Assume a one-year government coupon bond was just issued by a sovereign country. Face value F = $100. Coupon rate c = 5%, paid annually. One-year risk free rate is rf = 2%. (1) Assume this sovereign country is risk-free, please compute the bond price and yield today. (2) Assume today was in year 2012, and the bond issuer was the Greece government. The credit rating of Greek government bond was downgraded to “junk” bond. The estimated default probability in one year is π = 30%. If default, the recovery ratio ℓ = 40%. The expected bond market return is E(rM) = 3%. The beta of this bond β = 0.2. Please recompute the price and yield of this bond. In the above two cases, is the bond traded at discount, par or premium?

(1) If the country is risk-free, the bond price is calculated as the present value of the future cash flows. The cash flows consist of the coupon payment and the face value of the bond. The discount rate is the risk-free rate.

The bond price (P) is calculated as follows:

P = C / (1 + rf) + F / (1 + rf)

where: C = Coupon payment = F * c = $100 * 5$5 F = Face value = $100 rf = Risk-free rate = 2%

So,

P = $5 / (1 + 2$100 / (1 + 2%) = $4.90 +$98.04 = $102.94

The yield (Y) of the bond is the coupon rate, which is 5%.

(2) If the bond issuer is the Greece government and the bond is considered a "junk" bond, the bond price needs to be adjusted for the default risk. The expected cash flows are the coupon payment and the face value of the bond times the recovery ratio, discounted at the expected market return adjusted for the bond's beta.

The bond price (P) is calculated as follows:

P = [C * (1 - π) + F * ℓ * π] / (1 + E(rM) * β)

where: C = Coupon payment = $5 F = Face value =$100 π = Default probability = 30% ℓ = Recovery ratio = 40% E(rM) = Expected market return = 3% β = Beta of the bond = 0.2

So,

P = [$5 * (1 - 30$100 * 40% * 30%] / (1 + 3% * 0.2) = $3.50 +$12 / 1.006 = $15.44

The yield (Y) of the bond is the expected return, which is calculated as follows:

Y = [C * (1 - π) + F * ℓ * π] / P

So,

Y = [$5 * (1 - 30$100 * 40% * 30%] / $15.44 = 15.44%

In the first case, the bond is traded at a premium because the bond price ($102.94) is higher than the face value ($100). In the second case, the bond is traded at a discount because the bond price ($15.44) is lower than the face value ($100).