Three years ago, Wells Fargo made a fixed rate mortgage loan with the following terms. Loan amount: $$450,000$; Term: $30$ years; Interest rate: $5\%.$ Using a financial calculator or Excel, we can determine that the monthly payment on that loan is $$2,415\mathrm{.}70$ and the principal balance after $36$ payments have been applied is $$429,046\mathrm{.}07.$ Let's call that amount the "book value" of the loan.

Wells Fargo wants to take the loan off its books by selling it to Truist. In this exercise, I will show you how to determine the "market value" of the loan, which is the amount that Truist should be willing to pay Wells Fargo for the right to receive the future payments made on the loan.

Suppose interest rates have risen from $5\%$ to $7\%.$ Suppose also that Truist projects the loan will stay open for three more years after they buy it$,$ at which time the balance of the loan will be $$404,708\mathrm{.}67($i$.$e$.,$ the remaining balance after $72$ total payments have been applied$).$ Remember that mortgage borrowers in the US can prepay their loans for free at any time.

In buying the loan, Truist is making an investment. In return, they anticipate receiving $36$ monthly payments of $$2,415\mathrm{.}70$ from the borrower. And in the $36$th month, they will also receive the payoff balance of $$404,708\mathrm{.}67.$ Here's the timeline for the expected future cashflows from Truist's investment in buying the loan.

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Instead of buying the loan from Wells Fargo. Truist could simply make a loan themselves to a new borrower at $7\%.$ So$,$ in determining the market value of the existing loan, they will discount $($i$.$e$.,$ find PV$)$ the expected future cash flows using a discount rate of $7\%.$

What is the market value of the existing loan? To solve the problem, associate the values from the cashflow diagram with the TVM keys on your calculator and then compute PV$.$