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37

P = + Cv" = + re the interest functions are calculated at the yield rate /.

(7.1)

for bond coupons in the United States is semiannual. For example, an 8% bond with semiannual coupons has = .04. In international financial markets other coupon frequencies are commonly encountered, e.g. Eurobonds generally have annual coupons. It is assumed that coupons are constant. The case of varying coupons is considered in Section 7.9. The reader should note that the symbol was also used in Section 5.9 to represent the project return rate. Fr = the amount of the coupon.

g = the modified coupon me of a bond. The rate g is defined by Fr = Cg or g = FrIC, i.e. g is the coupon rate per unit of redemption value rather than per unit of par value, h should be noted that g will always be convertible at the same frequency as r. In practice, g will often equal r, which is the case whenever is equal to F.

i = the yield rate of a bond, often called the yield to maturity, i.e. the interest rate actually earned by the investor, assuming the bond is held until it is redeemed or matures. The yield to maturity is identical in concept to the internal rate of return as defined in Section 5.2. It is customary that yield rates are convertible at the same frequency as the coupon rate, and this will be assumed unless stated otherwise. The case in which the coupon payment period and the yield rate conversion period are not equal is considered in Section 7.9. It is also assumed that the yield rate is constant. The case in which this is not true is also discussed in Section 7.9. j

= the number of coupon payment periods from the date of calculation to the maturity date, or to a redemption date.

AT = the present value, computed at the yield rate, of the redemption value at the maftirity date, or a redemption date, i.e. = Cv" at the yield rate i. 1

G = the base amount of a bond. The amount G is defined by Gi = Fr G = Frti. Thus, G is the amount which, if invested at the yield rateJ i, would produce periodic interest payments equal to the coupons on the! bond. 1

The reader should be aware that in everyday business and financial usagej there are three different "yields" associated with a bond: J

1. Nominal yield is simply the annualized coupon rate on the bond. Fori example, if a $100 par value bond has coupons totalling $9 per year, then] the nominal yield on the bond is 9% per annum. The reader should notej that the use of the word "nominal" in this context is different than the]

meaning in Section 1.8 and yet another meaning to be introduced in Section 9.4, which is an unfortunate source of ambiguity.

Current yield is the ratio of the annualized coupon to the original price of the bond. For example, if the bond described above is selling for $90 in the market, then the current yield on the bond is 10% per annum. Note that the current yield does not reflect any gain or loss when the bond is sold, is redeemed, or matures.

Yield to maturity is the actual annualized yield rate, or internal rate of return, as defined in Section 5.2. The determination of the yield to maturity will be discussed in Section 7.6.

We will use the phrase "yield rate" to refer only to the third of these meanings. .However, again the reader is reminded that in practical applications not £,everyone uses terms with textbook precision, and it is important to ascertain iexactly what is meant by the user when encountering terms such as the "yield" Ion a bond.

The reader should be aware that F, C, r, g, and n are given by the terms a bond and remain fixed throughout its life. In essence, these parameters 3efine exactly what payments will be made by the borrower. On the other Ihand, P and i will vary throughout the life of the bond. The price and the yield ; rate have a precise inverse relationship to each other, i.e. as the price rises the iyield rate falls, and vice versa. Yield rates on bonds will fluctuate with iprevailing rates of interest in financial markets for securities of a similar type, ypluctuating market rates of interest will thus lead to fluctuating bond prices, iowever, these fluctuating bond prices do not generally reflect any increase or Becrease in the degree of safety or security attributed to the bond, but rather Biey merely reflect changing rates of interest in the securities markets. This pverse relationship between bond prices and yield rates has not always been toderstood by unsophisticated bondholders, who, in periods of rising rates of merest, have attributed the declining prices of their bonds to a deterioration in credit rating of the borrower, when this was not a factor at all.

There are four types of formulas which can be used to find the price of a i. The first of these, called the basic formula, is the most straightforward, cording to this method the price must be equal to the present value of fiiture Bupons plus the present value of the redemption value



= Gi + Cv"

= G(l - v") + Cv" = G + (C-G)v".

(7.3)

Formula (7.3) also requires only one value instead of two.

The fourth formula, called the Makeham formula (after a famous British actuary of the 19th century), can also be obtained from formula (7.1)

P = Cv" +

= Cv" + Cg

1 - v"

= Cv" + i(C- Cv") /

= K+liC-K). i

(7.4)

Formula (7.4) also requires only one value instead of two.

Makehams formula has an interesting verbal interpretation. From the basic formula, the price of a bond must be equal to the present value of the \ redemption value plus the present value of future coupons. is present value ; of the redemption value. If the modified coupon rate g is equal to the yield rate i, then the bond will sell for its redemption value C, so that - is the present value of future coupons. However, if the modified coupon rate g is different than the yield rate /, then the present value of future coupons will be proportionally higher or lower, i.e. it will be equal to {C - K).

Example 7.3 Find the price of a $1000 par value 10-year bond with coupons at 8.4% convertible semiannually, which will be redeemed at $1050,

1000

1050

08 - .042 2

;000(.042) =

1050

1,0 = .05 2

1050(1.05)

= 395.7340

2(1000) =

840.

Basic prmula:

P = +

= 42

551.05

+ 395.7340

= (42)(12.4622) + 395.7340 = $919.15. Premium/discount formula:

P = + { - )

= 1050 + (42 - 52.50) 2 1 o5

= 1050 + (42 - 52.50)(12.4622) = $919.15.

Base amount prmub:

P = G + (C-G)v"

= 840 + (1050 - 840) (1.05) = 840 + (1050 - 840)(.37689) = $919.15. Makeham formula:

= +1{ - ) i

= 395.7340 +-(1050-395.7340) = $919.15. .05

M PREMIUM AND DISCOUNT

If the purchase price of a bond exceeds its redemption value, i.e. if ? > C, en the bond is said to sell at a premium and the difference between P and /called the "premium." Similarly, if the purchase price is less than the Pedemption value, i.e. if P < C, then the bond is said to sell at a discount, and Ae difference between and P is called the "discount." The reader should note kfiiat this is yet another meaning for the overused word "discount" in addition to various meanings given in Chapter 1.

The bond is bought to yield 10% convertible semiannually. Use all four formulas.

In Uiis example we have:

The second formula, called the premium/discount formula, can be obtained from formula (7.1)

P = - + Cv"

= Fra C{l-ia )

= C+{Fr~ Ci) . (7-2)

The rationale for the name of this method will become evident in Section 7.4. Formula (7.2) is often more efficient than formula (7.1), since only one value need be computed or looked up in die interest tables instead of two.

The third formula, called the base amount formula, can also be obtained from formula (7.1)

P = - + Cv"



1 + ( - Oai] -(g-t)v = 1 + (? - )aJf= l.

The same reasoning applies for each successive line of the schedule.

Several additional observations are possible. First, it should be noted that the book values agree with the price given by formula (7.2) computed at the original yield rate. Second, the sum of the principal adjustment column is equal top, the amount of premium or discount, as the case may be. Third, the sum of the interest paid column is equal to the difference between the sum of the coupons and the sum of the principal adjustment column. An algebraic proof I of this result is left as an exercise. Fourth, the principal adjustment column is a geometric progression with common ratio 1 + /. Thus, it is a simple matter to find any one principal adjustment amount knowing any other principal ladjustment amount and the yield rate.

When a bond is bought at a premium, the book value will gradually be [adjusted downward. This process is called amortization of premium or writing Idown. In these cases the principal adjustment amount is often called the lamount for amortization of premium."

When a bond is bought at a discount, the book value will gradually be ladjusted upward. This process is called accumulation of discount or writing up. Iln these cases the principal adjustment amount is often called the "amount for I accumulation of discount."

It should be noted that Table 7.1 is based on a value of = 1. The values in the table are proportional for other values of C. In any particular case, the bond amortization schedule can be constructed from basic principles.

We can derive expressions for the premium and the discount from formula (7.2):

Premium = /> - = (Fr - 1) = Og - i)ai if g > i (7.5) Discount =C-P = (Ci -Fr)ai = C{i - g)aj\i if i > g (7.6) It is clear that premium and discount, although bearing different names, are essentially the same concept, since discount is merely a negative premium. It should also be noted that in many cases f = and therefore g = r. However, formulas (7.5) and (7.6) will handle the cases in which this is not true.

The price of a bond depends upon two quantities, the present value of future coupons and the present value of the redemption value. Since the purchase price of a bond is usually less than or greater than the redemption value, there will be a profit (equal to the discount) or a loss (equal to the premium) at the redemption date. This profit or loss is reflected in the yield rate for the bond, when calculated as the yield to maturity.

However, as a result of this profit or loss at the redemption date, the amount of each coupon cannot be considered as interest income to an investor. It will be necessary to divide each coupon into interest earned and principal adjustment portions similar to the separation of payments into interest and principal in Chapter 6.

When this approach is used, the value of the bond will be continually adjusted from the price on the purchase date to the redemption value on the redemption date. These adjusted values of the bond are called the book values of the bond. The book values provide a reasonable and smooth series of values for bonds and are used by many investors, such as insurance companies and pension fiinds, in reporting the asset values of bonds for financial statements. The determination of asset values for bonds and other securities will be discussed fiirther in Section 7.11.

The reader should be careful to note, however, that the book value after purchase will differ from the price of the bond if the bond were bought anew. The price of the bond in the market will vary as prevailing interest rates change. However, the book values will follow a smooth progression, since they are all based on the original yield rate at purchase.

When convenient, we will use the same notation for bonds as developed in Chapter 6 for loans. Thus, the book value t periods after purchase is denoted by Bf, the amount of interest earned in the rth coupon by /,, and the amount of principal adjustment in the rth coupon by P,. The level coupon will continue to be denoted by Fr. Note that the price P is equal to Bq , while the redemption value is equal to B .

A bond amortization schedule is a table which shows the division of each coupon into its interest earned and principal adjustment portions, together with the book value after each coupon is paid. Consider a bond for which is equal to 1, the coupon is equal to g, and the price is equal to I + p (where p can be either positive or negative). Table 7.1 is a bond amortization schedule for this bond.

Consider the first coupon period of the bond. At the end of the first coupon period the interest earned on the balance at the beginning of the period is

/, = /[1 +(g-i)ai .

The rest of the total coupon of g, i.e.

Pi = g-i[\ +i8-0ai] = {g-i)a-iai) = ig-Ov"

must be used to adjust the book value. The book value at the end of the period equals the book value at the beginning of the period less the principal adjustment amount, i.e.



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