Ch 6 HW

The following table shows the prices of a sample of Treasury strips. Each strip makes a single payment at maturity. Years to MaturityPrice, (% of face value) 1: 97.552% 2: 94.051 3: 90.244 4: 86.180 a. What is the 1-year interest rate? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) b. What is the 2-year interest rate? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) c. What is the 3-year interest rate? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) d. What is the 4-year interest rate? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) e. Is the yield curve upward-sloping, downward-sloping, or flat? f. Is this the usual shape of the yield curve?

a. 2.51 Bond price=FV / (1 + r)^t 97.552% × $1,000=$1,000 / (1 + r)^1 r=(1,000 / $975.52)^(1/1) - 1 r=0.0251, or 2.51% b. 3.1142 Strips are zero-coupon bonds in which single payment is made at maturity. 2-year interest rate = (future value of 2 year strip / PV of 2 year strip)1/2 - 1 2-year interest rate = (100/94.051)1/2 - 1 2-year interest rate = 3.1142% p.a. or Bond price=FV / (1 + r)^t 94.051% × $1,000=$1,000 / (1 + r)^2 r=(1,000 / $940.51)^(1/2) - 1 r=0.0311, or 3.11% c. 3.481 3-year interest rate = (future value of 3 year strip / PV of 3 year strip)1/3 - 1 3-year interest rate = (100/90.244)1/3 - 1 3-year interest rate = 3.481% p.a. Bond price=FV / (1 + r)^t 90.244% × $1,000=$1,000 / (1 + r)^3 r=(1,000 / $902.44)^(1/3) - 1 r=0.0348, or 3.48% d. 3.79 Bond price=FV / (1 + r)^t 86.180% × $1,000=$1,000 / (1 + r)^4 r=($1,000 / $861.80)^(1/4) - 1 r=0.0379 or 3.79% e. upward sloping The yield curve is upward-sloping because the yield on the longer-term bond is higher than the yield on the shorter-term bond. f. yes The yield curve is most commonly upward-sloping although it can be downward-sloping or even humped, where intermediate-term bonds have higher yields than either short-term or long-term bonds.

The following table shows some data for three zero-coupon bonds. The face value of each bond is $1,000. Bond: Price: Maturity (Years): Yield to Maturity A : 380 : 20 : — B : 380 : — : 10% C : — : 18 : 9 a. What is the yield to maturity of bond A? b. What is the maturity of B?

a. 4.957 Yield to maturity is calculated using the RATE function:- =RATE(20,0,-380,1000) =4.957% or Bond price=FV / (1 + r)t$380=$1,000 / (1 + r)20(1 + r) 20=$1,000 / $380r=2.6316(1 / 20) - 1r =0.0496, or 4.957% b. 10.15 Using NPER function:- =NPER(10%,0,-380,1000) =10.15 or Bond price=FV / (1 + r)t $380=$1,000 / 1.10t1.10t =$1,000 / $380t × ln1.10 =ln0.0496t =ln0.0496 / ln1.10t =10.15 years c. 211.99 Using PV function:- =PV(9%,18,0,1000) =211.99 or Bond price=FV / (1 + r)t =$1,000 / 1.0918 =$211.99

General Matter's outstanding bond issue has a coupon rate of 8.2%, and it sells at a yield to maturity of 7.25%. The firm wishes to issue additional bonds to the public. What coupon rate must the new bonds offer in order to sell at face value?

7.25% When the company has to sell new bonds at face value, the coupon rate must be equal to the yield to maturity (YTM). When coupon rate = YTM, the price of the bond = Face value. Hence the firm must issue new bonds at 7.25% Coupon rate which will make the price of the new bonds equivalent to face value When the bond is selling at face value, its yield to maturity equals its coupon rate. This firm's bonds are selling at a yield to maturity of 7.25%. So the coupon rate on the new bonds must be 7.25% if they are to sell at face value.

A bond with a face value of $1,000 has 10 years until maturity, carries a coupon rate of 8.4%, and sells for $1,160. Interest is paid annually. (Assume a face value of $1,000 and annual coupon payments.) a. If the bond has a yield to maturity of 9.6% 1 year from now, what will its price be at that time? b. What will be the rate of return on the bond? c. If the inflation rate during the year is 3%, what is the real rate of return on the bond? (Assume annual interest payments.)

a. $930 Bond Value = pv(rate, nper,pmt,fv) Nper (indicates the period left to maturiy) = 9 PV (indicates the price) = ? PMT (indicate the annual payment) = 1000*8.4% = 84 FV (indicates the face value) = 1000 Rate (indicates YTM) = 10% Bond Value = pv( 9.6%,9,84,1000) Bond Value = $ 929.78 or Bond price=PV of coupon payments + PV of face value =C × ((1 / r) - {1 / [r(1 + r)t]}) + FV / (1 + r)t =(0.0840 × $1,000) × ((1 / 0.096) - {1 / [0.096(1.0960)(10 - 1)]}) + $1,000 / 1.0960(10 - 1) =$930 Since yield to maturity equals the coupon rate, the bond must be priced at par. b. -12.61% Annual rate of return on the bond = (Price 1 year from now - Purchase Price + coupon)/Purchase Price Annual rate of return on the bond = (929.78 - 1160 + 84)/1160 Annual rate of return on the bond = -12.61% or Rate of return=[Annual interest + (Ending price - Beginning price)] / Beginning price =($84 + 930 - 1,160) / $1,160 =-0.1261, or -12.61% c. -15.16% Annual real rate of return on the bond = (1-12.61%)/(1+3%)-1 Annual real rate of return on the bond = - 15.16% or Rate of return=(1 + Nominal return) / (1 + Inflation rate) - 1 =[1 + (-0.1261)] / 1.03 - 1 =-0.1515, or -15.15%

A bond with face value $1,000 has a current yield of 6.2% and a coupon rate of 8.2%. a. If interest is paid annually, what is the bond's price? b. Is the bond's yield to maturity more or less than 8.2%?

a. 1322.58 Current yield = Coupon payments/Bond price 82/.062=1322.58 or Coupon payment=Coupon rate × Face value =0.082 × $1,000 =$82 Current yield=Coupon payment / Bond price 0.062=$82 / Bond price Bond price=$82 / 0.062 Bond price=$1,322.58 b. less Since the bond is selling at a premium, the YTM must be below the coupon rate of 8.2%.

A 2-year maturity bond with face value of $1,000 makes annual coupon payments of $112 and is selling at face value. What will be the rate of return on the bond if its yield to maturity at the end of the year is: a. 6% b. 11.2% c. 13.2%

a. 16.22% price of bond = (1000+112) / (1+6%) = 1049.06 rate of return = [112+(1049.06-1000)] / 1000 = 16.22% or Price=$1,112 / 1.06 =$1,049.06 Rate of return=[Annual interest + (Ending price - Beginning price)] / Beginning price =($112 + 1,049.06 - 1,000) / $1,000 =0.1611, or 16.11% b. 11.24% PoB=1112/(1+11.2%)=$1000 RoR=[112+(1000-100)] / 1000 = 11.24% or Price=$1,112 / 1.112 =$1,000 Rate of return=[Annual interest + (Ending price - Beginning price)] / Beginning price =($112 + 1,000 - 1,000) / $1,000 =0.1120, or 11.20% c. 9.43% PoB=1112 / (1+13.2%) = $982.33 RoR=[112+(982.33-1000)] / 1000 = 9.43% Price=$1,112 / 1.132 =$982.33 Rate of return=[Annual interest + (Ending price - Beginning price)] / Beginning price =($112 + 982.33 - 1,000) / $1,000 =0.0943, or 9.43%

A 5-year Circular File bond with a face value of $1,000 pays interest once a year of $60 and sells for $952. a. What are its coupon rate and yield to maturity? b. If Circular wants to issue a new 5-year bond at face value, what coupon rate must the bond offer?

a. 6% and 7.18% The bond pays annually $60, so the Coupon rate = 60/1000 = 6% Yield to maturity = YTM is given by rate formula in excel =rate(nper,pmt,pv,fv) where nper =5,pmt=60,pv=952,fv =1000 So YTM = rate(5,60,-952,1000) = 7.18% or n = 5, PV = (−)952, FV = 1,000, PMT = 60; compute i = 7.18%. b. 7.18 To sell a bond at its face value, the bond's coupon rate must be set equal to the yield to maturity of currently outstanding bonds. Thus, the new bond must offer a coupon rate of 7.18%.

a. Several years ago, Castles in the Sand Inc. issued bonds at face value of $1,000 at a yield to maturity of 8.8%. Now, with 7 years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 14%. What is the price of the bond now? (Assume semiannual coupon payments.) b. Suppose that investors believe that Castles can make good on the promised coupon payments but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 86% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive?

a. 772.62 a) coupon rate 8.8% b) par value 1000 c) years 7 d) coupon $ 44 : semi annual (1000 x 8.8%)/2 e) NO of periods 14 f) YTM 14% annual g) semi annual YTM 7% semi annual = price 772.62 PV( ) or Bond price=PV of coupon payments + PV of face value =C × ((1 / r) - {1 / [r(1 + r)t]}) + FV / (1 + r)^t =[(0.0880 × $1,000) / 2] × [[1 / (0.1400 / 2)] - (1 / {(0.1400 / 2)[1 + (0.1400 / 2)]^(7 × 2)})] + $1,000 / [1 + (0.1400 / 2)]^(7 × 2) =$772.62 b. 12.45% a) coupon rate 8.8% b) par value 1000 c) years 7 d) coupon $ 44 : semi annual (1000 x 8.8%)/2 e) NO of periods 14 f) Price 772.62 g) redemption value 860: 86% x 1000 h) semi annual YTM 6.22%: =RATE () = YTM (annual) 12.45% : 6.22% x 2 or $772.62=C × ((1 / r) - {1 / [r(1 + r)t]}) + FV / (1 + r)^t =[(0.0880 × $1,000) / 2] × [[1 / (r / 2)] - (1 / {(r / 2)[1 + (r / 2)]^(7 × 2)})] + (0.8600 × $1,000) / [1 + (r / 2)]^(7 × 2) YTM = 6.22% × 2 = 12.45%

A General Power bond carries a coupon rate of 8.1%, has 9 years until maturity, and sells at a yield to maturity of 7.1%. (Assume annual interest payments.) a. What interest payments do bondholders receive each year? b. At what price does the bond sell? (Do not round intermediate calculations. Round your answer to 2 decimal places.) c. What will happen to the bond price if the yield to maturity falls to 6.1%? (Do not round intermediate calculations. Round your answer to 2 decimal places.) d. If the yield to maturity falls to 6.1%, will the current yield be less, or more, than the yield to maturity?

a. 81 Interest payment: Interest payment=Coupon rate × Face value = $1,000*8.10% = $81 b.$ 1,064.88 1) Bond par value $ 1,0002 2) Coupon rate 8.10% 3) Number of compounding periods per year 1 4) (= 1*2/3) Interest per period (PMT)$ 81.00 5) Number of years to maturity 9 6) (= 3*5) Number of compounding periods till maturity (NPER)9 7) Market rate of return/Required rate of return 7.10% 8) (= 7/3) Market rate of return/Required rate of return per period (RATE) 7.10% Bond price at 7.10% yield: PV(RATE,NPER,PMT,FV)*-1 Bond price at 7.10% yield: $ 1,064.88 or Bond price=PV of coupon payments + PV of face value =C × ((1 / r) - {1 / [r(1 + r)t]}) + FV / (1 + r)t =$81 × {(1 / 0.071) - [1 / (0.071 × 1.0719)]} + $1,000 / 1.0719 =$1,064.88 c. rise by $70.56 1) Bond par value$ 1,000 2) Coupon rate8.10% 3) Number of compounding periods per year1 4) = 1*2/3Interest per period (PMT)$ 81.00 5) Number of years to maturity9 6) = 3*5Number of compounding periods till maturity (NPER)9 7) Market rate of return/Required rate of return6.10% 8) = 7/3Market rate of return/Required rate of return per period (RATE)6.10% Bond price at 6.10% yieldPV(RATE,NPER,PMT,FV)*-1 Bond price at 6.10% yield$ 1,135.44 or Bond price=PV of coupon payments + PV of face value =C × ((1 / r) - {1 / [r(1 + r)t]}) + FV / (1 + r)t =$81 × {(1 / 0.061) - [1 / (0.061 × 1.0619)]} + $1,000 / 1.0619 =$1,135.44 Price increase=$1,135.44 - 1,064.88 =$70.56 d. current yield is more than yield to maturity More. The current yield exceeds the yield to maturity on the bond because the bond is selling at a premium. At maturity the holder of the bond will receive only the $1,000 face value, reducing the total return on investment as measured by yield to maturity. There is an inverse relationship between the YTM and the bond price. As the YTM decreases, the price increases and vice versa.

Consider two bonds, a 3-year bond paying an annual coupon of 6.00% and a 10-year bond also with an annual coupon of 6.00%. Both currently sell at a face value of $1,000. Now suppose interest rates rise to 8%. a. What is the new price of the 3-year bonds? b. What is the new price of the 10-year bonds? c. Which bonds are more sensitive to a change in interest rates?

a. 948.46 a) Annual Coupon Amount $ 60.00 b) Present Value Annuity Factor for (3 Years,8%)2.577097 c) Present Value Of Annual Interest (a*b)$ 154.63 d) Redemption Value$ 1,000.00 e) Present Value Of (3 Years,8%)0.79383 g) Present Value Of Redemption Amount (d*e)$ 793.83 f) Bond Price(c+g)$ 948.46 or PV=$60×[1/0.08−1/0.08×(1.08)^3]+$1,0001.083=$948.46 b. 865.80 a) Annual Coupon Amount$ 60.00 b) Present Value Annuity Factor for (10 Years,8%)6.710081 c) Present Value Of Annual Interest (a*b)$ 402.60 d) Redemption Value$ 1,000.00 e) Present Value Of (10 Years,8%)0.46319 g) Present Value Of Redemption Amount (d*e)$ 463.19 f) Bond Price(c+g)$ 865.80 or PV=$60×[1/0.08−1/0.08×(1.08)^10]+$1,0001.0810=$865.80 c. long term bonds Long-term bonds are more sensitive to interest rate changes. The average cash flow is received later, thus the present value of those coupons are reduced by more than near-term cash flows