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Bond Pricing Theorems Spring, 2011 1

Bond Pricing Theorems

Floyd Vest The following Bond Pricing Theorems develop mathematically such facts as, when market interest rates rise, the price of existing bonds falls. If a person wants to sell a bond in this environment, they are likely to sell it for less than what it was previously worth. This is quite a disappointment for some people. For example, investors lost one trillion dollars on bonds in 1994 (Securities Industry Association). When market interest rates go up, the share prices of bond funds go down. In November 1994, investors pulled $10.9 billion out of bond funds. The mathematics of bonds and interest rate risk. If you buy an original issue bond, it is likely to pay you regular interest payments every six months, and at maturity pay the price of the bond back to you. The interest payments are referred to as coupons denoted by C, the price you pay is denoted by P, and the value at maturity is F. The yield or yield

to maturity (YTM) is the annual nominal rate y with a semiannual coupon rate of r =

y

2.

For example if you buy Bond A: a $10,000 bond paying a yield of y = 4% for ten years

with r =

1

2y with F = $10,000, then the coupon

C =1

2

!"#

$%&

0.04( ) 10,000( ) = $200 every

six months. Actually coupons are calculated by a day count to get the factor (number of

days in the six month period)/365 so the factor is close to

1

2 but seldom equal to

1

2.

The basic bond formulas. (See the Side Bar Notes for the derivation.) This arrangement

gives a neat mathematical formula with i =

1

2y where y = annual yield to maturity

(YTM):

(1)

P = C1

1+ i

!

"#

$

%&

j

+ F1

1+ i

!

"#

$

%&

N

j=1

N

' , so that

(2)

P = C1! (1+ i)!N

i

"

#$

%

&' + F(1+ i)!N by the formula for the Present Value of an

Ordinary Annuity. Also, let r be the semiannual coupon rate and C = rF =iF. (See the Exercises and Side Bar Notes.) The first coupon C is paid at the end of the first period. All of the bonds below are assumed to be semiannual coupon bonds unless stated otherwise.


Example. We can put numbers in this to see how Bond A above works:

P = 2001! (1+ 0.02)!20

0.02

"

#$

%

&' +10,000(1+ 0.02)!20 and when calculated gives P = $10,000.

You Try It #1 For a ten year bond with C = $500, F = $1000, at 10% per year, calculate the price P. Now watch what happens to Bond A, if the new y1 is 5% with the new i1 = 0.025 and N is still 20 periods, and C, r and F remain the same. The new market price (discount price)

P1= 2001! (1+ 0.025)!20

0.025

"

#$

%

&' +10,000(1+ 0.025)!20 = $9,220.54. The loss on the price of

the bond (the discount) = 10,000 – 9220.54 = $779.46. The percent loss (discount) is

779.46

10,000 = 0.0779 = 7.79% of the original $10,000 price. An increase in market interest

rate of one percentage point caused a (discount) loss in price of about 8%. Of course if the market interest rate falls, then the market price of the bond would rise yielding a premium. You Try #2 If for the above $10,000 ten-year bond, the market interest rate drops from 4% to 3%, what is the new price of the bond? How much money is gained if the bond is sold? What is the percent gain in the price? The bond pricing theorems. In stating and proving bond theorems, we will work with what we call Bond B, bought and sold on coupon dates, bought at par where P = F, with

an annual yield y (YTM), with a semiannual discount rate of i, i = r =

1

2y and a coupons

of C with N periods to maturity and P = F, C = rF = rP,

C

i=

C

r = F = P. From the

Standard Formula (1) from above

P = C1

1+ i

!

"#

$

%&

j

+ F1

1+ i

!

"#

$

%&

N

j=1

N

' and

By Formula (2) above we have

(3)

P =C

i1!

1

(1+ i)N

"

#$

%

&' +

F

(1+ i)N. (There are actually at least three more formulas for P,

see the Exercises.) If a bond sells at a new price P1, then C ≠ rP1, P1 ≠ F, or for a new discount rate i1≠ i, then C ≠ (i1)F. We will assume that for Bond B, F, C and r remain constant. We are interested in changes in P, i, and N and their effect on each other. New values resulting from changes in P, i, and N will be represented by P1, i1, and N1. For instance, in the above example,


it was illustrated that if i1 > i, then P1 < P for a given N. Also, bonds that are bought on a coupon date receive their first coupon at the end of the following period. Theorem 1: For the life of the bond (i.e., for any N), if P doesn’t change with C, r, F constant, then i doesn’t change.

Proof: Consider i1 as a new rate, i1 ≠ i. For any exponent j,

1

1+ i

!"#

$%&

j

'1

1+ i1

!"#

$%&

j

.

Therefore for i1, P1 ≠ P. This is a contradiction to the hypothesis. Thus the theorem is proven. Theorem 1.1: For the life of the bond (i.e., for any N), if i changes with C, N, F, and r constant, then P changes. This is the Contrapositive of Theorem 1, and thus follows from Theorem 1. Theorem 2: For the life of the bond with the original P, i, C, and F, the price doesn’t change. That is, if only N changes, the price doesn’t change. Consider N1 ≤ N as the remaining life of the bond. Substituting into Formula (3) we have

P for N1 =

P(N1) =C

i+

F !C

i

(1+ i)N1 =

C

i+

C

i!

C

i

(1+ i)N1= P + 0 = P.

Conclusion, for the life of the bond, the price doesn’t change if the only thing that changes is N. Since P = F, the bond is still at par. Theorem 2.1: For the life of Bond B, if the price changes, then i changes. Proof: This theorem is the contrapositive of Theorem 2 and thus follows from Theorem 2. Theorem 3: If i1 > i then P1 < P. If the market interest rate increases, then the price of the bond decreases. Proof: Let i1 be the new interest rate, with i1 > i, and with all numbers positive.

For any j,

1

1+ i1

!

"#

$

%&

j

<1

1+ i

!

"#

$

%&

j

. Therefore for any N,

C1

1+ i1

!

"#

$

%&

j

+ F1

1+ i1

!

"#

$

%&

N

j=1

N

' <

C1

1+ i

!

"#

$

%&

j

+ F1

1+ i

!

"#

$

%&

N

j=1

N

' so P1 < P. The new price is

less than the previous price. Theorem 4: If i1 < i then P1 > P. The proof is left as an exercise. Theorem 5: If P1 < P then i1 > i. Proof: This theorem is a consequence of the contrapositive of Theorem 4, and thus is proved by Theorem 4.


Theorem 6: If P1 > P then i1 < i. The proof is left as an exercise. Theorem 7: Over the life of Bond B, the size of its discount will decrease as its life gets shorter. Lemma for Theorem 7. We will first prove Lemma for Theorem 7: Over the life of Bond B, the size of its discount decreases as its life gets one period shorter. For a discount, it is required that the new i1 > i. Consider appropriate numbers positive. Proof: i1 > i. (i1)F > iF. F + (i1)F > F + iF. i = r. C = iF = rF F + (i1)F > F + C.

(1 + i1)F + C + F. F >

C

1+ i1+

F

1+ i1.

1

1+ i1

!

"#

$

%&

K

F >

C

1+ i1+

F

1+ i1

!

"#

$

%&

1

1+ i1

!

"#

$

%&

K

1

1+ i1

!

"#

$

%&

K

F +

C1

1+ I1

!

"#

$

%&

j=1

K

'( J

> C

1

1+ i1

!

"#

$

%&

K+1

+ F

1

1+ i1

!

"#

$

%&

K+1

+

C1

1+ i1

!

"#

$

%&

j=1

K

'( j

.

The left side is the discount price for K periods remaining called DPK. The right side is the discount price for K + 1 periods remaining called DP of (K + 1). So DPK > DP of (K + 1). –DPK < –DP of (K + 1). So the Discount for K = P – DPK < P – DP of (K + 1) = Discount for K + 1. Thus the Discount with K periods remaining < Discount with K + 1 periods remaining in the bond. So the size of the discount decreases when the life of the bond is one period shorter. Proof of Theorem 7: We will use the Lemma and a continuing induction to prove Theorem 7. From the Lemma, Discount with K + 1 left < Discount with K + 2 left thus the Discount with K left < Discount with K + 2. This continues until for any K < N, which gives the Discount with K left < Discount with N periods left. So as the life of the bond gets shorter, the discount decreases. Theorem 8: Over the life of Bond B, the premium decreases as the life gets shorter. Proof: To prove Theorem 8, use the same technique used to prove Theorem 7. You Try It #3 For Bond A in the above example, with N = 20 and i = 0.02, and a new i of i1 = 0.025, with i1 > i, the new Price P1 = $9220.54. We will call P1 the Discount price at 20 periods. Calculate the Discount price for i1 = 0.025 of N1 = 19 periods remaining in the life of the bond and compare the two Discount prices. What pattern does this suggest? You Try It #4 Calculate the Discount for Bond A at 19 periods arising from i = 2% and i1 = 2.5%. What was the Discount at 20 periods? What pattern does their comparison suggest?


For Theorem 9, we will paraphrase Theorem 4 from Sharpe p. 383. Theorem 9: A decrease in a bond’s yield will raise the bond’s price by an amount that is greater in size than the corresponding fall in the bond’s price that would occur if there were an equal sized increase in the bond’s yield. Discussion: A decrease in a bond’s yield means i1 < i and gives a new price of P1. The resulting rise in price = P1 – P, where by Theorem 4, P1 > P. If i2 > i then P2 < P by Theorem 3. The fall is P – P2. There is an equal sized change in bond yield means i2 – i = i – i1. We need to show that P1 – P > P – P2. The proof is left as an exercise. Bond calculations on the TI83/84. For bond calculations on the TI83/84, consider Formula 4 below, and the below Formula 5 for PV found in the Appendix to the TI83/84 Manual. We rewrite the PV Formula 5 as Formula 6 and compare it to Formula 4.

(4)

P = C1! (1+ i)!N

i

"

#$

%

&' + F(1+ i)!N

(5)

PV =PMT

i! FV

"

#$

%

&' (

1

(1+ i)N!

PMT

i

(6)

PV = PMT(1+ i)!N !1

i

"

#$

%

&' ! FV

1

1+ i

"

#$

%

&'

N

This comparison tells you that C = –PMT and F = –FV. For the TI83/84 TVM Solver, to calculate the price P = PV of a bond, for C enter PMT as negative, for F enter FV as negative. Also, enter the semiannual rate i as a percent in I%, enter C/Y as 1, select PMT:END. For a bond where C = $10, K = 5, M = $100, and i = 0.06, use the Code and commentary: 2nd Finance Enter 5 Enter for N 6 Enter for I% 0 Enter for PV (-)10 Enter for PMT (-)100 Enter for FV 1 Enter for P/Y Select Pmt:End Enter !!! to highlight PV Alpha Solve and read 116.85. If i increases from 6% to 7%, then for the new price when i increases to .07: Code: ! to highlight I% 7 Enter ! to highlight PV Alpha Solve and read 112.30. For an increase of i from 6% to 7%, there is a decline in price from $116.85 to $112.30.


Exercises

Show your work. Summarize. Put you calculator decimals on float so you can get the right answers. All of the bonds are assumed to be semiannual coupon bonds unless otherwise stated. 1. Do the calculations to show the following. You may use Financial Functions on a calculator. A five-year annual dividend bond with par value of $1000 has a yield of 8% per year. (a) If the price increases to $1100 then its yield falls to 5.649%. (b) If the price falls to $900, then its yield rises to 10.684%. 2. Write the following as a formula and explain the variables. The market price of a semiannual coupon bond is the present value of coupons and maturity value discounted at the semiannual rate i that is half the YTM. 3. Prove Corollary to Theorem 7: As the life of Bond B gets shorter, for a given i1 > i , the discount price gets greater. See the Proof of the Lemma. 4. Prove Corollary to Theorem 8: Over the life of the bond, for a given decrease in i, as the life of the bond decreases, the premium price decreases. 5. Prove Theorem 4. 6. Prove Theorem 6. 7. Prove Theorem 8. 8. Finish the proof of Theorem 9. 9. For a ten-year bond with semiannual coupons of $200 and YTM = 4% and maturity value of $1,000, and a price of $1,000, calculate its price at the same terms when sold one year later. 10. Prove in general that if a bond is sold one year later, with the same previous terms, the new price is equal to the former price. 11. Prove in general that if a two-year bond is sold on the second coupon date with the new price equal to the original price, then the new semiannual rate i1 is equal to the former rate i.


Prove the following three formulas for the new price P1 of a bond where F = value at maturity, k = the number of semiannual periods left in the bond, i = the semiannual rate paid by the buyer, r = the original coupon rate of the bond: (These are formulas that are often convenient for bond calculations. See Kellison p. 211.)

12. P1 = F + (Fr – Fi)

1

1+ i

!

"#

$

%&

j=1

n

'j

13. P1 =

Fr

i+ F !

Fr

i

"#$

%&'

1

1+ i

"#$

%&'

n

14. P1 =

F1

1+ i

!"#

$%&+

r

iF ' F

1

1+ i

!"#

$%&

n(

)**

+

,--

15. Consider a five-year bond with C = $60 per year, purchased at a discount rate of 3%, with value at maturity F = $700. Assume coupons are reinvested at 2% per year for five years. (a) What is the price of the bond? (b) What is the future value of the investment of the coupons at maturity? (c) What is the average rate of return on the investment of the bond and the coupons? (d) How much interest is earned on the coupons? (e) How much interest is earned on bond alone? (f) How much interest is earned on the total transaction? 16. For Bond A above, compare the price for N = 20, with the price for N = 19. What does this illustrate?

17. For Bond A, Graph 200

1! (1+ 0.02)!N

0.02

"

#$

%

&' and graph 10,000(1 + .02)–N for N from 0

to 20, and graph their sum. What do you observe about their sum? What do you observe about the two graphs individually? Are they straight segments, which way do they curve?

18. Given P =

Fr1! (1+ i)!N

i

"

#$

%

&' + F(1+ i)!N , solve explicitly for F, r, and N.


Side Bar Notes Derivation of basic bond pricing formula. Let the bond price P be such that P = v1 + v2 + … +vN + wN where v1(1 + i) = C, v2(1 + i)2 = C, …, vN(1 + i)N = C, and wN(1 + i)N = F. and

v1 =

C1

1+ i

!"#

$%&

, v2 =

C1

1+ i

!"#

$%&

2

, …, vN =

C1

1+ i

!"#

$%&

N

, wN =

F1

1+ i

!"#

$%&

N

so that

P =

C1

1+ i

!"#

$%&

1

+

C1

1+ i

!"#

$%&

2

+ … +

C1

1+ i

!"#

$%&

N

+

F1

1+ i

!"#

$%&

N

. This gives Formula 1,

which is in summation notation. For information on bonds see www.wikipedia.org, www.investopedia.com, and www.vanguard.com.


References For a copy of Chapter 14, Financial Functions, and the Appendix, Financial Formulas, in the TI84 manual, see http://www.ti.com/calc. For Bond Pricing Theorems see, Sharpe, William F. and Gordon J. Alexander, Investments, Fourth Edition, Prentice-Hall, 1990, pages 382 – 384 (Sharpe gives no proofs. This is customary in finance books. ) Beezer. Fobert A. 1996. “Closing in on the Internal Rate of Return,” UMAP Module 99750. Calculates Internal Rate of Return (IRR) [the same as YTM] on investments by various iteration methods including the bisection method. Gives examples. Discusses stocks, bonds, mutual finds. Has exercises and answers. Fabozzi, J. Frank, Fixed Income Mathematics, Probus Publishing Co., Chicago, IL. 1988. Also see Malkiel, Burton G., “Expectations, Bond Prices, and the Term Structure of Interest Rates,” Quarterly Journal of Economics, 76 No.2 (May, 1962):197-218. Gives Bond Pricing Theorems, some without proofs, and for some, uses calculus in proofs. Kellison, Stephen G., The Theory of Interest, Irwin, Boston, MA, 1991.


Answers to You Try Its 1. Price = $10,000 2. The new price = $10,858.43. The gain is $858.43. This is an 8.6% gain in price. 3. The discount price for N1 = 19 is $9251.06. The discount price at N = 20 is $9220.54. This illustrates that as the life of the bond decreases, the discount price increases. 4. The discount for N = 19 is 10,000 – 9251.05 = $748.94. For N = 20, the discount is 10,000 – 9220.54 = $779.46. This illustrates Theorem 7, as the life of the bond get shorter, the discount is less.


Answers to Exercises 1. (a) Checking these figures on a scientific calculator, we write the equation

1100 =

801! (1+ .05649)!5

.05649

"

#$

%

&' +1000(1+ .05649)!5 . Calculating on the right, we get

1100 = 340.23 + 759.75 . If the price is $1100, the yield has fallen to 5.649%. (b) Solving for I% on the TI83 TVM Solver: Code and comments: 2nd Finance Enter ! to PV 900 Enter for PMT (-)80 Enter for FV (-)10000 Enter Select P/Y 1 PMT:END !! to I% Alpha Solve and read 10.68424504 . With a price of $900, the yield is 10.684%. If the price falls, the yield rises.

2. The market price =

P = C1! (1+ i)!N

i

"

#$

%

&' + F(1+ i)!N , where F is the maturity value, C

is the coupon, N is the number of semiannual periods left in the bond and i is the semiannual rate. 6. Prove Theorem 6: If P1 > P then i1 < i. Proof: By Theorem 3, If i1 > i, then P1 < P. The Contrapositive of Theorem 3 is: If P1 is not < P, then i1 is not > i. A Consequence of this and Theorem 2.1 is: If P1 > P, the i1 < i. 9. For the bond with ten years remaining, price = $10,000. If the bond is sold one year later, at the same terms, there are 18 semiannual periods left, and the new price =

200

1! (1+ .02)!18

.02

"

#$

%

&' +10,000(1+ .02)!18

= $10,000 . If the only thing that changes is N,

then the price doesn’t change. 12. Prove the formula for the price of bond given in exercise 12. In the proofs of each of the three formulas in exercises 12, 13, and 14, the following Lemma can be used.

Lemma: 1 – i

1

1+ i

!

"#

$

%&

j=1

n

'j

=

1

1+ i

!"#

$%&

n

Proof: By the Formula for the Present Value of an Ordinary Annuity,

1

1+ i

!

"#

$

%&

j=1

n

'j

=

1! (1+ i)!n

i

"

#$

%

&'

i

1

1+ i

!

"#

$

%&

j=1

n

'j

= 1 –

1

1+ i

!"#

$%&

n

1 – i

1

1+ i

!

"#

$

%&

j=1

n

'j

=

1

1+ i

!"#

$%&

n


Formula for exercise 12: The formula is P = F + (Fr – Fi)

1

1+ i

!

"#

$

%&

j=1

n

'j

, where r is the

coupon rate and rF = C.

Proof: P1 = Fr

1

1+ i

!

"#

$

%&

j=1

n

'j

+ F

1

1+ i

!"#

$%&

n

from Formula 1 above.

P1 = Fr

1

1+ i

!

"#

$

%&

j=1

n

'j

+ F

1! i1

1+ i

"

#$

%

&'

j=1

n

(j"

#

$$

%

&

''

by the Lemma.

P1 = F + (Fr – Fi)

1

1+ i

!

"#

$

%&

j=1

n

'j

15. (a) The price of the bond is $878.60. (b) The future value of the reinvestment of the coupons is $312.24. (c) The average rate of return is 2.8%. (d) The interest earned on the investment of coupons is $12.24. (e) The interest earned on the bond alone is $121.40. (f) The interest earned on the total transaction is $133.64.

18. N =

! logiP ! Fr

iF ! Fr

"

#$

%

&'

log(1+ i)


Teachers’ Notes For introductory lessons on bonds, see Vest, Floyd, “The Mathematics of Bond Pricing and Interest Rate Risk,” HiMAP Pull-Out, Consortium 59, and Davis, Steve, “Zero-Coupon Bonds,” Consortium 20, HiMAP Pull-Out, in this course. Fabozzi on bonds. For the nominal rate y = yield to maturity (YTM), Fabozzi and others claim that “… to achieve the yield to maturity, on an investment one must reinvest the coupon payments at an interest rate equal to the yield to maturity” (p. 1). On page 89, (published 1981), Fobozzi says, “the computation of yield requires a trial-and-error procedure.” He then proceeds to demonstrate trial-and-error calculations. It was well known long before 1989 that the calculation was not limited to trial-and-error but there were simple iterative procedures that converge to the desired value. (See Vest, “Computerized Business Calculus Using Calculators,” in this course for exercises where students can rewrite and run programs on the TI83/84.) Have them put in a Run/Stop and Print so they can see the sequence converge. (For a more complete discussion, see Beezer, “Closing in on the Internal Rate of Return” in this course.)

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Bond Pricing Theorems Floyd Vest - comap.com · Bond Pricing Theorems Floyd Vest ... Lemma for Theorem 7. ... What was the Discount at 20 periods? What pattern does their comparison