Question

1. A $1000 par value bond has a coupon rate of 7% and matures in 4years. The bond makes semi-annual coupon payments.

a) If the bond is selling for $920, the YTM of the bond is?

b) If the bond is trading at $920, what is the current yield of the bond?

c) The capital gains yield that an investor expects to earn this year on the bond, if interest rates remain the same is?

d) The current price of the bond is $920. What is the modified duration in years of the bond?

2. Suppose that an investor with a three-year investment horizon is considering purchasing a 10 years 8% coupon bond for $875.38. The yield to maturity for this bond is 10%. The investor does not plan to reinvest the coupon interest payments but at the end of the three years he expects that the then 7years bond will be selling to offer a yield to maturity of 7%. What is the EAR of the investment if the expectations come true?

Answer 1

1.

a) we can use financial calculator for calculation of YTM of the bond.

coupons are paid semi-annually. so, maturity will be double and coupon payment will be half.

N = semi-annual maturity = 4*2 = 8; PMT = semi-annual coupon = $1,000*7%/2 = $35; FV = par value = $1,000; PV = current price = -$920 > CPT = compute > I/Y = semi-annual YTM = 4.72%

Annual YTM = semi-annual YTM*2 = 4.72%*2 = 9.44%

the YTM of the bond is 9.44%.

b) current yield = annual coupon/current price = ($1,000*7%)/$920 = $70/$920 = 0.0761 or 7.61%

c) If interest rates remain the same then selling price of the bond will remain the same as $920. so, capital gain yield will be zero because purchase price and selling price will be same as $920.

d) Modified duration = duration/(1+YTM/2)

Duration = [(1+YTM/2)/YTM] - [(1+YTM/2) + M(C-YTM)]/YTM + C[(1+YTM/2)^2M - 1]

C = annual coupon rate i.e.7%; YTM = yield to maturity i.e. 9.44%; M = bond maturity i.e. 4 years

Duration = [(1+0.0944/2)/0.0944] - [(1+0.0944/2) + 4(0.07-0.0944)]/0.0944 + 0.07[(1+0.0944/2)^2*4 - 1]

Duration = [(1+0.0472)/0.0944] - [(1+0.0472) + 4(-0.0244)]/0.0944 + 0.07[(1+0.0472)⁸ - 1]

Duration = (1.0472/0.0944) - (1.0472 - 0.0976)/0.0944 + 0.07[(1.0472)⁸ - 1]

Duration = 11.09322033898305 - (0.9496)/0.0944 + 0.07(1.4462290083870377065600425499034 - 1)

Duration = 11.09322033898305 - (0.9496)/0.0944 + 0.07*0.44622900838703770656004254990336 = 11.09322033898305 - 8.3792372881355932203389830508475 + 0.03123603058709263945920297849324 = 2.7452190814345494191202199276457 Years

Modified duration = 2.7452190814345494191202199276457/(1+0.0944/2) = 2.7452190814345494191202199276457/(1+0.0472) = 2.7452190814345494191202199276457/1.0472 = 2.62 years

the modified duration in years of the bond is 2.62.