Math Is Fun Forum

Upon revisiting my notes on the subject of bonds, serial bonds in particular, it occurred to me that I could perhaps reclaim several lines of bytes from my calculator’s memory if I could but derive a single plug and chug formula for the calculation of the purchase price (flat price) of a serial bond on an interest date to replace (hopefully) the byte guzzler BASIC like program that I wrote on my calculator for such a calculation (a program I used primarily for automating purposes and for checking the results I obtained in going through the motion of the traditional methods).

Eventually, I did somehow managed to derive such a formula, or formulas.  Judging by the nature of the formula(s), it turned out that I was dealing with something similar to the concept of the general annuity formulas. Accordingly, general annuity formulas are used to deal with the so-called complex or general cases where the payment interval is not the same as the interest period (including those simple or special cases where the payment interval is the same as the interest period).  In the case of serial bonds, there is apparently such a thing as a general-purpose formula that can deal with different cases.

In deriving these formulas, I was guided by three known (well known I dare say) statements which in turn somehow proved to be special cases.  To illustrate these three statements, let’s consider some examples.

Example 1.  On 2/1/77, Lawrence buys a farm worth $40,000 cash from Watkins, paying $10,000 cash and signing a contract to pay interest semiannually at the rate 7% and to pay the remaining principal in 6 equal semiannual installments.  What does an investor pay on buying this contract to yield 8% compounded semiannually on 8/1/78?

The contract is a serial bond involving six distinct bonds of $5,000 each.  At the end of a year and six months (on 8/1/78), three bonds remain alive. The price of the contract is the sum of the prices of three $5,000 bonds, to be redeemed at the ends of ½, 1, and 1½ years respectively.

The investor’s yield rate is greater than the bond rate; thus a discount is in order.  By use of the general method, we have

Price (on 8/1/78) = $4,975.96 + $4,952.85 + $4,930.62 = $14,859.43.

Those acquainted with annuity formulas will recognize that

≈ $13,875.45
On the other hand

can be summarized by use of a statement which says that

Thus,

This then gives us

= $13,875.45 + $983.98 =$14,859.43.
This is in fact the essence of another statement that is a direct consequence of (1).  It pretty much says that the price $P to yield the investor/buyer the rate i per period of a serial bond with the bond rate q per interest/coupon period, whose face value will be redeemed in equal installments of $H at the end of each period for n periods is given by

By use of the discount method, we have

Price (on 8/1/78) = $4,975.96 + $4,952.85 + $4,930.62 = $14,859.43.

By making use of statement (1), we arrive at the same sum by the following:

The next example serves to illustrate the third statement.
Example 2.  A man borrows $10,000, contracting to pay $2,000 of the principal at the end of each year for 5 years, and to pay interest semiannually at the rate 7%.  His contract is sold 2 years later to yield the investor 6% compounded semiannually.  Find the price he pays.

The contract is a serial bond involving five distinct bonds of $2,000 each.  At the end of 2 years, three bonds remain alive.  The price of the contract is the sum of the prices of three $2,000 bonds, to be redeemed at the ends of 1, 2, and 3 years respectively.

The bond rate is greater than the investor’s yield rate; thus a premium is in order.  By use of the general method, we have

Price (At the end of 2 years) = $2,019.135 + $2,037.171 + $2,054.172 = $6,110.478 or $6,110.48.

A compact expression for the sum of

is

On the other hand

can be summarized by use of a statement which says that

Thus,

≈ $773.343.  This then gives us

= $5,337.134 + $773.343 = $6,110.478 or $6,110.48.

By use of the premium method, we have

Price (At the end of 2 years) = $2,019.135 + $2,037.171 + $2,054.172 = $6,110.478 or $6,110.48.

By making use of statement (2), we duplicate the same sum by

≈ $6,110.4775954 or $6,110.48.

With these special cases aside, we now consider examples that don’t seem to exhibit the same format as those represented by (1) and (2).

Example 3.  A $1,000,000 serial issue of 7% bonds pays coupons on 2/1 and 8/1 and will be redeemed in 5 equal annual installments.  The bonds were issued on 2/1/76.  An insurance company buys all bonds outstanding on 8/1/78, to yield 8%.  Find the price paid.

This serial bond involves five distinct bonds of $200,000 each.  At the end of 2.5 years (on 8/1/78), three bonds remain alive. The price of the contract is the sum of the prices of three $200,000 bonds, to be redeemed at the ends of ½, 1½, and 2½ years respectively.

The investor’s yield rate is greater than the bond rate; thus a discount is in order.  By use of the general method, we have

Price (on 8/1/78) = $199,038.46 + $197,224.91 + $195,548.18 = $591,811.55.

A compact expression for the sum of

is

≈ $534,492.39
On the other hand, a compact expression for

is

≈ $57,319.16.  Thus, we duplicate the result obtained by the general method with

= $534,492.39 + $57,319.16 = $591,811.55.

By use of the discount method, we have

Price (on 8/1/78) = $199,038.46 + $197,224.91 + $195,548.18 = $591,811.55.
A compact expression for this method is of course given by

≈ $591,811.5482 or $591,811.55.

Example 4.  On 3/1/77, a corporation sells a $2,000,000 issue of 8% bonds paying coupons on 3/1 and 9/1.  The issue is redeemable in equal installments at the end of each 2 years for 10 years.  On 9/1/79, find the value of all outstanding bonds to yield 7%.

This serial bond involves five distinct bonds of $400,000 each.  At the end of 2.5 years (on 9/1/79), four bonds remain alive.  The value of all outstanding bonds is the sum of the prices of four $400,000 bonds, to be redeemed at the ends of 1½, 3½, 5½, and 7½ years respectively.

The bond rate is greater than the yield rate; thus a premium is in order.  By use of the general method, we have

Price (on 8/1/78) = $405,603.274 + $412,229.088 + $418,003.1021 + $423,034.8218
                          = $1,658,870.2859 or $1,658,870.29.

A compact expression for the sum of

is

On the other hand, a compact expression for the sum of

is

Thus, we duplicate the result obtained by the general method with

= $1,187,907.999 + $470,962.2863 = $1,658,870.2853 or $1,658,870.29.

By use of the premium method, we have

Price (on 8/1/78) = $405,603.274 + $412,229.088 + $418,003.1021 + $423,034.8218
                             = $1,658,870.2859 or $1,658,870.29.

A compact expression for this method is given by

≈ $1,658,870.286 or $1,658,870.29

Attempts on my part to discern empirically a pattern for a general formula from the preceding four examples and several others (notwithstanding their tantalizing germ of an idea/trend) have initially met with frustration after frustration.  Fortunately, a common denominator of these examples is the arithmetic progression format of the addends’ exponent when added vertically.  From this observation and after going through an “accelerated” subconscious assimilation, it becomes apparent that the key to linking these examples together lies in decomposing the elements of n (n being the number of interest periods) in the compound interest formula.

Consider an $M serial issue of j% bonds that pays interest/coupons every m periods and will be redeemed in equal installments.  Suppose that this serial issue is to be sold on an interest/coupon date.  We introduce the following notations:

H = equal installment of the serial bond
r = bond rate
j = yield rate or buyer’s/investor’s rate
m = coupon/interest period
b = time from purchase date (which is a coupon/interest date) to 1st or next redemption date (whichever
      applies)
d = common difference between redemption dates, i.e., time from 1st redemption date to 2nd redemption
      date, time from 2nd redemption date to 3rd redemption date, and so on and so forth.
h = number of terms (i.e., number of serial bond equal installment to be retired) as defined by

where

is the time from purchase date (which is a coupon/interest date) to last redemption date.

= flat price to yield j on an interest date using the general method

= flat price to yield j on an interest date using the premium method

= flat price to yield j on an interest date using the discount method

We then obtain the following results.

where X is

and where Y is

Using (3) and (4),

For the premium method version, we have:

For the discount method version we have:

From the looks of it, the general method form of (5.1) should be sufficient for counterchecking the results one obtains from the traditional methods.  For additional countercheck, depending upon the case on hand, I suppose (6) or (7) might be useful (it does require more steps though).

Revisiting Example 1, we get the following:
Example 1.
(Bond due in ½ year)      = {Bond due in [½ + (1-1)*½] year}
(Bond due in 1 year)       = {Bond due in [½ + (2-1)*½] year}
(Bond due in 1½ year)     = {Bond due in [½ + (3-1)*½] year}

With H = $5,000, r = .07, j = .08, m = 2, b = ½, d = ½, and h = 3, and using (5) to emphasize and highlight the link that example 1 has with the trend set forth by examples 3 and 4, we get

Notice that (1.05) and (8) are practically the same.

Similarly, using (5.1), we get

The breakdowns of the remaining examples are as follows:

Example 2.
(Bond due in 1 year)     = {Bond due in [1 + (1-1)*1] year}
(Bond due in 2 years)    = {Bond due in [1 + (2-1)*1] years}
(Bond due in 3 years)    = {Bond due in [1 + (3-1)*1] years}

H = $2,000, r = .07, j = .06, m = 2, b = 1, d = 1, and h = 3

Example 3.
(Bond due in ½ year)        = {Bond due in [½ + (1-1)* 1] year}
(Bond due in 1½ year)      = {Bond due in [½ + (2-1)* 1] years}
(Bond due in 2½ years)     = {Bond due in [½ + (3-1)* 1] years}

H = $200,000, r = .07, j = .08, m = 2, b = ½, d = 1, and h = 3

Example 4.
(Bond due in 1½ year)    = {Bond due in [1½ + (1-1)*2] years}
(Bond due in 3½ years)   = {Bond due in [1½ + (2-1)*2] years}
(Bond due in 5½ years)   = {Bond due in [1½ + (3-1)*2] years}
(Bond due in 7½ years)   = {Bond due in [1½ + (4-1)*2] years}

H = $400,000, r = .08, j = .07, m = 2, b = 1½, d = 2, and h = 4

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On a more personal note, I’m inclined to call formula (5.1) as the Campos-Hart general formula for the calculation of the purchase price of a serial bond on an interest date in honor of the late Mr. Jose Y. Campos, Founder and Chairman Emeritus of United Laboratories, Inc. (whose genius, vision, and generosity enabled my family and countless others to live prosperous lives, and whose charitable foundation to this day through his successors, namely his children and grandchildren, continues to help those who are in need of assistance) and the late Dr. William L. Hart, mathematician, book author and professor (whose book introduced me to the mathematics of finance).

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While searching the net for existing similar work on serial bonds, I came across a preview of two JSTOR webpages that pursued a different line (so it seemed to me), namely that of approximating the yield rate of serial bonds.  The notation used was similar to those used by an actuary. Unfortunately for me, I wasn’t granted access to these two fascinating documents (I’m not a member of JSTOR’s affiliated organizations) to see if the work of the author, a certain Mr. Ralph W. Snyder, has already anticipated the formulas that I’ve just derived.  The documents in question are as follows:

Direct Yield Formulas for Serial Bonds
Ralph W. Snyder
Geo. S. Olive & Co.
The Accounting Review, Vol. 30, No. 2 (Apr., 1955), pp. 257-267 (article consists of 11 pages)
Published by: American Accounting Association

Some More Notes on the Bond Yield Problem: Serial Bonds
Ralph W. Snyder
The Accounting Review, Vol. 28, No. 3 (Jul., 1953), pp. 412-421 (article consists of 10 pages)
Published by: American Accounting Association