# Question of CDS spread calculation

A risk analyst is valuing a 1-year credit default swap (CDS) contract that will pay the buyer 80% of the face value of a bond issued by a corporation immediately after a default by the corporation. To purchase this CDS, the buyer will pay the CDS spread, which is a percentage of the face value, once at the end of the year. The analyst estimates that the risk-neutral default probability for the corporation is 7% per year. The risk-free rate is 2.5% per year. Assuming defaults can only occur halfway through the year and that the accrued premium is paid immediately after a default, what is the estimate for the CDS spread? a. 560 basis points b. 570 basis points c. 580 basis points d. 590 basis points

Correct answer: d Explanation: The key to CDS valuation is to equate the present value (PV) of payments to the PV of expected payoff in the event of default. Let: r = risk-free rate = 2.5% s = CDS spread. π = probability of default during year 1 = 7% C = contingent payment in case of default = 80% d0.5 =discount factor for half-year = e-0.5*r = e-0.5*0.025 = 0.987578 d1.0 =discount factor for 1-year = e-1.0*r = e-0.025 = 0.975310 Therefore, to solve for the CDS spread (s): The PV of payments (premium leg, which includes the spread payment and accrual) is: s*[0.5*d0.5* π + d1.0*(1-π)] = s*[0.034565 + 0.907038] = s*0.941603 The payoff leg (in the event of default) = C * d0.5 * π = 0.8*0.987578*0.07= 0.055304 Equating the two PVs and solving for the spread: s*0.941603 = 0.055304 Thus, s = 0.058734 or a spread of approximately 587 basis points.

I’m not able to understand why there is one 0.5 in bold below:

s*[0.5 *d0.5* π + d1.0*(1-π)] = s*[0.034565 + 0.907038] = s*0.941603

with d0.5, we’ve already discounted a half year, why we must multiply a 0.5 again?