A zero-coupon, four-year corporate bond with a par value of $1,000. Assume that the risk-neutral marginal probability of default for the bond is 1.6%, the recovery rate is 30%, and the default-free spot rate curve is flat at 3%.

What is the present value (PV) of the expected loss under the risk-neutral probability at the end of first year?

(Round to 4 decimal places.)

Question

A zero-coupon, four-year corporate bond with a par value of $1,000. Assume that the risk-neutral marginal probability of default for the bond is 1.6%, the recovery rate is 30%, and the default-free spot rate curve is flat at 3%.

What is the present value (PV) of the expected loss under the risk-neutral probability at the end of first year?

(Round to 4 decimal places.)

BlackTom AI · Accepted Answer

First, restate the given data for clarity: a four-year zero-coupon corporate bond with par value 1000, risk-neutral probability of default in year 1 is 1.6%, recovery rate is 30%, and the default-free spot rate is flat at 3%. We are asked to find the present value of the expected loss under the risk-neutral measure at the end of year 1.

To compute the expected loss at the end of year 1, identify the components of the loss when default occurs in year 1. The loss given default (LGD) is (1 − recovery) × exposure. The recovery rate is 30%, so LGD = (1 − 0.30) × 1000 = 0.70 × 1000 = 700.

The probability of default in year 1 is 1.6%, i.e., 0.016. The expected loss in year 1 (in undiscounted terms) is then 0.016 × 700 = 11.2.

However, we must discount this expected loss back to present value using the default-free discount rate for one year, which is 3% (since the spot rate is flat at 3%). The present value factor is 1 / (1 + 0.03) = 1 / 1.03.

Thus, PV of expected loss = 11.2 / 1.03 ≈ 10.873786...

Rounding to four decimal places yields 10.8738.

Additional checks: note that the calculation assumes the loss occurs at the end of year 1 and uses risk-free discounting for the cash loss, consistent with the risk-neutral valuation framework for expected losses in the presence of default risk.

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