BOND VALUATION With Solutions

BOND VALUATION Bond Bond is a long term contract under which a borrower agrees to make payments of interest and principal, on specific dates, to the holders of the bond Key characteristics: VB = value of a bond/bond price M = par or maturity value of the bond; it is the stated face value of the bond and this is amount that must be paid off at maturity and it is often equal to $ 1.000 INT = coupon payment or dollars of interest paid each year; (Coupon rate x Par value) rk = coupon interest rate; (coupon payment / par value) rd = the bond's required rate of the return; that is the market rate of interest for that type of bond; it is also called the yield N = number of years before the bond matures; maturity date is a date on which the par value must be repaid m = number of discounting periods per year The value of any financial asset - a stock, a bond, a lease, or even a physical asset such as an apartment building is simply the present value of the cash flows the asset is expected to produce.

Bond Valuation The cash flows from a specific bond depend on the contractual features meaning the type of the bond. The following general equation, written in several forms, can be used to find the value of any bond, VB.

∫¿ (1+r d )

N

+

∫¿ (1+ r d )2 ∫¿

M N ( 1+r d )

+…+ ¿

+¿ (1+r d )1 V B =¿

∫¿ t

(1+r d )

+

M =∫ ∙ PVIFA Nr + M ∙ PVIFrN N ( 1+r d ) d

d

N

¿∑ ¿ t =1

1

¿∫ ∙

[

]

1 1 M − + r d r d (1+r d ) N ( 1+ r d ) N

So, the cash flows consist of an annuity of N years plus a lump sum payment at the end of Year N. 1. Standard coupon-bearing bond Standard coupon-bearing bond =the cash flows consist of interest payment during the life of the bond, plus the amount borrowed when the bond matures Types:  Bond with a fixed coupon rate = coupon payments are constant. It can be:  Floating-rate bond. = if a bond's coupon payment vary over time, meaning that the coupon rate is set for, say, the initial six month period, after which it is adjusted every six months based on some market rate. 1.1. Bond with a fixed coupon rate Bond with a fixed coupon rate = coupon payments are constant. It can be: a) the bond is selling at a price equal to its par value; required rate of return (rd)=coupon rate (rk) b) the bond is selling at a price below its par value; required rate of return (rd) coupon rate (rk); it is called a discount bond c) the bond is selling at a price above its par value; required rate of return (rd) coupon rate (rk); it is called a premium bond. Valuation of bonds on the date of their issue =valuation of bonds at the beginning of an interest payment dates Bonds with annual coupons Example: (Tool Kit 4 Chapter) A bond has a 15-year maturity, a 10% annual coupon and a $ 1,000 par value. The required rate of return or the yield to maturity on the bond is 10%, given its risk, maturity, liquidity and other rates in the economy. What is a fair value for the bond i.e. its market price? M (or FV)=$ 1,000 rk=10% rd=10% INT= $ 100 N=15 VB=?

2

15

15

V B =100∙ PVIFA 10 +1,000∙ PVIF 10 =100∙ 7.606+1,000 ∙ 0.239=1,000 or

V B =100∙

[

]

1 1 1,000 − + =1,000 15 0.10 0.10 ∙(1+0.10) ( 1+ 0.10 )15

The bond is selling at a price equal to its par value. Example: (Tool Kit 4.3.) A bond matures in 6 years has a par value of $ 1.000, an annual coupon payment of $ 80 and a market interest rate of 9%. What is its price? M (or FV)=$ 1,000 rd=9% INT= $ 80  rk=8% N=6 VB=? 6

6

V B =80 ∙ PVIFA 9 +1,000 ∙ PVIF 9=80∙ 4.48592+1,000 ∙ 0.59627=955.14 The bond is selling at a price below its par value, meaning that the required rate of return (rd) coupon rate (rk).It is called a discount bond. Example: (Tool Kit 4.3.) A bond matures in 18 years has a par value of $ 1,000, an annual coupon of 10% and a market interest rate of 7%. What is its price? M (or FV)=$ 1,000 rd=7% rk=10%

INT= $ 100

N=18 VB=? 18 V B =100∙ PVIFA 18 7 +1,000∙ PVIF 7 =100∙ 10.05909+1,000 ∙0.29586=1,3011.77

The bond is selling at a price above its par value, meaning that the required rate of return (rd) coupon rate (rk). It is called a premium bond. Exercise: 1 Jackson Corporation's bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $1.000 par value, and the coupon interest rate is 8%. The bonds have a YTM of 9%. What is the current market price of these bonds?

3

∫¿ t

(1+r d )

+

M =∫ ∙ PVIFA Nr + M ∙ PVIFrN N ( 1+r d ) d

d

N

¿∑ ¿ t =1

¿∫ ∙

[

]

1 1 M − + r d r d (1+r d ) N ( 1+ r d ) N

M (or FV)=$1,000 rk=8% rd=9% INT=$80 N=12 VB=?

12 V B =80 ∙ PVIFA 12 9 +1.000 ∙ PVIF 9 =80∙ 7.16073+1.000 ∙ 0.35553=928.39

V B =80 ∙

[

]

1 1 1000 − + =928.39 12 0.09 0.09 ∙(1+0.09) ( 1+ 0.09 )12

Bonds with semiannual coupons

∫ ¿ ∙ PVIFA 2 ∙ N + M ∙ PVIF 2 ∙N 2

2∙ N

V B =∑ t=1

rd 2

rd 2

∫ ¿2 t

(1+r d /2)

+

M =¿ 2∙ N ( 1+r d /2 )

Example: (Tool Kit 4.6.)

4

A bond has a 25-year maturity, an 8% annual coupon paid semiannually, and a face value of $1,000. The going nominal annual interest rate is 6%. What is the bond's price? M (or FV)=$ 1,000 rd=6% rk=8%

INT= $ 80

N=25 m=2 VB=?

V B=

80 50 ∙ PVIFA 50 3 +1,000 ∙ PVIF 3 =40 ∙ 25.72976+1,000 ∙ 0.22811=1,257.30 2

Exercise: Rentro Rentals has issued bonds that have a 10% coupon rate, payable semiannually. The bonds mature is 8 years, have a face value of $1,000 and a YTM of 8.5%. What is the price of the bonds?

M (or FV)=$1,000 rd=8.5% rk=10%

INT= $ 100

N=8 m=2 VB=?

r r (¿¿ d / 2)(1+ r d /2)2 N 1 (¿ ¿ d / 2 )− ¿ 1 M ¿ + ( 1+r ) 2 N d/2

∫¿ ∙¿

m V B=¿ 5

V B=

[

]

100 1 1 1000 ∙ − + =1,085.80 2 (0.085/2) (0.085/2)(1+0.085 /2)16 ( 1+0.085 /2 )16

Exercise: Suppose HM sold an issue of bonds with a 10-year maturity, a $1,000 par value, a 10% coupon rate, and semiannual interest payments. a) two years after the bonds were issued, the going rate of interest on bonds such as these fell to 6%; at what price would the bonds sell? b) suppose that 2 years after the initial offering, the going interest rate had risen to 12%; at what price would the bonds sell? c) suppose that the conditions in part a existed-2 years after the issue date, interest rates fell to 6%. Suppose further that the interest rate remained at 6% for the next 8 years. What would happen to the price of bonds over time? a) M (or FV)=$1,000 rk=10% m=2 INT=$100 N=10 rd fall to 6% VB=?

V 1=

[

]

[

]

100 1 1 1,000 ∙ − + =1,251.22 16 2 (0.06 /2) (0.06 /2)∙(1+ 0.06/ 2) ( 1+ 0.06/ 2 )16

b)

V 1=

100 1 1 1,000 ∙ − + =898.94 16 2 (0.12/ 2) (0.12/ 2) ∙(1+0.12/2) ( 1+ 0.12/ 2 )16 6

c) The price of the bond will decline toward $1,000, hitting $1,000 (plus accrued interest) at the maturity date 8 years (16 six-month periods) hence. 2. Zero-coupon bond Zero-coupon bond = bonds that pay no interest but are offered at a substantial discount below their par values and hence provide capital appreciation rather than interest income. These securities are called zero coupon bonds (“zeros”), or original issue discount bonds (OIDs). N

V B =M ∙ PVIF r

d

Bond yields Unlike the coupon interest rate, which is fixed, the bond's yields vary from day to day depending on current market conditions. The yield can be calculated in three different ways, and three "answers" can be obtained. These different yields are described in the following sections. 1. Yield to Maturity (YTM) The YTM is defined as the rate of return that will be earned if a bond makes all scheduled payments and is held to maturity. It can be viewed as the bond's promised rate of return which is the return that investors will receive if all the promised payments are made. YTM equals to expected rate of return only if the probability of default is zero and the bond cannot be called. If there is some default risk, or if the bond may be called, then there is some probability that the promised payments to maturity will not be received, in which case the calculated YTM will differ from expected return, meaning that the expected rate of return will be less than the promised YTM. The YTM for a bond that sells at pas consists entirely of an interest yield, but if the bond sells at a price other than its par value, the YTM will consist of the interest yield plus a positive or negative capital gains yield. It cannot be solved for directly. It generally must be determined using trial and error or an iterative technique. Steps: 1. Given that it is difficult to guess the YTM of the first attempt, we are assuming at least two rates: one that gives a lower price than current market price and the other which results in a higher price of current market price 2. When we have those two prices we are approaching to the procedure for linear interpolation: 7

YTM =rd1 +(rd2−rd1) ∙

(P1−P) (P1−P2)

An Approximation: If you are not inclined to follow the trial-and-error approach, you can employ the following formula to find the approximate YTM on a bond:

∫ +M −V B YTM =

N 0.4 ∙ M +0,6 ∙ V B

This formula was suggested by Gabriel A. Hawawini and Ashok Vora, in the article published in the Journal for Finance March 1982 issue.

Example: (Tool Kit 4 Chapter) Suppose that you are offered a 14-year, 10% annual coupon, $1,000 par value bond at a price of $1,494.93. What is the YTM of the bond? M (or FV) =$ 1,000 rk=10%

INT= $ 100

N=14 VB=$ 1,494.93 rd/YTM=? VB  M  YTM  rk So, YTM has to be less than a coupon rate, meaning less than 10%. rd1=8% P1=

100 ∙ PVIFA (8 ,14) +1,000 ∙ PVIF(8 ,14) =100∙ 7.53608+1,000 ∙ 0.39711=1,150.72

1,150.72  1,494. 93  P1 P  8% is too high rate rd 2=5% P2=

100 ∙ PVIFA (5 ,14) +1,000 ∙ PVIF (5 ,14) =100∙ 9.89864+ 1,000∙ 0.50507=1,494.934

1,494.93=1,494.93  P2=P  5% is YTM Exercise: 4-2 Wilson Wonders' bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $1.000 par value, and the coupon interest rate is 10%. The bonds sell at price of $850. What is their yield to maturity? M (or FV)=$1,000 8

rk=10% INT=$100 N=12 VB=$850 YTM=? VB  M  YTM  rk So, YTM has to be higher than a coupon rate, meaning more than 10% I way: rd1=12% P1=

100 ∙ PVIFA (12 , 12)+ 1.000∙ PVIF (12 ,12)=100 ∙ 6.19437+1000 ∙0.25668=876.117

876.117  850.00  P1  P  12% is too low rate rd2=13% P2=

100 ∙ PVIFA (13 ,12) +1.000∙ PVIF(13 ,12)=100 ∙ 5.91765+1000 ∙0.23071=822.475

822.475  850  P2  P  13% is too high rate YTM=12%+x

P −P x = 1 k 2−k 1 P1−P2 x 876.117−850.00 = 0,13−0,12 876.117−822.475 x 26.117 = 0.01 53.642 x=0,01 ∙0.486=0.00486 → 0.486 YTM=12%+0.486%=12.486% II way:

YTM =k 1 +( k 2−k 1)∙

( P1−P) ( P1 −P 2)

9

YTM =0.12+ ( 0.13−0.12 ) ∙

(25.117 ) =0.12486 12.486 ( 53.642 )

III way: Approximation

1,000−850 12 YTM = =0.1236 12.36 0.60∙ 850+0.40 ∙ 1,000 100+

Exercise: 4-10 The BC's bonds have 5 years remaining to maturity and interest is paid annually, the bonds have a $1,000 par value, and the coupon rate is 9%. a) what is the YTM at a current market price of $829 b) would you pay $829 for one of these bonds if you thought the appropriate rate of interest was 12% a) YR = 13.98%. b) Yes. At a price of $829, the yield to maturity, 13.98 percent, is greater than your required rate of return of 12 percent. If your required rate of return were 12 percent, you should be willing to buy the bond at any price below $891.86.

2.Yield to Call (YTC) If you purchased a bond that was callable and the company called it, you would not have the option of holding the bond until it matured. Therefore, the YTM would not be earned.

∫¿ t

(1+r d )

+

Call price N (1+r d ) N

Price of callable bond (if called at N )=∑ ¿ t =1

N = the number of years until the company can call the bond Call price = the price the company must pay in order to call the bond (it is often set equal to the par value plus 1 year's interest rd = YTC The YTC is the rate of return investors will receive if their bonds are called. If the issuer has the right to call the bonds and if interest rate fall, then it would be logical for the issuer to call the bonds and replace them with new bonds that carry a lower coupon.

10

The YTC is find similarly to the YTM. The same formula is used, but years to maturity are replaced with years to call, and the maturity value is replaced with the call price. Example: (Tool Kit 4 Chapter) Suppose you purchase a 15-year, 10% annual coupon, $ 1.000 par value bond with a call provision after 10 years at a call price of $ 1.100. One year later, interest rates have fallen from 10% to 5% causing the value of the bond to rise to $ 1.494,93. What is the bond's YTC? Call price =$ 1,100 rk=10%

INT= $ 100

N=10-1=9 (years to call) VB=$ 1,494.93 rd/YTC=? VB  Call price  YTC  rk So, YTC has to be less than a coupon rate, meaning less than 10%. rd 1=5% P1=

100 ∙ PVIFA (5 ,9) +1,100 ∙ PVIF (5 ,9 )=100 ∙7,10782+1,100 ∙0,64461=1,419.853

1.419,853  1,494.93  P1 P  5% is too high rate rd 2=4% P2=

100 ∙ PVIFA (4 , 9) +1,100 ∙ PVIF (4 ,9) =100 ∙7,4352+1,100 ∙0,70259=1.516,369

1.516,369  1.494,93  P2 P  4% is too low rate YTM=4%+x

P −P x = 1 r d 2−r d 1 P1 −P 2 x 1,516.369−1,494.93 = 0.05−0.04 1,516.369−1,419.853 x 21.439 = 0.01 96.516 x=0.01 ∙0.222=0.00222→ 0.222 YTM=4%+0.222%=4.22% 11

Exercise: 4-8 Thatcher Corporation's bonds will mature in 10 years. The bonds have a face value of $1,000 and an 8% coupon rate, paid semiannually. The price of the bond is $1,100. The bonds are callable in 5 years at a price of $1,050. What is their YTM? What is their yield to call? M (or FV)=$1,000 rk=8% m=2 INT=$80 N=10 VB=$1,100 Callable price in 5 years=$1,050 YTM=? VB  M  YTM  rk So, YTM has to be lower than a coupon rate, meaning less than 8% I way: rd1=7% m=2

P1=

[

]

80 1 1 1000 ∙ − + =1,071.062 2 (0.07 / 2) (0.07 / 2)∙(1+0.07/ 2)20 ( 1+0.07/ 2 )20

1,071.062  1,100.00  P1 P  7% is too high rate rd 2=6% m=2

P 2=

[

]

80 1 1 1000 ∙ − + =1,148.775 20 2 (0.06 / 2) (0.06 / 2)∙(1+0.06/ 2) ( 1+ 0.06 )20

1,148.775  1,100 P2 P  6% is too low rate

P −P x = 1 k 2−k 1 P1−P2

YTM=6%+x

x 1,148.775−1,100 = 0.07−0.06 1,148.775−1,071.062 x 48.775 = 0.01 77.713 12

x=0.01 ∙0.06276=0.006276 → 0.6276 YTM=6%+0.62%=6.62% YTC=?

∫¿ t

(1+r d )

+

Call price N (1+r d ) N

Price of callable bond (if called at N )=∑ ¿ t =1

N=5 Callable price=1,050

YTC=? VB  Callable price  YTC  rk

So, YTC has to be lower than a coupon rate, meaning less than 8%

I way: rd1=7% m=2

P 1=

[

]

80 1 1 1,050 ∙ − + =1,077.029 10 2 (0.07 / 2) (0.07 / 2)∙(1+0.07/ 2) ( 1+0.07/ 2 )10

1,077.029  1,100.00  P1 P  7% is too high rate rd 2=6% m=2

P 2=

[

]

80 1 1 1,050 ∙ − + =1,122.507 2 (0.06 / 2) (0.06 / 2)∙(1+0.06/ 2)10 ( 1+ 0.06 )10

1,122.507  1,100 P2P  6% is too low rate YTM=6%+x

P −P x = 1 k 2−k 1 P1−P2

13

x 1,122.507−1,100 = 0.07−0.06 1,122.507−1,077.029

x 22.507 = 0.01 45.478 x=0.01 ∙0.4948=0.00494 → 0.494 YTM=6%+0.49%=6.49% Exercise: 4-22 AI' bonds have a current market price of $1.200. The bonds have an 11% annual coupon payment, a $1,000 face value, and 10 years left until maturity. The bond may be called in 5 years at 109% of face value (call price=$1,090). a)what is the YTM? b)what is the YTC if they are called in 5 years? c) which yield might investors expects to earn on these bonds and why? d) the bond's indenture indicates that the call provision gives the firm the right to call them at the end of each year beginning in Year 5. In Year 5 they must be called at 109% of face value, but in each of the next 4 years the cal percentage will decline by 1 percentage point. Thus, in Year 6 they may be called at 108% of face value, in Year 7 they may be called at 107% of face value and so on. If the yield curve is horizontal and interest rate remains at their current level, when is the latest that investors might expect the firm to call the bonds? a) M (or FV)=$1,000 rk=11% INT=$110 N=10 VB=$1,200 Callable price in 5 years=$1,090 YTM=? VB  M  YTM  rk So, YTM has to be lower than a coupon rate, meaning less than 11%. I way: rd1=10%

P1=110 ∙

[

]

1 1 1,000 − + =1,061.446 10 (0.10) (0.10)∙(1+ 0.10) ( 1+ 0.10 )10

1,061.446  1,200  P1 P  10% is too high rate 14

rd 2=8%

P2=110 ∙

[

]

1 1 1,000 − + =¿ 10 1,201.302 (0.08) (0.08)∙(1+ 0.08) ( 1+ 0.08 )10

1,201.302  1,200 P2 P  8% is too low rate YTM=8%+x

P −P x = 1 k 2−k 1 P1−P2 x 1,201.302−1,200 = 0.10−0.08 1,201.302−1,061.446 x 1.302 = 0.02 139.856 x=0.02 ∙0.0093095=0.000186 → 0.02

YTM=8%+0.02%=8.02% b) YTC=? VB  Callable price  YTC rk So, YTC has to be lower than a coupon rate, meaning lower than 11% I/YR = YTC = 7.59% rd1=10%

P1=110 ∙

[

]

1 1 1,090 − + =1,093.791 (0.10) (0.10)∙(1+ 0.10)5 ( 1+ 0.10 )5

1,093.791  1,200  P1 P  10% is too high rate rd1=7%

P2=110 ∙

[

]

1 1 1,090 − + =1,228.177 5 (0.07) (0.07)∙(1+0.07) ( 1+0.07 )5

1,228.177  1,200  P2 P  7% is too low rate 15

YTM=7%+x

P −P x = 1 k 2−k 1 P1−P2 x 1,228.177−1,200.00 = 0.10−0.07 1,228.177−1,093.791 x 28.177 = 0.03 134.386 x=0.03 ∙ 0.2096=0.006290 → 0.62 YTM=7%+0.62%=7.62% c) The bonds are selling at a premium which indicates that interest rates have fallen since the bonds were originally issued. Assuming that interest rates do not change from the present level, investors would expect to earn the yield to call. (Note that the YTC is less than the YTM). d) Similarly from above, YTC can be found, if called in each subsequent year. If called in Year 6: YTC = 7.80%. If called in Year 7: YTC = 7.95%. If called in Year 8: YTC = 8.07%. If called in Year 9: YTC = 8.17%. According to these calculations, the latest investors might expect a call of the bonds is in Year 7. This is the last year that the expected YTC will be less than the expected YTM. At this time, the firm still finds an advantage to calling the bonds, rather than seeing them to maturity.

3. Current Yield The current yield provides information regarding the amount of cash income that a bond will generate in a given year, but since it does not take account of capital gains or losses that will be realized if the bond is held until maturity or call, it does not provide an accurate measure of the bond's total expected return. 16

Current yield=

Annual interest payment Bonds current price

Example: (Tool Kit 4.4.) A bond currently sells for $850. It has an 8-year maturity, an annual coupon of $80 and a par value of $1,000. What is its YTM? What is its current yield? A bond currently sells for $1,250. It pays a $110 annual coupon and has a 20-year maturity, but it can be called in 5 years at $ 1,100. What is its YTM and its YTC? M (or FV) =$ 1,000 rk=8%

INT= $ 80

N=8 VB=$ 850 rd/YTM=?

Current yield=

Annual interest payment 80 = =0.0941 9.41 Bonds current price 850

M (or FV) =$ 1,000 rk=11%

INT= $ 110

N=20 VB=$ 1,250 rd/YTM=? VB  M  YTM  rk So, YTM has to be less than a coupon rate, meaning less than 11%. rd1=9% P1=

110 ∙ PVIFA (9 , 20)+ 1,000∙ PVIF(9 , 20)=110 ∙ 9.12855+1,000 ∙0.17843=1,182.57

1,182.57  1,250.00  P1 P  9% is too high rate rd 2=8% P2=

110 ∙ PVIFA (9 , 20)+ 1,000∙ PVIF(9 , 20)=110 ∙ 9.81815+1,000 ∙0.21455=1,294.5465

1,294.54651,250.00  P2P  8% is too low rate YTM=8%+x

P −P x = 1 rd 2−rd 1 P1−P2 17

x 1,294.5465−1,250.00 = 0.09−0.08 1,294.5465−1,182.57 x 44.5465 = 0.01 111,9765 x=0.01 ∙0.39782=0.0039782→ 0.39782 YTM=8%+0.39782%=8.39% Call price =$ 1,100 rk=10%

INT= $ 100

N=5 (years to call) VB=$ 1,250.00 rd/YTC=? VB  Call price  YTC  rk So, YTC has to be less than a coupon rate, meaning less than 11%. rd 1=8% P1=

110 ∙ PVIFA (8 , 5) +1,100∙ PVIF (8 , 5)=110 ∙ 3.99271+1,100∙ 0.68058=¿

1,187.84

1,187.84  1,250.00  P1 P  8% is too high rate rd 2=6% P2=

110 ∙ PVIFA (6 , 5) +1,100 ∙ PVIF (6 , 5)=110 ∙ 4.21236+ 1,100∙ 0.74726=1,285.35

1,285.35 1,250.00  P2 P  6% is too low rate YTM=6%+x

P −P x = 1 rd 2−rd 1 P1−P2 x 1,285.35−1,250.00 = 0.08−0.06 1,285.35−1,187.84 x 35.35 = 0.02 97.51

18

x=0.02 ∙0.3625=0.007251→ 0.725 YTM=6%+0.725%=6.725% Exercise: 4-3 Heath Foods's bonds have 7 years remaining to maturity. The bonds have face value of $1.000 and a yield to maturity of 8%. They pay interest annually and have a 9% coupon rate. What is the current yield?

Current yiel d=

Annual interest payment Bonds current price

M (or FV)=$1,000 rk=9 % YTM=8% INT=$90 N=7 Current yield =? 1. have to find the current value of the bonds and then calculate their current yield:

V B =90 ∙

[

]

1 1 1,000 − + =1,052.06 7 0.09 0.09 ∙(1+0.09) ( 1+0.09 )7

2.

Current yield=

Annual interest payment 90 = =0.0855 8.55 Bonds current price 1,052.06

Exercise: 4-14 A bond that matures in 7 years sells for $1,020. The bond has a face value of $1,000 and a YTM of 10.5883%. The bonds pay coupons semiannually. What is the bond's current yield? M (or FV)=$1,000 N=7 VB=$1,020 YTM=10.5883% Current yield=?

19

∫¿ ∙ 2

[

]

1 1 1,000 − + 14 (0.105883/ 2) (0.105883/2) ∙(1+0.105883/2) ( 1+0.105883/2 )14 1,020=¿

∫ ¿ 110 In order to solve for the current yield we need to find INT. Current yield = Annual interest/Current Price = $110/$1,020 = 10.78%. Exercise: 4-12 A 10 year, 12% semiannually coupon bond with a par value of $1,000 may be called in 4 years at a call price of $1,060. The bond sells for $ 1,100. Assume that the bond has just been issued. a) what is the bond's yield to maturity? b) what is the bond's current yield? c) what is the bond's capital gain or loss yield d) what is the bonds' yield to call? a) M (or FV)=$1,000 rk=12% m=2 INT=$120 N=4 VB=$1,100 Callable price in 4 years=$1,000 YTM=? VB  M  YTM rk So, YTM has to be lower than a coupon rate, meaning less than 12% I way: rd1=10% m=2

P 1=

[

]

120 1 1 1,000 ∙ − + =1,124.62 20 2 (0.10/ 2) (0.10/2)∙(1+0.10 /2) ( 1+0.10 /2 )20

1,124.62  1,1000  P1 P  10% is too low rate rd =11% m=2

P 2=

[

]

120 1 1 1,000 ∙ − + =1,059.75 20 2 2 (0.10/ 2) (0.10/2)∙(1+0.10 /2) ( 1+0.10 /2 )20 20

1,069.75  1,100

P2

P  11% is too high rate YTM=10%+x

P −P x = 1 k 2−k 1 P1−P2 x 1,124.62−1,100 = 0.11−0.10 1,124.62−1,059.75 x 24.62 = 0.01 64.87

x=0.01 ∙0.37=0.0037 → 0.37 YTM=10%+0.37%=10.37%

b)

Current yield=

Annual interest payment 120 = =0.109110.91 Bonds current price 1,100

c)

YTM =Current Yield +Capital Gains ( Loss ) Yield 10.37% = 10.91% + Capital Loss Yield -0.54% = Capital Loss Yield

d) YTC N = 8, PV = -1,100, PMT = 60, FV = 1,060, and solve for I/YR = 5.0748%.

M (or FV)=$1.000 rk=12% m=2 INT=$120 N=10 VB=$1,100 21

Callable price in 4 years=$1,000

However, this is a periodic rate. 10.1495% ≈ 10.15%.

The nominal annual rate = 5.0748%∙(2) =

YTC=? VB Callable price  YTC rk So, YTC has to be lower than a coupon rate, meaning lower than 12% I way: rd1=10%

P 1=

m=2

[

]

120 1 1 1,000 ∙ − + =1,064.632 8 2 (0.10/ 2) (0.10/ 2)∙(1+0.10 /2) (1+0.10 /2 )8

1,064.632  1,100.00  P1 P  10% is too high rate rd2=11%

P 2=

m=2

[

]

120 1 1 1,000 ∙ − + =1,059.75 20 2 2 (0.10/ 2) (0.10/ 2)∙(1+0.10 /2) ( 1+0.10 /2 )20

1,069.75  1,100 P2

P  11% is too high rate YTM=10%+x

P −P x = 1 k 2−k 1 P1−P2 x 1,124.62−1,100 = 0.11−0.10 1,124.62−1,059.75 x 24.62 = 0.01 64.87 x=0.01 ∙0.37=0.0037 → 0.37 YTM=10%+0.37%=10.37% Exercise: 4-12

22

A 10 year, 12% semiannually coupon bond with a par value of $1,000 may be called in 4 years at a call price of $1,060. The bond sells for $ 1,100. Assume that the bond has just been issued. a) what is the bond's yield to maturity? b) what is the bond's current yield? c) what is the bond's capital gain or loss yield d) what is the bonds' yield to call? a) M (or FV)=$1,000 rk=12% m=2 INT=$120 N=4 VB=$1,100 Callable price in 4 years=$1,000 YTM=? VB  M  YTM rk So, YTM has to be lower than a coupon rate, meaning less than 12% I way: rd1=10% m=2

P 1=

[

]

120 1 1 1,000 ∙ − + =1,124.62 20 2 (0.10/ 2) (0.10/ 2)∙(1+0.10 /2) ( 1+0.10 /2 )20

1,124.62  1,1000  P1 P  10% is too low rate rd =11% m=2

P 2=

[

]

120 1 1 1,000 ∙ − + =1,059.75 20 2 2 (0.10/ 2) (0.10/ 2)∙(1+0.10 /2) ( 1+0.10 /2 )20

1,069.75  1,100

P2

P  11% is too high rate

P −P x = 1 k 2−k 1 P1−P2

YTM=10%+x

x 1,124.62−1,100 = 0.11−0.10 1,124.62−1,059.75

23

x 24.62 = 0.01 64.87 x=0.01 ∙0.37=0.0037 → 0.37 YTM=10%+0.37%=10.37% b)

Current yield=

Annual interest payment 120 = =0.109110.91 Bonds current price 1,100

c)

YTM =Current Yield +Capital Gains ( Loss ) Yield 10.37% = 10.91% + Capital Loss Yield -0.54% = Capital Loss Yield

d) YTC N = 8, PV = -1,100, PMT = 60, FV = 1,060, and solve for I/YR = 5.0748%. M (or FV)=$1.000 rk=12% m=2 INT=$120 N=10 VB=$1,100 Callable price in 4 years=$1,000 However, this is a periodic rate. 10.1495% ≈ 10.15%.

The nominal annual rate = 5.0748%∙(2) =

YTC=? VB Callable price  YTC rk So, YTC has to be lower than a coupon rate, meaning lower than 12% I way: rd1=10%

P 1=

m=2

[

]

120 1 1 1,000 ∙ − + =1,064.632 2 (0.10/ 2) (0.10/ 2)∙(1+0.10 /2)8 (1+0.10 /2 )8

1,064.632  1,100.00  P1 P  10% is too high rate 24

rd2=11%

P 2=

m=2

[

]

120 1 1 1,000 ∙ − + =1,059.75 20 2 2 (0.10/ 2) (0.10/2)∙(1+0.10 /2) ( 1+0.10 /2 )20

1,069.75  1,100 P2

P  11% is too high rate YTM=10%+x

P −P x = 1 k 2−k 1 P1−P2 x 1,124.62−1,100 = 0.11−0.10 1,124.62−1,059.75 x 24.62 = 0.01 64.87 x=0.01 ∙0.37=0.0037 → 0.37

YTM=10%+0.37%=10.37% Exercise: 4-13 You just purchased a bond that matures in 5 years. The bond has a face value of $1,000 and has an 8% annual coupon. The bond has a current yield of 8.21%. What is the bond's YTM? M (or FV)=$1,000 rk=8% INT=$80 N=5 Current yield= 8.21% YTM=? However, you are also given that the current yield is equal to 8.21%. Given this information, we can find PV. Current yield 0.0821 PV

= Annual interest/Current price = $80/PV = $80/0.0821 = $974.42. YTM = 8.65%.

The Determinants of Market Interest Rates (rd) 25

The quoted/nominal interest rate on a debt security, rd, is composed of a real riskfree rate of interest r*, plus several premiums that reflect inflation, the risk of the security and the security's marketability or liquidity. rd = r*+IP+DRP+LP+MRP = rRF + DRP + LP + MRP rd = quoted or nominal rate of interest r*= real risk-free rate of interest - the rate that would exist on a riskless security if zero inflation were expected ; inflation-indexed Treasury bond (TIPS) is a good estimate of the real interest rate IP = inflation premium - it is equal to the average expected inflation rate over the life of the security In inflation rate built into a 1-year bond is expected inflation rate for the next year, but the inflation rate built into a 3-year bond is average rate of inflation expected over the next 30 years. If It is the expected inflation during the year t, then the inflation premium for an Nyear bond's yield (IPN) can be approximated as:

IP N =

I 1+ I 2 +…+ I N N

DRP = default risk premium - this premium reflects the possibility that the issuer will not pay interest or principal at the stated time and in the stated amount. Bond's rating as an indicator of its default risk; the rating has a direct, measurable influence on the bond's yield. A bond spread is the difference between a bond's yield and the yield on some other security of the same maturity, mostly a similar maturity T-bond. LP = liquidity or marketability premium - a premium charged by lenders to reflect the fact that some securities cannot be converted to cash on short notice at a "reasonable" price MRP = maturity risk premium - all bonds, even T-bonds, are exposed to two additional sources of risk: interest rate risk and reinvestment risk; the net effect of these two sources of risk upon a bond's yield is called the maturity risk premium Example: (Tool Kit 4.13.) Assume that the real risk-free rate is r* = 3% and the average expected inflation rate is 2.5% for the foreseeable future. The DRP and LP for a bond are each 1%, and the applicable MRP is 2%.What is the bond's yield? rd = r*+IP+DRP+LP+MRP = 3+2.5+1+1+2=9.5% Example: In investors expect inflation to average 3% during Year 1 and 5% during Year 2. Then the inflation premium built into a 2-year bond's yield can be approximated by:

26

IP2=

3 +5 =4 2

Example: In March 2008 the yield on a 5-year non-indexed T-bond was 2.46% and the yield on 5-year TIPS was -0.05%. IP5=2.46%-(-0.05%)=2.51% This implies that investors expected inflation to average 2.51% over the next 5 years. Example: (Tool Kit 4.9.) The yield on 15-year TIPS is 3% and the yield on a 15-year Treasury bond is 5%. What is the inflation premium for a 15-year security? IP15=5%-2%=3% rRF=r*+IP ( it is either quoted U.S. T-bill rate or the quoted T-bond rate) Example: (Tool Kit 4.11.) A 10-year T-bond has a yield of 6%. A corporate bond with a rating of AA has a yield of 7.5%. If the corporate bond has excellent liquidity, what is an estimate of the corporate bond's default risk premium? Yield on T-Bond - Yield on corporate bond = Default risk premium

6.0% 7.5% 1.5%

Exercise: 4-5 A T-bond that matures in 10 years has a yield of 6%. A 10-year corporate bond has a yield of 9%. Assume that the liquidity premium on the corporate bond is 0.5%. What is the default risk premium on the corporate bond? rT-10 = 6%; rC-10 = 9%; LP = 0.5%; DRP = ? r = r* + IP + DRP + LP + MRP rT-10 = 6% = r* + IP + MRP; DRP = LP = 0 rC-10 = 8% = r* + IP + DRP + 0.5% + MRP

27

Because both bonds are 10-year bonds the inflation premium and maturity risk premium on both bonds are equal. The only difference between them is the liquidity and default risk premiums. rC-10 = 9% = r* + IP + MRP + 0.5% + DRP But we know from above that r* + IP + MRP = 6%; therefore, rC-10 = 9% = 6% + 0.5% + DRP 2.5% = DRP Exercise: 4-6 The real risk-free rate is 3% and inflation is expected to be 3% for the next 2 years. A 2-year Treasury security yields 6.3%. What is the maturity risk premium for the 2year security? r* = 3%; IP = 3%; rT-2 = 6.3%; MRP2 = ? rT-2 = r* + IP + MRP = 6.3% rT-2 = 3% + 3% + MRP = 6.3% MRP = 0.3%. Exercise: 4-19 Assume that the real risk-free rate r* is 3% and that inflation is expected to be 8% in Year 1, 5% in Year 2 and 4% thereafter. Assume also that all Treasury securities are highly liquid and free of default risk. If 2-year and 5-year Treasury notes both yield 10%, what is the difference in the maturity risk premiums (MRPs) on the two notes, that is, what is MRP5 minus MRP2? First, note that we will use the equation rt = 3% + IPt + MRPt We have the data needed to find the IPs:

IP5 =

8% + 5% + 4% + 4% + 4% 5

=

25% 5

= 5%

28

IP2 =

8% + 5% 2

= 6.5%

Now we can substitute into the equation: r2 = 3% + 6.5% + MRP2 = 10% r5 = 3% + 5% + MRP5 = 10% Now we can solve for the MRPs, and find the difference: MRP5 = 10% - 8% = 2% MRP2 = 10% - 9.5% = 0.5% Difference = (2% - 0.5%) = 1.5%

Exercise: 4-4 The real risk-free rate of interest is 4%. Inflation is expected to be 2% this year and 4% during the next 2 years. Assume that he maturity risk premium is zero. What is the yield on 2-year Treasury securities? What is the yield on 3-year Treasury securities? r* = 4%; I1 = 2%; I2 = 4%; I3 = 4%; MRP = 0; rT-2 = ?; rT-3 = ? r = r* + IP + DRP + LP + MRP

IP N =

I 1+ I 2 +…+ I N N

29

 Since these are Treasury securities, DRP = LP = 0. rT-2 = r* + IP2 IP2 = (2% + 4%)/2 = 3% rT-2 = 4% + 3% = 7%

rT-3 = r* + IP3 IP3 = (2% + 4% + 4%)/3 = 3.33% rT-3 = 4% + 3.33% = 7.33% Exercise: 4-18 The real risk-free rate is 2%. Inflation is expected to be 3% this year, 4% next year and then 3.5% thereafter. The maturity risk premium is estimated to be 0.0005 x (t1), where t = number of years to maturity. What is the nominal interest rate on a 7year Treasury security? r = r* + IP + MRP + DRP + LP r* = 0.02 IP = [0.03 + 0.04 + (5)∙(0.035)]/7 = 0.035 MRP = 0.0005∙(6) = 0.003 DRP = 0 LP = 0 r = 0.02 + 0.035 + 0.003 = 0.058 = 5.8%.

30