1. CHAPTER 14, 15 and 16 Ch 14 Bond Prices and Yields
Ch 15 The Term Structure of Interest Rates
Ch 16 Managing the Bond Portfolios

2. Comparative Performance of Stocks and Bonds (1997-2008)

3. Bond Prices and Yields
Chapter 14

4. Bond Characteristics
Bonds are debt – “IOU” securities. Issuers are borrowers and holders are creditors.
The indenture is the contract between the issuer and the bondholder.
The indenture gives the coupon rate, maturity date, and par value.
Annual trading volume in US government bonds alone is about 10 times the total amount of national debt.
Mortgage-backed securities market is about $3.8 trillion.

5. Bond Characteristics
Face or par value is typically $1000; this is the principal repaid at maturity.
The coupon rate determines the interest payment.
Interest is usually paid semiannually.
The coupon rate can be zero.
Interest payments are called “coupon payments”.

6. U.S. Treasury Bonds
Bonds and notes may be purchased directly from the Treasury.
Maturities vary.
Note maturity is 1-10 years
Bond maturity is years
Denomination can be as small as $100, but $1,000 is more common.
Bid price of 100:08 means 100 8/32 or $

8. Corporate Bonds
Callable bonds can be repurchased before the maturity date.
Convertible bonds can be exchanged for shares of the firm’s common stock.
Puttable bonds give the bondholder the option to retire or extend the bond.
Floating rate bonds have an adjustable coupon rate

9. List of Corporate Bonds

10. Callable Bonds
Thus far, we have calculated bond prices assuming that the actual bond maturity is the original stated maturity.
However, most bonds are callable bonds.
A callable bond gives the issuer the option to buy back the bond at a specified call price anytime after an initial call protection period.
Therefore, for callable bonds, YTM may not be useful.

11. YTM and YTC
Suppose a 20-year bond has a coupon of 8 percent, a price of , and is callable in 10years. The call prices 105. What are its yield to maturity and yield to call?
YTM = 8.12 percent
YTC = 8.50 percent

12. Inflation-Indexed Treasury Securities
Recently, the U.S. Treasury has issued securities called Treasury Inflation Protected Securities (TIPS) that guarantee a fixed rate of return in excess of realized inflation rates .
That is, they are indexed to inflation in order to protect investors from the negative effects of inflation.
TIPS can be purchased directly from the government through the TreasuryDirect system in $100 increments with a minimum investment of $100 and are available with 5-, 10-, and 20-year maturities.
You can hold a TIPS until it matures or sell it in the secondary market before it matures.

13. Example: TIPS
Their par value rises with inflation and falls with deflation, as measured by the CPI.
TIPS pay interest twice a year, at a fixed coupon rate on their current principal
But, principal are adjusted semiannually according to the most recent inflation rate.
So, like the principal, interest payments rise with inflation and fall with deflation.
When a TIP matures, you are paid the adjusted principal or original principal, whichever is greater.
Suppose an inflation-indexed note is issued with a coupon rate of 3.5% and an initial principal of $1,000.
Six months later, the note will pay a coupon of $1,000 × (3.5%/2) = $17.50.
Assuming 2 percent inflation over the six months since issuance, the note’s principal is then increased to $1,000 × 102% = $1,020.
Six months later, the note pays $1,020 × (3.5%/2) = $17.85
Its principal is again adjusted to compensate for recent inflation.

14. Principal and Interest Payments for a Treasury Inflation Protected Security

15. Bond Pricing PB = Price of the bond Ct = interest or coupon payments
T = number of periods to maturity
r = semi-annual discount rate or the semi-annual yield to maturity

16. Bond Pricing
Price of a 30 year, 8% coupon bond with semi-annual coupon payments.
Market rate of interest is 10%.

17. Bond Prices and Yields
Prices and yields (required rates of return) have an inverse relationship
The bond price curve is convex.
For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield.
The longer the maturity, the more sensitive the bond’s price to changes in market interest rates.
The lower the coupon rate, the more sensitive the bond’s price to changes in market interest rates.

18. The Inverse Relationship Between Bond Prices and Yields

19. Bond Prices at Different Interest Rates

20. Yield to Maturity
Interest rate that makes the present value of the bond’s payments equal to its price is the YTM.
Solve the bond formula for r

21. Yield to Maturity Example
Suppose an 8% coupon paid semiannually, 30 year bond is selling for $ What is its average rate of return?
r = 3% per half year
Bond equivalent yield = 6%
EAR = (1.03) = 6.09%

22. YTM vs. Current Yield The YTM is the bond’s internal rate of return.
YTM is the interest rate that makes the present value of a bond’s payments equal to its price.
YTM assumes that all bond coupons can be reinvested at the YTM rate.
The current yield is the bond’s annual coupon payment divided by the bond price.
For bonds selling at a premium, coupon rate > current yield>YTM.
For discount bonds, relationships are reversed.

23. Yield to Call
If interest rates fall, price of straight bond can rise considerably.
The price of the callable bond is flat over a range of low interest rates because the risk of repurchase or call is high.
When interest rates are high, the risk of call is negligible and the values of the straight and the callable bond converge.

24. Bond Prices: Callable and Straight Debt

25. Realized (Horizon) Yield vs. Promised YTM
YTM is a promised yield, which may not equal the realized yield.
Two assumptions with YTM
YTM assumes the investor’s holding bond until maturity.
YTM assumes that coupons are reinvested at YTM.
That is, YTM will equal the realized return only if all coupons are reinvested at YTM and investors hold the bond until maturity..
Two questions on the Promised YTM.
What if the investor does not hold the bond until maturity?
What if the investor is unable to reinvest coupons at the promised YTM?
Horizon Yield or Holding Period Return (HPR) measure the expected rate of return of a bond that you expect to sell prior to its maturity with varying reinvestment rates.

26. YTM vs. HPR
YTM
HPR
YTM is the average return if the bond is held to maturity.
YTM depends on coupon rate, maturity, and par value.
All of these are readily observable.
HPR is the rate of return over a particular investment period.
HPR depends on the bond’s price at the end of the holding period, an unknown future value.
HPR can only be forecasted.

27. Growth of Invested Funds

28. Example Suppose we bought a par bond with 14%, 20-year bond.
This means that YTM is 14%.
Assume the holding period is 2 years.
Assumes that the market interest rate is expected to decline to 10 percent.
Expected price at the end of year 2
N = 36, I = 10 (2P), PMT = 70, FV = 1000
Price at the end of year 2 = $1,330.94
Horizon Yield
N = 4, PV = -1000, PMT = 70, FV = ,
I = 27.50%

29. Prices over Time of 30-Year Maturity, 6.5% Coupon Bonds

30. The Price of a 30-Year Zero-Coupon Bond over Time

31. Default Risk and Bond Pricing
Rating companies:
Moody’s Investor Service, Standard & Poor’s, Fitch
Rating Categories
Highest rating is AAA or Aaa
Investment grade bonds are rated BBB or Baa and above
Speculative grade/junk bonds have ratings below BBB or Baa.

32. Factors Used by Rating Companies
Coverage ratios
Leverage ratios
Liquidity ratios
Profitability ratios
Cash flow to debt

33. Financial Ratios and Default Risk by Rating Class, Long-Term Debt

34. Protection Against Default
Sinking funds – a way to call bonds early
Subordination of future debt– restrict additional borrowing
Dividend restrictions– force firm to retain assets rather than paying them out to shareholders
Collateral – a particular asset bondholders receive if the firm defaults

35. Default Risk and Yield
The risk structure of interest rates refers to the pattern of default premiums.
There is a difference between the yield based on expected cash flows and yield based on promised cash flows.
The difference between the expected YTM and the promised YTM is the default risk premium.

36. Yield Spreads

37. Credit Default Swaps
A credit default swap (CDS) acts like an insurance policy on the default risk of a corporate bond or loan.
CDS buyer pays annual premiums.
CDS issuer agrees to buy the bond in a default or pay the difference between par and market values to the CDS buyer.
Institutional bondholders, e.g. banks, used CDS to enhance creditworthiness of their loan portfolios, to manufacture AAA debt.
CDS can also be used to speculate that bond prices will fall.
This means there can be more CDS outstanding than there are bonds to insure!

38. Prices of Credit Default Swaps

39. Credit Risk and Collateralized Debt Obligations (CDOs)
Major mechanism to reallocate credit risk in the fixed-income markets
Structured Investment Vehicle (SIV) often used to create the CDO
Loans are pooled together and split into tranches with different levels of default risk.
Mortgage-backed CDOs were an investment disaster in 2007

40. Collateralized Debt Obligations

41. The Term Structure of Interest Rates
CHAPTER 15

42. Overview of Term Structure
The yield curve is a graph that displays the relationship between yield and maturity.
Information on expected future short term rates can be implied from the yield curve.

43. Money Market Interest Rates

44. The Treasury Yield Curve
Because of its sheer size, the leading world market for debt securities is the market to U.S. Treasury securities.
The Treasury yield curve is a plot of Treasury yields against maturities.
It is fundamental to bond market analysis, because it represents the interest rates for default-free lending across the maturity spectrum.

45. Example: The Treasury Yield Curve

46. Treasury Yield Curves

47. The Term Structure of Interest Rates
The term structure of interest rates is the relationship between time to maturity and the interest rates for default-free, pure discount instruments.
The term structure is sometimes called the “zero-coupon yield curve” to distinguish it from the Treasury yield curve, which is based on coupon bonds.

48. Two Types of Yield Curves
Pure Yield Curve
On-the-run Yield Curve
The pure yield curve uses stripped or zero coupon Treasuries.
The pure yield curve may differ significantly from the on-the-run yield curve.
The on-the-run yield curve uses recently issued coupon bonds selling at or near par.
The financial press typically publishes on-the-run yield curves.

49. Bond Pricing
Yields on different maturity bonds are not all equal.
We need to consider each bond cash flow as a stand-alone zero-coupon bond.
Bond stripping and bond reconstitution offer opportunities for arbitrage.
The value of the bond should be the sum of the values of its parts.

50. Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value)

51. Valuing Coupon Bonds
Value a 3 year, 10% coupon bond using discount rates from Table 15.1:
Price = $ and YTM = 6.88%
6.88% is less than the 3-year rate of 7%.

52. Yield Curve Under Certainty
Suppose you want to invest for 2 years.
Buy and hold a 2-year zero
- or
Rollover a series of 1-year bonds
Equilibrium requires that both strategies provide the same return.

53. Two 2-Year Investment Programs

54. Yield Curve Under Certainty
Buy and hold vs. rollover:
Next year’s 1-year rate (r2) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2-year bond.

55. Spot Rates vs. Short Rates
Spot rate – the rate that prevails today for a given maturity
Short rate – the rate for a given maturity (e.g. one year) at different points in time.
A spot rate is the geometric average of its component short rates.

56. Short Rates and Yield Curve Slope
When next year’s short rate, r2 , is greater than this year’s short rate, r1, the yield curve slopes up.
May indicate rates are expected to rise.
When next year’s short rate, r2 , is less than this year’s short rate, r1, the yield curve slopes down.
May indicate rates are expected to fall.

57. Short Rates versus Spot Rates

58. Forward Rates from Observed Rates
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n

59. Forward Rates
The forward interest rate is a forecast of a future short rate.
Rate for 4-year maturity = 8%, rate for 3-year maturity = 7%.

60. Interest Rate Uncertainty
Suppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6%. The value of a 2-year zero is:
The value of a 1-year zero is:

61. Interest Rate Uncertainty
The investor wants to invest for 1 year.
Buy the 2-year bond today and plan to sell it at the end of the first year for $1000/1.06 =$
Or,
Buy the 1-year bond today and hold to maturity.

62. Interest Rate Uncertainty
What if next year’s interest rate is more (or less) than 6%?
The actual return on the 2-year bond is uncertain!

63. Interest Rate Uncertainty
Investors require a risk premium to hold a longer-term bond.
This liquidity premium compensates short-term investors for the uncertainty about future prices.

64. Theories of Term Structure
Expectations
Liquidity Preference
Upward bias over expectations

65. Expectations Theory fn = E(rn) and liquidity premiums are zero.
Observed long-term rate is a function of today’s short-term rate and expected future short-term rates.
The term structure of interest rates reflects financial market beliefs about future interest rates.
Long-term and short-term securities are perfect substitutes.
Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.
fn = E(rn) and liquidity premiums are zero.

66. Liquidity Premium Theory
Long-term bonds are more risky, and therefore, long-term interest rates contain a maturity premium necessary to induce lenders into making longer term loans.
Long-term bonds are more risky; therefore, fn generally exceeds E(rn)
The excess of fn over E(rn) is the liquidity premium.
Investors will demand a premium for the risk associated with long-term bonds.
Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
The yield curve has an upward bias built into the long-term rates because of the liquidity premium.

67. Yield Curves

68. Interpreting the Term Structure
The yield curve reflects expectations of future interest rates.
The forecasts of future rates are clouded by other factors, such as liquidity premiums.
An upward sloping curve could indicate:
Rates are expected to rise
And/or
Investors require large liquidity premiums to hold long term bonds.

69. Interpreting the Term Structure
The yield curve is a good predictor of the business cycle.
Long term rates tend to rise in anticipation of economic expansion.
Inverted yield curve may indicate that interest rates are expected to fall and signal a recession.

70. Term Spread: Yields on 10-year vs. 90-day Treasury Securities

71. Dynamic Yield Curve

72. Inverted Yield Curve = Prognosis for Recession?
The Interest Rate Spread and Annual % Change in Real GDP The diagram relates the interest rate spread (10-year T-bond yield minus Fed funds rate) to the rate of economic growth. Notice how the interest rate spread became negative just before each of the recessions shown in the table, thus serving as an important leading economic indicator. The interest rate spread has correctly forecasted all U.S. recessions since World War II. Note that the interest rate spread again became negative during the last two quarters of 1998, and again in 2000 and the first quarter of While the first of these inversions may have occurred as a consequence of the Asian financial crisis, the latter inversion correctly indicated the recession that began in March While GDP growth is strong, the interest rate spread has been declining since the first quarter of While the Fed has been raising short-term rates to head off inflationary fears, long-term rates have remained relatively low. The latter is due in part to China and other countries with a trade surplus with the US using their dollar holdings to buy US treasury bonds. The Fed doesn’t consider the current inversion in the interest rates spread to be a leading indicator of a coming recession. Not all economists agree though.

73. Inverted Yield Curve = Prognosis for Recession?

74. Managing the Bond Portfolios
Chapter 16

75. Bond Pricing Relationships
Bond prices and yields are inversely related.
An increase in a bond’s yield to maturity results in a smaller price change than a decrease of equal magnitude.
Long-term bonds tend to be more price sensitive than short-term bonds.

76. Bond Pricing Relationships
As maturity increases, price sensitivity increases at a decreasing rate.
Interest rate risk is inversely related to the bond’s coupon rate.
Price sensitivity is inversely related to the yield to maturity at which the bond is selling.

77. Change in Bond Price as a Function of Change in Yield to Maturity

78. Prices of 8% Coupon Bond (Coupons Paid Semiannually)

79. Prices of Zero-Coupon Bond (Semiannually Compounding)

80. Duration
Bondholders know that the price of their bonds change when interest rates change. But,
How big is this change?
How is this change in price estimated?
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 our objective.
A composite measure considering both coupon and maturity would be beneficial.
Macaulay Duration, or Duration, is the name of concept that helps bondholders measure the sensitivity of a bond price to changes in bond yields. That is:
Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.

81. Macaulay’s Duration or Duration
Macaulay’s Duration values are stated in years, and are often described as a bond’s effective maturity.
The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment
For a zero-coupon bond, duration = maturity.
For a coupon bond, duration = a weighted average of individual maturities of all the bond’s separate cash flows
Duration is shorter than maturity for all bonds except zero coupon bonds.
Duration is a measure of interest rate sensitivity or elasticity of a liability or asset.

82. Macaulay Duration Where CFt = cash flow to be delivered in t periods
DMAC = duration
t = number of periods in the future
CFt = cash flow to be delivered in t periods
T= time-to-maturity
y = yield to maturity
PV = present value

83. Example: Annual Bond

84. Example: Semi-annual Bond

85. Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%

86. Durations of Coupon and Zero coupon Bonds

87. Duration/Price Relationship
Price change is proportional to duration and not to maturity
Some analysts prefer to use a variation of Macaulay’s Duration, D*, known as Modified Duration.
D* = DMAC / (1+y/2)

88. Modified Duration
The relationship between percentage changes in bond prices and changes in bond yields is approximately:
Duration is a measure of interest rate sensitivity or elasticity of a liability or asset.

89. Example 1
Example: Suppose a bond has a Macaulay Duration of 11 years, and a current yield to maturity of 8%.
If the yield to maturity increases to 8.50%, what is the resulting percentage change in the price of the bond?
The bond’s price will decline by approximately 5.29% in response to 50 basis point increase in yields.

90. Example 2
Consider two bonds that have the same durations of years.
One is a 2-year, 8% coupon bond with YTM=10%.
The other bond is a zero coupon bond with maturity of years.
Duration of both bonds is x 2 = semiannual periods.
Alternatively, keep DMAC = years
Modified D = /1.05 = semiannual periods.
Alternatively, DMOD = / 1.05= years

91. Example 2, cont’d
Suppose the semiannual interest rate increases by 0.01% (This means 0.02% annual rate). Bond prices fall by:
= x 0.01% = %
Alternatively, x 0.02% = %
Both bonds with equal duration have the same interest rate sensitivity.

92. Example 2, cont’d
Coupon Bond
Zero
The coupon bond, which initially sells at $ , falls to $ when its yield increases to 5.01%
percentage decline of %.
The zero-coupon bond initially sells for $1,000/ = $
At the higher yield, it sells for $1,000/ = $ This price also falls by %.

93. Example 3
Suppose DMAC = 8 year, YTM = Assume that you expect the bond’s YTM to decline by 75 basis point .
DMOD = 8/(1+0.10/2) = 7.62
Percent change in bond price = - DMOD * Change in YTM
%∆P = -7.62*(-.0075) = or 5.72%
The bond price should increase approximately 5.72 percent.

94. Example 4
The 6-year Eurobond with an 8% coupon and 8% yield, had a duration of DMAC = 4.99 years. If yields rose 1 basis point, then:
Since C= YTM, it’s a par bond priced at $1,000.
DMOD = 4.99 / (1+ 8%) = , assuming that it is an annual bond.
dP/P = -(4.6204) (.0001) = or %
To calculate the dollar change in value, rewrite the equation,
dP = -DMOD * P *dR = ( )($1,000)(.0001)= -$0.462
The bond price falls to $ after a one basis point increase in yields.

95. Examples 5

96. Calculating Macaulay’s Duration
In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is:
In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.

97. Calculating Macaulay’s Duration for Par Bonds
If a bond is selling for par value, the duration formula can be simplified to:

98. Example

99. Calculating Duration Using Excel
We can use the DURATION and MDURATION functions in Excel to calculate Macaulay Duration and Modified Duration.
The Excel functions use arguments like we saw before:
=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)
You can verify that a 5-year bond, with a 9% coupon and a 7% YTM has a Duration of 4.17 and a Modified Duration of 4.03.

100. Calculating Macaulay and Modified Duration

101. Rules for Duration
Rule 1 The duration of a zero-coupon bond equals its time to maturity
Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower
Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity

102. Rules for Duration
Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

103. Bond Duration versus Bond Maturity

104. Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)