1. Chapter 6 Bonds, Bond Prices, and the Determination of Interest Rates
Chapter Six
Chapter 6
Bonds, Bond Prices, and the
Determination of Interest Rates

2. Goals of the Chapter Present value and bond pricing
Relationship of bond prices, interest rates (yields), and returns.
Key drivers of bond prices - supply and demand in the bond market determine bond prices .
Why bonds are risky – default, inflation, and interest changes.

3. Bond Prices
A standard bond specifies the fixed amounts to be paid and the exact dates of the payments.
We will examine four basic types.

4. Bond Prices 1. Zero-coupon or discount bond
Promise a single payment on a future date
Example: Treasury bill
2. Fixed-payment loan (just skim)
Sequence of fixed payments
Example: Mortgage or car loan
3. Coupon bond
periodic interest payments + principal repayment at maturity
Example: U.S. Treasury Bonds and most corporate bonds
4. Consol
periodic interest payments forever, principal never repaid
Example: U.K. government has some outstanding

5. Zero-Coupon Bonds
Straightforward type of bond. U.S. Treasury bills (T-bills) are a good example - money market instruments that mature is less than a year.
Each T-bill represents a promise by the U.S. government to pay $1000 at maturity on a fixed future date.
No coupon payments - zero-coupon bonds
Also called discount bonds since the bond price is less than face value - they sell at a discount.

6. Zero-Coupon Bonds Assume i = 5%: Price of a One-Year Treasury Bill
How to price a $1000 face value zero-coupon bond -
Assume i = 5%:
Price of a One-Year Treasury Bill
Price of a Six-Month Treasury Bill
Remember units must match: annual interest rate, therefore n must be annual – in the case ½ of a year.

7. Zero-Coupon Bonds
We can compute the interest rate from the price using the present value formula.
Suppose a T-Bill zero has a face value of $1000 and the price is $950.
i = ($1000/$950) - 1 = = 5.26%.

8. Fixed-Payment Loans
Conventional home mortgages and car loans are fixed-payment loans.
They promise a fixed number of equal payments at regular intervals.
Amortized loans - the borrower pays off part of the principal along with the interest for the life of the loan.
Value of a Fixed Payment Loan =
The sum of the present value of the payments.

9. Coupon Bond Price
A simple contract-

10. Present Value of Coupon Payments
Price of a Coupon Bond
Present Value of Principal Payment
Present Value of Coupon Payments
Price of Coupon Bond (PB) =
Present Value of yearly coupon payments (C)
+ Present Value of the Face Value (FV),
where: i = interest rate and n = time to maturity

11. Example: Price of a 10%, n-year Coupon Bond
Coupon Payment (C) =$100, Face value (FV) = $1,000,
and n = time to maturity
C is contractually fixed.
Given values for i and n, we can determine the bond price PB
Definition: Coupon Rate = Coupon Payment / Face Value

12. Price of a 10-year Coupon Bond
If n = 10, i = 0.10, C = $100 and FV = $1000.
What’s the Coupon Rate?
Excel Example

13. Price of a 10-year Coupon Bond
If n = 10, i = 0.12, C = $100 and FV = $1000.
i goes up and PB goes down
What’s the coupon rate for this bond?

14. Yield to Maturity -YTM yield to maturity:
The yield bondholders receive if they hold the bond to its maturity when the final principal payment is made.
Simple example:
Suppose a $100 face value, 1-yr, 5% coupon bond sells for $100:
The value of i that solves the equation is the yield to maturity (YTM).

15. Yield to Maturity -YTM
Example: 1 year, $100 FV, 5% coupon bond selling for $99 Yield to maturity for this bond is 6.06 percent found as the solution to:

16. Yield to Maturity - YTM This relationship should make sense.
If you pay $99 for a $100 face value bond, you will receive both the interest payments and the increase in value from $99 to $100.
This rise in value is referred to as a capital gain and is part of the return on your investment.
When the price of a bond is higher than face value, the bondholder incurs a capital loss.

17. Current Yield and YTM
Example: 1 year, $100 FV, 5% coupon bond selling for $99: Current Yield = , or 5.05%
, or 6.06%

18. YTM: 10-year Coupon Bond
Suppose n = 10, PB = $950, C = $100 and FV = $1000.
What is the Coupon Rate?
What is the Current Yield?
What’s the YTM? - more complicated
YTM = or 10.85%

19. C = the coupon payment
FV = Face Value
P = Price
n = years to maturity

20. Using the Approximation Formula
Previous example:
n = 10, PB = $950, C = $100 and FV = $1000.

21. For our 10-year, $1000 FV bond with a $100 coupon payment selling at $950:
Coupon rate = 10%
Current Yield = 10.52%
YTM = 10.85%
Approx. YTM = 10.77%

22. When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate
The price of a coupon bond and the yield to maturity are negatively related
The yield to maturity is greater than the coupon rate when the bond price is below its face value

23. More on Zero Coupon or Discount Bonds
Definition: A discount bond is sold at some price P, and pays a larger amount (FV) after t years. There is no periodic interest payment.
Let PB = price of the bond, i= interest rate, n = years to maturity, and FV = Face Value (the value at maturity):

24. Zero Coupon Bonds - Price
Price of a One-Year Treasury Bill at 4% and FV = $1,000:
Price of a Six-Month Treasury Bill at 4% and FV = $1,000:
Price of a 20-Year zero coupon bond at 8% and FV = $20,000:

25. YTM - Zero Coupon Bonds

26. Zero Coupon Bonds - YTM i = 1.0657 -1=> i =6.57%
For a discount bond with FV = $15,000 and P = $4,200, and n = 20, the interest rate (or yield to maturity) would be:
i = => i =6.57%
Note: This is the formula for compound annual rate of growth

27. Zero Coupon Bonds - YTM i = 1.06368 – 1 = .06368 or 6.368%
For a discount bond with FV = $10,000 and P = $6,491, and n = 7, the interest rate (or yield to maturity) would be:
i = – 1 = or 6.368%

28. Create n+1 discount bonds
From a Coupon Bond to Zero Coupon Bonds Zero Coupon Bonds are called Strips – here’s why.
Create n+1 discount bonds

29. Consols
Consols or perpetuities, are like coupon bonds whose coupon payments last forever.
The borrower pays only interest, never repaying the principal.
The U.S. government sold consols once in 1900, but the Treasury has bought them all back.
The price of a consol is the present value of all future coupon payments.

30. Holding Period Return
The holding period return is the return to holding a bond and selling it before maturity.
The holding period return can differ from the yield to maturity.

31. One Year Holding Period Return
Example:
10 year bond
6% coupon rate
Purchase at face value, $100
Hold for one year and then sell it

32. Holding Period Return
What if market interest rates at the time of the sale fall to 5%?
1-yr Holding Period Return =
Current Yield + Capital Gain
The investor earned $13.11 on a $100 investment.
The 1-yr Holding Period Return = 13.11%

33. Holding Period Return
What if market interest rates at the time of the sale rise to 7%?
1yr Holding Period Return =
1-yr Holding Period Return = %

34. Holding Period Return = Current Yield + Capital Gain (as a %)
The one-year holding period return is the sum of the yearly coupon payment divided by the price paid for the bond and the change in the price divided by the price paid.
= Current Yield Capital Gain (as a %)

35. Holding Period Return
You purchase coupon bond and sell one year later.
Current Yield
Capital Gain

36. One year holding period return =
You purchase a 10-year, 10% coupon bond, face value = $1,000. If the interest rate one year later is the same at 10%:
One year holding period return =
or 10.0%

37. Holding Period Return
If the interest rate one year later is lower, say at 8%:
One year holding period return =
or 22.5%
Where did the $1125 come from?
In the previous example, where did the $ come from?

38. Holding Period Returns
If the interest rate in one year is higher at 12%:
One year holding period return =
or -.70%
Where did the $893 come from. You need to know how to calculate the $1125 and $893

40. Key Conclusions From Table
The return equals the yield to maturity (YTM) only if the holding period equals the time to maturity.
A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the holding period is less than the time to maturity.
The greater the percentage price change associated with an interest-rate change, the more distant the maturity.

41. Interest-Rate Risk
Change in bond price due to change in interest rate
Prices and returns for long-term bonds are more volatile than those for shorter-term bonds
There is no interest-rate risk for a bond whose time to maturity matches the holding period

42. Reinvestment (interest rate) Risk
If investor’s holding period exceeds the term to maturity
 proceeds from sale of bond are reinvested
at new interest rate
 the investor is exposed to
reinvestment risk
The investor benefits from rising interest rates, and suffers from falling interest rates

43. Bond Supply, Bond Demand and Equilibrium in the Bond Market
Supply and demand determine bond prices and bond yields.
The bond supply curve is the relationship between the price and the quantity of bonds investors, corporations and governments are willing to sell, all else equal.
The higher the price of a bond, the larger the quantity supplied.
The bond supply curve slopes upward.
Why?

44. Bond Supply Investors:
The higher the price, the more tempting it is to sell a bond they currently hold – quantity supplier increases.
Companies seeking to finance projects and that finance spending:
The higher the price at which they can sell bonds, the cheaper it is to borrow - quantity supplied increases.
For a $100 one-year zero-coupon bond, the quantity supplied will be higher at $95 than it will be at $90, all other things being equal.

45. Bond Demand
The bond demand curve is the relationship between the price and the quantity of bonds that investors demand, all else equal.
The bond demand curve slopes downward.
Why?
Investors:
The lower the price bond buyers must pay for a fixed-dollar payment on a future date, the more likely they are to buy a bond.
For a $100 one-year zero-coupon bond, the quantity demanded will be higher at $90 than it will be at $95, all other things being equal.

46. Bond Supply, Bond Demand and Equilibrium in the Bond Market
Equilibrium is the point at which supply equals demand.

47. Bond Supply, Bond Demand and Equilibrium in the Bond Market
If bond prices are above equilibrium?
Quantity supplied > quantity demanded.
Excess supply puts downward pressure on the price until supply equals demand.
If the price is below equilibrium?
Quantity demanded > quantity supplied.
Excess demand puts upward pressure on the price until supply equals demand.

48. Bond Supply, Bond Demand and Equilibrium in the Bond Market
To understand bond prices, need to understand what determines supply and demand.
Remember the distinction between moving along a curve (change in quantity supplied or demanded) versus a shift in the curve (change in supply or demand).
A shift in either supply or demand changes the price of bonds, so it changes yields as well.

49. Factors That Shift Bond Supply
Changes in Government Borrowing
Any increase in the government’s borrowing needs increases the quantity of bonds outstanding, shifting the bond supply curve to the right.
Change in General Business Conditions
As business conditions improve, the bond supply curve shifts to the right.
Changes in Expected Inflation
When expected inflation rises, the cost of borrowing in real terms falls, shifting the bond supply curve to the right.

50. Factors That Shift Bond Supply
When borrowers’ desire for funds increases, the supply curve shifts to the right. This lowers bond prices, thereby raising interest rates.

52. Factors That Shift Bond Demand
Wealth
Increase in wealth increases bond demand, shifts the demand for bonds to the right.
Expected Inflation
Declining inflation means promised payments have higher value - bond demand shifts right.
Expected Returns and Expected Interest Rates
If the expected return on bonds rises relative to the return on alternative investments, bond demand will shift right.
When interest rates are expected to fall, price prices are expected to rise shifting bond demand to the right.

53. Factors That Shift Bond Demand
Risk Relative to Alternatives
If bonds become less risky relative to alternative investments, demand for bonds shifts right.
Liquidity Relative to Alternatives
Investors like liquidity: the more liquid the bond, the higher the demand.

54. Factors That Shift Bond Demand
When bonds become more attractive for investors, the demand curve shifts to the right.
This raises bond prices, lowering interest rates.

57. Case Study: Increase in Expected Rate of Inflation
Recall expected inflation affects both bond supply and bond demand.
Increasing expected inflation
Reduces the real cost of borrowing shifting bond supply to the right.
But, lowers real return on lending (also possible capital loss), shifting bond demand to the left
What happens to bond prices and interest rates?

58. Response to an Increase in the Expected Rate of Inflation – the Fisher Effect58

59. Expected Inflation and Interest Rates (Three-Month Treasury Bills), 1953–2011

60. Case Study: Interest Rates and the Business Cycle
A business cycle downturn (recession):
Reduces business investment opportunities shifting bond supply to the left.
Reduces wealth, shifting bond demand to the left also.
In theory, both shifts could give an ambiguous answer.
The data is not ambiguous. In recessions, interest rates tend to fall meaning that bond prices rise.

61. FIGURE 7 Business Cycle and Interest Rates (Three-Month Treasury Bills), 1951–200861

62. Understanding Changes in Equilibrium Bond Business Cycle Downturn
Shift S left S0
Price per Bond
Quantity of Bonds D0 E0 D1
Shift D left E1
We know price increases so supply shifts more than demand.

63. Bonds Are Risky!!!
Bonds are a promise to pay a certain amount in the future. How can that be risky?
Default risk - the chance the bond’s issuer may not make payment.
Inflation risk - investor cannot be aware of the real of the payments made.
Interest rate risk - a rise in interest rates before bond is sold could mean a capital loss.

64. Case Study: What Happened Here?
Investors’ concerns about risk affect their demand for U.S. Treasuries.
1998, Russia defaulted and people lost confidence in emerging market countries.
Safest assets are U.S. Treasuries.
Demand increased, price increased, and yields declined.

65. Why Bonds are Risky
Risk arises because an investment has many possible payoffs during the holding horizon.
We need to look at the risk the bondholder faces, what are the possible payoffs, and how likely each is to occur.
We will compare the risk on a bond’s return relative to the risk-free rate.

66. Default Risk
Although there is little to no default risk with U.S. Treasury bonds, there is with other government and corporate bonds.
We can look at an example of a corporate bond.
Assume the one-year risk-free interest rate is 5 percent.
A company has issued a 5 percent coupon bond with a face value of $100.

67. Default Risk
If this bond was considered default risk-free, the price of the bond would be the present value of the $105 payment.
Price of risk free bond = ($100 + $5)/$1.05 = $100
Suppose, however, there is a 10% probability that the company will go bankrupt before paying back the loan.
Assume the outcome is either $105 or $0.
Expected value equals $94.50
Price of bond = $94.50/1.05 = $90

68. Default Risk
Suppose 10% probability firm goes bankrupt – you get nothing
Expected Value of bond payment
Possibilities
Payoff
Probability
Payoff x Probability
Full Payment
$105
.90
$94.50
Default $0
.10
Expect to receive $94.50 one-year from now.
Discount at risk-free rate =
P = $90

69. Default Risk
If the price of the bond is $90, what yield to maturity does this price imply?
YTM = $105/$ =
Default risk premium is the promised yield to maturity minus the risk-free rate:
Risk Premium = percent - 5 percent
= percent.
The higher the default risk, the higher the yield and risk premium.

70. Default Risk Premium
We can calculate the probability of repayment from the interest rates.
Let 1+k be the return on a one-year corporate debt and 1+ i be the return on a one-year default risk-free treasury.
At equilibrium: 1+i = p(1+k)
The probability of repayment is
the probability of default is 1 – p
The probability of repayment:

71. Yields on 9/20/16 3mo 6mo 9mo 1yr 2yr 3yr 5yr 10yr 20yr 30yr+
BONDS
U.S. Treasury
0.29%
0.51%
0.60%
0.68%
0.82%
0.94%
1.23%
1.67%
2.01%
2.40%
U.S. Treasury Zeros --
0.56%
0.73%
0.89%
1.25%
1.82%
2.32%
Agency/GSE
0.71%
0.81%
0.96%
1.01%
1.13%
1.42%
1.62%
2.42%
3.13%
3.20%
Corporate (Aaa/AAA)
0.39%
1.05%
1.40%
1.59%
2.45%
3.35%
3.91%
Corporate (Aa/AA)
0.74%
1.15%
1.08%
1.22%
1.44%
1.96%
2.75%
3.60%
4.49%
Corporate (A/A)
0.93%
1.32%
1.84%
1.89%
2.65%
3.44%
4.25%
4.32%
Corporate (Baa/BBB)
1.21%
1.65%
2.15%
2.54%
4.48%
4.88%
5.13%
6.08%
6.42%
Municipal (Aaa/AAA)
0.77%
0.75%
0.97%
1.12%
1.43%
2.85%
2.93%
Municipal (Aa/AA)
0.86%
0.85%
0.88%
1.63%
1.61%
2.59%
3.07%
3.28%
Municipal (A/A)
0.80%
1.00%
0.99%
1.20%
1.68%
2.52%
3.45%
3.50%
Taxable Municipal*
1.11%
1.10%
1.72%
2.05%
2.09%
4.30%
3.94%
3.36%

72. Inflation-indexed bonds are structured so that the government promises to pay you a fixed interest rate plus the change in the consumer price index (CPI).
The U.S. Treasury sells two types of bonds that are adjusted for inflation.
Series I savings bonds
Treasury Inflation Protected Securities (TIPS)

73. Inflation Risk
With few exceptions, bonds promise to make fixed-dollar payments.
Remember that we care about the purchasing power of our money, not the number of dollars.
This means bondholders care about the real interest rate.
How does inflation risk affect the interest rate?

74. Inflation Risk Think of the interest rate having three components:
The real interest rate
Expected inflation, and
Compensation for inflation risk.
Example:
Real interest rate is 3 percent.
Inflation could be either 1 percent or 3 percent.
Expected inflation is 2 percent, with a standard deviation of 1.0 percent.

75. Inflation Risk Nominal interest rate should equal:
i = 3 percent real interest rate +
2 percent expected inflation +
Compensation for inflation risk
The greater the inflation risk, the larger the compensation for it.

76. Interest-Rate Risk
Interest-rate risk arises from the fact that investors don’t know the holding period return of a long-term bond.
The longer the term of the bond, the larger the price change for a given change in the interest rate.
For investors with holding periods shorter than the maturity of the bond, the potential for a change in interest rates creates risk.
The more likely the interest rates are to change during the bondholder’s investment horizon, the larger the risk of holding a bond.