# A general relationship between prices of bonds and their yields.

Introduction

As a precondition to his five bond theorems, Malkiel (1962) derives the relationship between a bond's coupon rate, yield, and price. In particular, he shows that if the coupon rate equals the yield, then immediately after the coupon is paid, a bond will sell for its par value. Further, a bond will sell for discount (premium) if the yield is greater (less) than coupon rate.  This relationship is important in understanding the behavior of bond prices and is stated in many textbooks, including Brealey and Myers (2003), Ehrhardt and Brigham (2006), Fabozzi (2010), Jones (1998), Kolb (1992), and Sharpe and Alexander (1990).

This relationship has two major limitations. First, it applies only to straight bonds, wherein the principal is returned only at maturity. This is a serious limitation because many securities, such as mortgages and collateralized mortgage obligations, distribute principal value over time. Second, the relationship applies only when the next coupon is paid in exactly one period, which is a somewhat rare occurrence. This paper overcomes these limitations by generalizing the relationship between a bond's coupon rate, yield, and price to securities that distribute principal over time and to the common situation in which the next coupon is paid in less than one period.

Notation

P [equivalent to] Market price (invoice or dirty price);

CP [equivalent to] Clean price (quoted price):

Y [equivalent to] Annual yield to maturity compounded n times per year;

R [equivalent to] Annual contract rate (or coupon rate) compounded n times per year;

n [equivalent to] Number of periods per year;

y [equivalent to] Periodic yield to maturity (Y/n);

r [equivalent to] Periodic contract rate to maturity (R/n);

T [equivalent to] Number of periods;

[m.sub.1], [m.sub.2], ..., [m.sub.T] [equivalent to] Par (or principal) value distributed at the end of periods 1, 2, ..., T; and

M [equivalent to] [m.sub.1], + [m.sub.2], + ... + [m.sub.T].

For example, the periodic rate (y or r) for a treasury bond is the annual rate (Y or R) divided by two, and the periodic rate for a mortgage is the annual rate divided by 12.

The Relationship between Price, Yield, and Coupon Rate for a Straight Bond

The periodic yield on a bond is defined as the positive real number y that satisfies:

P = [N.summation over (i = l)] rM/[(1 + y)/sup.i] + M/[(1 + y).sup.N]. (1)

The yield as an annual percentage rate Y equals the periodic yield y times the number of payments n. The closed-form of the above equation is given by:

P = Mr/y (1 - 1/[(1 + y).sup.n]) + M/[(1 + y).sup.N] (2)

which can be expressed as:

P = M (r/y + l - r/y/[(1 + y).sup.N]). (3)

Based upon equation (3), Malkiel (1962) (hereafter, MK) demonstrates the following relationships between bond prices and yields:

MK(4) and(4a):r = y [??] P = M

MK(4b): y > r [??] P < M

MK(4c): y < r [??] P > M

Bremmer and Kesselring (1992) point out that MK did not prove the converse of relationships (4b) and (4c) and further that the converse is not an obvious consequence of equation (3). They complete the price-yield relationship by proving that P < M [??] y > r and P > M [??] y < r.

Here we present a simple proof. If P < M, then based upon (2):

M > Mr/y (1 - 1/[(1 + y).sup.N]) + M/[(1 + y).sup.N]

or

M(1 - 1/[(1 + y).sup.N]) > Mr/y (1 - 1/[(1 + y).sup.N]). (4)

Given that by definition M, y > 0, (4) can be rewritten as r/y < 1. Likewise, if P > M, then

M < Mr/y(1 - 1/[(1 + y).sup.N]) + M/[(1 + y).sup.N]

which implies for M, y > 0 that r/y > 1. Therefore, we can conclude for straight bonds the following relationships hold:

1. r = y [??] P = M

2. y > r [??] P < M

3. y < r [??] P > M

Results at the Ex Coupon Date

This section extends the relationship between the price, yield, and coupon rate to securities in which the principal is paid out over time, often unpredictably, including mortgage-backed securities and collateralized mortgage obligations. The results in this section hold at the ex coupon date. This means that the coupon has just been paid and the next coupon is due in exactly one period (1/n years). The next section allows for accrued interest. For now, we focus on the simple, textbook case wherein the next coupon is paid in exactly one period.

Theorem 1

Par Bonds. Regardless of how the par value is distributed, at the ex coupon date a bond will sell for the remaining par value if and only if the contract rate equals the yield to maturity.

Proof. Par values [m.sub.1], [m.sub.2], ..., [m.sub.T] are distributed at dates 1, 2, ..., T. The instrument promises a coupon equal to the remaining par value times the periodic coupon rate, defined as the annual rate divided by the number of payments per year. So the bond promises ([m.sub.1] + ... [m.sub.T])r + [m.sub.1] at date 1, ([m.sub.2] + ... [m.sub.T)]r + [m.sub.2] at date 2, and finally [m.sub.T]r + [m.sub.T] at date T. We want to show that the price always will equal the remaining par value if and only if the cash flows are discounted at the rate r. Clearly, at date T the bond will sell for [m.sub.T]r + [m.sub.T] immediately prior to the payment of interest and principal. Therefore, the price at date T - 1 will equal the promised value at date T discounted by the periodic yield to maturity:

[m.sub.T]r + [m.sub.T]/1 + y (5)

which equals [m.sub.T] if and only if y - r (or Y - R). Further, at date T - 1 the price will equal the par value if and only if y = r:

[m.sub.T] + ([m.sub.T-1] + [m.sub.T])r + [m.sub.T-1]/1 + r = [m.sub.T-1] + [m.sub.T]. (6)

Continuing, we see for dates T - 2, ... , 0 the price always equals the remaining principal, so that the price at date 0 equals:

[m.sub.1] + [m.sub.2] + ... + [m.sub.T]

Theorem 2

Discount and Premium Bonds. Regardless of how the par value is distributed, a bond will sell at a discount (premium) at the ex coupon date if and only if the contract rate is less than (greater than) the yield to maturity.

Proof. If and only if y > r (or Y > R), the bond will sell for less than the remaining par value at date T by equation (5):

[m.sub.T]r + [m.sub.T]/1 + y < [m.sub.T].

The same is true at date T - 1 by equation (6). Continuing, we see for dates T - 2, ..., 0 the price is always less than the remaining principal. Likewise, if and only if y < r, the bond will sell for more than its par value on dates 0, 1, ..., T - 1.

The following two examples illustrate Theorems 1 and 2 for a mortgage-backed security and a collateralized mortgage obligation. In particular, they show that the instrument will sell for its par value if the yield equals the coupon rate, a discount if the yield is greater, and a premium if the yield is less.

Example 1

Mortgage with Prepayments. For simplicity consider a three-year mortgage with a contract rate of 10 percent and a principal value of \$100. Based upon the contract rate and the initial principal value of \$100, the scheduled payment the first year equals \$40.21. This is the payment on a three-year, l0 percent fixed rate mortgage for \$100. Suppose the mortgage has prepayment rates of 20 percent and 30 percent in each of the next two years, respectively. (2) Table 1 shows the projected cash flows, where PMT [equivalent to] scheduled payment of interest and principal, INT [equivalent to] interest payment, SP [equivalent to] scheduled principal payment (adjusted for cumulative prepayments), and PP [equivalent to] principal prepayment. See Fabozzi (2010) for the prepayment convention and the calculation of prepayments. The last row of Table 1 demonstrates that the present value of the cash flows, including scheduled and prepayments of principal, at the contract rate of 10 percent equals the total par value of \$100. As all the cash flows are positive, the bond will sell for less than \$100 if the yield is greater than the contract rate of 10 percent and more than \$100 if the yield is less than the contract rate.

Example 2

Collateralized Mortgage Obligation (CMO). Suppose the mortgage in the last example is used as collateral for a CMO with two tranches (A and B) with principal values of \$44.17 and \$55.83. Table 2 shows the payments to the two tranches. Principal payments are first directed to the A tranche. Once the A tranche is exhausted, principal payments are directed to the B tranche. Consequently, the first payment to B consists entirely of interest on the initial principal of \$55.83. Tranche A's first payment equals the interest of \$4.417 on the initial balance of \$44.17 plus scheduled and prepaid principal of \$44.17 (sum of Table 1 columns SP and PP at date 1). For simplicity, the initial balance of tranche A is chosen to equal the total principal payment in the first year, so that tranche A is exhausted at the end of one period. The last row of Table 2 demonstrates that the present value of the cash flows, which include modified prepayments of principal, at the contract rate of 10 percent for each tranche equals the total principal value of each tranche. At a yield higher than 10 percent (lower), each tranche would sell at a discount (premium) to its principal value.

Accrued Interest and The Clean Price

This section generalizes the results of the last section to the more common situation in which the first coupon is not paid in exactly one period. Let n be the number of times per year the bond makes payments. Assume the bond yield is an annual rate compounded n times per year. For a treasury bond, the yield is the annual rate compounded twice per year, and for a mortgage the yield is the annual rate compounded monthly.

Define the simple price S of a bond as the price assuming the next coupon is paid in exactly 1 period that equals l/n years (when, in fact, it typically is not). By Theorem 1, the yield is equal to the contract rate if and only if the simple price equals the par value. Let us find the relationship between the simple price and the actual market price (invoice or dirty price). Suppose the next payment is made in a period 0 < [theta] < 1 of length 1/n years. The simple price is the value of the bond immediately after the last coupon payment date [theta] - 1 (Figure 1). Therefore, the market price P is the simple price carried forward 1 - [theta] periods to the present:

P = S[(1+y).sup.1-[theta]] (7)

where y is the periodic yield to maturity (over l/n years).

Theorem 3

Yield and Contract Rate. Regardless of how the par value is distributed, the yield on a bond:

1. is less than the contract rate if and only if P > ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]]

2. equals the contract rate if and only if P = ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]]

3. is greater than contract rate if and only if P < ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]]

Proof. If and only if y = r (or Y = R), the simple price equals the par value by Theorem 1 and the market price is determined by (7): P = ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]]. Therefore, if P < ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]], it follows that the simple price is less than par value and the periodic yield is greater than the periodic coupon rate r. Further, if P > ([m.sub.l] + ... + [m.sub.T])[(1 + r).sup.1-[theta]], it follows that the simple price is less than par and the yield is less than the coupon rate.

Example 3

Bond Prices Between Payment Dates. A bond has a coupon rate of l0 percent, a par value of \$100, and total of 20 semiannual payments strating with the first payment in 91 days. The number of days between payments is 182, and so [theta] = 91/182. The yield is equal to 10 percent if and only if:

P = 100[(1 + 0.05).sup.91/182] = 102.47.

The yield is greater than 10 percent if and only if P < 102.47 and less than 10 percent if P > 102.47.

Theorem 3 shows that the relationship between the yield and the coupon rate depends upon the relationship of the market price to the par value increased by the true accrual of interest:

([m.sub.l] + ... + [m.sub.T])[(l + r).sup.1-[theta]] - ([m.sub.1] + ... + [m.sub.T])

Because the quoted price on a bond is the clean price, it is more convenient to work with the clean price than the simple price. If the yield equals the coupon rate, the clean price CP closely approximates the simple price, which in this case equals the par value. The clean price CP of a bond is the market price P minus the accounting accrued interest, defined as:

Mr(l - [theta])

where M = [m.sub.1] + [m.sub.2] + ... + [m.sub.T]. So the clean price is determined by:

CP = P - Mr(1 - [theta]).

If the yield equals the coupon rate (S = M), then by equation (7) the market price is determined by:

P = M[(1 + r).sup.1-[theta]].

Expand P in a first-order Taylor polynomial about r = 0:

P = M + Mr(1 - [theta]) + Q(r) (8)

where Q(r) is the remainder term of order O([r.sup.2]). Therefore, in this case, the clean price approximately equals the par value:

CP = P - Mr(1 - [theta]) = M + Q(r).

As a practical matter, because r is typically small r [much less than] 1, the approximation is good (Table 3). We can determine the direction of the error Q(r) by examining the sign of the remainder term of the Taylor series (Rade and Westergren, 2000, p. 171):

Q(r) = P"(z)[r.sup.2]/2 for z between 0 and r. (9)

Observe that

P"(z) = - M[theta](1 - [theta])/[(l + z).sup.[theta]+l] [less than or equal to] 0

for all z when [theta] [member of] [0, 1]. Further, for any given [theta] [member of] [0, 1] the expression for P"(z) has a minimum at z = 0 in the region z [greater than or equal to] 0:

min P"(z) = -M[theta](l - [theta]).

Finally, the above expression has a minimum at [theta] = 1/2 of -M/4. (3) Putting this all together with equation (9), we can put a tight bound on the error term Q:

- [Mr.sup.2]/8 [less than or equal to] Q(r)[less than or equal to] 0

or in terms of the annual coupon rate:

- M/8 [(R/n).sup.2] [less than or equal to] Q(R/n) [less than or equal to] 0. (10)

For example, if the annual coupon rate is 10 percent on a bond that makes semiannual payments (R/2 = 5 percent), the error as a percentage of the par value is in the range -0.031 percent to 0, depending on the value of [theta]. Because the periodic rate equals R/n, where R is the annual coupon rate, as n increases the magnitude of Q(r) decreases to the limiting value of zero.

In the last example, the clean price is determined by

CP = 102.47 = 5 x (91/182) = 99.97

which is slightly less than the simple price of \$100. Figure 2 shows the clean price as a function of the time in half-years to the first coupon payment. The maximum difference of \$0.03 occurs at roughly the midpoint between the payment dates. Table 3 lists the maximum difference between the simple price of \$100 and the clean price versus the coupon rate (assuming the yield equals the coupon rate). The difference is independent of the maturity of the bond. For realistic coupon rates, say less than 20 percent, the maximum error is less than twelve cents.

[FIGURE 2 OMITTED]

Theorem 4

Yield and Contract Rate Based upon the Clean Price. Regardless of how the par value is distributed, the yield on a bond:

1. is less than the contract rate if and if only CP > 100 + Q(r)

2. equals the contract rate if and only if CP = 100 + Q(r)

3. is greater than the contract rate if only if CP < 100 + Q(r)

where the error term is bounded as follows:

- M/8 [(r).sup.2] [is less than or equal to] Q(r) [less than or equal to] 0.

Proof. By equation (8):

M[(l+r).sup.l-[theta]] = M + Mr(1-[theta]) + Q(r).

Therefore,

P > M[(1+r).sup.1-[theta]] [??] P > M + Mr(1-[theta]) + Q(r) [??] CP [equivalent to] P - Mr(1-[theta]) > M + Q(r)

The rest follows from Theorem 3 and equation (10).

Because the error term Q(r) is typically small, as indicated by equation (10), the relationship between the clean price and yield for a bond between payment dates is roughly the same as the relationship between the price and yield on the ex coupon date. So if the clean price is close to par, the bond yield is approximately equal to the coupon rate. If the clean price is at reasonable discount, the yield is greater than the coupon rate. If the clean price is at premium, the yield is less than the coupon rate. Further, as the payment frequency n increases the magnitude of Q decreases and in the limit equals zero.

Summary of Properties

At the ex coupon date, regardless of how the par or principal value is distributed, a security will sell for the remaining par value if the contract rate equals the yield to maturity. Further, the security will sell for a discount to par (premium) if the contract rate less (greater) than yield. For a security between payment dates, the relationship must be modified to account for the accrual of interest. The relationship is roughly the same with the clean price substituted for the market price. The exact relationship depends upon the relation between the market price and the par value increased by the true accrual of interest. These are general results that apply regardless of how the principal value is distributed and which apply even if the future distribution of principal is random. Consequently, these results apply to a wide range of fixed income securities and structured products, such as mortgage-backed securities and collateralized mortgage obligations.

References

[1.] Brealey, Richard A., and Stewart C. Myers, Principles of Corporate Finance, seventh edition (McGraw-Hill/Irwin, 2003).

[2.] Bremmer, Dale, and Randall Kesselring, "The Relationship Between Interest Rates and Bond Prices: A. 1992," American Economist 36 (1992), pp. 85-86.

[3.] Ehrhardt, Michael C., and Eugene F. Brigham, Corporate Finance: A Focused Approach, second edition (Thomson South-Western Ohio USA, 2006).

[4.] Fabozzi, Frank J., Bond Market Analysis and Strategies, seventh edition, (Prentice-Hall, 2010).

[5.] Jones, Charles P., Investments, Analysis and Management, sixth edition (John Wiley & Sons, 1998).

[6.] Kolb, Robert, Investments. third edition, (Miami, Florida: Kolb Publishing Company, 1992).

[7.] Lawrence, Edward, and Siddharth Shankar, "A Simple and Student-Friendly Approach to the Mathematics of Bond Prices," Quarterly Journal of Finance and Accounting, 46, no. 4 (2007), pp. 91-99.

[8.] Malkiel, Burton, "Expectations, Bond Prices and the Term Structure of Interest Rates," Quarterly Journal Economics, 76 (1962), pp. 197-218.

[9.] Rade, Lennart, and Bertil Westergren, Beta Mathematics Handbook, second edition (CRC Press, Inc., 2000), page 171.

[10.] Sharpe, William, and Gordon Alexander, Investments, fourth edition (Prentice Hall, 1990).

Joel R. Barber

Florida International University

(1) This relationship is essential in the proofs of Malkiel's Theorems 2, 3, and 4. See Lawrence and Shankar (2007) for a simplified proof.

(2) We do not need a prepayment rate in year 3 as the remaining principal is returned. As the principal is returned, the scheduled payments will decline.

(3) This can be verified by checking the first- and second-order conditions for a minimum.
```Table 1--Mortgage Cash Flows and Present Value

Balance

Date    Before    After PMT  After PP    PMT    SP     INT

0     100
1     110 (1)     69.79 (2)  55.83 (4)  40.21  30.21  10.00
2      61.41 (6)  29.24      20.45      32.17  26.59  5.58
3      22.52       0          0         22.52  20.48  2.04

Present Value at 10%

Date     PP      Cash Flow

0
1     13.96 (3)   54.17 (5)
2      8.79       40.96
3      0          22.52

Present Value at 100.00

(1.) Balance Before Scheduled Payment (PMT) at date 1: 100 x 1.1

(2.) Balance After Scheduled Payment (PMT): 110-40.21

(3.) Prepayment (PP): 69.79 x 0.2

(4.) Balance After Scheduled and Prepayment: 69.79 - 13.96

(5.) Cash Flow: 40.21 + 13.96 or 30.21 + 10.00 + 13.96

(6.) Balance Before Scheduled Payment at Date 2: 55.83 x 1.1

Table 2--CDO Cash Flows and Present Value

Balance           Cash Flow

Date     PMT       A        B       A        B

0                44.17    55.83
1       40.21      0      55.83   48.587   5.583
2       32.17      0      20.45     0      40.94
3       22.52      0        0       0      22.52

Present Value at 10%              44.17    55.83

Table 3--Maximum Difference Between a Simple and Clean Price
when Yield Equals the Coupon Rate

Annual Rate (paid      Minimum     Maximum
semiannually)      Clean Price    Error

10%              99.97       0.03
20%              99.88       0.12
40%              99.54       0.46
80%              98.32       1.68

Figure 1--Timing of Bond Coupons

Price P is observed today at time 0 and the next coupon
C is paid in [theta] periods

S       P      C          C           C

1       1      1          1           1
[theta]-1   0   [theta]   [theta]-1   [theta]-2
```
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