1. 3
Valuing Bonds

2. 3-1 Using the present value formula to value bonds
This topic represents the first application of concepts presented in this chapter. Make sure to explain the bond terminology: coupon rate, maturity, par value (face value), and yield to maturity.
PV(bond) = PV(coupon payments) + PV(final payment)
PV = Σ(Ct)/[(1 + r)t] + F/([(1 + r)N] (where: t varies from 1 to N)
Where: Ct = coupon interest payment [Ct = (coupon rate) × (face value)]
F = face value
r = yield to maturity
N = maturity

3. 3-1 Using the present value formula to value bonds
Example
Today is October 1, 2010; what is the value of the following bond? An IBM bond pays $115 every September 30 for five years. In September 2015 it pays an additional $1,000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%).
This is an example of the valuation of a AAA rated bond. The par value = $1,000; coupon rate = 11.5%; maturity = 5 years. PMT = 115; FV = 1000; N = 5; yield to maturity I = 7.5%. Compute the price of the bond.
There are two types of problems in bond valuation:
 Given the yield to maturity calculate the price of the bond;
 Given the price of a bond calculate the yield to maturity.
The par value = $1,000; coupon rate = 11.5%; maturity = 5 years; PMT = 115; FV = 1000; N = 5; I = 7.5%. Compute PV = $1, [Note: this is a premium bond.]
Additional example: A bond pays $100 interest payment per year. Its maturity is 5 years and the face value = $1,000. Calculate the price at 12% yield to maturity (YTM).
PV = $927.90, [Financial calculator inputs: PMT = 100; FV = 1000; N = 5; I = 12%.]
[Note: this is a discount bond]

4. 3-1 Using the present value formula to value bonds
Example: France
In October 2011 you purchase 100 euros of bonds in France which pay a 5% coupon every year. If the bond matures in 2016 and the YTM is 3.0%, what is the value of the bond?
French government bond example: the par value = €100; coupon rate = 8.5%; maturity = 4 years; YTM = 3.0%. Compute: PV = €
[Financial calculator inputs: PMT = 8.5; FV = 100; N = 4; I = 3.]

5. 3-1 Using the present value formula to value bonds
Another Example: Japan
In July 2010 you purchase 200 yen of bonds in Japan which pay an 8% coupon every year. If the bond matures in 2015 and the YTM is 4.5%, what is the value of the bond?
The face value = ¥200; coupon rate= 8%; maturity = 5 years; YTM = 4.5%. Compute: PV = ¥
[Financial calculator inputs: PMT = 16; FV = 200; N = 5; I = 4.5%.]

6. 3-1 Using the present value formula to value bonds
Example: USA
In February 2012 you purchase a three-year U.S. government bond. The bond has an annual coupon rate of 11.25%, paid semiannually. If investors demand a 0.085% semiannual return, what is the price of the bond?
The par value = $1,000; coupon rate = 4.875% (makes semiannual interest payments); maturity = 3 years; YTM = % (semiannual rate). Compute PV = $
[Financial calculator inputs: PMT = ; FV = 1000; N = 6; I = Note: this is a discount bond.]

7. 3-2 how bond prices vary with interest rates
Example, Continued: USA
Take the same three-year U.S. government bond. If investors demand a 4.0% semiannual return, what is the new price of the bond?
The par value = $1,000; coupon rate = 4.875% (makes semiannual interest payments); maturity = 3 years; YTM = 2.0 % (semiannual rate). Compute: PV = $
[Financial calculator inputs: PMT = ; FV = 1000; N = 6; I = Note: this is a premium bond. Observe that the relationship between bond price and yield to maturity is an inverse one.]

8. Figure 3.1 Interest rate on 10-year treasuries
Yield, %
Discuss the relationship between bond prices and YTM, while showing the historical Treasury yield. This leads into the next slide where we see a graphic representation of the relationship.
Year

9. 3-2 how bond prices vary with interest rates
The relationship between bond price and yield to maturity is an inverse one. As the yield to maturity (interest rate) increases, the bond price decreases, and vice versa.
Interest rate, %

10. Figure 3.2 Maturity and Prices
When interest rate = 11.25% coupon, both bonds sell for face value
Bond price
This is a graphical representation of the bond price vs. the interest rate (YTM). Here two bonds, a long-term (30-year) bond and a short-term (3-year) bond, are presented. Long-term (30-year) bond prices fluctuate much more than the short-term (3-year) bond prices for a given change in interest rates.
Interest rate, %

11. 3-2 how bond prices vary with interest rates
The duration formula is presented by itself. Since there are no other slides to discuss its uses, now may be a good time to explain how duration is used and the logic behind the math. The formula can be intimidating, but presenting an example should help alleviate this.

12. 3-2 Duration calculation
Year
Payment Ct
PV(Ct) at 4.0%
Fraction of Total Value
[PV(Ct)/V]
Year × fraction of total value
[t × PV(Ct)/PV] 1
$90
$86.54
0.0666 2 90
83.21
0.0640
0.1280 3
80.01
0.0615
0.1846 4
76.93
0.0592
0.2367 5
73.97
0.0569
0.2845 6
71.13
0.0547
0.3283 7
1090
828.31
0.6371
4.4598
PV = $
Total = duration = 5.60
This is a complicated procedure; make sure to go over it thoroughly. Sum of a geometric series procedure may be used to derive the formula for duration.
Duration = [(1)Cf1/(1 + r) + (2)Cf 2/(1 + r) (n)Cfn /(1+r)n] / PV
Example: Face value = $1,000; coupon rate = 10%; maturity = 3 years; YTM = 5%;
Calculate the bond’s duration.
PV = $
[Financial calculator inputs: PMT = 100; FV = 1000; N = 3; I = 5%.]
Duration = [(1)(100)/( ) + (2)(100)/( )2 + (3)(1100)/(1+0.05)3]/( )
= years
Another example: Face value = $1,000; coupon rate = 5.5%; maturity = 4 years; YTM = 2.75%
Calculate the duration of the bond.
  PV = $
[Financial calculator inputs: PMT = 55; FV = 1000; N = 4; I =2.75%.]
Duration = [55(1)/(1.0275) + 55(2)/(1.0275^2) + 55(3)/(1.0275^3) (4)/(1.0275^4)]/[ ]
= years.
Duration is thought of as the weighted average maturity where the weights are the ratio of the present value of the cash flows to the total PV. For coupon bonds the duration is always less than the maturity. For a zero-coupon bond, duration and maturity are the same. Spreadsheet programs are very useful for calculating the duration.

13. 3-3 term structure of Interest Rates
Short- and long-term rates are not always parallel
September 1992–April 2000: U.S. short-term rates rose sharply while long-term rates declined
As a precursor to the topic of term structure, show how interest rates do not necessarily move in lockstep. This can cause unusual complications in pricing bonds. The topic is necessary to prevent complacency on the part of students when pricing bonds and doing bond analysis.

14. 3-3 Term structure of interest rates
YTM (r)
Year
1981
1987 & Normal
1976
Spot Rate: Actual interest rate today (t = 0)
Yield To Maturity (YTM): IRR on interest-bearing instrument
Term structure of interest rates and duration are two very complicated concepts that need detailed explanation. The term structure of interest rates depicts the relationship between the yield to maturity and maturity of bonds with the same risk. The shape of the yield curve changes over a business cycle. Yield curve could be downward sloping as in 1981, normal as in 1987, and relatively flat as in Upward sloping yield curve is considered “normal” yield curve.

15. Figure 3.4 spot rates on u.s. treasury strips, 02/2012
Shows the yield curve as of February The graph depicts the term structure of Treasury strips. Data for daily yield curve for U.S. Treasury securities can be obtained at the following website:

16. 3-3 Law of One Price
All interest-bearing instruments priced to fit term structure
Accomplished by modifying asset price
Modified price creates new yield, which fits term structure
New yield called yield to maturity (YTM)
Yield to maturity is a complex concept. It can be thought of as a weighted average of spot and forward rates that make discount factors that give the same present value. It can also be thought of as the IRR irate of return) on an interest-bearing instrument.

17. 3-3 Yield to Maturity Example
$1,000 Treasury bond expires in 5 years. Pays coupon rate of 10.5%. What is YTM if market price is ?
Calculate IRR = 8.5%
There are two types of problems in bond valuation:
 Given the yield to maturity calculate the price of the bond;
 Given the price of a bond calculate the yield to maturity.
Here is an example where the price of the bond is given and you are asked to calculate the yield to maturity.
Face value = $1,000; maturity = 5 years; coupon rate = 10.5%; price of the bond = $1,078.80, (107.88% of the face value). Calculate the yield to maturity of the bond.
PMT = 105; N = 5; PV = ; FV = 1,000.
Compute I = 8.5%.
IRR function in the calculator can also be used for calculating the YTM of the bond.
[Use the cash-flow register. CF0 = ; C1 = 105, F1 = 4; C2 = 1105, F2 = 1; IRR→Compute→8.5%.] C0 C1 C2 C3 C4 C5 −
105
1105

18. 3-4 Term Structure Expectations Theory
Term Structure and Capital Budgeting
CF should be discounted using term structure info
When rate incorporates all forward rates, use spot rate that equals project term
Take advantage of arbitrage
Many theories try to explain the shape of the yield curve. The most popular theory is the unbiased expectations theory. This theory states that forward interest rates are unbiased estimates of expected future spot rates. Term structure implies that for capital budgeting CF should be discounted to include term structure information. The spot rate used for discounting cash flows should be equal to the term of the project.

19. 3-5 Debt and Interest Rates
Classical Theory of Interest Rates (Economics)
Developed by Irving Fisher:
Nominal Interest Rate = Actual rate paid when borrowing money
Real Interest Rate = Theoretical rate paid when borrowing money; determined by supply and demand
Real interest rate is the theoretical rate you pay when you borrow money. It is determined by the supply and demand. Real interest rate cannot be observed; it can only be derived.
Supply
Demand
$ Qty r
Real r

20. Figure 3.5 annual u.s. Inflation Rates, 1900-2011
Annual U.S. inflation rates from 1900–2011 are shown.

21. Figure 3.6 Global Inflation Rates, 1900-2011
The U.S. has, on average, low inflation.

22. 3-5 Debt and Interest Rates
Nominal r = Real r + expected inflation (approximation)
Real r theoretically somewhat stable
Inflation is a large variable
Term structure of interest rates shows cost of debt
Nominal rate = real rate + expected inflation.
Real rate r is theoretically somewhat stable and inflation is a large variable. This theory helps us understand the term structure of interest rates which in turn helps us understand the cost of debt.

23. 3-5 Debt and Interest Rates
Debt and Interest Formula:
Exact formula: (1 + rnominal) = (1 + rreal)(1 + expected inflation).
Exact formula should be used for higher values.

24. Figure 3.7 UK Bond Yields Interest rate, %
10-year nominal interest rate
Interest rate, %
The graph shows the nominal and real interest rates over time for the UK.
10-year real interest rate

25. Figure 3.8 Govt. Bills vs. Inflation, 1953-2011
The graph shows the T-bill rates and inflation over time for the UK.

26. Figure 3.8 Govt. Bills vs. Inflation, 1953-2011
The graph shows the T-bill rates and inflation over time for the U.S.

27. Figure 3.8 Govt. Bills vs. Inflation, 1953-2011
The graph shows the T-bill rates and inflation over time for Germany.

28. 3-6 The risk of default Corporate Bonds and Default Risk
Payments promised to bondholders represent best-case scenario
Most bonds’ safety judged by bond ratings
This slide introduces the concept of default risk with respect to corporate bonds. Promised returns on corporate bonds represent a best-case scenario. Most bonds’ safety can be judged by their rating.

29. Table 3.6 Prices and yields of corporate bonds, 01/2011
Issuer
Coupon
Maturity
S&P Rating
Price, % of Face Value
Yield to Maturity
Johnson & Johnson
5.15%
2017
AAA
122.88%
1.27%
Walmart
5.38 AA
117.99
1.74
Walt Disney
5.88 A
121.00
2.07
Suntrust Banks
7.13
BBB
109.76
4.04
U.S. Steel
6.05 BB
97.80
6.54
American Stores
7.90 B
97.50
8.49
Caesars Entertainment
5.75
CCC
41.95
25.70
Sample listing of corporate bonds and the corresponding yields.

30. Investment grade bonds
Table 3.7 Bond ratings
Moody's
Standard & Poor's
and Fitch
Investment grade bonds
Aaa
AAA Aa AA A
Baa
BBB
Junk bonds Ba BB B
Caa
CCC Ca CC C
Bond ratings are shown in rank order. Explain the difference in the numbering scale and the similarity in the meanings.

31. 3-6 the risk of default Sovereign Bonds and Default Risk
Sovereign debt is generally less risky than corporate debt
Inflationary policies can reduce real value of debts
This slide introduces the concept of default risk regarding sovereign bonds. Countries are less likely to default on their debt, but when they do the effects can be catastrophic.

32. 3-6 The risk of default Sovereign Bonds and Default Risk
Foreign Currency Debt
Default occurs when foreign government borrows dollars
If crisis occurs, governments may run out of taxing capacity and default
Affects bond prices, yield to maturity
This slide introduces foreign currency debt, which is the most common type of debt to cause countries to default. The book includes real-world examples of foreign currency default.

33. 3-6 The risk of default Sovereign Bonds and Default Risk
Own Currency Debt
Less risky than foreign currency debt
Governments can print money to repay bonds
This slide introduces own currency debt, which is less likely to cause countries to default. The book includes real-world examples of own currency default.

34. 3-6 The risk of default Sovereign Bonds and Default Risk Eurozone Debt
Can’t print money to service domestic debts
Money supply controlled by European Central Bank
This slide examines the monetary policies of the Eurozone, the member countries of which cannot generate own-currency debt. Eurozone countries have ceded control of their money supply to the European Central Bank.