FIN 350 CH 7 HW
A 10-year bond, $1,000 face value bond with a 11% coupon rate and semi-annual coupons has a yield to maturity of 8%. The bond should be trading at a price of
(1) With an 11% coupon rate and semiannual coupons, the coupon payment per six months is: CPN = FV × Coupon Rate / Number of Payments per year = $1,000 × 11% / 2 = $55. (2) YTM is 8% per year, so YTM per six months is 4%. N = 10 years × 2 =20. t=0 t=1 t=2 t=19 t=20 |----------------|----------------|--------- ··········-----|----------------| $55 $55 $55 $55 + $1000 Use this formula to find the price: 𝑃 = 𝐶𝑃𝑁/𝑌𝑇𝑀 × (1−1/(1+𝑌𝑇𝑀)^𝑁)+𝐹𝑉/(1+𝑌𝑇𝑀)^𝑁, where P is the price of the bond, CPN is the coupon payment, N is the number of payments, FV is the face value paid at maturity, and y is the yield to maturity. So in this case: P = $55/0.04 × (1 - (1/(1+0.04)^20)) + $1,000 / (1+0.04)^20
Andrew Industries is contemplating issuing a 30-year bond with a coupon rate of 7.50% (annual coupon payments) and a face value of $1,000. Andrew believes it can get a rating of A from Standard Poor's. However, due to recent financial difficulties at the company, Standard and Poor's is warning that it may downgrade Andrew Industries bonds to BBB. Yields on A-rated long-term bonds are currently 7.00%, and yields on BBB-rated bonds are 7.40%. (a) What is the price of the bond if Andrew maintains the A rating for the bond issue? (b) What will the price of the bond be if it is downgraded?
a) $1062.1 b) $1011.9 a) If Andrew Industries maintains a rating of A, it should have a YTM = 7%. With an 7.5% coupon rate and annual coupons, the coupon payment per year is: CPN = FV × Coupon Rate / Number of Payments per year = $1,000 × 7.5% / 1 = $75. YTM is 7% per year. N = 30 years. t=0 t=1 t=2 t=29 t=30 |----------------|----------------|--------- ··········-----|----------------| $75 $75 $75 $75 + $1000 Use this formula to find the price: 𝑃 = 𝐶𝑃𝑁/𝑌𝑇𝑀 × (1−1/(1+𝑌𝑇𝑀)^𝑁)+𝐹𝑉/(1+𝑌𝑇𝑀)^𝑁, where P is the price of the bond, CPN is the coupon payment, N is the number of payments, FV is the face value paid at maturity, and y is the yield to maturity. So in this case: P = $75/0.07 × (1 - (1/(1+0.07)^30)) + $1,000 / (1+0.07)^30 (b) If Andrew Industries maintains a rating of BBB, it should have a YTM = 7.4%. So in this case: P = $75/0.074 × (1 - (1/(1+0.074)^30)) + $1,000 / (1+0.074)^30
Suppose the current, zero-coupon, yield curve for risk-free bonds is as follows: Maturity (years) 1 2 3 4 5 Yield to Maturity 4.70% 5.20% 5.45% 5.65% 5.80% (a) What is the price per $100 face value of a 3-year, zero-coupon risk-free bond? (b) What is the price per $100 face value of a 5-year, zero-coupon, risk-free bond? (c) What is the risk-free interest rate for 3-year term? The risk-free interest rate for 3-year maturity is
a) $85.28 b) $75.43 c) 5.45% (a) The formula for the price of a zero-coupon bond is as follows: 𝑃=𝐹𝑉/(1+𝑌𝑇𝑀𝑛)^𝑛, where FV is the face value of $100, YTM is the yield to maturity and n is the number of periods. Therefore: P = $100 ÷ (1+5.45%)^3 (b) The formula for the price of a zero-coupon bond is as follows: 𝑃=𝐹𝑉/(1+𝑌𝑇𝑀𝑛)^𝑛, where FV is the face value of $100, YTM is the yield to maturity and n is the number of periods. Therefore: P = $100 ÷ (1+5.80%)^5 (c) The risk-free interest rate for a 3-year term should equal the YTM on a risk-free bond with 3 year maturity: r3 = YTM3 = 5.45%
A bond will make payments every six months as shown on the following timeline (using six-month periods): t=0 t=1 t=2 t=27 t=28 |----------------|----------------|--------- ··········-----|----------------| $60 $60 $60 $1,060 (a) What is the maturity of the bond (in years)? (b) What is the face value? (c) What is the coupon rate (in percent)?
a) 14 years b) $1000 c) 12.00% (a) To determine the maturity of the bond, find the number of periods on timeline and divided by two because there are two payments in each year. (b) The face value is the amount paid at maturity, so the last payment is made up of the semi-annual coupon payment and the face value (= $60 coupon + $1,000 face value). (c) To determine the coupon rate, use the formula: Coupon Payment= (Coupon Rate × Face Value) / Number of Payments per year. This implies: $60 = $1,000 × Coupon Rate / 2