1. FIXED-INCOME ANALYSIS
Chapter 18
FIXED-INCOME ANALYSIS

2. 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?

3. 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?

4. 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?

5. 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 = SCFt/(1+k)t

6. The Fundamentals of Bond Valuation
To incorporate the specifics of bonds:
Pm = S(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 bond
n is the number of years to maturity
Ci is the annual coupon payment
Ym is the yield to maturity of the bond
Pp is the par value of the bond

7. 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

8. The Yield Model
The yield on the bond may be computed when we know the market price
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

9. Computing Bond Yields Yield Measure Purpose Coupon 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.

10. 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

11. Promised Yield to Maturity
Widely used bond yield figure
Assumes
Investor holds bond to maturity
All the bond’s cash flow is reinvested at the computed yield to maturity
Solve for Y that will equate the current price to all cash flows from the bond to maturity, similar to IRR

12. 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.

13. 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 maturity
As 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

14. 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

15. Realized Yield
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

16. 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 price
When issues are quoted on a promised yield, as with municipals
For portfolio managers who frequently trade bonds

17. Calculating Future Bond Prices
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
Ym = expected semiannual rate at the end of the holding period

18. 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

19. 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 returns
FTEY = Tax-Free Annual Return/(1-T)
Where T is the investor’s marginal tax rate

20. What Determines Interest Rates?
Inverse relationship with bond prices
Changes 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

21. 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

22. Determinants of Interest Rates
Nominal interest rates (i) can be broken down into the following components:
i = RFR + I + RP
where:
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

23. 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)

24. 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

25. 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

26. Determinants of Interest Rates
Term Structure Theories
Liquidity 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

27. Determinants of Interest Rates
Term Structure Theories
Segmented market hypothesis
Asserts 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

28. Determinants of Interest Rates
Term Structure and Trading
Knowledge of the term structure can aid in bond market trading strategies
For 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

29. Determinants of Interest Rates
Yield Spreads
Bond 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

30. 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

31. Determinants of Bond Price Volatility
Four factors determine a bond’s price volatility to changing interest rates:
Par value
Coupon
Years to maturity
Prevailing level of market interest rate

32. 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

33. Determinants of Bond Price Volatility
The maturity effect
The longer the time to maturity, the greater a bond’s price sensitivity
Price volatility increases at a decreasing rate with maturity
The coupon effect
The greater the coupon rate, the lower a bond’s price sensitivity

34. Determinants of Bond Price Volatility
The yield level effect
For 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

35. Determinants of Bond Price Volatility
The Duration Measure
Since 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

36. 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

37. 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

38. Determinants of Bond Price Volatility
Characteristics of Macaulay Duration
There is an inverse relation between YTM and duration
Sinking funds and call provisions can have a dramatic effect on a bond’s duration

39. Duration and Bond Price Volatility
An adjusted measure of duration can be used to approximate the price volatility of a bond
Where:
m = number of payments a year
Ym = nominal YTM

40. Duration and Bond Price Volatility
Bond price movements will vary proportionally with modified duration for small changes in yields:
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

41. 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

42. 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

43. 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

44. 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

45. Bond Convexity Determinants of Convexity
Inverse relationship between coupon and convexity
Direct relationship between maturity and convexity
Inverse relationship between yield and convexity

46. 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 change
Convexity is desirable
Greater price appreciation if interest rates fall, smaller price drop if interest rates rise