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

Where:

P = market price of the bond

Ct = annual coupon payment

nc = number of years to first call

Pc = call price of the bond

ncc

cnc

tt

c

t

Y

P

Y

CP

2

2

1 )2/1()2/1(

2/

Realized Yield

hpR

fhp

tt

R

tm Y

P

Y

CP

2

2

1 )2/1()2/1(

2/

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

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 priceWhen issues are quoted on a promised

yield, as with municipalsFor portfolio managers who frequently

trade bonds

Calculating Future Bond Prices

Where:

Pf = estimated future price of the bond

Ci = annual coupon paymentn = number of years to maturityhp = holding period of the bond in years

Ym = expected semiannual rate at the end of the holding period

hpnm

hpn

tt

m

i

f YY

CP

22

p22

1 )2/1(

P

)2/1(

2/

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 Tax-Exempt Bonds

In order to compare taxable and tax-exempt bonds on an “equal playing field” for an investor, we calculate the fully taxable equivalent yield (FTEY) for tax-free bonds based on their returnsFTEY = Tax-Free Annual Return/(1-T)Where T is the investor’s marginal tax rate

What Determines Interest Rates?

Inverse relationship with bond pricesChanges 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 well-positioned 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 + RPwhere: 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


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)


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


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


Term Structure TheoriesLiquidity 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


Term Structure TheoriesSegmented market hypothesisAsserts 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


Term Structure and TradingKnowledge of the term structure can aid in

bond market trading strategiesFor 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


Yield SpreadsBond 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


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


The maturity effectThe longer the time to maturity, the greater

a bond’s price sensitivityPrice volatility increases at a decreasing

rate with maturity

The coupon effectThe greater the coupon rate, the lower a

bond’s price sensitivity


The yield level effectFor 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


The Duration MeasureSince 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


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

Price

)(

)1(

)1()(

1

1

1

n

tt

n

tt

m

t

n

tt

m

t CPVt

YCYtC

D


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


Characteristics of Macaulay DurationThere is an inverse relation between YTM

and durationSinking 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

mY

1

durationMacaulay duration Modified

m

Where:

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:

mmod Y100

DP

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-value-weighted 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

Price changes are not linear, but a curvilinear (convex) function

mmod Y100

DP

P

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

Pdi

Pd2

2

Convexity

Bond Convexity

Determinants of Convexity Inverse relationship between coupon and

convexityDirect 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 changeConvexity is desirable Greater price appreciation if interest rates fall,

smaller price drop if interest rates rise

Chapter 18

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