Question

Assume you are pricing a 5 year bullet loan to a client. The loan will have equal annual interest payments in each year, unless the loan defaults in that year, in which case the payment received will equal the recovery value of the collateral, which is assumed to be 70 percent of the loan principal amount. A bullet repayment of principal will occur in year 5 if there is no default. Also, assume that the probability of default in each of the 5 years is 2 percent. If the loan is priced at 8.8 percent, what is the expected realized interest rate on the loan?

Guidelines: calculate the expected cash flow in each period using the proposed interest rate and the probabilities and recovery rate provided, where the loan may default in any of the 5 periods. The internal rate of return on those cash flows is the expected return on a portfolio of such loans. The alternative approach calculates the IRR on each possible outcome and then calculates the probability-weighted IRR of those outcomes; this is the expected return on a single loan.

How does the expected realized loan interest rate vary with the probability of default and expected recovery rate? How does the result change depending on taking the portfolio view or the single asset view?

Answer 1

Default in	Year 1	Year 2	Year 3	Year 4	Year 5	No default
		8.8	8.8	8.8	8.8	8.8
			8.8	8.8	8.8	8.8
				8.8	8.8	8.8
					8.8	8.8
						8.8
	70	70	70	70	70	100
Total	70	78.8	87.6	96.4	105.2	144
98%	2.00%	1.96%	1.92%	1.88%	1.84%	90.39%
	1.4	1.54448	1.6826208	1.814618176	1.940662609	130.1645947
Default in	Year 1	Year 2	Year 3	Year 4	Year 5	No default
		8.8	8.8	8.8	8.8	8.8
			8.8	8.8	8.8	8.8
				8.8	8.8	8.8
					8.8	8.8
						8.8
	70	70	70	70	70	100
Total	70	78.8	87.6	96.4	105.2	144
98%	2.00%	1.96%	1.92%	1.88%	1.84%	90.39%
	1.4	1.54448	1.6826208	1.814618176	1.940662609	130.1645947