Fixed Income. ECE 695 Financial Engineering Ilya Pollak Spring 2012

Transcription

1 Fixed Income ECE 695 Financial Engineering Spring 2012

2 Fixed Income Securi>es Owning a share = par>al ownership of the company. Owning a bond = loaning money to the company. Company obligated to pay principal and interest. 2

3 Fixed Income Securi>es Owning a share = par>al ownership of the company. Owning a bond = loaning money to the company. Company obligated to pay principal and interest. Risks: default (inability to keep paying interest and/or pay back principal) if you sell before maturity, market price could be lower than the price you paid 3

4 Bonds vs Stocks Bond prices reflect market s opinion on credit - worthiness of the issuer. Most stocks trade on exchanges whereas most bonds trade over the counter. Countries and states can issue bonds but not stocks. Typically, bonds (especially government bonds) are less vola>le and have a smaller expected return than stocks. 4

5 Callable and PuWable Bonds Callable: issuer has the right to redeem before maturity. Example: any mortgage or car loan in US PuWable: holder has the right to demand repayment before maturity. A bond may be both callable and puwable. In this course, we will only consider plain bonds with no call or put op>ons. 5

6 Perpetual Bonds No maturity, i.e., the principal is never repaid. Usually callable. Pay regular coupons. We will only consider bonds with finite maturity date. 6

7 Types of US Treasurys Treasury bills (T- bills) maturity up to a year (currently, 1, 3, 6, 12 months) zero coupon Treasury notes (T- notes) maturity between 1 and 10 years (currently, 2, 3, 5, 7, 10 years) pay coupons every six months Treasury bonds (T- bonds) maturity above 10 years (currently, 20 and 30 years) pay coupons every six months Treasury Infla>on- Protected Securi>es (TIPS) principal >ed to the CPI 5, 10, 30 years pay coupon every six months US Savings Bonds non- marketable 7

8 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond 8

9 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond E.g., suppose a $10000 bond matures on Nov 15, 2012 and pays 4% interest per year, every six months. 9

10 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond E.g., suppose a $10000 bond matures on Nov 15, 2012 and pays 4% interest per year, every six months. This implies the following remaining payments as of now (March 29, 2012): coupon payment of $200 on May 15, 2012 coupon payment of $200 on November 15, 2012 principal repayment of $10000 on November 15,

11 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond E.g., suppose a $10000 bond matures on Nov 15, 2012 and pays 4% interest per year, every six months. This implies the following remaining payments as of now (March 29, 2012): coupon payment of $200 on May 15, 2012 coupon payment of $200 on November 15, 2012 principal repayment of $10000 on November 15, 2012 This can be made into three zero- coupon STRIPS bonds: a $200 bond with maturity May 15, 2012 a $200 bond with maturity November 15, 2012 a $10000 bond with maturity November 15,

12 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond E.g., suppose a $10000 bond matures on Nov 15, 2012 and pays 4% interest per year, every six months. This implies the following remaining payments as of now (March 29, 2012): coupon payment of $200 on May 15, 2012 coupon payment of $200 on November 15, 2012 principal repayment of $10000 on November 15, 2012 This can be made into three zero- coupon STRIPS bonds: a $200 bond with maturity May 15, 2012 a $200 bond with maturity November 15, 2012 a $10000 bond with maturity November 15, 2012 Also, three such zero- coupon bonds can be made into a 4% bond with $10000 face value, maturing on November 15,

13 US Treasurys: STRIPS Separate Trading of Registered Interest and Principal Securi>es zero- coupon bonds obtained by par>>oning a coupon bond into mul>ple bonds, each corresponding to a single coupon payment or a principal repayment constructed by financial companies STRIPS components can be recons>tuted back into a coupon bond E.g., suppose a $10000 bond matures on Nov 15, 2012 and pays 4% interest per year, every six months. This implies the following remaining payments as of now (March 29, 2012): coupon payment of $200 on May 15, 2012 coupon payment of $200 on November 15, 2012 principal repayment of $10000 on November 15, 2012 This can be made into three zero- coupon STRIPS bonds: a $200 bond with maturity May 15, 2012 a $200 bond with maturity November 15, 2012 a $10000 bond with maturity November 15, 2012 Also, three such zero- coupon bonds can be made into a 4% bond with $10000 face value, maturing on November 15, Allows investors to generate zero- coupon bonds of many different maturi>es. E.g., useful when saving for college tui>on. 13

14 Example: March 15, 2012 quotes coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/

15 Discount factors d(t) = present value of $1 to be received t years from now E.g., if d(0.5) = 0.9 then the value of $1 to be received in six months is 90 cents.

16 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 16

17 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 Assuming the mid- price of is what the bond was worth on 3/15, we have: d(0.5) =

18 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 Assuming the mid- price of is what the bond was worth on 3/15, we have: d(0.5) = , and d(0.5) =

19 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 Assuming the mid- price of is what the bond was worth on 3/15, we have: d(0.5) = , and d(0.5) = The maturing on 3/15/13 will pay: on 9/15/ on 3/15/13 19

20 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 Assuming the mid- price of is what the bond was worth on 3/15, we have: d(0.5) = , and d(0.5) = The maturing on 3/15/13 will pay: on 9/15/ on 3/15/ = d(0.5) d(1) = * d(1) 20

21 Discount factors on 3/15/12 from mid- prices coupon maturity bid ask mid- price 1.375% 9/15/ % 3/15/ % 9/15/ % 3/15/ The 1.375% maturing on 9/15/12 will pay (assuming $100 face value): = (100 principal coupon) on 9/15/12 Assuming the mid- price of is what the bond was worth on 3/15, we have: d(0.5) = , and d(0.5) = The maturing on 3/15/13 will pay: on 9/15/ on 3/15/ = d(0.5) d(1) = * d(1) d(1) =

22 Compounding There is a difference between annual and semi- annual coupon payments. 22

23 Compounding There is a difference between annual and semi- annual coupon payments. A 5% $100 bond purchased on 1/1/12 that pays interest annually, will pay $5 on 1/1/13. 23

24 Compounding There is a difference between annual and semi- annual coupon payments. A 5% $100 bond purchased on 1/1/12 that pays interest annually, will pay $5 on 1/1/13. A 5% $100 bond purchased on 1/1/12 that pays interest semi- annually, will pay $2.50 on 7/1/12 and $2.50 on 1/1/13. 24

25 Compounding There is a difference between annual and semi- annual coupon payments. A 5% $100 bond purchased on 1/1/12 that pays interest annually, will pay $5 on 1/1/13. A 5% $100 bond purchased on 1/1/12 that pays interest semi- annually, will pay $2.50 on 7/1/12 and $2.50 on 1/1/13. This is more valuable than a single payment of $5 on 1/1/13, because $2.50 paid on 7/1/12 can be reinvested! 25

26 Compounding $1 invested at 5%/yr simple interest => $1.05 aier one year 26

27 Compounding $1 invested at 5%/yr simple interest => $1.05 aier one year 5%/yr, compounded semiannually: $1.025 aier 6 months, (1.025) 2 = aier 1 year % annual return 27

28 Compounding $1 invested at 5%/yr simple interest => $1.05 aier one year 5%/yr, compounded semiannually: $1.025 aier 6 months, (1.025) 2 = aier 1 year % annual return 5%/yr, compounded daily: $(1+0.05/365) 365 = aier 1 year 28

29 Compounding $1 invested at 5%/yr simple interest => $1.05 aier one year 5%/yr, compounded semiannually: $1.025 aier 6 months, (1.025) 2 = aier 1 year % annual return 5%/yr, compounded daily: $(1+0.05/365) 365 = aier 1 year 5%/yr, compounded N >mes per year: $(1+0.05/N) N aier 1 year 29

30 Compounding $1 invested at 5%/yr simple interest => $1.05 aier one year 5%/yr, compounded semiannually: $1.025 aier 6 months, (1.025) 2 = aier 1 year % annual return 5%/yr, compounded daily: $(1+0.05/365) 365 = aier 1 year 5%/yr, compounded N >mes per year: $(1+0.05/N) N aier 1 year 5%/yr, compounded con>nuously: lim N- > (1+0.05/N) N = exp(0.05) = aier 1 year 30

31 Zero- Coupon Bonds Aka pure discount bonds or zeros Pay no principal or interest un>l maturity Par value or face value = amount paid to holder at maturity 31

32 Spot rate n- year spot rate is defined as a rate of return by some authors and yield by others. 32

33 Spot rate n- year spot rate is defined as a rate of return by some authors and yield by others. We will call the n- year spot yield the annual yield to maturity of a zero- coupon bond maturing in n years. 33

34 Spot rate n- year spot rate is defined as a rate of return by some authors and yield by others. We will call the n- year spot yield the annual yield to maturity of a zero- coupon bond maturing in n years. We will call the n- year spot interest rate the semi- annually compounded annual rate of return of a zero- coupon bond maturing in n years. 34

35 Zero- coupon example 20- yr, par = $1000 Suppose the bond sells for $ when issued. 35

36 Zero- coupon example 20- yr, par = $1000 Suppose the bond sells for $ when issued. This implies 6% 20- year spot interest rate, because $306.56(1.03) 40 = $

37 Zero- coupon example 20- yr, par = $1000 Suppose the bond sells for $ when issued. This implies 6% 20- year spot interest rate, because $306.56(1.03) 40 = $1000. This also implies 20- year spot yield of (1.03) 2 1 = 6.09%. 37

38 Zero- coupon example, con>nued 20- yr, par = $ year spot interest rate when issued 6%. Buy at $1000/(1.03) 40 =$1000/(1.0609) 20 = $

39 Zero- coupon example, con>nued 20- yr, par = $ year spot interest rate when issued 6%. Buy at $1000/(1.03) 40 =$1000/(1.0609) 20 = $ Suppose in 6 months, investors demand a year spot interest rate of 7%. 39

40 Zero- coupon example, con>nued 20- yr, par = $ year spot interest rate when issued 6%. Buy at $1000/(1.03) 40 =$1000/(1.0609) 20 = $ Suppose in 6 months, investors demand a year spot interest rate of 7%. This is equivalent to saying that the price has gone down to $1000/(1.035) 39 = $261.41, i.e., we ve lost $

41 Zero- coupon example, con>nued 20- yr, par = $ year spot interest rate when issued 6%. Buy at $1000/(1.03) 40 =$1000/(1.0609) 20 = $ Suppose in 6 months, investors demand a year spot interest rate of 7%. This is equivalent to saying that the price has gone down to $1000/(1.035) 39 = $261.41, i.e., we ve lost $ This could happen due to Larger perceived risk of default. The fact that similar bonds can now be obtained at 7%. 41

42 Zero- coupon example, con>nued 20- yr, par = $ year spot interest rate when issued 6%. Buy at $1000/(1.03) 40 =$1000/(1.0609) 20 = $ Suppose in 6 months, investors demand a year spot interest rate of 7%. This is equivalent to saying that the price has gone down to $1000/(1.035) 39 = $261.41, i.e., we ve lost $ This could happen due to Larger perceived risk of default. The fact that similar bonds can now be obtained at 7%. 6.09% annual yield to maturity at which we bought is only guaranteed if we hold to maturity, and there is no default! 42

43 Price, interest, and yield for a zero Price = PAR*(1+r) NT = PAR*(1+y) T where T = number of years to maturity compounding is assumed N >mes per year (usually, N=2) Nr = T- year spot interest rate y = (1+r) N 1 = T- year spot yield 43

44 Coupon Bonds Make regular payments Sold at par when issued At maturity, pay final interest payment + principal 44

45 Coupon Bond Example 20- yr, par=$1000, 40 semi- annual coupon payments of $30, and principal re- payment of $1000 aier 20 years. Implies holding period return of 3% per 6 months, because with this return the present value of this income stream is: 40 t = = 1000 (1.03) t 40 (1.03) (Present value of a $30 payment 6 months from now is $30/1.03. This is because if we invest $30/1.03 at 3% per six months, we will have $30 in six months.) 45

46 NT t =1 Deriva>on of the formula from the PAR r + PAR ( 1 + r) t 1+ r ( ) NT previous slide More generally, if the coupon payments are equal to PAR*r and the bond is trading at par, then Nr is the implied annual interest rate (compounded N >mes per year), because then the present value is: = PAR 1 + r NT ( ) NT r ( 1+ r) NT t + 1 = PAR 1+ r = PAR 1+ r t =1 NT 1 ( ) NT r ( 1+ r) k + 1 t =0 ( ) NT r 1+ r ( )NT r 1 = PAR( 1+ r) NT ( 1 + r) NT = PAR (k = NT t) 46

47 If coupon is C per 1/N- th of a year, and Nr is the implied annual interest rate, compounded N >mes per year, then the value of the bond is: NT C + PAR NT = C ( NT 1 1+ r 1+ r 1+ r ) 1 1+ r t =1 ( ) t ( ) = C 1+ r t =0 ( ) t ( ( ) 1 1+ r) NT 1 ( 1+ r) 1 1 = C ( 1+ r NT ) r ( ) ( ) NT = C 1 1+ r r = C r + PAR C r + PAR ( 1 + r) NT NT + PAR ( 1+ r) + PAR ( 1 + r) NT NT + PAR ( 1+ r) ( 1+ r) NT 47

48 Coupon Example, con>nued 20- yr, par=$1000, 40 semi- annual coupon payments of $30, and principal re- payment of $1000 aier 20 years. Buy for 1000, hold for 6 months. If the price is such that the implied holding period return is s>ll 3% per 6 months, the present value immediately before the first coupon payment is: = 1.03 (1.03) t (1.03) 39 t =0 40 t = (1.03) t (1.03) 40 = % semi- annual return, as expected 48

49 Coupon Example, con>nued 20- yr, par=$1000, 40 semi- annual coupon payments of $30, and principal re- payment of $1000 aier 20 years. Buy for 1000, hold for 6 months. If the price is such that the implied holding period return is now 2.5% per 6 months, the present value immediately before the first coupon payment is: = (1.025) t (1.025) 39 t = % semi- annual return 49

50 Coupon Example, con>nued 20- yr, par=$1000, 40 semi- annual coupon payments of $30, and principal re- payment of $1000 aier 20 years. Buy for 1000, hold for 6 months. If the price is such that the implied holding period return is now 3.5% per six months, the present value just before the first coupon payment is: = (1.035) t (1.035) 39 t =0 7.55% semi- annual return 50

51 Yields Coupon yield (aka coupon rate, or nominal yield) is C/PAR where C is the coupon payment. Current yield = C/(current price). Yield to maturity (aka yield) = effec>ve rate of return earned if the bond is held to maturity, and all the coupon payments and principal repayment are made as promised. 51

52 Example Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. 52

53 Example Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 1: it is selling at PAR 53

54 Example Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 1: if it is selling at PAR, the effec>ve rate of return if we buy and hold to maturity, is 4% per 6 months, yielding 8.16% per year. 54

55 Example Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 1: if it is selling at PAR, the effec>ve rate of return if we buy and hold to maturity, is 4% per 6 months, yielding 8.16% per year. This is the coupon yield. In this case, it also happens to be the current yield and the yield to maturity. 55

56 Example, con>nued Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 2: if it is selling for $1200 and we buy it and hold un>l maturity, we will make less than 4% per six months for two reasons 56

57 Example, con>nued Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 2: if it is selling for $1200 and we buy it and hold un>l maturity, we will make less than 4% per six months for two reasons: Coupon payments are only 40/1200 = 3.33% per six months. This is the current yield, and it is smaller than the coupon rate. 57

58 Example, con>nued Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 2: if it is selling for $1200 and we buy it and hold un>l maturity, we will make less than 4% per six months for two reasons: Coupon payments are only 40/1200 = 3.33% per six months. This is the current yield, and it is smaller than the coupon rate. At maturity, we only get $1000 back, not $1200! 58

59 Example, con>nued So our overall effec>ve rate of return per six months is determined from PRICE = C r + Note that, for a zero- coupon bond, this becomes In our example, we have 1200 = 40 r PAR C r ( 1+ r) NT PRICE = PAR ( 1 + r) NT r 1+ r r 3.24% per six months ( )

60 Example, con>nued Suppose a bond matures in T=30 years, has PAR = $1000, and pays C=$40 coupon every 6 months. Case 3: if it is selling for $900 and we buy it and hold un>l maturity, we will make more than 4% per six months for two reasons: Coupon payments are 40/900 = 4.44% per six months. This is the current yield, and it is larger than the coupon rate. At maturity, we get $1000 back, not $900! Yield to maturity is 4.48%. 60

61 Rela>onships among yields If a bond is selling at a premium, i.e., PRICE > PAR, then coupon rate > current yield > yield to maturity. If a bond is selling at par, i.e., PRICE = PAR, then coupon rate = current yield = yield to maturity. If a bond is selling at a discount, i.e., PRICE < PAR, then coupon rate < current yield < yield to maturity. 61

62 Coupon bond = porqolio of zeros Consider a 1- yr coupon bond with semi- annual $40 coupon and PAR=$

63 Coupon bond = porqolio of zeros Consider a 1- yr coupon bond with semi- annual $40 coupon and PAR=$1000. This is like two zero- coupon bonds: one with PAR =$40 and maturity 6 months, and another one with PAR=$1040 and maturity 1 year. 63

64 Coupon bond = porqolio of zeros Consider a 1- yr coupon bond with semi- annual $40 coupon and PAR=$1000. This is like two zero- coupon bonds: one with PAR =$40 and maturity 6 months, and another one with PAR=$1040 and maturity 1 year. Suppose 6- month spot interest rate is 5% and 1- yr spot interest rate is 6%. Price of our 1- yr coupon bond is: =

65 Spot rates and yield to maturity of a coupon bond The yield to maturity of the coupon bond from the previous slide, y, is found from = = y (1 + y) 2 y 2.99% per six months 65

66 Spot rates and yield to maturity of a coupon bond: general formula Suppose we have a coupon bond with coupon payments C per 1/N years T years un>l maturity face value PAR yield to maturity y per 1/N years Let Nr n = n/n- year spot interest rate, for n=1,,nt NT C (1+ r n ) + PAR = n (1 + r NT ) NT n=1 NT n=1 C (1 + y) + PAR n (1 + y) NT 66

67 Term Structure of Interest Rates Aka the yield curve. = the yield to maturity as a func>on of maturity. Enough to do this for zero- coupon bonds, as yields of coupon bonds can be expressed through yields of zero- coupon bonds. Typically, higher yields for longer maturi>es due to risk premium, although there are excep>ons. Credit spread curve: difference of yields of two types of securi>es. 67

68 Forward rates Let y 1, y 2 be (annual) spot yields (= yields to maturity for zero- coupon bonds) for maturi>es T 1 and T 2, respec>vely, such that T 1 < T 2. The forward rate r 12 for [T 1, T 2 ] is defined by Hence, ( ) T 1 1+ y 1 1+ r 12 Gross return over [0,T 1 ] for maturity T 1 ( ) T2 T 1 = ( 1+ y ) T 2 2 Gross return over [0,T 2 ] for maturity T 2 r 12 = ( ) T y T2 T1 2 1 ( 1+ y 1 ) T 1 68

69 Spot yields from forward rates If r k,k+1 is forward rate for [T k, T k+1 ] for k=0,,n 1, then spot yield for maturity T n is found from: ( 1+ r 01 ) T 1 y n = 1+ r 01 ( 1 + r 12 ) T 2 T 1 ( 1 + r n 1,n ) T n T n 1 = ( 1 + yn ) T n ( ) T 1 ( 1 + r 12 ) T 2 T 1 ( 1 + r n 1,n ) T n T n 1 1 T n 1 69

70 Example: forward rates from spot yields Suppose spot yields for 1, 2, and 3 years are y 1 =6%, y 2 =6.5%, and y 3 =7%, respec>vely. 70

71 Example: forward rates from spot yields Suppose spot yields for 1, 2, and 3 years are y 1 =6%, y 2 =6.5%, and y 3 =7%, respec>vely. The forward rate r 1 for the first year [0,1] is the same as the spot yield: r 1 =y 1 =6%/yr. 71

72 Example: forward rates from spot yields Suppose spot yields for 1, 2, and 3 years are y 1 =6%, y 2 =6.5%, and y 3 =7%, respec>vely. The forward rate r 1 for the first year [0,1] is the same as the spot yield: r 1 =y 1 =6%/yr. The forward rate r 2 for the second year [1,2] is determined from ( 1 + y 1 ) 1 + r 2 ( ) = ( 1 + y 2 ) 2 ( ) 2 ( ) 1 = r 2 = 1 + y 2 1+ y

73 Example, con>nued Suppose spot yields for 1, 2, and 3 years are y 1 =6%, y 2 =6.5%, and y 3 =7%, respec>vely. The forward rate r 3 for the third year [2,3] is determined from ( 1 + y 2 ) r 3 ( ) = ( 1+ y 3 ) 3 ( ) 3 ( ) 1 = r 3 = 1 + y 3 1+ y

74 Example: spot yields from forward rates Suppose we know r 1 =y 1 =6%, r 2 =7%, r 3 =8%. Then ( 1+ y 2 ) 2 = ( 1+ r 1 )( 1+ r 2 ) Hence y 2 = ( 1+ r 1 )( 1 + r 2 ) 1 = Also, ( 1+ y 3 ) 3 = ( 1 + r 1 )( 1 + r 2 )( 1 + r 3 ) Hence y 3 = ( 1+ r 1 )( 1 + r 2 )( 1+ r 3 ) In general, = ( ) ( 1+ r n ) y n = n 1 + r

75 Equivalent representa>ons of the term structure Forward rates as a func>on of maturity. Prices of zero- coupon bonds as a func>on of maturity. 75

76 Prices and spot yields If y T is the spot yield for maturity T, and P(T) is the price of a zero with maturity T and face value PAR, then P(T) = PAR/(1+y T ) T y T = [PAR/P(T)] 1/T 1 76

77 Example: Prices from forward rates Suppose we know r 1 =y 1 =6%, r 2 =7%, r 3 =8%. Then the price P(1) for a zero- coupon bond maturing in 1 yr with PAR $1000 is The price P(2) for a zero maturing in 2 yrs with PAR $1000 is 1000 (1+ r 1 )(1+ r 2 ) = = r The price P(3) for a zero maturing in 3 yrs with PAR $1000 is 1000 (1+ r 1 )(1+ r 2 )(1+ r 3 ) =

78 Example: forward rates from prices Suppose we know P(1) = $920, P(2) = $830, P(3) = $760 for PAR = $1000. Then 920 = P(1) = 1000 and r 1 + r 1 = P(1) 1 = = P(2) = 1000 (1+ r 1 )(1+ r 2 ) = P(1) = r 2 1+ r 2 r 2 = P(1) P(2) 1 = = P(3) = 1000 (1 + r 1 )(1 + r 2 )(1 + r 3 ) = P(2) = r 3 1+ r 3 r 3 = P(2) P(3) 1 =

79 Prices and forward rates If r k,k+1 is forward rate for [T k, T k+1 ] for k=0,,n 1, then price P(T k ) for maturity T k and face value PAR is found from: P( T k ) = ( 1 + r 01 ) T 1 PAR ( ) T k T k 1 ( 1+ r 12 ) T 2 T 1 1+ r k 1,k 79

80 Prices and forward rates If r k,k+1 is forward rate for [T k, T k+1 ] for k=0,,n 1, then price P(T k ) for maturity T k and face value PAR is found from: P( T k ) = ( 1 + r 01 ) T 1 PAR ( ) T k T k 1 ( 1+ r 12 ) T 2 T 1 1+ r k 1,k Hence, P( T k ) = ( ) P T k 1 ( 1 + r k 1,k ) T k T k 1 80

81 Prices and forward rates If r k,k+1 is forward rate for [T k, T k+1 ] for k=0,,n 1, then price P(T k ) for maturity T k and face value PAR is found from: P( T k ) = ( 1 + r 01 ) T 1 PAR ( ) T k T k 1 ( 1+ r 12 ) T 2 T 1 1+ r k 1,k Hence, P( T k ) = r k 1,k = ( ) P T k 1 ( 1 + r k 1,k ) T k T k 1 ( ) ( ) P T k 1 P T k 1 Tk Tk

82 Con>nuously compounded spot rate Con>nuously compounded spot rate is the effec>ve yearly rate under con>nuous compounding. If con>nuously compounded spot rate for [0, T k ] is r k then the overall gross return during [0, T k ] is exp(r k T k ). current price is P(T k ) = PAR/exp(r k T k ) 82

83 Con>nuously compounded forward rate Con>nuously compounded forward rate r k,k+1 is the yearly interest rate prevailing during [T k, T k+1 ] assuming con>nuous compounding. Hence, the gross return during the >me interval [T k, T k+1 ] is exp[r k,k+1 (T k+1 T k )]. The price of a zero maturing in k years is P( T k ) = Therefore, ( ) exp r k 1,k T k T k 1 exp r 01 T 1 + r 1,2 T 2 T 1 ( ) ( ) = P T k 1 P T k PAR ( ) + + r k 1,k ( T k T k 1 ) and so r k 1,k = 1 ln P T k 1 T k T k 1 P T k ( ) ( ) 83

84 Con>nuously compounded forward rate and spot rate Matching the two expressions for the price, we have: P( T k ) = PAR = exp r 01 T 1 + r 1,2 ( T 2 T 1 ) + + r k 1,k ( T k T k 1 ) r k = r 01 T 1 + r 1,2 T 2 T 1 ( ) + + r k 1,k ( T k T k 1 ) T k PAR [ ] exp r k T k r k 1,k ( T k T k 1 ) = y k T k y k 1 T k 1 r k 1,k = r kt k r k 1 T k 1 T k T k 1 84

85 Instantaneous forward rate Instantaneous yearly forward rate is a func>on r(t) such that the price of a zero with par = 1 maturing T years from now is T D(T ) = exp r(t)dt = exp Tr T 0 ( ) D(T) is called the discount func>on r T is the annual con>nuously compounded spot rate. r T = 1 T T r(t)dt 0 The price of a zero with par value PAR is P(T) = PAR * D(T) 85

86 If instantaneous forward rate is piecewise constant, If r(t) = r k,k+1 for T k t < T k+1 and k=0,,n 1, then D( T k ) = 1 = exp r 01 T 1 + r 1,2 ( T 2 T 1 ) + + r k 1,k ( T k T k 1 ) 1 [ ] exp r k T k 86

87 Interest- rate risk Interest- rate risk is the sensi>vity of bond prices to interest rates. ( ) D(T ) = exp Tr T d dr T D(T ) = T exp Tr T d dr T D(T ) D(T ) = T Δ bond price bond price ( ) = TD(T ) T Δ spot rate Longer- term bonds have higher interest- rate risks than shorter- term bonds 87