1. Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Eighth Edition by Frank K. Reilly & Keith C. Brown
Chapter 18

2. Chapter 18 - The Analysis and Valuation of Bonds
Questions to be answered:
How do you determine the value of a bond based on the present value formula?
What are the alternative bond yields that are important to investors?

3. Chapter 18 - The Analysis and Valuation of Bonds
How do you compute the following yields on bonds: current yield, yield to maturity, yield to call, and compound realized (horizon) yield?
What are spot rates and forward rates and how do you calculate these rates from a yield to maturity curve?
What is the spot rate yield curve and forward rate curve?

4. Chapter 18 - The Analysis and Valuation of Bonds
How and why do you use the spot rate curve to determine the value of a bond?
What are the alternative theories that attempt to explain the shape of the term structure of interest rates?
What factors affect the level of bond yields at a point in time?
What economic forces cause changes in bond yields over time?

5. Chapter 18 - The Analysis and Valuation of Bonds
When yields change, what characteristics of a bond cause differential price changes for individual bonds?
What is meant by the duration of a bond, how do you compute it, and what factors affect it?
What is modified duration and what is the relationship between a bond’s modified duration and its volatility?

6. Chapter 18 - The Analysis and Valuation of Bonds
What is effective duration and when is it useful?
What is the convexity for a bond, how do you compute it, and what factors affect it?
Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?

7. Chapter 18 - The Analysis and Valuation of Bonds
What happens to the duration and convexity of bonds that have embedded call options?
What are effective duration and effective convexity and when are they useful?
What is empirical duration and how is it used with common stocks and other assets?
What are the static yield spread and the option-adjusted spread?

8. Chapter 18 - The Analysis and Valuation of Bonds
What are effective duration and effective convexity and when are they useful?
What is empirical duration and how is it used with common stocks and other assets?
What are the static yield spread and the option-adjusted spread?

9. The Fundamentals of Bond Valuation
The present-value model
Where:
Pm=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
Pp=the par value of the bond

10. The Fundamentals of Bond Valuation
If yield < coupon rate, bond will be priced at a premium to its par value
If yield > coupon rate, bond will be priced at a discount to its par value
Price-yield relationship is convex (not a straight line)

11. The Present Value Model
The value of the bond equals the present value of its expected cash flows
where:
Pm = 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
Pp = the par value of the bond

12. The Yield Model
The expected yield on the bond may be computed from the market price
where:
i = the discount rate that will discount the cash flows to equal the current market price of the bond

13. 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
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.

14. Nominal Yield
Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

15. Current Yield Similar to dividend yield for stocks
Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond

16. 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

17. Computing the Promised Yield to Maturity
Solve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

18. Computing Promised Yield to Call
where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond

19. Realized (Horizon) Yield Present-Value Method

20. 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
i = expected semiannual rate at the end of the holding period

21. Yield Adjustments for Tax-Exempt Bonds
Where:
FTEY = fully taxable yield equivalent
i = the promised yield on the tax exempt bond
T = the amount and type of tax exemption (i.e., the investor’s marginal tax rate)

22. Bond Valuation Using Spot Rates
where:
Pm = the market price of the bond
Ct = the cash flow at time t
n = the number of years
it = the spot rate for Treasury securities at maturity t

23. 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

24. What Determines Interest Rates
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
foreign bond risk including exchange rate risk and country risk

25. Term Structure of Interest Rates
It is a static function that relates the term to maturity to the yield to maturity for a sample of bonds at a given point in time.
Term Structure Theories
Expectations hypothesis
Liquidity preference hypothesis
Segmented market hypothesis
Trading implications of the term structure

26. Spot Rates and Forward Rates
Creating the Theoretical Spot Rate Curve
Calculating Forward Rates from the Spot Rate Curve

27. Expectations Hypothesis
Any long-term interest rate simply represents the geometric mean of current and future one-year interest rates expected to prevail over the maturity of the issue

28. Liquidity Preference Theory
Long-term securities should provide higher returns than short-term obligations because investors are willing to sacrifice some yields to invest in short-maturity obligations to avoid the higher price volatility of long-maturity bonds

29. Segmented-Market Hypothesis
Different institutional investors have different maturity needs that lead them to confine their security selections to specific maturity segments

30. Trading Implications of the Term Structure
Information on maturities can help you formulate yield expectations by simply observing the shape of the yield curve

31. Yield Spreads
Segments: government bonds, agency bonds, and corporate bonds
Sectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities versus BBB utilities
Coupons or seasoning within a segment or sector
Maturities within a given market segment or sector

32. Yield Spreads
Magnitudes and direction of yield spreads can change over time

33. What Determines the Price Volatility for Bonds
Bond price change is measured as the percentage change in the price of the bond
Where:
EPB = the ending price of the bond
BPB = the beginning price of the bond

34. What Determines the Price Volatility for Bonds
Four Factors
1. Par value
2. Coupon
3. Years to maturity
4. Prevailing market interest rate

35. What Determines the Price Volatility for Bonds
Five observed behaviors
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

36. What Determines the Price Volatility for Bonds
The maturity effect
The coupon effect
The yield level effect
Some trading strategies

37. 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

38. The Duration Measure 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

39. 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 relationship 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 bond’s duration

40. Modified Duration and Bond Price Volatility
An adjusted measure of duration can be used to approximate the price volatility of an option-free (straight) bond
Where:
m = number of payments a year
YTM = nominal YTM

41. 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 time 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

42. Trading Strategies Using Modified Duration
Longest-duration security provides the maximum price variation
If you expect a decline in interest rates, increase the average modified duration of your bond portfolio to experience maximum price volatility
If you expect an increase in interest rates, reduce the average modified duration to minimize your price decline
Note that the modified duration of your portfolio is the market-value-weighted average of the modified durations of the individual bonds in the portfolio

43. Bond Duration in Years for Bonds Yielding 6 Percent Under Different Terms

44. Bond Convexity
Modified duration is a linear approximation of bond price change for small changes in market yields
However, price changes are not linear, but a curvilinear (convex) function

45. 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
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

46. Modified Duration
For small changes this will give a good estimate, but this is a linear estimate on the tangent line

47. Determinants of Convexity
The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by price
Convexity is the percentage change in dP/di for a given change in yield

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

49. 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

50. Duration and Convexity for Callable Bonds
Issuer has option to call bond and pay off with proceeds from a new issue sold at a lower yield
Embedded option
Difference in duration to maturity and duration to first call
Combination of a noncallable bond plus a call option that was sold to the issuer
Any increase in value of the call option reduces the value of the callable bond

51. 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

52. Convexity of Callable Bonds
Noncallable bond has positive convexity
Callable bond has negative convexity

53. Limitations of Macaulay and Modified Duration
Percentage change estimates using modified duration only are good 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
Initial assumption that cash flows from the bond are not affected by yield changes

54. Effective Duration
Measure of the interest rate sensitivity of an asset
Use a pricing model to estimate the market prices surrounding a change in interest rates
Effective Duration Effective Convexity
P- = the estimated price after a downward shift in interest rates
P+ = the estimated price after a upward shift in interest rates
P = the current price
S = the assumed shift in the term structure

55. Effective Duration Effective duration greater than maturity
Negative effective duration
Empirical duration

56. Empirical Duration
Actual percent change for an asset in response to a change in yield during a specified time period

57. Yield Spreads With Embedded Options
Static Yield Spreads
Consider the total term structure
Option-Adjusted Spreads
Consider changes in the term structure and alternative estimates of the volatility of interest rates

58. The Internet Investments Online

59. End of Chapter 18
The Analysis and Valuation of Bonds

60. Future topics Chapter 19
Bond Portfolio Management Strategies