1. Financial and Investment Mathematics Dr. Eva Cipovova
2. Lecture
Financial and Investment Mathematics
Dr. Eva Cipovova

2. Bonds
Bonds are a form of financing the operations of a company. Bonds are a debt security because the principle must eventually be returned to the bondholders

3. Bond Terms Every bond will feature a set of terms which include:
Principle Amount
Number of Years Until Maturity
Contract Interest Rate
Market Interest Rate
Interest Payment Schedule

4. Bond Principle
The bond principle represents the investment made by the bondholder
Eventually this principle must be paid back with interest
Bond principle can be used by the company to finance operations
Known as Face value - FV

5. Years Until Maturity Every bond will feature a maturity date
The maturity date is the date that the bond principle must be paid back
Usually there will be a period of several years between the issue date and the maturity date

6. Contract Interest Rate
Every bond will feature a contract rate
This is the rate of interest that must be paid every year to the bondholders
The interest paid out represents the cost of financing for the company
The interest received by the bondholder represents a return on the investment
It is called also coupon rate (can be divided on fixed and variable coupon rate)

7. Market Interest Rate
On the open market, there is always a prevailing interest rate
Most of the time this rate will be different than the contract rate on any given bond
The relationship between the market rate and contract rate will help to determine the issue of the bond
Know as yield of the bond

8. Interest Payment Schedule
The company must pay each bondholder the contract rate of interest each year. The payment can be scheduled a variety of ways including:
Annual (once per year)
Semi-annual (twice per year)
Quarterly (four time each year)
Monthly (twelve times each year)

9. Interest Rate Relationships
Contract Rate = Market Rate:
The bond sells at par
Contract Rate < Market Rate:
The bond will sell at a discount price
Contract Rate > Market Rate:
The bond will sell at a premium price

10. Bond Issued at Par
A company sells a $200,000 bond with a 5 % contract rate which matures in 10 years. The interest payments are to be made semi-annually and the market rate for a similar bond is 5 %. The journal entry required to record the sale is:
Cash $200,000
Bond Payable $200,000

11. Interest Payment
The journal entry to record the first semi-annual interest payment follows:
Bond Interest Expense $ 5,000
Cash $5,000
Ultimately, this journal entry will be recorded 20times over 10 years

12. Bond Maturity
Once the bond matures in 10 years, the following entry will be made:
Bond Payable $200,000
Cash $200,000

13. Bond Sold at Discount
A company sells a $100,000 bond with a 8 % contract rate which matures in 2 years. The interest payments are to be made semi-annually and the market rate for a similar bond is 10 %. The journal entry to sell the bond is:
Cash $96,454
Discount $3,546
Bond Payable $100,000

14. Bond Interest Payment
The journal entry to record the first semi-annual interest payment and amortize the discount follows:
Bond Interest Expense $4,886.50
Discount $883,5
Cash $4,000

15. Bond Maturity The journal entry to retire the bond at maturity is:
Bond Payable $100,000
Cash $100,000

16. Bond Sold at Premium
A company sells a $100,000 bond with a 12 % contract rate which matures in 2 years. The interest payments are to be made semi-annually and the market rate for a similar bond is 10 %. The journal entry to sell the bond is:
Cash $103,546
Premium $3,546
Bond Payable $100,000

17. Bond Interest Payment
This journal entry is required for the first semi-annual interest payment and amortization of the premium:
Bond Interest Expense $5,113.50
Premium $886.50
Cash $6,000

18. Secured/Unsecured Bonds
Secured bonds are backed up by an asset which serves as collateral
Unsecured bonds are only backed up by the reputation of the company
Typically, the interest rate is higher on an unsecured bond

19. Registered/Bearer Bonds
A registered bond is registered in the name of the specific bondholder for security purposes
A bearer bond is not registered and could be redeemed by anyone
In the United States, bearer bonds are no longer issued

20. Risk/Reward A relationship exist between risk and reward
The greater the risk, the greater the potential reward
The lower the risk, the lower the potential reward

21. Debt Investment Classification
No we will purchasing our investment, not selling anymore
Debt investments (bonds) can be classified in the following categories:
Held to maturity (HTM) – this classification signals the intent of the company to hold the bond as an investment until it has fully matured
Trading – signals the intent to actively trade the bond as soon as a profit could be made
Available for Sales (AFS) – signals that the company does not intend to hold the investment to maturity or actively trade it. In other words, this is a catch-all category

22. Classification of bonds
Government Bonds
In general, fixed-income securities are classified according to the length of time before maturity. These are the three main categories:
Treasury Bills. T-bills have maturities of up to 12 months. They are zero coupon bonds, so the only cash flow is the face value received at maturity.
Treasury Notes. Notes have maturities between one year and ten years. They are straight bonds and pay coupons twice per year, with the principal paid in full at maturity.
Treasury Bonds. T-Bonds may be issued with any maturity, but usually have maturities of ten years or more. They are straight bonds and pay coupons twice per year, with the principal paid in full at maturity.

23. Classification of bonds
Corporate Bonds
We will consider three major types of corporate bonds:
Mortgage Bonds. These bonds are secured by real property such as real estate or buildings. In the event of default, the property can be sold and the bondholders repaid.
Debentures. These are the normal types of bonds. It is unsecured debt, backed only by the name and goodwill of the corporation. In the event of the liquidation of the corporation, holders of debentures are repaid before stockholders, but after holders of mortgage bonds.
Convertible Bonds. These are bonds that can be exchanged for stock in the corporation.
In the United States, most corporate bonds pay two coupon payments per year until the bond matures, when the principal payment is made with the last coupon payment.

24. Classification of bonds
Bonds according of length of maturity
Short-term bonds (up to 1 year)
Medium-term bonds (up to 10 years)
Long-term bonds
Domestic bonds, foreign bonds, eurobonds….

25. Classification of bonds
Bonds with the possibility of early repayment
Callable bonds - A bond that can be redeemed by the issuer prior to its maturity. Usually a premium is paid to the bond owner when the bond is called.  In other words, on the call date(s), the issuer has the right, but not the obligation, to buy back the bonds from the bond holders at a defined call price. Technically speaking, the bonds are not really bought and held by the issuer but are instead cancelled immediately.
Also known as a "redeemable bond.“
Puttable bonds - (put bond, putable or retractable bond) is a bonds with an embedded  put option. The holder of the puttable bond has the right, but not the obligation, to demand early repayment of the principal. The put option  is exercisable on one or more specified dates.
Convertible bonds

26. Classification of bonds
Bonds according to the repayment of bonds
Perpetuity bonds (Consol bonds)
Zero-coupon bonds (discount bonds)
Coupon bonds
See attached .pdf file with formulas

27. Summary
Rule 1: If the yield y is equal to the coupon rate c, then the bond price P is equal to face value FV, if yield y is higher, resp. less than the coupon rate c, then the bond price P is smaller, resp. greater than the face value FV.
Rule 2: If the price of the bond increases, resp. decreases, this results in a decrease, resp. increase of the yield of the bond. Reverse: decrease, resp. rise in interest rates (yields) results in an increase, resp. decrease in bond prices.

28. Rule 3:
If the bond comes closer to its maturity, then the bond price comes closer to the face value of bond.
Rule 4:
The closer is the bond to its maturity the higher is the velocity of approaching the face value by the price of the bond.
Relationship of the
bond price and time
to maturity of the
bond

29. Rule 5: The decrease in a bond yield leads to an increase in bond
price by an amount higher than is the amount
corresponding to the decrease (in absolute value) in the
price of the bond if the yield increases by same percentage
as previously decreased.
Example: Assume a 5-year bond with a face value FV = CZK, coupon rate c = 10 % and yield y = 14 %.
Yield
12 %
13 %
14 %
15 %
16 %
Price
927,90
894,48
862,68
832,39
803,54
Price change
65,22
31,80
-30,29
-59,14

30. Examples
Zero-coupon bond with a principal $1,000 has a maturity 10 years and yield 8 %. Calculate bond’s price
P= 1000/1,08^10 = $463.2
2. Coupon bond with the principal $1,000 has a maturity 10 years, coupon rate = 14.9 % and yield = 8%. Calculate bond’s price
P= $1,463

31. Bond pricing

32. The aliquot interest yield is a part of the nominal yield which pertaining to the bond holder for the period from the bond issue or from the last coupon payment calculated until the transaction settlement date. Aliquot interest yield is added automatically to the bond price and can be positive or negative. For a client buying a bond, a positive aliquot interest yield means compensation that he has to pay when s/he becomes entitled to the whole coupon without holding the bond all the time. For a client buying a bond, a negative aliquot interest yield represents compensation for the period when the bond is held without entitlement to the coupon.

33. A + B = 360

34. Example: Assume a 5-year bond with a face value FV = 10
Example: Assume a 5-year bond with a face value FV = CZK issued at with maturity and with coupon rate c = 14.85%. The yield of this bond was y = 7% on Calculate the clean price PCL of the bond.

35. Results: