1 Lecture notes Loan repayment Math 177 UCLA Winter 2023 Introduction A simple description of loan repayment The lender or investor gives capital to the borrower The borrower gives the lender repayment Each repayment of a loan must repay the principal plus interest But how do we choose when to make payments and for each payment how much to pay towards the interest and the principal There are di erent approaches The main one is amortization Amortized loan structure An amortized loan of amount L at time t 0 has the following repayment 2 Important convention unless speci ed otherwise we will use compound interest We will often be interested in how much the borrower still owes to the lender after a certain amount of payments De nition 1 The outstanding loan balance at time t is how much the borrower must further pay to close the loan at that moment We denote the outstanding loan balance at time t 3 OBt Usually we measure t in payment periods in this case for an integer k we also have OBk outstanding loan balance after the kth payment Let s nd formulas for OB0 OB1 4 5 De nition 2 We introduce the notation It the interest accumulated from time t 1 to time t and PRt amount payed towards the principal by the tth payment We can now write OBt both in terms of It and PRt If PRt Kt It 0 then the payment results in a reduction of the principal 6 On the other hand if PRT Kt It 0 then the payment is not even enough to cover the interest from that period The excess interest becomes part of the principal owed capitalization of interest Usually the borrower does not want capitalization of interest to occur Calculating OBt in actual examples We have two methods retrospective and prospective 7 Which of the above methods we use depends on the information we have access to in a given problem 8 Info we have Method Schedule of remaining payments Original capital and payments so far Everything 9 Example 3 Consider a loan of 3000 with an repayment schedule that with each payment reduces the outstanding balance by 250 The e ec tive interest rate is 0 02 How much is each payment 10 Both retrospective and prospective approaches become easy to express when Amortization with level payments we have level payments This is because we can use annuities Consider a loan of amount L which is to be payed back with n equally spaced payments of amount K with e ective interest rate i per period Above we found an expression for OB0 L when we have level payments 11 What about the other OBt 12 Example 4 A home buyer borrows 250 000 to be repaid over 30 years with level monthly payments beginning one month after the loan is made The nominal annual rate convertible monthly is 9 a What s the amount of interest and principal payed in the rst year b What s the amount of interest and principal payed in the last year 13