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

2. Lecture

Financial and Investment Mathematics

Dr. Eva Cipovova

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

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

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

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

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)

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

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)

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

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

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

Bond Maturity

• Once the bond matures in 10 years, the following entry will be made:

Bond Payable $200,000

Cash $200,000

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

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

Bond Maturity

• The journal entry to retire the bond at maturity is:

Bond Payable $100,000

Cash $100,000

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

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

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

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

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

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

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:

1. 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.

2. 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.

3. 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.

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.

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….

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

Classification of bonds

• Bonds according to the repayment of bonds

1. Perpetuity bonds (Consol bonds)

2. Zero-coupon bonds (discount bonds)

3. Coupon bonds

4. See attached .pdf file with formulas

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.

• 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

• 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 = 1.000 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 0 -30,29 -59,14

Examples1. 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

Bond pricing

• 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.

• A + B = 360

• Example:Assume a 5-year bond with a face value FV = 10.000 CZK issued at 6. 2. 1998 with maturity 6. 2. 2003 and with coupon rate c = 14.85%. The yield of this bond was y = 7% on 9. 11. 1999. Calculate the clean price PCL of the bond.

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