Chapter 18 FIXED-INCOME ANALYSIS. Chapter 18 Questions How is the value of a bond determined, based...
-
date post
21-Dec-2015 -
Category
Documents
-
view
235 -
download
2
Transcript of Chapter 18 FIXED-INCOME ANALYSIS. Chapter 18 Questions How is the value of a bond determined, based...
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?
Chapter 18 Questions
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?
Chapter 18 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 = 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
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 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
Determinants of Interest Rates
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
Determinants of Interest Rates
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
Determinants of Interest Rates
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
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 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
Determinants of Bond Price Volatility
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
Determinants of Bond Price Volatility
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
Determinants of Bond Price Volatility
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
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 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