1. Bond Valuation
Applying time value of money and annuity concept in order to value bond and determine bond yield.
Importantly, we will examine the bond terminology and characteristics and understand bond relationships.

2. What is Bond?
When a corporation or government wishes to borrow money from public, it usually does so by issuing, or selling bonds
When investors buy a bond, they lend money to the bond issuer, the government or corporation
As a lender, investors expect to be paid back the original amount (principle) and interest over some specified period time
As a borrower, the bond issuer must repay principle and interest to buyers over some specified period time
A bond is a type of security instrument used to raise money by an issuing party.

3. Definition of Bond
A security that the government or corporation issue or sell to borrow money from investors today and pay promised payments later
A loan that the borrower (issuer) promises to repay the lender (investor) the amount borrowed (principle) plus interest over some specified period time

4. Key Features Par value (face value, F) Annual Coupon (C)
Amount borrowed by issuers (sellers) from investors (buyers) at the beginning
Re-paid at the end of loan
Assume $1,000 for corporate bonds
Annual Coupon (C)
Annual interest payments to buyers
Coupon Rate = Annual coupon/Par Value = C/F

5. Key Features Maturity (T): Yield to maturity (YTM):
Number of years until par value is repaid by issuers (sellers)
Yield to maturity (YTM):
Discount rate used to value a bond
Quoted as an annual rate
Market rate of return

6. Annual Coupon Bond Price
Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1,000. If the yield to maturity is 11%, the current price is?
Par value: 𝐹=$1,000
Maturity: 𝑇=15 years
Yield to maturity: 𝑌𝑇𝑀=11%
Coupon Rate =10%
Annual coupon: 𝐶=10%×$1000=$100
Price = ?

7. Annual Coupon Bond Price
The cash flows from investing on this bond are 15 years $100 plus $1,000 in the end of year 15
How much will investors pay to buy the bond?
Present value of this stream of cash flows 14 …
100 1 2 15
1,000
P=?

8. Annual Coupon Bond Price
The present value of 15 years $100 is the present value of the annuity that pays $100 at the end of each year for 15 years, so we can make use of annuity formula
𝑃𝑉 𝑐𝑜𝑢𝑝𝑜𝑛𝑠 = 𝐶 𝑟 1− 𝑟 𝑇
= − =$
𝑃= 1000×10% 11% × 1− % % 15 =$𝟗𝟐𝟖.𝟎𝟗
A stream of cash flows that investors will receive after buy a bond is the annual coupons ($100) and face value ($1,000) at maturity (15 years)

9. Annual Coupon Bond Price
The present value of $1,000 is the present value of one lump-sum future value at year 15, so we can make use of present value formula
𝑃𝑉 𝑓𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 = 𝐷 1+𝑟 𝑇 = =$
Bond price is the present value of annual coupons plus the present value of face value
𝑃= =$928.09

10. Annual Coupon Bond Price
𝑃𝑟𝑖𝑐𝑒=𝑃𝑉 𝑐𝑜𝑢𝑝𝑜𝑛𝑠 +𝑃𝑉 𝑓𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 𝑷𝒓𝒊𝒄𝒆= 𝑪 𝒀𝑻𝑴 𝟏− 𝟏 (𝟏+𝒀𝑻𝑴) 𝑻 + 𝑭 𝟏+𝒀𝑻𝑴 𝑻

11. Problem 6-3 Bond Prices
Lycan, Inc., has 7.6% coupon bonds on the market that have 9 years left to maturity. The bonds make annual payments. If the YTM on these bonds is 9.6%, what is the current bond price?
882.97

12. Semiannual Coupon Bond Price
Bonds issued in US usually make coupon payments twice a year
Semiannual coupon bond price:
𝑷= 𝑪 𝟐 𝒀𝑻𝑴 𝟐 𝟏− 𝟏 𝟏+ 𝒀𝑻𝑴 𝟐 𝟐𝑻 + 𝑭 𝟏+ 𝒀𝑻𝑴 𝟐 𝟐𝑻
C is the annual coupon

13. Semiannual Coupon Bond Price
Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1,000. This bond makes semiannual payments. If the yield to maturity is 11%, the current price is?
𝑃𝑟𝑖𝑐𝑒= −
=$927.33

14. Problem 6-6 Bond Prices
App Store Co. issued 20-year bonds one year ago at a coupon rate of 6.1%. The bonds make semiannual payments. If the YTM on these bonds is 5.3%, what is the current bond price?

15. Bond YTM
Suppose we have a bond with the following characteristics. What is the YTM of the bond?
Par value: 𝐹=$1,000
Maturity: 𝑇=30 years
Coupon Rate =9.8%
Annual coupon: 𝐶=98
Price: 𝑃=$1,278
If the bond make annual coupon payment, 𝑌𝑇𝑀= ?
If the bond make semiannual coupon payment, 𝑌𝑇𝑀= ?

16. Annual Coupon Annual coupon bond
𝑪= 𝑷𝒓𝒊𝒄𝒆− 𝑭 𝟏+𝒀𝑻𝑴 𝑻 ×𝒀𝑻𝑴 𝟏− 𝟏 (𝟏+𝒀𝑻𝑴) 𝑻

17. Problem 6-5 Coupon Rate
Merton Enterprises has bonds on the market making annual payments, with 14 years to maturity, and selling for $953. At this price, the bonds yield 9.4%. What must the coupon rate be on Merton’s bonds?
8.78%

18. Annual Coupon Semiannual coupon bond
𝑪= 𝑷𝒓𝒊𝒄𝒆− 𝑭 𝟏+ 𝒀𝑻𝑴 𝟐 𝟐𝑻 ×𝒀𝑻𝑴 𝟏− 𝟏 (𝟏+ 𝒀𝑻𝑴 𝟐 ) 𝟐𝑻

19. Problem 6-8 Coupon Rate
Volbeat Corporation has bonds on the market with 12 years to maturity, a YTM of 9.7%, and a current price of $948. The bonds make semiannual payments. What must the coupon rate be on the bonds?
8.96%

20. Bond Sold at Par Price = Par value When are bonds sold at par?
30-year annual coupon bonds with $100 face value and 5% coupon rate
If YTM is 5%, how much is the bond?
Issuer promises $5 annual coupon (interest payment)
Bondholders expect $5 return
Actual interest payment = Expected return
Investors are willing to lend $100 to the issuer, so pay $100 to buy the bond

21. Discount Bond Price < Par value
If YTM is 10%, how much is the bond?
Issuer promises $5 annual coupon (interest payment)
Bondholders expect $10 return
Actual interest payment < Expected return
Investors are not willing to lend $100 to the issuer
Investors will pay less than $100 to buy
Bond price is $52.87 < $100

22. Premium Bond Price > Par value If YTM is 3%, how much is the bond?
Issuer promises $5 annual coupon (interest payment)
Bondholders expect $3 return
Actual interest payment > Expected return
Investors would like to pay more than $100 to buy
Investors will lend more than $100 to the issuer
Bond price is $ > $100
they will pay $139.20

23. Current Yield Coupon portion of bond return
Current Yield = Annual Coupon / Bond Price
Discount bound
Current yield: =9.46%<10%=𝑌𝑇𝑀
Current yield: 9.46%>5%=𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒
Premium bound
Current yield: =3.59%>3%=YTM
Current yield: 3.59%<5%=𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒

24. Relations Bond sold at par Discount bond Premium bond
𝑌𝑇𝑀=𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑=𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒
Discount bond
𝑌𝑇𝑀>𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑>𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒
Premium bond
𝑌𝑇𝑀<𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑<𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒
Discount bond current yield ignores the gain from price discount. Premium bond YTM fails to take into account the loss from price premium.

25. Interest Rate Risk
Interest rate movements affect investors’ expectation on bond return, so do bond price
Interest rate risk refers to the sensitivity of bond price to interest rate variations
Two features determine interest rate risk
Maturity
Coupon Rate

26. Interest Rate Risk
The longer the maturity, the greater the interest rate risk
Bond with longer maturity has more distant cash flows, which are more adversely affected by the increasing of interest rate
The lower the coupon rate, the greater the interest rate risk
Bond with lower coupon rate proportionally depends more on the present value of par value, so its price is more adversely affected by the increasing of interest rate

27. Problem 6-16 Interest Rate Risk
Both Bond Bill and Bond Ted have 12.4% coupons, make semiannual payments, and are priced at par value. Bond Bill has 5 years to maturity, whereas Bond Ted has 22 years to maturity. If interest rates suddenly rise by 3%, what is the percentage change in the price of these bonds?

28. Interest Rates Rise by 3%
YTM = 12.4%
Both Bond Bill and Ted are worth par value $1,000
YTM = 15.4%
Bond Bill is worth $897.97
Bond Ted is worth $812.64
% Change
Bond Bill: − =−10.20%
Bond Ted: − =−18.74%

29. Maturity
The only difference between Bond Bill and Ted is coupon rate: Ted has a longer maturity
% decline in price for Bond Ted is greater than % decline for Bill, so Bond Ted price is more sensitive to interest rate changes

30. Problem 6-17 Interest Rate Risk
Bond J has a coupon rate of 4.5%. Bond S has a coupon rate of 14.5%. Both bonds have eight years to maturity, make semiannual payments, and have a YTM of 10%. If interest rates suddenly rise by 3%, what is the percentage change in the price of these bonds? What does this problem tell you about the interest rate risk of lower-coupon bonds?

31. Interest Rates Rise by 3%
YTM = 10%
Bond J is worth $701.96
Bond S is worth $1,243.85
YTM = 13%
Bond J is worth $584.87
Bond S is worth $1,073.26
% Change
Bond J: − =−16.68%
Bond S: − =−13.71%

32. Coupon Rate
The only difference between Bond J and S is coupon rate: S has a greater coupon rate
% decline in price for Bond S is smaller than % decline for S, so Bond S price is less sensitive to interest rate changes