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I think what I'd do is just run a few scenarios and average them together. You're not going to know precisely how fast and how far interest rates will rise so I'd make a couple duplicate spreadsheets in your Excel workbook and experiment. Do a best and a worst case and see where the loan portfolio is trading relative to that. Value the loan portfolio with today's cash flows and discount rates and see what things look like with the cash flows and discount rates a couple percentage points higher.

It sounds like in a rising interest rate environment your discount rate would go up (because the risk free rate would go up), and your cash flows would also go up since they're tied to LIBOR. So you get more cash but it's also discounted at a higher rate. The increased risk of default due to the higher payments should be handled by the higher discount rate.

The discount rate is a combination of the risk free rate and the risk premium. The discount rate will definitely go up since the risk free rate will rise, but it's possible it may also need to go up due to needing a higher risk premium if some of those companies won't be able to afford the higher interest payments. For a bond the risk premium is basically the odds of default risk and that's about it: easier stuff than with equities. You could look at the weighted average bond rating of the loan book and make some guesses there as to how easily the companies will be able to afford the increased payments. Aswath Damodaran has done some nice work codifying risk premiums with regard to bond ratings by doing statistical regressions:

What's the average maturity of the loan book? How far out do you have to model? Just a few years probably?

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